{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 "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Dark Red Emphasis" -1 259 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Purple Emphasis" -1 260 "Times " 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 261 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Emphasis" -1 262 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Magenta E mphasis" -1 263 "Times" 1 12 200 0 200 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Orange Emphasis" -1 264 "Times" 1 12 225 100 10 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 275 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 261 277 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 1 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 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Times" 1 12 128 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Ti mes" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "T imes" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "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 16 stage, combined \+ order 8 and 9 Runge-Kutta schemes" }}{PARA 0 "" 0 "" {TEXT -1 46 "by P eter Stone, Gabriola Island, B.C., Canada " }}{PARA 0 "" 0 "" {TEXT -1 18 "Version: 19.2.2012" }}{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 30 "Relations between the nodes A " }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 84 "The Runge- Kutta schemes considered in this worksheet are such that the nodes nod es " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\"&" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6]" "6# &%\"cG6#\"\"'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7]" "6#&%\"cG6#\"\" (" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[8]" "6#&%\"cG6#\"\")" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[9];" "6#&%\"cG6#\"\"*" }{TEXT -1 31 " satisfy the certain relations." }}{PARA 0 "" 0 "" {TEXT -1 63 "These \+ arise from the stage-order conditions for stages 4 to 9. " }}{PARA 0 " " 0 "" {TEXT -1 54 "\nWe assume that certain linking coefficients are \+ zero." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[i,2]=0" "6 #/&%\"aG6$%\"iG\"\"#\"\"!" }{TEXT -1 5 ", " }{XPPEDIT 18 0 "i = 4; " "6#/%\"iG\"\"%" }{TEXT -1 8 " . . 9. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[i,3]=0" "6#/&%\"aG6$%\"iG\"\"$\"\"!" }{TEXT -1 5 ", " }{XPPEDIT 18 0 "i = 6;" "6#/%\"iG\"\"'" }{TEXT -1 8 " . . 9. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[i,4]=a[i,5] " "6#/&%\"aG6$%\"iG\"\"%&F%6$F'\"\"&" }{XPPEDIT 18 0 "``=0" "6#/%!G\" \"!" }{TEXT -1 5 ", " }{XPPEDIT 18 0 "i = 8;" "6#/%\"iG\"\")" } {TEXT -1 8 " . . 9. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 57 "The stages of the scheme have the following stage-order s." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "matrix([[stage, `|`, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [`stage-order`, `|`, 1, 2, 3, 3, 4, 4, 5, 5, 5, \+ 5, 5, 5, 5, 5, 5]])" "6#-%'matrixG6#7$73%&stageG%\"|grG\"\"#\"\"$\"\"% \"\"&\"\"'\"\"(\"\")\"\"*\"#5\"#6\"#7\"#8\"#9\"#:\"#;73%,stage-orderGF )\"\"\"F*F+F+F,F,F-F-F-F-F-F-F-F-F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "We also have zero linking coefficients as indicated by th e following tableau." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[c[2], a[2,1], ``, ``, ``, ``, ``, ``, ``], [c[3], a[3,1], a[3,2], ``, ``, ``, ``, ``, ``], [c[4 ], a[4,1], 0, a[4,3], ``, ``, ``, ``, ``], [c[5], a[5,1], 0, a[5,3], a [5,4], ``, ``, ``, ``], [c[6], a[6,1], 0, 0, a[6,4], a[6,5], ``, ``, ` `], [c[7], a[7,1], 0, 0, a[7,4], a[7,5], a[7,6], ``, ``], [c[8], a[8,1 ], 0, 0, 0, 0, a[8,6], a[8,7], ``], [c[9], a[9,1], 0, 0, 0, 0, a[9,6], a[9,7], a[9,8]]])" "6#-%'matrixG6#7*7+&%\"cG6#\"\"#&%\"aG6$F+\"\"\"%! GF0F0F0F0F0F07+&F)6#\"\"$&F-6$F4F/&F-6$F4F+F0F0F0F0F0F07+&F)6#\"\"%&F- 6$F " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 5 "" 0 " " {TEXT -1 20 "tableau construction" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 274 "matrix([[c[2],a[2,1],``$7],[c[3],a[3,1],a[3,2],``$6] ,[c[4],a[4,1],0,a[4,3],``$5],[c[5],a[5,1],0,a[5,3],a[5,4],``$4],[c[6], a[6,1],0$2,a[6,4],a[6,5],``$3],[c[7],a[7,1],0$2,seq(a[7,i],i=4..6),``$ 2],[c[8],a[8,1],0$4,seq(a[8,i],i=6..7),``],[c[9],a[9,1],0$4,seq(a[9,i] ,i=6..8)]]);" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "We specify the zero coefficients in stages 3 to 9." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "e0 := \{seq(a[i,2]=0,i=4..9 ),seq(a[i,3]=0,i=6..9),a[8,4]=0,a[8,5]=0,a[9,4]=0,a[9,5]=0\}:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "Since sta ge 4 has stage-order 3, we have: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[4, 3]*c[3] = 1/2;" "6#/*&&%\"aG6$\"\"%\"\"$\"\"\"&% \"cG6#F)F**&F*F*\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[4]^2;" " 6#*$&%\"cG6#\"\"%\"\"#" }{TEXT -1 8 ", " }{XPPEDIT 18 0 "a[4,3]* c[3]^2 = 1/3;" "6#/*&&%\"aG6$\"\"%\"\"$\"\"\"*$&%\"cG6#F)\"\"#F**&F*F* F)!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[4]^3;" "6#*$&%\"cG6#\"\"%\" \"$" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 20 "If we suppose tha t " }{XPPEDIT 18 0 "c[3] <> c[4];" "6#0&%\"cG6#\"\"$&F%6#\"\"%" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "c[3] <> 0;" "6#0&%\"cG6#\"\"$\"\" !" }{TEXT -1 106 " then, in order for these two equations to be consis tent with a unique value for the linking coefficient " }{XPPEDIT 18 0 "a[4,3];" "6#&%\"aG6$\"\"%\"\"$" }{TEXT -1 32 " the determinant of \+ the matrix " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix ([[c[3], c[4]^2/2], [c[3]^2, c[4]^3/3]]);" "6#-%'matrixG6#7$7$&%\"cG6# \"\"$*&&F)6#\"\"%\"\"#F0!\"\"7$*$&F)6#F+F0*&&F)6#F/F+F+F1" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 39 "must be zero. This gives the rela tion " }{XPPEDIT 18 0 "c[3] = 2/3;" "6#/&%\"cG6#\"\"$*&\"\"#\"\"\"F'! \"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[4];" "6#&%\"cG6#\"\"%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 161 "subs(e0,[seq(op(StageOrderConditions(i,4..4,'expande d')),i=2..3)]);\nlinalg[genmatrix](%,[a[4,3]],flag);\nlinalg[det](%)=0 :\nop(solve(\{%,c[3]<>c[4],c[3]<>0\},c[3]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$/*&&%\"aG6$\"\"%\"\"$\"\"\"&%\"cG6#F*F+,$*&#F+\"\"#F+ *$)&F-6#F)F2F+F+F+/*&F&F+)F,F2F+,$*&#F+F*F+*$)F5F*F+F+F+" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#K%'matrixG6#7$7$&%\"cG6#\"\"$,$*&#\"\"\"\"\"#F/* $)&F)6#\"\"%F0F/F/F/7$*$)F(F0F/,$*&#F/F+F/*$)F3F+F/F/F/Q(pprint06\"" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"$,$*&#\"\"#F'\"\"\"&F%6# \"\"%F,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 55 ": This relation can be expressed in the int egral form: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x *(x-c[3]),x=0..c[4])=0" "6#/-%$IntG6$*&%\"xG\"\"\",&F(F)&%\"cG6#\"\"$! \"\"F)/F(;\"\"!&F,6#\"\"%F2" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "Int(x*(x-c[3]),x=0..c[4])=0;\nc[3]=solve(value(%), c[3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&%\"xG\"\"\",&F(F )&%\"cG6#\"\"$!\"\"F)/F(;\"\"!&F,6#\"\"%F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"$,$*&#\"\"#F'\"\"\"&F%6#\"\"%F,F," }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "#-------- -----------------------------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 42 "We obtain a relation involving the nodes " } {XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5]" "6#&%\"cG6#\"\"&" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[6] " "6#&%\"cG6#\"\"'" }{TEXT -1 47 " from the fact that stage 6 has sta ge-order 4." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "subs(e0,[seq (op(StageOrderConditions(i,6..6,'expanded')),i=2..4)]):\nlinalg[genmat rix](%,[a[6,4],a[6,5]],flag);\nop(solve(\{linalg[det](%)=0,c[4]<>c[5], c[4]<>0\},c[4]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7%7%& %\"cG6#\"\"%&F)6#\"\"&,$*&#\"\"\"\"\"#F2*$)&F)6#\"\"'F3F2F2F27%*$)F(F3 F2*$)F,F3F2,$*&#F2\"\"$F2*$)F6FAF2F2F27%*$)F(FAF2*$)F,FAF2,$*&#F2F+F2* $)F6F+F2F2F2Q(pprint16\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6# \"\"%,$*&#\"\"\"\"\"#F+*(&F%6#\"\"'F+,&*&\"\"$F+F.F+!\"\"*&F'F+&F%6#\" \"&F+F+F+,&*&F,F+F.F+F4*&F3F+F6F+F+F4F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "This equation can be expressed \+ in integral form." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "Int(x*( x-c[4])*(x-c[5]),x=0..c[6])=0;\nc[4]=solve(value(%),c[4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*(%\"xG\"\"\",&F(F)&%\"cG6#\"\"%! \"\"F),&F(F)&F,6#\"\"&F/F)/F(;\"\"!&F,6#\"\"'F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"%,$*&#\"\"\"\"\"#F+*(&F%6#\"\"'F+,&*&\"\"$ F+F.F+F+*&F'F+&F%6#\"\"&F+!\"\"F+,&*&F,F+F.F+F+*&F3F+F5F+F8F8F+F+" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "#-------- -----------------------------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 45 "We obtain two equations involving the nodes " }{XPPEDIT 18 0 "c[6];" "6#&%\"cG6#\"\"'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7];" "6#&%\"cG6#\"\"(" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c [8];" "6#&%\"cG6#\"\")" }{TEXT -1 47 " from the fact that stage 8 has stage-order 5." }}{PARA 0 "" 0 "" {TEXT -1 5 "( I )" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 176 "subs(e0,[seq(op(StageOrderConditions(i,8 ..8,'expanded')),i=2..4)]);\nlinalg[genmatrix](%,[a[8,6],a[8,7]],flag) ;\nop(solve(\{linalg[det](%)=0,c[6]<>c[7],c[6]<>c[8],c[6]<>0\},c[6])); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%/,&*&&%\"aG6$\"\")\"\"'\"\"\"&% \"cG6#F+F,F,*&&F(6$F*\"\"(F,&F.6#F3F,F,,$*&#F,\"\"#F,*$)&F.6#F*F9F,F,F ,/,&*&F'F,)F-F9F,F,*&F1F,)F4F9F,F,,$*&#F,\"\"$F,*$)F " 0 "" {MPLTEXT 1 0 64 "Int(x*(x-c[6])*(x-c[7]),x=0. .c[8])=0;\nc[6]=solve(value(%),c[6]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*(%\"xG\"\"\",&F(F)&%\"cG6#\"\"'!\"\"F),&F(F)&F,6#\"\"(F/ F)/F(;\"\"!&F,6#\"\")F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\" \"',$*&#\"\"\"\"\"#F+*(&F%6#\"\")F+,&*&\"\"$F+F.F+!\"\"*&\"\"%F+&F%6# \"\"(F+F+F+,&*&F,F+F.F+F4*&F3F+F7F+F+F4F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "( II )" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 176 "subs(e0,[seq(op(StageOrderConditions(i,8..8,'expa nded')),i=3..5)]);\nlinalg[genmatrix](%,[a[8,6],a[8,7]],flag);\nop(sol ve(\{linalg[det](%)=0,c[6]<>c[7],c[6]<>c[8],c[6]<>0\},c[6]));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7%/,&*&&%\"aG6$\"\")\"\"'\"\"\")&%\"cG 6#F+\"\"#F,F,*&&F(6$F*\"\"(F,)&F/6#F5F1F,F,,$*&#F,\"\"$F,*$)&F/6#F*F " 0 "" {MPLTEXT 1 0 66 "Int(x^2*(x-c[6])* (x-c[7]),x=0..c[8])=0;\nc[6]=solve(value(%),c[6]);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%$IntG6$*()%\"xG\"\"#\"\"\",&F)F+&%\"cG6#\"\"'!\"\" F+,&F)F+&F.6#\"\"(F1F+/F);\"\"!&F.6#\"\")F8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"',$*&#\"\"$\"\"&\"\"\"*(&F%6#\"\")F-,&*&\" \"%F-F/F-!\"\"*&F,F-&F%6#\"\"(F-F-F-,&*&F+F-F/F-F5*&F4F-F7F-F-F5F-F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "These \+ two equations to numerical ratios between the nodes " }{XPPEDIT 18 0 "c[6]" "6#&%\"cG6#\"\"'" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[8]" " 6#&%\"cG6#\"\")" }{TEXT -1 30 " and also between the nodes " } {XPPEDIT 18 0 "c[7]" "6#&%\"cG6#\"\"(" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "c[8]" "6#&%\"cG6#\"\")" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "eq1 := int(x*(x-c[6])*(x-c[7]),x=0..c[8]) =0:\neq2 := int(x^2*(x-c[6])*(x-c[7]),x=0..c[8])=0:" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "sol := sol ve(\{eq1,eq2\},\{c[6],c[7]\}):\nop(factor(expand(rationalize([allvalue s(sol)]))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$<$/&%\"cG6#\"\"(,$*&# \"\"\"\"#5F,*&,&\"\"'F,*$F0#F,\"\"#F,F,&F&6#\"\")F,F,F,/&F&6#F0,$*&#F, F-F,*&,&F0!\"\"F1F,F,F4F,F,F?<$/F%F:/F8F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "Thus we have either " } {XPPEDIT 18 0 "c[6]=(6-sqrt(6))/10" "6#/&%\"cG6#\"\"'*&,&F'\"\"\"-%%sq rtG6#F'!\"\"F*\"#5F." }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[8]" "6#&%\"cG6 #\"\")" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[7]=(6+sqrt(6))/10" "6# /&%\"cG6#\"\"(*&,&\"\"'\"\"\"-%%sqrtG6#F*F+F+\"#5!\"\"" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "c[8]" "6#&%\"cG6#\"\")" }{TEXT -1 10 " or " } {XPPEDIT 18 0 "c[6] = (6+sqrt(6))/10;" "6#/&%\"cG6#\"\"'*&,&F'\"\"\"-% %sqrtG6#F'F*F*\"#5!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[8]" "6#&%\" cG6#\"\")" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[7] = (6-sqrt(6))/10 ;" "6#/&%\"cG6#\"\"(*&,&\"\"'\"\"\"-%%sqrtG6#F*!\"\"F+\"#5F/" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[8]" "6#&%\"cG6#\"\")" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "#-------- -----------------------------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 43 "We obtain an equation involving the nodes " } {XPPEDIT 18 0 "c[6]" "6#&%\"cG6#\"\"'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7];" "6#&%\"cG6#\"\"(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[8];" " 6#&%\"cG6#\"\")" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[9];" "6#&%\"c G6#\"\"*" }{TEXT -1 47 " from the fact that stage 9 has stage-order 5 ." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 183 "subs(e0,[seq(op(StageO rderConditions(i,9..9,'expanded')),i=2..5)]):\nlinalg[genmatrix](%,[a[ 9,6],a[9,7],a[9,8]],flag);\nop(solve(\{linalg[det](%)=0,c[8]<>c[7],c[6 ]<>c[8],c[8]<>0\},c[8]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrix G6#7&7&&%\"cG6#\"\"'&F)6#\"\"(&F)6#\"\"),$*&#\"\"\"\"\"#F5*$)&F)6#\"\" *F6F5F5F57&*$)F(F6F5*$)F,F6F5*$)F/F6F5,$*&#F5\"\"$F5*$)F9FFF5F5F57&*$) F(FFF5*$)F,FFF5*$)F/FFF5,$*&#F5\"\"%F5*$)F9FSF5F5F57&*$)F(FSF5*$)F,FSF 5*$)F/FSF5,$*&#F5\"\"&F5*$)F9FjnF5F5F5Q(pprint56\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"cG6#\"\"),$*&#\"\"\"\"\"&F+*(,**(\"#?F+&F%6#\"\" 'F+&F%6#\"\"(F+F+*(\"#:F+F1F+&F%6#\"\"*F+!\"\"*&\"#7F+)F9\"\"#F+F+*(F8 F+F4F+F9F+F " 0 "" {MPLTEXT 1 0 73 "Int(x*(x-c[6])*(x-c[7])*(x-c [8]),x=0..c[9])=0;\nc[8]=solve(value(%),c[8]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$**%\"xG\"\"\",&F(F)&%\"cG6#\"\"'!\"\"F),&F(F) &F,6#\"\"(F/F),&F(F)&F,6#\"\")F/F)/F(;\"\"!&F,6#\"\"*F:" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"),$*&#\"\"\"\"\"&F+*(&F%6#\"\"*F+,* *&\"#7F+)F.\"\"#F+F+*(\"#:F+F.F+&F%6#\"\"'F+!\"\"*(F7F+&F%6#\"\"(F+F.F +F;*(\"#?F+F8F+F=F+F+F+,**&\"\"$F+F4F+F+*(\"\"%F+F.F+F8F+F;*(FFF+F=F+F .F+F;*(F:F+F8F+F=F+F+F;F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "Substituting \+ " }{XPPEDIT 18 0 "c[6]=(6-sqrt(6))/10" "6#/&%\"cG6#\"\"'*&,&F'\"\"\"-% %sqrtG6#F'!\"\"F*\"#5F." }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[8]" "6#&%\" cG6#\"\")" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[7]=(6+sqrt(6))/10" "6#/&%\"cG6#\"\"(*&,&\"\"'\"\"\"-%%sqrtG6#F*F+F+\"#5!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[8]" "6#&%\"cG6#\"\")" }{TEXT -1 10 " or \+ " }{XPPEDIT 18 0 "c[6] = (6+sqrt(6))/10;" "6#/&%\"cG6#\"\"'*&,&F'\"\" \"-%%sqrtG6#F'F*F*\"#5!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[8]" "6# &%\"cG6#\"\")" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[7] = (6-sqrt(6) )/10;" "6#/&%\"cG6#\"\"(*&,&\"\"'\"\"\"-%%sqrtG6#F*!\"\"F+\"#5F/" } {TEXT -1 1 " " }{XPPEDIT 18 0 "c[8]" "6#&%\"cG6#\"\")" }{TEXT -1 26 " \+ in this equation gives " }{XPPEDIT 18 0 "c[8]=4/3" "6#/&%\"cG6#\"\") *&\"\"%\"\"\"\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[9]" "6#&%\" cG6#\"\"*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 170 "eq3 := int(x*(x-c[6])*(x-c[7])*(x- c[8]),x=0..c[9])=0:\nfactor(simplify(subs(\{c[6]=1/10*(6-6^(1/2))*c[8] ,c[7]=1/10*(6+6^(1/2))*c[8]\},eq3)));\nop(solve(\{%,c[8]<>c[9]\},c[8]) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*&#\"\"\"\"#?F'*()&%\"cG6#\" \"*\"\"#F',&*&\"\"%F'F+F'!\"\"*&\"\"$F'&F,6#\"\")F'F'F'),&F6F'F+F3F/F' F'F3\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"),$*&#\"\"% \"\"$\"\"\"&F%6#\"\"*F-F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 170 "eq3 := int(x*(x-c[6])*(x-c[7])*(x- c[8]),x=0..c[9])=0:\nfactor(simplify(subs(\{c[6]=1/10*(6+6^(1/2))*c[8] ,c[7]=1/10*(6-6^(1/2))*c[8]\},eq3)));\nop(solve(\{%,c[8]<>c[9]\},c[8]) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*&#\"\"\"\"#?F'*()&%\"cG6#\" \"*\"\"#F',&*&\"\"%F'F+F'!\"\"*&\"\"$F'&F,6#\"\")F'F'F'),&F6F'F+F3F/F' F'F3\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"),$*&#\"\"% \"\"$\"\"\"&F%6#\"\"*F-F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "We can express " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG 6#\"\"%" }{TEXT -1 15 " in terms of " }{XPPEDIT 18 0 "c[5]" "6#&%\"c G6#\"\"&" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[8]" "6#&%\"cG6#\"\") " }{TEXT -1 22 " by substituting for " }{XPPEDIT 18 0 "c[6]" "6#&%\"c G6#\"\"'" }{TEXT -1 19 " in the equation " }{XPPEDIT 18 0 "c[4] = c[ 6]*(3*c[6]-4*c[5])/(2*(2*c[6]-3*c[5]))" "6#/&%\"cG6#\"\"%*(&F%6#\"\"' \"\"\",&*&\"\"$F,&F%6#F+F,F,*&F'F,&F%6#\"\"&F,!\"\"F,*&\"\"#F,,&*&F8F, &F%6#F+F,F,*&F/F,&F%6#F5F,F6F,F6" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 33 "There are the two possibilities: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "c[6]=1/10*(6-6^(1/2))*c[8];\nsubs(%,c[4]=c[6]*( 3*c[6]-4*c[5])/(2*(2*c[6]-3*c[5]))):\nfactor(rationalize(%));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"',$*&#\"\"\"\"#5F+*&,&F'F +*$F'#F+\"\"#!\"\"F+&F%6#\"\")F+F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%\"cG6#\"\"%,$*&#\"\"\"\"#gF+**,&\"\"'!\"\"*$F/#F+\"\"#F+F+,(*&\" \"*F+&F%6#\"\")F+F0*&\"#CF+&F%6#\"\"&F+F+*(F'F+F " 0 "" {MPLTEXT 1 0 106 "c[6]=1/10*(6+6^(1/2))*c [8];\nsubs(%,c[4]=c[6]*(3*c[6]-4*c[5])/(2*(2*c[6]-3*c[5]))):\nfactor(r ationalize(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"',$*&# \"\"\"\"#5F+*&,&F'F+*$F'#F+\"\"#F+F+&F%6#\"\")F+F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"%,$*&#\"\"\"\"#gF+**,&\"\"'F+*$F/#F+\" \"#F+F+,(*&\"\"*F+&F%6#\"\")F+F+*&\"#CF+&F%6#\"\"&F+!\"\"*(F'F+F;F+F/F 1F+F+F6F+,(*&F/F+F;F+F>*&F;F+F/F1F+*&F2F+F6F+F+F>F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Thus we have:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[3]=2/3" "6#/&%\"cG6#\"\"$*&\"\"#\"\" \"F'!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" } {TEXT -1 8 ", " }{XPPEDIT 18 0 "c[9]=3/4" "6#/&%\"cG6#\"\"**&\" \"$\"\"\"\"\"%!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[8]" "6#&%\"cG6# \"\")" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 13 "and either ( " }{TEXT 260 1 "A" }{TEXT 276 1 " " }{TEXT -1 1 ")" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[6]=((6-sqrt(6))*c[8])/10" "6#/&%\"cG6 #\"\"'*(,&F'\"\"\"-%%sqrtG6#F'!\"\"F*&F%6#\"\")F*\"#5F." }{TEXT -1 8 " , " }{XPPEDIT 18 0 "c[7]=((6+sqrt(6))*c[8])/10" "6#/&%\"cG6#\"\" (*(,&\"\"'\"\"\"-%%sqrtG6#F*F+F+&F%6#\"\")F+\"#5!\"\"" }{TEXT -1 8 ", \+ " }{XPPEDIT 18 0 "c[4] = (6-sqrt(6))*(9*c[8]-24*c[5]-4*c[5]*sqrt (6))*c[8]/(60*(2*c[8]-6*c[5]-c[5]*sqrt(6)));" "6#/&%\"cG6#\"\"%**,&\" \"'\"\"\"-%%sqrtG6#F*!\"\"F+,(*&\"\"*F+&F%6#\"\")F+F+*&\"#CF+&F%6#\"\" &F+F/*(F'F+&F%6#F:F+-F-6#F*F+F/F+&F%6#F5F+*&\"#gF+,(*&\"\"#F+&F%6#F5F+ F+*&F*F+&F%6#F:F+F/*&&F%6#F:F+-F-6#F*F+F/F+F/" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 6 "or ( " }{TEXT 260 1 "B" }{TEXT -1 2 " )" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[6] = (6+sqrt(6))* c[8]/10;" "6#/&%\"cG6#\"\"'*(,&F'\"\"\"-%%sqrtG6#F'F*F*&F%6#\"\")F*\"# 5!\"\"" }{TEXT -1 8 ", " }{XPPEDIT 18 0 "c[7] = (6-sqrt(6))*c[8] /10;" "6#/&%\"cG6#\"\"(*(,&\"\"'\"\"\"-%%sqrtG6#F*!\"\"F+&F%6#\"\")F+ \"#5F/" }{TEXT -1 8 ", " }{XPPEDIT 18 0 "c[4] = (6+sqrt(6))*(9*c [8]-24*c[5]+4*c[5]*sqrt(6))*c[8]/(60*(2*c[8]-6*c[5]+c[5]*sqrt(6)));" " 6#/&%\"cG6#\"\"%**,&\"\"'\"\"\"-%%sqrtG6#F*F+F+,(*&\"\"*F+&F%6#\"\")F+ F+*&\"#CF+&F%6#\"\"&F+!\"\"*(F'F+&F%6#F9F+-F-6#F*F+F+F+&F%6#F4F+*&\"#g F+,(*&\"\"#F+&F%6#F4F+F+*&F*F+&F%6#F9F+F:*&&F%6#F9F+-F-6#F*F+F+F+F:" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "The first case (A) gives " }{XPPEDIT 18 0 "c[6] " 0 "" {MPLTEXT 1 0 200 "node_eqsA := [c[3]=2/3*c[4] ,c[9]=3/4*c[8],c[6]=1/10*(6-6^(1/2))*c[8],c[7]=1/10*(6+6^(1/2))*c[8], \n c[4]=1/60*(6-6^(1/2))*(9*c[8]-24*c[5]-4*c[5]*6^(1/2))* c[8]/(2*c[8]-6*c[5]-c[5]*6^(1/2))]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 200 "node_eqsB := [c[3]=2/3*c[ 4],c[9]=3/4*c[8],c[6]=1/10*(6+6^(1/2))*c[8],c[7]=1/10*(6-6^(1/2))*c[8] ,\n c[4]=1/60*(6+6^(1/2))*(9*c[8]-24*c[5]+4*c[5]*6^(1/2)) *c[8]/(2*c[8]-6*c[5]+c[5]*6^(1/2))]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 29 "#============================" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 29 "Relations between the nodes B" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 122 "The relations between the node s obtained in the previous section can be obtained directly from the s tage-order conditions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 857 "SO_eqs := [op(StageOrderConditions (2,4..9,'expanded')),\n op(StageOrderConditions(3,4..9,' expanded')),\n op(StageOrderConditions(4,6..9,'expanded' )),\n op(StageOrderConditions(5,8..9,'expanded'))]:\ne_1 := \{seq(a[i,2]=0,i=4..9),seq(a[i,3]=0,i=6..9),\n seq(a[i, 4]=0,i=8..9),seq(a[i,5]=0,i=8..9)\}:\nnode_eqs := subs(e_1,SO_eqs):\ns ol_1 := op(remove(u_->subs(u_,a[9,8])=0,[solve(\{op(node_eqs)\},indets (node_eqs) minus \{c[2],c[5],c[8]\})])):\ns_1 := factor(rationalize([a llvalues(\{seq(c[i]=subs(sol_1,c[i]),i=[4,6,7,9])\})])):op(1,%);op(2,% %);\nsA,sB := selectremove(u->evalf(coeff(subs(u,c[6]),c[8])%*node_eqsAG7'/&%\"cG6#\"\"(, $*&#\"\"\"\"#5F.*&,&\"\"'F.*$F2#F.\"\"#F.F.&F(6#\"\")F.F.F./&F(6#\"\"$ ,$*&#F5F%*node_eqsBG7'/&%\" cG6#\"\"%,$*&#\"\"\"\"#gF.**,&\"\"'F.*$F2#F.\"\"#F.F.,(*&\"\"*F.&F(6# \"\")F.F.*&\"#CF.&F(6#\"\"&F.!\"\"*(F*F.F>F.F2F4F.F.F9F.,(*&F2F.F>F.FA *&F>F.F2F4F.*&F5F.F9F.F.FAF.F./&F(6#F2,$*&#F.\"#5F.*&F1F.F9F.F.F./&F(6 #\"\"$,$*&#F5FRF.F'F.F./&F(6#F8,$*&#FRF*F.F9F.F./&F(6#\"\"(,$*&#F.FMF. *&,&F2FAF3F.F.F9F.F.FA" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Thus we have:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[3]=2/3 " "6#/&%\"cG6#\"\"$*&\"\"#\"\"\"F'!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 8 ", " }{XPPEDIT 18 0 "c[9] =3/4" "6#/&%\"cG6#\"\"**&\"\"$\"\"\"\"\"%!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "c[8]" "6#&%\"cG6#\"\")" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 13 "and either ( " }{TEXT 260 1 "A" }{TEXT 267 1 " " } {TEXT -1 1 ")" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[6] =((6-sqrt(6))*c[8])/10" "6#/&%\"cG6#\"\"'*(,&F'\"\"\"-%%sqrtG6#F'!\"\" F*&F%6#\"\")F*\"#5F." }{TEXT -1 8 ", " }{XPPEDIT 18 0 "c[7]=((6+ sqrt(6))*c[8])/10" "6#/&%\"cG6#\"\"(*(,&\"\"'\"\"\"-%%sqrtG6#F*F+F+&F% 6#\"\")F+\"#5!\"\"" }{TEXT -1 8 ", " }{XPPEDIT 18 0 "c[4] = (6-s qrt(6))*(9*c[8]-24*c[5]-4*c[5]*sqrt(6))*c[8]/(60*(2*c[8]-6*c[5]-c[5]*s qrt(6)));" "6#/&%\"cG6#\"\"%**,&\"\"'\"\"\"-%%sqrtG6#F*!\"\"F+,(*&\"\" *F+&F%6#\"\")F+F+*&\"#CF+&F%6#\"\"&F+F/*(F'F+&F%6#F:F+-F-6#F*F+F/F+&F% 6#F5F+*&\"#gF+,(*&\"\"#F+&F%6#F5F+F+*&F*F+&F%6#F:F+F/*&&F%6#F:F+-F-6#F *F+F/F+F/" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "or ( " } {TEXT 260 1 "B" }{TEXT -1 2 " )" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "c[6] = (6+sqrt(6))*c[8]/10;" "6#/&%\"cG6#\"\"'*(,&F'\" \"\"-%%sqrtG6#F'F*F*&F%6#\"\")F*\"#5!\"\"" }{TEXT -1 8 ", " } {XPPEDIT 18 0 "c[7] = (6-sqrt(6))*c[8]/10;" "6#/&%\"cG6#\"\"(*(,&\"\"' \"\"\"-%%sqrtG6#F*!\"\"F+&F%6#\"\")F+\"#5F/" }{TEXT -1 8 ", " } {XPPEDIT 18 0 "c[4] = (6+sqrt(6))*(9*c[8]-24*c[5]+4*c[5]*sqrt(6))*c[8] /(60*(2*c[8]-6*c[5]+c[5]*sqrt(6)));" "6#/&%\"cG6#\"\"%**,&\"\"'\"\"\"- %%sqrtG6#F*F+F+,(*&\"\"*F+&F%6#\"\")F+F+*&\"#CF+&F%6#\"\"&F+!\"\"*(F'F +&F%6#F9F+-F-6#F*F+F+F+&F%6#F4F+*&\"#gF+,(*&\"\"#F+&F%6#F4F+F+*&F*F+&F %6#F9F+F:*&&F%6#F9F+-F-6#F*F+F+F+F:" }{TEXT -1 1 "," }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "The first case (A) gi ves " }{XPPEDIT 18 0 "c[6] " 0 "" {MPLTEXT 1 0 200 "node_eqsA := [c[3]=2/3*c[4],c[9]=3/4*c[8],c[6]=1/10*(6-6^(1/2))*c [8],c[7]=1/10*(6+6^(1/2))*c[8],\n c[4]=1/60*(6-6^(1/2))*( 9*c[8]-24*c[5]-4*c[5]*6^(1/2))*c[8]/(2*c[8]-6*c[5]-c[5]*6^(1/2))]:" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 200 "node_eqsB := [c[3]=2/3*c[4],c[9]=3/4*c[8],c[6]=1/10*(6+6^(1/2)) *c[8],c[7]=1/10*(6-6^(1/2))*c[8],\n c[4]=1/60*(6+6^(1/2)) *(9*c[8]-24*c[5]+4*c[5]*6^(1/2))*c[8]/(2*c[8]-6*c[5]+c[5]*6^(1/2))]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 260 0 "" }{TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT 277 21 "_____________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "#======== =====================" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 29 "Verner's \"most robust\" scheme" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 63 "The following ( \"most robust\" ) sc heme comes from his website: " }{URLLINK 17 "http://www.math.sfu.ca/~j verner/" 4 "http://www.math.sfu.ca/~jverner/" "" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 144 "See: J.H. Verner, SIAM Journal of Numeri cal Analysis 1978, 772-790, \"Explicit Runge-Kutta methods with estima tes of the Local Truncation Error.\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "The nodes are: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[2] = 1/25" "6#/&%\"cG6#\"\"#*&\"\"\"F )\"#D!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3] = 48/335-32/1675*sq rt(6)" "6#/&%\"cG6#\"\"$,&*&\"#[\"\"\"\"$N$!\"\"F+*(\"#KF+\"%v;F--%%sq rtG6#\"\"'F+F-" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 72/335-48/167 5*sqrt(6)" "6#/&%\"cG6#\"\"%,&*&\"#s\"\"\"\"$N$!\"\"F+*(\"#[F+\"%v;F-- %%sqrtG6#\"\"'F+F-" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5] = 72/125" " 6#/&%\"cG6#\"\"&*&\"#s\"\"\"\"$D\"!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6] = 48/125-8/125*sqrt(6)" "6#/&%\"cG6#\"\"',&*&\"#[\"\"\"\"$D \"!\"\"F+*(\"\")F+F,F--%%sqrtG6#F'F+F-" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7] = 48/125+8/125*sqrt(6)" "6#/&%\"cG6#\"\"(,&*&\"#[\"\"\"\"$D \"!\"\"F+*(\"\")F+F,F--%%sqrtG6#\"\"'F+F+" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[8] = 16/25" "6#/&%\"cG6#\"\")*&\"#;\"\"\"\"#D!\"\"" } {TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "c[ 9] = 12/25" "6#/&%\"cG6#\"\"**&\"#7\"\"\"\"#D!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[10] = 3377/50000" "6#/&%\"cG6#\"#5*&\"%xL\"\"\"\"&++ &!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[11] = 1/4" "6#/&%\"cG6#\"# 6*&\"\"\"F)\"\"%!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[12] = 57617 028499878/85094827871699" "6#/&%\"cG6#\"#7*&\"/y)*\\Gqhd\"\"\"\"/*pry# [4&)!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[13] = 1623/2000" "6#/&% \"cG6#\"#8*&\"%B;\"\"\"\"%+?!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c [14] = 453/500" "6#/&%\"cG6#\"#9*&\"$`%\"\"\"\"$+&!\"\"" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "c[15] = 1" "6#/&%\"cG6#\"#:\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[16] = 1" "6#/&%\"cG6#\"#;\"\"\"" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "#---- -----------------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "checking the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "coefficients of the combined scheme " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13413 "ee := \{c[2]=1/25,\nc[3]=48/335-32/1675*6^(1/2),\nc[4]=72/335 -48/1675*6^(1/2),\nc[5]=72/125,\nc[6]=48/125-8/125*6^(1/2),\nc[7]=48/1 25+8/125*6^(1/2),\nc[8]=16/25,\nc[9]=12/25,\nc[10]=3377/50000,\nc[11]= 1/4,\nc[12]=57617028499878/85094827871699,\nc[13]=1623/2000,\nc[14]=45 3/500,\nc[15]=1,\nc[16]=1,\n\na[2,1]=1/25,\na[3,1]=-15792/112225+5536/ 112225*6^(1/2),\na[3,2]=31872/112225-1536/22445*6^(1/2),\na[4,1]=18/33 5-12/1675*6^(1/2),\na[4,2]=0,\na[4,3]=54/335-36/1675*6^(1/2),\na[5,1]= 4014/3125+252/625*6^(1/2),\na[5,2]=0,\na[5,3]=-14742/3125-972/625*6^(1 /2),\na[5,4]=12528/3125+144/125*6^(1/2),\na[6,1]=1232/16875-152/16875* 6^(1/2),\na[6,2]=0,\na[6,3]=0,\na[6,4]=29684/106875-13372/320625*6^(1/ 2),\na[6,5]=2132/64125-284/21375*6^(1/2),\na[7,1]=2032/16875+152/16875 *6^(1/2),\na[7,2]=0,\na[7,3]=0,\na[7,4]=-7348/98325-33652/294975*6^(1/ 2),\na[7,5]=10132/64125-716/21375*6^(1/2),\na[7,6]=2592/14375+2912/143 75*6^(1/2),\na[8,1]=16/225,\na[8,2]=0,\na[8,3]=0,\na[8,4]=0,\na[8,5]=0 ,\na[8,6]=64/225+4/225*6^(1/2),\na[8,7]=64/225-4/225*6^(1/2),\na[9,1]= 57/800,\na[9,2]=0,\na[9,3]=0,\na[9,4]=0,\na[9,5]=0,\na[9,6]=177/800+69 /1600*6^(1/2),\na[9,7]=177/800-69/1600*6^(1/2),\na[9,8]=-27/800,\na[10 ,1]=2844530829046074022657/58982400000000000000000,\na[10,2]=0,\na[10, 3]=0,\na[10,4]=0,\na[10,5]=0,\na[10,6]=4287156859652598464203/58982400 000000000000000-1598864762333658025459/117964800000000000000000*6^(1/2 ),\na[10,7]=4287156859652598464203/58982400000000000000000+15988647623 33658025459/117964800000000000000000*6^(1/2),\na[10,8]=-14103388621860 4337343/6553600000000000000000,\na[10,9]=-21409264848554971927/2048000 00000000000000,\na[11,1]=-72189389771/9959178240000-459663572789/59755 069440000*6^(1/2),\na[11,2]=0,\na[11,3]=0,\na[11,4]=0,\na[11,5]=0,\na[ 11,6]=1/30,\na[11,7]=-14201240926266911/557169364500480000-31790792357 660029/557169364500480000*6^(1/2),\na[11,8]=22414436941/1563197440000+ 459663572789/56275107840000*6^(1/2),\na[11,9]=154180604903/25341542400 00+459663572789/11403694080000*6^(1/2),\na[11,10]=21871487332435000000 /125536952879579583419+18386542911560000000/1129832575916216250771*6^( 1/2),\na[12,1]=-178144571353393080183496267158614821877982611914666395 752937745405391408707734804982447062502773/124771801011299405474614551 6410425353598134947600568397156373491324203879120405304413829778240000 +352591194575569317115651180991223097026568880384478484650944931412648 046608828531034562538513357/811016706573446135584994585666776479838787 7159403694581516427693607325214282634478689893558560000*6^(1/2),\na[12 ,2]=0,\na[12,3]=0,\na[12,4]=0,\na[12,5]=0,\na[12,6]=-39115022545645779 6885858319889751408825022451618311227032014355910365202109457518505831 37867/\n38425428798268102367071108589483043166777031255688218331140907 046999467997785653437677908480-490826700396287454540598331961129757400 186839154291202145347161815801053957874498846640625/\n1537017151930724 0946828443435793217266710812502275287332456362818799787199114261375071 163392*6^(1/2),\na[12,7]=-62206429668093251619352586547306026465246348 6635595667447855699920023546211607797417635623004817316459/90745170356 7059050763665262359998604536360756710836393258911136378360869155327677 312519401429106240000+643210041535328932923955834959360270277930780334 485030265105750796567439186095378077346484617022694073/181490340713411 8101527330524719997209072721513421672786517822272756721738310655354625 038802858212480000*6^(1/2),\na[12,8]=120394354672838529464426818685476 9106596033156989737459970836017592684437096940490927391123081521/65467 3243150492794078974012593317347953963670018788019816813203813253436012 2730704444373657280000-15111051196095827876385050613909561301138666302 1919350564690684891134877118069370443383945077153/\n327336621575246397 0394870062966586739769818350093940099084066019066267180061365352222186 828640000*6^(1/2),\na[12,9]=636814661567013784495259031609289617901004 34377497936201427140303316098275608362465676059840449/1190576349318934 9085278056977778076911485016760607844450768012710791316534229770595631 4829715000-35259119457556931711565118099122309702656888038447848465094 4931412648046608828531034562538513357/15477492541146153810861474071111 49998493052178879019778599841652402871149449870177432092786295000*6^(1 /2),\na[12,10]=1027893404876323970566863593065627643827980253781207669 1952406255660602976082835598179622579836784640000000/20706607333558650 0045638539160747510779946083475530875873014324650020277019795123510531 14927376343490460197-2475794627968340439266926692582942523578559086951 1266396518921812563538267024485653444025561440747520000000/26918589533 6262450059330100908971764013929908518190138634918622045026360125733660 563690494055892465375982561*6^(1/2),\na[12,11]=19635000096509466380843 956455829094932847113883632439882926745444809822658280742880787456/150 0993312432347748713715179276681373702227783425321028560191681523416718 6635020874092933,\na[13,1]=-254885119509487666021637618429662720370053 87677568247/25343364340644945281003771622773961523200000000000000+2525 608241949563386308964624438617443527/124160178977563548308078592000000 00000000*6^(1/2),\na[13,2]=0,\na[13,3]=0,\na[13,4]=0,\na[13,5]=0,\na[1 3,6]=-2617546081675469247418718340204655213/43139258710988420693360640 0000000000-552470350996365859393640759989/2400793528263703412500070400 000*6^(1/2),\na[13,7]=-45969294618407232267578352626581642155421231201 /10187731154133781589406192186163200000000000000+154049353658281830373 8906021531546510663696889/88588966557685057299184279879680000000000000 0*6^(1/2),\na[13,8]=20301387341801440158880077748955015368218592139771 9/6003783517480134311881093736968119910400000000000000-396881295163502 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" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "for ct from 2 to 6 do\n so ||ct||_16 := StageOrderConditions(ct,16,'expanded');\nend do:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 138 "Stage 3 \+ has stage-order 2, stages 4 and 5 have stage-order 3, stages 6 and 7 h ave stage-order 4, while stages 8 to 16 have stage-order 5. " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 175 "[seq([seq(expand(subs(ee,so ||i||_16[j])),i=2..6)],j=1..14)]:\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,%):\nsim plify(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#70\"\"#\"\"$F%\"\"%F&\" \"&F'F'F'F'F'F'F'F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 27 "The simplifying conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j+1 .. 16) = b[j]*(1-c[j] );" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F, \"#;*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 9 " . . 15, " }}{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 0 151 "[Sum(b[i]*a[i,1],i=1+1..16) =b[1],seq(Sum(b[i]*a[i,j],i=j+1..16)=b[j]*(1-c[j]),j=2..15)]:\nmap(u-> lhs(u)-rhs(u),subs(ee,eval(subs(Sum=add,%))));\nnops(%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#71\"\"!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 27 "The simplifying conditions:" }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(`b*`[i]*a[i,j],i = j+1 .. 16) = ` b*`[j]*(1-c[j]);" "6#/-%$SumG6$*&&%#b*G6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/ F+;,&F0F,F,F,\"#;*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 9 " . . 15, " }} {PARA 0 "" 0 "" {TEXT -1 8 "(where " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\" cG6#\"\"\"\"\"!" }{TEXT -1 17 " ) are satisfied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 163 "[Sum(`b*`[i ]*a[i,1],i=1+1..16)=`b*`[1],seq(Sum(`b*`[i]*a[i,j],i=j+1..16)=`b*`[j]* (1-c[j]),j=2..15)]:\nmap(u->lhs(u)-rhs(u),subs(ee,eval(subs(Sum=add,%) )));\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#71\"\"!F$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 93 "We can calculate t he principal error norm, that is, the 2-norm of the principal error te rms. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "errterms9_16 := Pr incipalErrorTerms(9,16,'expanded'):\nsm := 0:\nfor ct to nops(errterms 9_16) do\n sm := sm+(evalf(subs(ee,errterms9_16[ct])))^2;\nend do:\n sqrt(sm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+5k!=^$!#;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "We can also cal culate the principal error norm of the order 8 embedded scheme." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "`errterms8_16*` := subs(b=` b*`,PrincipalErrorTerms(8,16,'expanded')):\nsm := 0:\nfor ct to nops(` errterms8_16*`) do\n sm := sm+(evalf(subs(ee,`errterms8_16*`[ct])))^ 2;\nend do:\nsqrt(sm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"++71^T!#: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#----------------- ----------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 34 "construction of the order 9 scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 94 "First we construct a 15 stage orde r 9 scheme starting with a consideration of stages 2 to 11. " }}{PARA 0 "" 0 "" {TEXT -1 22 "We specify the nodes: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "c[2]=1/25" "6#/&%\"cG6#\"\"#*&\"\"\"F) \"#D!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5]=72/125" "6#/&%\"cG6# \"\"&*&\"#s\"\"\"\"$D\"!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[8]=1 6/25" "6#/&%\"cG6#\"\")*&\"#;\"\"\"\"#D!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[10]=3377/50000" "6#/&%\"cG6#\"#5*&\"%xL\"\"\"\"&++&! \"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[11] = 1/4;" "6#/&%\"cG6#\"#6 *&\"\"\"F)\"\"%!\"\"" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 35 " and the zero linking coefficients: " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "a[4,2]=0" "6#/&%\"aG6$\"\"%\"\"#\"\"!" }{TEXT -1 2 " , " }{XPPEDIT 18 0 "a[5,2]=0" "6#/&%\"aG6$\"\"&\"\"#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[6,2]=0" "6#/&%\"aG6$\"\"'\"\"#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[6,3]=0" "6#/&%\"aG6$\"\"'\"\"$\"\"!" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "a[7,2]=0" "6#/&%\"aG6$\"\"(\"\"#\"\"! " }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[7,3]=0" "6#/&%\"aG6$\"\"(\"\"$\" \"!" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[8,2]=0" "6#/&%\"aG6$\"\")\"\"#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[8,3]=0" "6#/&%\"aG6$\"\")\"\"$\"\"!" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "a[8,4]=0" "6#/&%\"aG6$\"\")\"\"%\"\"!" }{TEXT -1 2 ", \+ " }{XPPEDIT 18 0 "a[8,5]=0" "6#/&%\"aG6$\"\")\"\"&\"\"!" }{TEXT -1 2 " , " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[9,2]=0" "6#/& %\"aG6$\"\"*\"\"#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[9,3]=0" "6# /&%\"aG6$\"\"*\"\"$\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[9,4]=0" " 6#/&%\"aG6$\"\"*\"\"%\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[9,5]=0 " "6#/&%\"aG6$\"\"*\"\"&\"\"!" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[10,2]=0" "6#/&%\"aG6$\"#5\"\"#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[10,3]=0" "6#/&%\"aG6$\"#5\"\"$\"\"! " }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[10,4]=0" "6#/&%\"aG6$\"#5\"\"%\" \"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[10,5]=0" "6#/&%\"aG6$\"#5\"\"& \"\"!" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[11,2]=0" "6#/&%\"aG6$\"#6\"\"#\"\"!" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "a[11,3]=0" "6#/&%\"aG6$\"#6\"\"$\"\"!" }{TEXT -1 2 ", \+ " }{XPPEDIT 18 0 "a[11,4]=0" "6#/&%\"aG6$\"#6\"\"%\"\"!" }{TEXT -1 2 " , " }{XPPEDIT 18 0 "a[11,5]=0" "6#/&%\"aG6$\"#6\"\"&\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 20 " in stages 2 to 11. " }}{PARA 0 " " 0 "" {TEXT -1 17 "We also specify " }{XPPEDIT 18 0 "a[11,6] = 1/30; " "6#/&%\"aG6$\"#6\"\"'*&\"\"\"F*\"#I!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "The stage-order e quations relating to these stages, such that stages " }{XPPEDIT 18 0 " 3,4,5,6,7,8,9,10,11;" "6+\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#5\"#6" }{TEXT -1 19 " have stage orders " }{XPPEDIT 18 0 "2,3,3,4,4,5,5,5,5; " "6+\"\"#\"\"$F$\"\"%F%\"\"&F&F&F&" }{TEXT -1 96 " respectively taken together with the row-sum conditions can then be solved to obtain the nodes " }{XPPEDIT 18 0 "c[3],c[4],c[6],c[7],c[9],c[11];" "6(&%\"cG6# \"\"$&F$6#\"\"%&F$6#\"\"'&F$6#\"\"(&F$6#\"\"*&F$6#\"#6" }{TEXT -1 67 " and the remaining non-zero linking coefficients for these stages. " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "The comp utation is made more efficient by " }{TEXT 260 48 "including explicitl y relations between the nodes" }{TEXT -1 42 " arising from the stage-o rder conditions: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "c[3]=2/3" "6#/&%\"cG6#\"\"$*&\"\"#\" \"\"F'!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" } {TEXT -1 8 ", " }{XPPEDIT 18 0 "c[4] = (6-sqrt(6))*(9*c[8]-24*c[ 5]-4*c[5]*sqrt(6))*c[8]/(60*(2*c[8]-6*c[5]-c[5]*sqrt(6)));" "6#/&%\"cG 6#\"\"%**,&\"\"'\"\"\"-%%sqrtG6#F*!\"\"F+,(*&\"\"*F+&F%6#\"\")F+F+*&\" #CF+&F%6#\"\"&F+F/*(F'F+&F%6#F:F+-F-6#F*F+F/F+&F%6#F5F+*&\"#gF+,(*&\" \"#F+&F%6#F5F+F+*&F*F+&F%6#F:F+F/*&&F%6#F:F+-F-6#F*F+F/F+F/" }{TEXT -1 2 ", " }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[6]=((6-sqrt(6))*c[8])/10" "6#/&%\"cG6#\"\"' *(,&F'\"\"\"-%%sqrtG6#F'!\"\"F*&F%6#\"\")F*\"#5F." }{TEXT -1 8 ", \+ " }{XPPEDIT 18 0 "c[7]=((6+sqrt(6))*c[8])/10" "6#/&%\"cG6#\"\"(*(,& \"\"'\"\"\"-%%sqrtG6#F*F+F+&F%6#\"\")F+\"#5!\"\"" }{TEXT -1 8 ", \+ " }{XPPEDIT 18 0 "c[9]=3/4" "6#/&%\"cG6#\"\"**&\"\"$\"\"\"\"\"%!\"\" " }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[8]" "6#&%\"cG6#\"\")" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Th e equations that lead to " }{XPPEDIT 18 0 "c[6] < c[7]" "6#2&%\"cG6# \"\"'&F%6#\"\"(" }{TEXT -1 19 ", have been chosen." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 812 "RSeqs := Ro wSumConditions(11,'expanded'):\nSOeqs := [op(StageOrderConditions(2,11 ,'expanded')),\n op(StageOrderConditions(3,4..11,'expand ed')),\n op(StageOrderConditions(4,6..11,'expanded')),\n op(StageOrderConditions(5,8..11,'expanded'))]:\n\nnode_ eqsA := [c[3]=2/3*c[4],c[9]=3/4*c[8],c[6]=1/10*(6-6^(1/2))*c[8],c[7]=1 /10*(6+6^(1/2))*c[8],\n c[4]=1/60*(6-6^(1/2))*(9*c[8]-24*c[5]-4*c [5]*6^(1/2))*c[8]/(2*c[8]-6*c[5]-c[5]*6^(1/2))]:\n\ne1 := \{c[2]=1/25, c[5]=72/125,c[8]=16/25,c[10]=3377/50000,c[11]=1/4,\n seq(a[ i,2]=0,i=4..11),seq(a[i,3]=0,i=6..11),\n seq(a[i,4]=0,i=8.. 11),seq(a[i,5]=0,i=8..11),a[11,6]=1/30\}:\neqns := expand(rationalize( subs(e1,[op(RSeqs),op(SOeqs),op(node_eqsA)]))):\nconvert(ListTools[Enu merate](%),matrix);\n``;\nindets(eqns);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7L7$\"\"\"/&%\"aG6$\"\"#F(#F(\"#D7$F-/,&&F+ 6$\"\"$F(F(&F+6$F5F-F(&%\"cG6#F57$F5/,&&F+6$\"\"%F(F(&F+6$F@F5F(&F96#F @7$F@/,(&F+6$\"\"&F(F(&F+6$FJF5F(&F+6$FJF@F(#\"#s\"$D\"7$FJ/,(&F+6$\" \"'F(F(&F+6$FWF@F(&F+6$FWFJF(&F96#FW7$FW/,*&F+6$\"\"(F(F(&F+6$F]oF@F(& F+6$F]oFJF(&F+6$F]oFWF(&F96#F]o7$F]o/,(&F+6$\"\")F(F(&F+6$F[pFWF(&F+6$ F[pF]oF(#\"#;F/7$F[p/,*&F+6$\"\"*F(F(&F+6$FgpFWF(&F+6$FgpF]oF(&F+6$Fgp F[pF(&F96#Fgp7$Fgp/,,&F+6$\"#5F(F(&F+6$FeqFWF(&F+6$FeqF]oF(&F+6$FeqF[p F(&F+6$FeqFgpF(#\"%xL\"&++&7$Feq/,.&F+6$\"#6F(F(#F(\"#IF(&F+6$FfrF]oF( &F+6$FfrF[pF(&F+6$FfrFgpF(&F+6$FfrFeqF(#F(F@7$Ffr/,$*&F.F(F6F(F(,$*&#F (F-F(*$)F8F-F(F(F(7$\"#7/*&FAF(F8F(,$*&FhsF(*$)FCF-F(F(F(7$\"#8/,&*&FK F(F8F(F(*&FMF(FCF(F(#\"%#f#\"&Dc\"7$\"#9/,&*&FXF(FCF(F(*&FOF(FZF(F(,$* &FhsF(*$)FfnF-F(F(F(7$\"#:/,(*&F^oF(FCF(F(*&FOF(F`oF(F(*&FboF(FfnF(F(, $*&FhsF(*$)FdoF-F(F(F(7$Fap/,&*&F\\pF(FfnF(F(*&F^pF(FdoF(F(#\"$G\"\"$D '7$\"#/,,*&FgrF(FfnF(F(*&FirF(FdoF(F(*&F`pF(F[sF(F(* &F]sF(F^qF(F(*&F^rF(F_sF(F(#F(\"#K7$\"#?/*&FAF(FjsF(,$*&#F(F5F(*$)FCF5 F(F(F(7$\"#@/,&*&FKF(FjsF(F(*&FMF(FbtF(F(#\"';W7\"(DJ&>7$\"#A/,&*&FXF( FbtF(F(*&#\"%%=&F[uF(FZF(F(,$*&F`yF(*$)FfnF5F(F(F(7$\"#B/,(*&F^oF(FbtF (F(*&FbzF(F`oF(F(*&FboF(FeuF(F(,$*&F`yF(*$)FdoF5F(F(F(7$\"#C/,&*&F\\pF (FeuF(F(*&F^pF(F`vF(F(#\"%'4%\"&vo%7$F//,(*&FhpF(FeuF(F(*&FjpF(F`vF(F( *&#\"$c#FhvF(F\\qF(F(,$*&F`yF(*$)F^qF5F(F(F(7$\"#E/,**&FfqF(FeuF(F(*&F hqF(F`vF(F(*&Fb\\lF(FjqF(F(*&F\\rF(FcwF(F(#\",LOu6&Q\"0++++++v$7$\"#F/ ,,*&FgrF(FduF(F(*&FirF(F`vF(F(*&Fb\\lF(F[sF(F(*&F]sF(FcwF(F(*&#F]x\"++ +++DF(F_sF(F(#F(\"$#>7$\"#G/,&*&FXF(FbyF(F(*&#\"'[KPF[zF(FZF(F(,$*&Fas F(*$)FfnF@F(F(F(7$\"#H/,(*&F^oF(FbyF(F(*&Ff^lF(F`oF(F(*&FboF(FgzF(F(,$ *&FasF(*$)FdoF@F(F(F(7$Fhr/,&*&F\\pF(FgzF(F(*&F^pF(Fb[lF(F(#\"&%Q;\"'D 1R7$\"#J/,(*&FhpF(FgzF(F(*&FjpF(Fb[lF(F(*&#Fj[lF[uF(F\\qF(F(,$*&FasF(* $)F^qF@F(F(F(7$Fix/,**&FfqF(FgzF(F(*&FhqF(Fb[lF(F(*&Ff`lF(FjqF(F(*&F\\ rF(Fg\\lF(F(#\"0T'[#eT0I\"\"5+++++++++D7$\"#L/,,*&FgrF(FfzF(F(*&FirF(F b[lF(F(*&Ff`lF(F[sF(F(*&F]sF(Fg\\lF(F(*&#Fa]l\"0++++++D\"F(F_sF(F(#F( \"%C57$\"#M/,&*&F\\pF(F[_lF(F(*&F^pF(Ff_lF(F(#\"(w&[5\")D\"G)[7$\"#N/, (*&FhpF(F[_lF(F(*&FjpF(Ff_lF(F(*&#\"&Ob'F^`lF(F\\qF(F(,$*&#F(FJF(*$)F^ qFJF(F(F(7$\"#O/,**&FfqF(F[_lF(F(*&FhqF(Ff_lF(F(*&FbclF(FjqF(F(*&F\\rF (Fj`lF(F(#\"3d1m0C*G>R%\":++++++++++Dc\"7$\"#P/,,*&FgrF(Fj^lF(F(*&FirF (Ff_lF(F(*&FbclF(F[sF(F(*&F]sF(Fj`lF(F(*&#Fcal\"4++++++++D'F(F_sF(F(#F (\"%?^7$\"#Q/F8,$*&#F-F5F(FCF(F(7$\"#R/F^q#F\\tF/7$\"#S/Ffn,&#\"#[FQF( *(F[pF(FQ!\"\"FWFhsFbfl7$\"#T/Fdo,&F_flF(*(F[pF(FQFbflFWFhsF(7$\"#U/FC ,&#FP\"$N$F(*(F`flF(\"%v;FbflFWFhsFbflQ(pprint66\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "e2 := expand(rationalize(solve(\{op (eqns)\}))):\ne3 := `union`(e1,e2):\ninfolevel[solve] := 0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 106 " If the equations giving the relations between the nodes are omitted we need to select the solutio n with " }{XPPEDIT 18 0 "c[6] < c[7]" "6#2&%\"cG6#\"\"'&F%6#\"\"(" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 23 " Thus we require that \+ " }{XPPEDIT 18 0 "c[6]=48/125-8/125*sqrt(6)" "6#/&%\"cG6#\"\"',&*&\"#[ \"\"\"\"$D\"!\"\"F+*(\"\")F+F,F--%%sqrtG6#F'F+F-" }{TEXT -1 1 " " } {TEXT 265 1 "~" }{TEXT -1 20 " 0.22723266 and " }{XPPEDIT 18 0 "c[ 7]=48/125+8/125*sqrt(6)" "6#/&%\"cG6#\"\"(,&*&\"#[\"\"\"\"$D\"!\"\"F+* (\"\")F+F,F--%%sqrtG6#\"\"'F+F+" }{TEXT -1 1 " " }{TEXT 266 1 "~" } {TEXT -1 47 " 0.54076734, rather than the other way round. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "The following c ommands achieve this." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 185 "eqns := subs(e1,[op(RSeqs),op(SOeqs)]): \nsol := solve(\{op(eqns)\}):\ne2 := op(select(u_->evalf(subs(u_,c[6]) " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e3 " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2068 "e3 := \{a[6,1] = 1232/16875-152/16875*6^(1/2), a[7,6] = 2592/143 75+2912/14375*6^(1/2), a[11,9] = 154180604903/2534154240000+4596635727 89/11403694080000*6^(1/2), c[7] = 48/125+8/125*6^(1/2), a[3,2] = 31872 /112225-1536/22445*6^(1/2), a[5,1] = 4014/3125+252/625*6^(1/2), a[8,1] = 16/225, a[6,4] = 29684/106875-13372/320625*6^(1/2), c[3] = 48/335-3 2/1675*6^(1/2), c[4] = 72/335-48/1675*6^(1/2), a[11,8] = 22414436941/1 563197440000+459663572789/56275107840000*6^(1/2), a[6,5] = 2132/64125- 284/21375*6^(1/2), a[11,10] = 21871487332435000000/1255369528795795834 19+18386542911560000000/1129832575916216250771*6^(1/2), a[9,8] = -27/8 00, a[9,6] = 177/800+69/1600*6^(1/2), a[3,1] = -15792/112225+5536/1122 25*6^(1/2), a[4,1] = 18/335-12/1675*6^(1/2), c[6] = 48/125-8/125*6^(1/ 2), a[5,3] = -14742/3125-972/625*6^(1/2), a[10,8] = -14103388621860433 7343/6553600000000000000000, c[9] = 12/25, a[8,6] = 64/225+4/225*6^(1/ 2), a[9,7] = 177/800-69/1600*6^(1/2), a[11,6] = 1/30, a[9,5] = 0, a[10 ,2] = 0, a[10,3] = 0, a[10,4] = 0, a[10,5] = 0, a[11,7] = -14201240926 266911/557169364500480000-31790792357660029/557169364500480000*6^(1/2) , a[11,2] = 0, a[11,3] = 0, a[11,4] = 0, a[11,5] = 0, a[8,4] = 0, a[8, 5] = 0, a[9,2] = 0, a[9,3] = 0, a[9,4] = 0, c[10] = 3377/50000, c[11] \+ = 1/4, a[4,2] = 0, a[5,2] = 0, a[6,2] = 0, a[6,3] = 0, c[8] = 16/25, a [7,2] = 0, a[7,3] = 0, a[8,2] = 0, a[8,3] = 0, c[5] = 72/125, a[2,1] = 1/25, a[10,9] = -21409264848554971927/204800000000000000000, a[8,7] = 64/225-4/225*6^(1/2), a[10,1] = 2844530829046074022657/58982400000000 000000000, c[2] = 1/25, a[9,1] = 57/800, a[10,6] = 4287156859652598464 203/58982400000000000000000-1598864762333658025459/1179648000000000000 00000*6^(1/2), a[4,3] = 54/335-36/1675*6^(1/2), a[10,7] = 428715685965 2598464203/58982400000000000000000+1598864762333658025459/117964800000 000000000000*6^(1/2), a[7,1] = 2032/16875+152/16875*6^(1/2), a[7,4] = \+ -7348/98325-33652/294975*6^(1/2), a[5,4] = 12528/3125+144/125*6^(1/2), a[7,5] = 10132/64125-716/21375*6^(1/2), a[11,1] = -72189389771/995917 8240000-459663572789/59755069440000*6^(1/2)\}:" }{TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "printed coe fficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 220 "subs(e3,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(9 -i)],i=2..9)]));\nfor ii from 10 to 11 do\n print(``);\n print(c[i i]=subs(e3,c[ii])); \n for jj to ii-1 do\n print(a[ii,jj]=subs( e3,a[ii,jj]));\n end do:\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #K%'matrixG6#7*7+#\"\"\"\"#DF(%!GF+F+F+F+F+F+7+,&#\"#[\"$N$F)*(\"#KF) \"%v;!\"\"\"\"'#F)\"\"#F4,&#\"&#z:\"'DA6F4*(\"%ObF)F;F4F5F6F),&#\"&s=$ F;F)*(\"%O:F)\"&XC#F4F5F6F4F+F+F+F+F+F+7+,&#\"#sF0F)*(F/F)F3F4F5F6F4,& #\"#=F0F)*(\"#7F)F3F4F5F6F4\"\"!,&#\"#aF0F)*(\"#OF)F3F4F5F6F4F+F+F+F+F +7+#FG\"$D\",&#\"%9S\"%DJF)*(\"$_#F)\"$D'F4F5F6F)FN,&#\"&UZ\"FZF4*(\"$ s*F)FgnF4F5F6F4,&#\"&GD\"FZF)*(\"$W\"F)FVF4F5F6F)F+F+F+F+7+,&#F/FVF)*( \"\")F)FVF4F5F6F4,&#\"%K7\"&vo\"F)*(\"$_\"F)FjoF4F5F6F4FNFN,&#\"&%oH\" 'vo5F)*(\"&sL\"F)\"'D1KF4F5F6F4,&#\"%K@\"&DT'F)*(\"$%GF)\"&v8#F4F5F6F4 F+F+F+7+,&FdoF)*(FfoF)FVF4F5F6F),&#\"%K?FjoF)*(F\\pF)FjoF4F5F6F)FNFN,& #\"%[t\"&D$)*F4*(\"&_O$F)\"'v\\HF4F5F6F4,&#\"&K,\"FgpF)*(\"$;(F)FjpF4F 5F6F4,&#\"%#f#\"&vV\"F)*(\"%7HF)FarF4F5F6F)F+F+7+#\"#;F*#Ffr\"$D#FNFNF NFN,&#\"#kFhrF)*(\"\"%F)FhrF4F5F6F),&FjrF)*(F]sF)FhrF4F5F6F4F+7+#FMF*# \"#d\"$+)FNFNFNFN,&#\"$x\"FdsF)*(\"#pF)\"%+;F4F5F6F),&FfsF)*(FisF)FjsF 4F5F6F4#!#FFdsQ)pprint156\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#5#\"%xL\"&++&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"\"#\"7dE-ug/H3`WG\"8++++++++S# )*e" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"#\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"$\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"',&#\"7.UY)f_'fo:(G%\"8++++++++S#)*e\"\"\"*(\"7 fa-eOLiZ'))f\"F-\"9++++++++!['z6!\"\"F(#F-\"\"#F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"(,&#\"7.UY)f_'fo:(G%\"8++++++++S#)*e\" \"\"*(\"7fa-eOLiZ'))f\"F-\"9++++++++!['z6!\"\"\"\"'#F-\"\"#F-" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\")#!6VtL/'=i)Q.T\"\"7+ +++++++g`l" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"*#!5F>( \\b[[E49#\"6++++++++![?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#6#\"\"\"\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"\",&#\",r(*Q*=s\".++Cy\"f**!\" \"*(\"-*ysNmf%F(\"/++Wp]vfF-\"\"'#F(\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"#6\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\" \"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"'#\"\"\" \"#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"(,&#\"26pEE4C ,U\"\"3++[+XOprb!\"\"*(\"2H+mdBz!zJ\"\"\"F,F-\"\"'#F0\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"),&#\",TpV9C#\".++W(>j:\"\" \"*(\"-*ysNmf%F-\"/++%y5vi&!\"\"\"\"'#F-\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"*,&#\"-.\\g!=a\"\".++CaT`#\"\"\"*(\"-* ysNmf%F-\"/++3%p.9\"!\"\"\"\"'#F-\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"#5,&#\"5+++NCL([r=#\"6>Mez&zG&p`D\"\"\"\"*(\"5+++ g:\"Ha'Q=F-\"7r2D;i\"fdK)H6!\"\"\"\"'#F-\"\"#F-" }}}{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 27 "We do not need \+ to specify " }{XPPEDIT 18 0 "c[12]" "6#&%\"cG6#\"#7" }{TEXT -1 43 ", \+ because, according to Verner, the nodes " }{XPPEDIT 18 0 "c[8]" "6#&% \"cG6#\"\")" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "c[9]" "6#&%\"cG6#\"\"*" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "c[10]" "6#&%\"cG6#\"#5" }{TEXT -1 2 " , " }{XPPEDIT 18 0 "c[11]" "6#&%\"cG6#\"#6" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "c[12]" "6#&%\"cG6#\"#7" }{TEXT -1 30 " 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),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,!\"\"\"\"$-%*factorialG 6#F4F3F-/F,;\"\"!F-F--F&6$*&-%\"qG6#F,F-,&F-F-F,F3F-/F,;F:F-F-*&-F&6$* &-F*6#F,F--F/6#*&,&F-F-F,F3\"\"#-F66#FNF3F-/F,;F:F-F--F&6$*&-F?6#F,F-- F/6#*&,&F-F-F,F3FN-F66#FNF3F-/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[8])*(x-c[9])*(x-c[10])*(x-c[11])" "6#/-%\"pG6#%\"xG* ,F'\"\"\",&F'F)&%\"cG6#\"\")!\"\"F),&F'F)&F,6#\"\"*F/F),&F'F)&F,6#\"#5 F/F),&F'F)&F,6#\"#6F/F)" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "q(x)=(x -c[12])*p(x)" "6#/-%\"qG6#%\"xG*&,&F'\"\"\"&%\"cG6#\"#7!\"\"F*-%\"pG6# F'F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 155 "See: J.H. Verner, SIAM Journal of Numerical Analysi s 1978, 772-790, \"Explicit Runge-Kutta methods with estimates of the \+ Local Truncation Error.\" (page 780)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "p := x -> x*(x-c[8])*(x-c [9])*(x-c[10])*(x-c[11]):\n'p(x)'=p(x);\nq := x -> (x-c[12])*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),&F'F)&F,6#\"#5F/F),&F'F )&F,6#\"#6F/F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"qG6#%\"xG*.,&F' \"\"\"&%\"cG6#\"#7!\"\"F*F'F*,&F'F*&F,6#\"\")F/F*,&F'F*&F,6#\"\"*F/F*, &F'F*&F,6#\"#5F/F*,&F'F*&F,6#\"#6F/F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "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);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %$IeqG/*&-%$IntG6$,$*&#\"\"\"\"\"'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-F8F--F(6$,$*&FEF-*&F>F-FHF-F-F-F8F-" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "e4 := solv e(subs(e3,value(Ieq)),\{c[12]\}):\nc[12]=subs(e4,c[12]);\ne5 := `union `(e3,e4):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#7#\"/y)*\\Gqh d\"/*pry#[4&)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Thus " }{XPPEDIT 18 0 "c[12]=57617028499878/850948278716 99" "6#/&%\"cG6#\"#7*&\"/y)*\\Gqhd\"\"\"\"/*pry#[4&)!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 99 "Now we can use the quadrature eq uations to find the weights once the remaining nodes once the nodes" } }{PARA 256 "" 0 "" {XPPEDIT 18 0 "c[13]=1623/2000" "6#/&%\"cG6#\"#8*& \"%B;\"\"\"\"%+?!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[14]=453/500 " "6#/&%\"cG6#\"#9*&\"$`%\"\"\"\"$+&!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[15]=1" "6#/&%\"cG6#\"#:\"\"\"" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 38 "are specified along with the weights " } {XPPEDIT 18 0 "b[2]=0" "6#/&%\"bG6#\"\"#\"\"!" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "b[3]=0" "6#/&%\"bG6#\"\"$\"\"!" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "b[4]=0" "6#/&%\"bG6#\"\"%\"\"!" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "b[5]=0" "6#/&%\"bG6#\"\"&\"\"!" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "b[6]=0" "6#/&%\"bG6#\"\"'\"\"!" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "b[7]=0" "6#/&%\"bG6#\"\"(\"\"!" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 294 "Qeqs := QuadratureConditions(9,15,'expanded'):\ne6 := \{seq(b[i]= 0,i=2..7),c[13]=1623/2000,c[14]=453/500,c[15]=1\}:\ne7 := `union`(e5,e 6):\nquadeqns := subs(e7,Qeqs):\nfor ct to nops(quadeqns) do\n print (`equation `||ct); print(``);print(quadeqns[ct]);print(``);\nend do: \nindets(quadeqns);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equ ation~~~1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,4&%\"bG6#\"\"\"F(&F&6#\"\")F(&F&6#\"\"*F(&F&6#\"#5F(& F&6#\"#6F(&F&6#\"#7F(&F&6#\"#8F(&F&6#\"#9F(&F&6#\"#:F(F(" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equatio n~~~2G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,2*&#\"#;\"#D\"\"\"&%\"bG6#\"\")F)F)*&#\"#7F(F)&F+6#\" \"*F)F)*&#\"%xL\"&++&F)&F+6#\"#5F)F)*&#F)\"\"%F)&F+6#\"#6F)F)*&#\"/y)* \\Gqhd\"/*pry#[4&)F)&F+6#F0F)F)*&#\"%B;\"%+?F)&F+6#\"#8F)F)*&#\"$`%\"$ +&F)&F+6#\"#9F)F)&F+6#\"#:F)#F)\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~3G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,2*&#\"$ c#\"$D'\"\"\"&%\"bG6#\"\")F)F)*&#\"$W\"F(F)&F+6#\"\"*F)F)*&#\")HTS6\"+ ++++DF)&F+6#\"#5F)F)*&#F)\"#;F)&F+6#\"#6F)F)*&#\"=%)[,Y]p`d:t>s>L\"=,m 90?'4390tH6C(F)&F+6#\"#7F)F)*&#\"(HTj#\"(+++%F)&F+6#\"#8F)F)*&#\"'4_? \"'++DF)&F+6#\"#9F)F)&F+6#\"#:F)#F)\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~4G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,2*& #\"%'4%\"&Dc\"\"\"\"&%\"bG6#\"\")F)F)*&#\"%G\"K*4XRJd1i$Q:_[&)yt9!)o#=;'F)&F+6#\"#7F)F)*&#\"+n8>vU \"+++++!)F)&F+6#\"#8F)F)*&#\")x'fH*\"*+++D\"F)&F+6#\"#9F)F)&F+6#\"#:F) #F)\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~5G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,2*&#\"&Ob'\"'D1R\"\"\"&%\"bG6#\" \")F)F)*&#\"&O2#F(F)&F+6#\"\"*F)F)*&#\"0T'[#eT0I\"\"4++++++++D'F)&F+6# \"#5F)F)*&#F)\"$c#F)&F+6#\"#6F)F)*&#\"YcM`\\&H,6=aEz,_EHlkmlIJ0zRb?5\" \"Y,K&=]6oC'=VCB:l;9#paxE\\8u(fRV_F)&F+6#\"#7F)F)*&#\".T')eN'Qp\"/++++ ++;F)&F+6#\"#8F)F)*&#\",\"oL26U\",++++D'F)&F+6#\"#9F)F)&F+6#\"#:F)#F) \"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~6G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,2*&#\"(w&[5\"(Dcw*\"\"\"&%\"bG6# \"\")F)F)*&#\"'K)[#F(F)&F+6#\"\"*F)F)*&#\"3d1m0C*G>R%\"9++++++++++DJF) &F+6#\"#5F)F)*&#F)\"%C5F)&F+6#\"#6F)F)*&#\"`oo$=HX]-'=:8+)Q&)oO!p[3)p1 aEQZC\"[bps:(\\j\"ao*\\e/;Xyln*fqp1E)[N<9Iv&e@')p(p4g6;ye=Y%F)&F+6#\"# 7F)F)*&#\"2VVOgbSh7\"\"2+++++++?$F)&F+6#\"#8F)F)*&#\"/$\\dB;w!>\"/++++ +DJF)&F+6#\"#9F)F)&F+6#\"#:F)#F)\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~7G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,2*&#\" );sx;\"*D19W#\"\"\"&%\"bG6#\"\")F)F)*&#\"(%)f)HF(F)&F+6#\"\"*F)F)*&#\" 7*oQg\"Rl(RaJ[\"\">++++++++++++Dc\"F)&F+6#\"#5F)F)*&#F)\"%'4%F)&F+6#\" #6F)F)*&#\"^p/\"f>Bumc3%en^L4v0[;t%[)3:%y&HkEW:h!z:v61?v^eO\"_p,)>h=6; =z$peS!pJF@\"z;rn1A3Hz5#\"]q**=T r))pc0OFu`i6ScgM8y4'QDHW[W)*p6C9$G#yKQz)\\x4$)oF)*)3B$F)&F+6#\"#7F)F)* &#\"8ZAYmpJn\\*RmH\"9++++++++++!G\"F)&F+6#\"#8F)F)*&#\"4P5y=7?+Y\"R\"4 +++++++D\"yF)&F+6#\"#9F)F)&F+6#\"#:F)#F)F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~9G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,2*&#\"+'Hn\\H%\"-D1*ye_\"\"\"\"&%\"bG6#\"\")F)F)*&#\"*'p\")*H% F(F)&F+6#\"\"*F)F)*&#\">\")oMy.'ebix2%39p\"\"G++++++++++++++++D1RF)&F+ 6#\"#5F)F)*&#F)\"&Ob'F)&F+6#\"#6F)F)*&#\"jqORIjsmQzlHt!>!4yi;'Q)=0%=Kx ?N5C'*\\oH(Q@.$oH&4FAqyU!)4zB_+5EX@\"\"[r,k%RNd1k/Q#p!=Na#*4SF6&QQq*yC ^r#)GadY**RRB5\"zR'*R;[z58i*f&fP,K\\FF)&F+6#\"#7F)F)*&#\";\")oAolVb>$Q mW\"[\"<++++++++++++c#F)&F+6#\"#8F)F)*&#\"7h(4y5_6*QJt<\"7++++++++D1RF )&F+6#\"#9F)F)&F+6#\"#:F)#F)F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<+&%\"bG6#\"\"\"&F%6#\"\")&F%6#\" \"*&F%6#\"#5&F%6#\"#6&F%6#\"#7&F%6#\"#8&F%6#\"#9&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 73 "e8 := solve(\{op(quadeqns)\} ):\ne9 := `union`(e7,e8):\ninfolevel[solve] := 0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "for ii to 1 5 do\n wt_val := subs(e9,b[ii]);\n if wt_val<>0 then print(b[ii]=w t_val) end if;\nend do:\n``;\nevalf[8](subs(e9,[seq(b[i],i=1..15)])); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"\"#\"<(yv%pY)f,'*\"@w@w4o.:ve/br\"Sxy ,*GHad3R!>8^=2V\\&)>J'[OJa\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"b G6#\"#6#\"=OvXbo![(p2_KrBB\">\"396.Gp7?U^yBN5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#7#\"er>OOE7j%)pEhgc1=nATj-Q*pj/&**p0!4)zf)) 3@\"e^KaSET6RP*Qr6R>JapXTKH+'3$G\"frks%)HH%35'=_N?#ohXY'fcBhe(H!)*3QpJ @#)z)HBe)4D8Jw(e)3*Q<2qvl!o43q1aY:Re\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#8#\"E++++++!3AMl(>TBNOhH\"FR9$4bP!yZRo4F];cK%**Q" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#9#\"B++++D\"GX]q<\"*=l[q6 \"B,w\"Q]$[k)3S(y!eC?!#5$\")r/y@!\")$\")`*[F\"F.$\")xhWAF.$\")XD(y\"F.$\")[ M%f(F&$\")f%[H\"F.$\")[uZHF&" }}}{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 3068 "e9 := \{a[4,2] = 0, c[11] = 1/4, a[2,1] = 1/ 25, a[10,1] = 2844530829046074022657/58982400000000000000000, c[4] = 7 2/335-48/1675*6^(1/2), a[10,7] = 4287156859652598464203/58982400000000 000000000+1598864762333658025459/117964800000000000000000*6^(1/2), a[1 1,9] = 154180604903/2534154240000+459663572789/11403694080000*6^(1/2), a[10,8] = -141033886218604337343/6553600000000000000000, a[9,1] = 57/ 800, c[2] = 1/25, c[5] = 72/125, c[8] = 16/25, c[10] = 3377/50000, a[5 ,2] = 0, a[6,2] = 0, a[6,3] = 0, a[7,2] = 0, a[7,3] = 0, a[8,2] = 0, a [8,3] = 0, a[8,4] = 0, a[8,5] = 0, a[9,2] = 0, a[9,3] = 0, a[9,4] = 0, a[9,5] = 0, a[10,2] = 0, a[10,3] = 0, a[10,4] = 0, a[10,5] = 0, a[11, 2] = 0, a[11,3] = 0, a[11,4] = 0, a[11,5] = 0, a[11,6] = 1/30, b[14] = 117048651891177050452812500000000/903958175807874008864483503817601, \+ b[10] = 1967337516701564001434375000000000000000000000000/154313648631 19854943071851131903908575429289017877, a[11,1] = -72189389771/9959178 240000-459663572789/59755069440000*6^(1/2), a[6,1] = 1232/16875-152/16 875*6^(1/2), a[5,3] = -14742/3125-972/625*6^(1/2), c[13] = 1623/2000, \+ c[14] = 453/500, c[15] = 1, a[11,7] = -14201240926266911/5571693645004 80000-31790792357660029/557169364500480000*6^(1/2), b[11] = 2323713252 076974806855457536/10352378514220126928031114081, b[2] = 0, b[3] = 0, \+ b[4] = 0, b[5] = 0, c[3] = 48/335-32/1675*6^(1/2), a[4,3] = 54/335-36/ 1675*6^(1/2), c[7] = 48/125+8/125*6^(1/2), a[7,4] = -7348/98325-33652/ 294975*6^(1/2), a[6,4] = 29684/106875-13372/320625*6^(1/2), b[6] = 0, \+ b[7] = 0, c[9] = 12/25, b[15] = 145778296653275182685983/4945417885871 057962703934, b[1] = 100976787617015984669475787/692150295240326231043 7464576, b[9] = 961916572949681511747758515625/44164177155045875150368 09762176, a[9,8] = -27/800, c[12] = 57617028499878/85094827871699, a[8 ,1] = 16/225, b[13] = 296136352341197653422080000000000000/38994325616 50270968394778037550931439, a[10,9] = -21409264848554971927/2048000000 00000000000, b[8] = 8877148253451235588984375/438551403820878248218763 8272, b[12] = 28308600293241456954311939117138937391141264054325158121 088859798090056999504636993802634122671806566061266984631226363619/158 3915465406700809680657570071738908858776311325098582329879822131693808 98029758612356596464561682203552186100842929847264, a[6,5] = 2132/6412 5-284/21375*6^(1/2), a[11,10] = 21871487332435000000/12553695287957958 3419+18386542911560000000/1129832575916216250771*6^(1/2), a[7,1] = 203 2/16875+152/16875*6^(1/2), a[4,1] = 18/335-12/1675*6^(1/2), a[11,8] = \+ 22414436941/1563197440000+459663572789/56275107840000*6^(1/2), a[8,6] \+ = 64/225+4/225*6^(1/2), a[5,4] = 12528/3125+144/125*6^(1/2), a[5,1] = \+ 4014/3125+252/625*6^(1/2), a[3,2] = 31872/112225-1536/22445*6^(1/2), a [3,1] = -15792/112225+5536/112225*6^(1/2), a[7,5] = 10132/64125-716/21 375*6^(1/2), c[6] = 48/125-8/125*6^(1/2), a[9,7] = 177/800-69/1600*6^( 1/2), a[7,6] = 2592/14375+2912/14375*6^(1/2), a[9,6] = 177/800+69/1600 *6^(1/2), a[10,6] = 4287156859652598464203/58982400000000000000000-159 8864762333658025459/117964800000000000000000*6^(1/2), a[8,7] = 64/225- 4/225*6^(1/2)\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 109 "It remains to determine the li nking coefficients in stages 12 to 15. We have the following zero coef ficients." }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "a[12,2] =0" "6#/&%\"aG6$\"#7\"\"#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[12, 3]=0" "6#/&%\"aG6$\"#7\"\"$\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[1 2,4]=0" "6#/&%\"aG6$\"#7\"\"%\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 " a[12,5]=0" "6#/&%\"aG6$\"#7\"\"&\"\"!" }{TEXT -1 2 ", " }}{PARA 256 " " 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "a[13,2]=0" "6#/&%\"aG6$\"#8\"\" #\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[13,3]=0" "6#/&%\"aG6$\"#8\" \"$\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[13,4]=0" "6#/&%\"aG6$\"#8 \"\"%\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[13,5]=0" "6#/&%\"aG6$\" #8\"\"&\"\"!" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "a[14,2]=0" "6#/&%\"aG6$\"#9\"\"#\"\"!" }{TEXT -1 2 ", \+ " }{XPPEDIT 18 0 "a[14,3]=0" "6#/&%\"aG6$\"#9\"\"$\"\"!" }{TEXT -1 2 " , " }{XPPEDIT 18 0 "a[14,4]=0" "6#/&%\"aG6$\"#9\"\"%\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[14,5]=0" "6#/&%\"aG6$\"#9\"\"&\"\"!" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[15,2]=0 " "6#/&%\"aG6$\"#:\"\"#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[15,3] =0" "6#/&%\"aG6$\"#:\"\"$\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[15, 4]=0" "6#/&%\"aG6$\"#:\"\"%\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[1 5,5]=0" "6#/&%\"aG6$\"#:\"\"&\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 154 "We make use of the stage -order conditions for stages 12 to 15 so that all these stages all hav e stage-order 5 and incorporate the simplifying conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j+1 .. \+ 15) = b[j]*(1-c[j]);" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+%\"j GF,/F+;,&F0F,F,F,\"#:*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "for " }{XPPEDIT 18 0 "j=1" "6#/%\"jG \"\"\"" }{TEXT -1 10 " ( with " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6# \"\"\"\"\"!" }{TEXT -1 6 " ), " }{XPPEDIT 18 0 "j=7" "6#/%\"jG\"\"( " }{TEXT -1 8 " . . 13." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 185 "Then it turns out that the following collection of \+ \"simple\" order conditions (given in abreviated form) is sufficient t o determine the remaining linking coefficients in stages 11 to 15." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 194 "SO9 := SimpleOrderConditions(9):\n[seq([i,SO9[i]],i=[102,106,125, 212,223,239,245,251,253])]:\nlinalg[augment](linalg[delcols](%,2..2),m atrix([[` `]$(linalg[rowdim](%))]),linalg[delcols](%,1..1));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7+7%\"$-\"%#~~G/*(%\"bG\" \"\"%\"cGF--%!G6#*&%\"aGF--F06#*&F3F--F06#*&)F.\"\"$F-F3F-F-F-F-#F-\"$ g*7%\"$1\"F)/*&F,F--F06#*&F3F--F06#*&F3F--F06#*&)F.\"\"%F-F3F-F-F-F-#F -\"%!o\"7%\"$D\"F)/*(F,F-F.F--F06#*&)F.\"\"&F-F3F-F-#F-\"#[7%\"$7#F)/* (F,F-)F.\"\"#F-F/F-#F-\"%!3\"7%\"$B#F)/*(F,F-F.F-FBF-#F-\"%!*=7%\"$R#F )/*(F,F-FhnF-FEF-#F-\"$q#7%\"$X#F)/*(F,F-F.F--F06#*&F3F-FSF-F-#F-\"$y$ 7%\"$^#F)/*(F,F-FhnF-FSF-#F-\"#a7%\"$`#F)/*(F,F-F.F--F06#*&)F.\"\"'F-F 3F-F-#F-\"#jQ(pprint16\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "The associated trees" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 456 "AST9 := AllSimpleTrees(9):\nwhch : = [102,106,125,212,223,239,245,251,253]:\nm := 3: ## number of trees p er row\nnn := nops(whch): q := iquo(nn,m,'r'):\nfor i to nn do \n p|| i := DrawTree(AST9[whch[i]],height=4,width=2,show_ordercondition=true, \n color=COLOR(RGB,.5,0,.9),font_color=black);\nend do:\npp := plot ([[1,1]],style=line,axes=none):\nplots[display](convert([seq([p||((k-1 )*m+1..m*k)],k=1..q),\n `if`(r>0,[p||(m*q+1..nn),pp$(m*(q+1)-nn)],NUL L)],array));" }}{PARA 13 "" 1 "" {GLPLOT2D 782 948 948 {PLOTDATA 2 "6f u-%%TEXTG6&7$$\"#g\"\"!$!+E!G3\"R!\")Q5b~c~~(a~(a~c~))~=~~~6\"-%'COLOU RG6&%$RGBGF)F)F)-%%FONTG6$%(COURIERG\"#5-F$6&7$F'$!+uu)G%QF,Q7~~~~~~~~ ~~~~~~~~~~~~~1F.F/F3-F$6&7$F'$!+(*)=T'QF,Q9~~~~~~~~~~~~~~~~~~~~~___F.F /F3-F$6&7$F'$!+,+++SF,Q9~~~~~~~~~~~~~~~~~~~~~270F.F/F3-F$6&7$F'$!+0\\R MQF,Q7~~~~2~~~~~~~~4~~~~~~~~F.F/F3-%'CURVESG6&7+7$F'$!+,Q^)p$F,7$$\"++ +++bF,$!+^`)QF$F,7$F'FZ7$$\"+++++lF,FZ7$Fhn$!+,pD\\GF,7$$\"+LLLLjF,$!+ ]%GYU#F,7$$\"+WWWWkF,F`o7$$\"+cbbblF,F`o7$$\"+nmmmmF,F`o-%'SYMBOLG6#%' CIRCLEG-%&COLORG6&F2$F)F)FbpFbp-%&STYLEG6#%&POINTG-FQ6&FS-F\\p6#%(DIAM ONDGF_pFcp-FQ6&FS-F\\p6#%&CROSSGF_pFcp-FQ6%7$FTFW-F`p6&F2$\"\"&!\"\"Fb p$\"\"*Fhq-%*THICKNESSG6#\"\"#-FQ6%7$FTFfnFdqF[r-FQ6%7$FTFgnFdqF[r-FQ6 %7$FgnFjnFdqF[r-FQ6%7$FjnF]oFdqF[r-FQ6%7$FjnFboFdqF[r-FQ6%7$FjnFeoFdqF [r-FQ6%7$FjnFhoFdqF[r-FQ6%7#7$F'$!+++++?F,-F06&F2$F7FhqFbpFbp-Fdp6#%%L INEG-FQ6%7#7$$\"+++++]F,FUFjsF]t-FQ6%7#7$$\"+++++qF,FUFjsF]t-F$6&7$$\" #IF)$!+d-G3\"*!\"*Q7b~(a~(a~(a~c~)))~=~~~~F.F/F3-F$6&7$F_u$!+PZ()G%)Fc uQ9~~~~~~~~~~~~~~~~~~~~~~~1F.F/F3-F$6&7$F_u$!+i*)=T')FcuQ<~~~~~~~~~~~~ ~~~~~~~~~~~____F.F/F3-F$6&7$F_u$!+++++5F,Q<~~~~~~~~~~~~~~~~~~~~~~~1680 F.F/F3-F$6&7$F_u$!+Z!\\RM)FcuQ9~~~~~~~~~~~~~4~~~~~~~~~~F.F/F3-FQ6&7*7$ F_u$!+1!Q^)pFcu7$F_u$!+/N&)QFFcu7$F_u$\"+)*4V2:Fcu7$F_u$\"+/br`dFcu7$$ \"++++DEF,$\"+++++5F,7$$\"++++vGF,F_x7$$\"++++DJF,F_x7$$\"++++vLF,F_xF [pF_pFcp-FQ6&F_wFipF_pFcp-FQ6&F_wF^qF_pFcp-FQ6%7$F`wFcwFdqF[r-FQ6%7$Fc wFfwFdqF[r-FQ6%7$FfwFiwFdqF[r-FQ6%7$FiwF\\xFdqF[r-FQ6%7$FiwFaxFdqF[r-F Q6%7$FiwFdxFdqF[r-FQ6%7$FiwFgxFdqF[r-FQ6%7#7$F_uF_xFjsF]t-FQ6%7#7$$\"+ ++++?F,FawFjsF]t-FQ6%7#7$$\"+++++SF,FawFjsF]t-FQ6#-%'LEGENDG6#QB__neve r_display_this_legend_entryF.-F$6&7$FbpF*Q:b~c~~(a~(a~(a~c~)))~=~~~~F. 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66" "Curve 67" "Curve 68" "Curve 69" "Curve 70" "Curve 71 " "Curve 72" "Curve 73" "Curve 74" "Curve 75" "Curve 76" "Curve 77" "C urve 78" "Curve 79" "Curve 80" "Curve 81" "Curve 82" "Curve 83" "Curve 84" "Curve 85" "Curve 86" "Curve 87" "Curve 88" "Curve 89" "Curve 90 " "Curve 91" "Curve 92" "Curve 93" "Curve 94" "Curve 95" "Curve 96" "C urve 97" "Curve 98" "Curve 99" "Curve 100" "Curve 101" "Curve 102" "Cu rve 103" "Curve 104" "Curve 105" "Curve 106" "Curve 107" "Curve 108" " Curve 109" "Curve 110" "Curve 111" "Curve 112" "Curve 113" "Curve 114 " "Curve 115" "Curve 116" "Curve 117" "Curve 118" "Curve 119" "Curve 1 20" "Curve 121" "Curve 122" "Curve 123" "Curve 124" "Curve 125" "Curve 126" "Curve 127" "Curve 128" "Curve 129" "Curve 130" "Curve 131" "Cur ve 132" "Curve 133" "Curve 134" "Curve 135" "Curve 136" "Curve 137" "C urve 138" "Curve 139" "Curve 140" "Curve 141" "Curve 142" "Curve 143" "Curve 144" "Curve 145" "Curve 146" "Curve 147" "Curve 148" "Curve 149 " "Curve 150" "Curve 151" "Curve 152" "Curve 153" "Curve 154" "Curve 1 55" "Curve 156" "Curve 157" "Curve 158" "Curve 159" "Curve 160" "Curve 161" "Curve 162" "Curve 163" "Curve 164" "Curve 165" "Curve 166" "Cur ve 167" "Curve 168" "Curve 169" }}}}{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 49 "SO9_15 := SimpleOr derConditions(9,15,'expanded'):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 485 "SOeqs2 := [seq(add(a[i,j],j= 1..i-1)=c[i],i=12..15),op(StageO rderConditions(2,12..15,'expanded')),\n op(StageOrderCondition s(3,12..15,'expanded')),op(StageOrderConditions(4,12..15,'expanded')), \n op(StageOrderConditions(5,12..15,'expanded'))]:\nord_ cdns := [seq(SO9_15[i],i=[102,106,125,212,223,239,245,251,253])]:\nsim p_eqs := [add(b[i]*a[i,1],i=1+1..15)=b[1],seq(add(b[i]*a[i,j],i=j+1..1 5)=b[j]*(1-c[j]),j=[$7..13])]:\ncdns := [op(SOeqs2),op(ord_cdns),op(si mp_eqs)]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 127 "It is po ssible to manage with fewer equations, but the computation may be less efficient if the number of equations is reduced." }}{PARA 0 "" 0 "" {TEXT -1 50 "For example, the simplifying conditions given by " } {XPPEDIT 18 0 "j=8" "6#/%\"jG\"\")" }{TEXT -1 3 ", " }{XPPEDIT 18 0 " j=10" "6#/%\"jG\"#5" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "j=12" "6#/% \"jG\"#7" }{TEXT -1 17 " may be omitted." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "e10 :=\{seq(seq(a[i,j ]=0,i=12..15),j=2..5)\}:\ne11 := `union`(e9,e10):\neqns2 := subs(e11,c dns):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "There are 37 equations and 34 unknowns." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "nops(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 74 "e12 := solve(\{op(eqns2)\}): \ninfolevel[solve] := 0:\ne13 := `union`(e11,e12):" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "e13 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11088 "e13 := \{a[14,2] = 0, a[12,3] = 0, a[15,2] = 0, a[14,3] = 0, a[13,3] = 0, a[12,4] = 0, a [15,3] = 0, a[13,4] = 0, a[12,5] = 0, a[15,4] = 0, a[14,4] = 0, a[13,5 ] = 0, a[15,5] = 0, a[14,5] = 0, a[4,2] = 0, c[11] = 1/4, a[14,12] = - 3599885339347525438133682259450581379615044406751677715705637405791754 84411998973149563531328527853237660249760109643/2341281374456682407360 6771166244237989248347093174069661571660234572993334330800875873259019 4294958617150200000000000, a[2,1] = 1/25, a[10,1] = 284453082904607402 2657/58982400000000000000000, c[4] = 72/335-48/1675*6^(1/2), a[10,7] = 4287156859652598464203/58982400000000000000000+1598864762333658025459 /117964800000000000000000*6^(1/2), a[11,9] = 154180604903/253415424000 0+459663572789/11403694080000*6^(1/2), a[10,8] = -14103388621860433734 3/6553600000000000000000, a[9,1] = 57/800, c[2] = 1/25, c[5] = 72/125, c[8] = 16/25, c[10] = 3377/50000, a[5,2] = 0, a[6,2] = 0, a[6,3] = 0, a[7,2] = 0, a[7,3] = 0, a[8,2] = 0, a[8,3] = 0, a[8,4] = 0, a[8,5] = \+ 0, a[9,2] = 0, a[9,3] = 0, a[9,4] = 0, a[9,5] = 0, a[10,2] = 0, a[10,3 ] = 0, a[10,4] = 0, a[10,5] = 0, a[11,2] = 0, a[11,3] = 0, a[11,4] = 0 , a[11,5] = 0, a[11,6] = 1/30, a[11,1] = -72189389771/9959178240000-45 9663572789/59755069440000*6^(1/2), a[6,1] = 1232/16875-152/16875*6^(1/ 2), a[5,3] = -14742/3125-972/625*6^(1/2), c[13] = 1623/2000, c[15] = 1 , a[11,7] = -14201240926266911/557169364500480000-31790792357660029/55 7169364500480000*6^(1/2), b[2] = 0, b[3] = 0, b[4] = 0, b[5] = 0, c[3] = 48/335-32/1675*6^(1/2), a[4,3] = 54/335-36/1675*6^(1/2), c[7] = 48/ 125+8/125*6^(1/2), a[7,4] = -7348/98325-33652/294975*6^(1/2), a[6,4] = 29684/106875-13372/320625*6^(1/2), b[6] = 0, b[7] = 0, c[9] = 12/25, \+ c[14] = 453/500, b[14] = 117048651891177050452812500000000/90395817580 7874008864483503817601, b[10] = 19673375167015640014343750000000000000 00000000000/15431364863119854943071851131903908575429289017877, b[9] = 961916572949681511747758515625/4416417715504587515036809762176, b[15] = 145778296653275182685983/4945417885871057962703934, b[1] = 10097678 7617015984669475787/6921502952403262310437464576, b[11] = 232371325207 6974806855457536/10352378514220126928031114081, b[8] = 887714825345123 5588984375/4385514038208782482187638272, b[12] = 283086002932414569543 1193911713893739114126405432515812108885979809005699950463699380263412 2671806566061266984631226363619/15839154654067008096806575700717389088 5877631132509858232987982213169380898029758612356596464561682203552186 100842929847264, b[13] = 296136352341197653422080000000000000/38994325 61650270968394778037550931439, a[9,8] = -27/800, c[12] = 5761702849987 8/85094827871699, a[8,1] = 16/225, a[10,9] = -21409264848554971927/204 800000000000000000, a[6,5] = 2132/64125-284/21375*6^(1/2), a[11,10] = \+ 21871487332435000000/125536952879579583419+18386542911560000000/112983 2575916216250771*6^(1/2), a[7,1] = 2032/16875+152/16875*6^(1/2), a[4,1 ] = 18/335-12/1675*6^(1/2), a[11,8] = 22414436941/1563197440000+459663 572789/56275107840000*6^(1/2), a[8,6] = 64/225+4/225*6^(1/2), a[5,4] = 12528/3125+144/125*6^(1/2), a[5,1] = 4014/3125+252/625*6^(1/2), a[3,2 ] = 31872/112225-1536/22445*6^(1/2), a[3,1] = -15792/112225+5536/11222 5*6^(1/2), a[7,5] = 10132/64125-716/21375*6^(1/2), c[6] = 48/125-8/125 *6^(1/2), a[9,7] = 177/800-69/1600*6^(1/2), a[7,6] = 2592/14375+2912/1 4375*6^(1/2), a[9,6] = 177/800+69/1600*6^(1/2), a[10,6] = 428715685965 2598464203/58982400000000000000000-1598864762333658025459/117964800000 000000000000*6^(1/2), a[8,7] = 64/225-4/225*6^(1/2), a[12,2] = 0, a[13 ,2] = 0, a[14,11] = -4486663212896126915859959582534794915351772517208 0973/5429607175821308141232283339099249860726338769531250, a[12,8] = 1 2039435467283852946442681868547691065960331569897374599708360175926844 37096940490927391123081521/6546732431504927940789740125933173479539636 700187880198168132038132534360122730704444373657280000-151110511960958 2787638505061390956130113866630219193505646906848911348771180693704433 83945077153/3273366215752463970394870062966586739769818350093940099084 066019066267180061365352222186828640000*6^(1/2), a[15,13] = -475944389 2077050695292772766912275498897121280000000000/46140261584864893044261 98097242323507384603506591241137, a[13,8] = 20301387341801440158880077 7489550153682185921397719/60037835174801343118810937369681199104000000 00000000-3968812951635028178485515838403541696971/18374628007211630439 078297600000000000000*6^(1/2), a[14,10] = -803142860359718096414775032 1204281053557209400385686643427859804352/25135864401783913069001781649 09088113749933981949109368563389778875+2149529255301345308782931502056 3569618432/29973653446264566614372595657785433238875*6^(1/2), a[13,9] \+ = -677602211254760908521917338264019314117/635714271434389910412902400 000000000000*6^(1/2)+5734007279191463783949220415640081522844951901139 2389/19744238839693126186283635091213610188800000000000000, a[13,12] = 594716139297486674475082356103592029886330739685357945223223985387172 161751824241784555387316314439/818019847119678359631725769101238313386 585208743795596694052943582031156000597320370585600000000000, a[13,6] \+ = -2617546081675469247418718340204655213/43139258710988420693360640000 0000000-552470350996365859393640759989/2400793528263703412500070400000 *6^(1/2), a[13,7] = -45969294618407232267578352626581642155421231201/1 0187731154133781589406192186163200000000000000+15404935365828183037389 06021531546510663696889/885889665576850572991842798796800000000000000* 6^(1/2), a[12,6] = -39115022545645779688585831988975140882502245161831 122703201435591036520210945751850583137867/384254287982681023670711085 89483043166777031255688218331140907046999467997785653437677908480-4908 2670039628745454059833196112975740018683915429120214534716181580105395 7874498846640625/15370171519307240946828443435793217266710812502275287 332456362818799787199114261375071163392*6^(1/2), a[12,7] = -6220642966 8093251619352586547306026465246348663559566744785569992002354621160779 7417635623004817316459/90745170356705905076366526235999860453636075671 0836393258911136378360869155327677312519401429106240000+64321004153532 8932923955834959360270277930780334485030265105750796567439186095378077 346484617022694073/181490340713411810152733052471999720907272151342167 2786517822272756721738310655354625038802858212480000*6^(1/2), a[13,1] \+ = -25488511950948766602163761842966272037005387677568247/2534336434064 4945281003771622773961523200000000000000+25256082419495633863089646244 38617443527/12416017897756354830807859200000000000000*6^(1/2), a[13,10 ] = 24100788715039192225758197856261740786803059875758180717/966271700 1210818114953908042643919942065136939286292000-88000287175942975132716 53743688562521/20449362746897197076153911415852248500*6^(1/2), a[12,1] = -178144571353393080183496267158614821877982611914666395752937745405 391408707734804982447062502773/124771801011299405474614551641042535359 8134947600568397156373491324203879120405304413829778240000+35259119457 5569317115651180991223097026568880384478484650944931412648046608828531 034562538513357/811016706573446135584994585666776479838787715940369458 1516427693607325214282634478689893558560000*6^(1/2), a[13,11] = 124358 916033523439225154730110040589545064737/199359124499456694850025101787 04400000000000, a[12,11] = 1963500009650946638084395645582909493284711 3883632439882926745444809822658280742880787456/15009933124323477487137 151792766813737022277834253210285601916815234167186635020874092933, a[ 12,10] = 1027893404876323970566863593065627643827980253781207669195240 6255660602976082835598179622579836784640000000/20706607333558650004563 8539160747510779946083475530875873014324650020277019795123510531149273 76343490460197-2475794627968340439266926692582942523578559086951126639 6518921812563538267024485653444025561440747520000000/26918589533626245 0059330100908971764013929908518190138634918622045026360125733660563690 494055892465375982561*6^(1/2), a[12,9] = 63681466156701378449525903160 928961790100434377497936201427140303316098275608362465676059840449/119 0576349318934908527805697777807691148501676060784445076801271079131653 42297705956314829715000-3525911945755693171156511809912230970265688803 84478484650944931412648046608828531034562538513357/1547749254114615381 0861474071111499984930521788790197785998416524028711494498701774320927 86295000*6^(1/2), a[15,7] = -60242025722206083647640080390153021797730 167/4671870346091448695506240355969515576320000+3122923069829044501615 82808948983992497619279/54031196176535884913246084116864832317440000*6 ^(1/2), a[15,11] = 209814213871916216679569640811090009729840605120/11 651974660960738096987230622554824075229159217, a[15,12] = 394155089286 3952281184122715485813150214519858148244578543766661999079625453432169 420046111603168071904019820197053706993310801913/576348806945747835561 7493981609669453909166107652101824938575939967327573639769764715055303 37022074730630638475548489174498085584, a[15,1] = -4311793849461244922 3384237106139955697243762787772847383/14030104862294690648579566521727 488445420965260989440000+4220334940168717525563719481764309232553/5553 268895030690235789996003491120640000*6^(1/2), a[15,10] = -432162297873 27667461772487493266526541342720000000/2687992206790662400529710662148 7120800647611685231*6^(1/2)+822220421784353922416106346105112190310210 366765138782108892460117760000000/112539961436930436130682455392076582 934435837857314380718907401253008522853, a[15,9] = 9016439677522482531 7874747969895939573106537263617688583/99321945821955728204114927159105 30662743409752296080000-4220334940168717525563719481764309232553/10597 89239915401287502749181781142480000*6^(1/2), a[14,1] = 184592679361470 753674594239208189459317466722387929369596771/127891217091830656233364 014205731655064571321600000000000000-293880952873230803935166416296767 553377/866938749696854853908947200000000000000*6^(1/2), a[14,13] = 101 1136807359189181222571916914688/2927578889427871359661497484969729, a[ 15,6] = -492170504431044248358505817476691843/263110163815490209572519 79880325120-65372649360291914372644744384375/4575828935921568862130779 10962176*6^(1/2), a[14,7] = 990092406144467276115942254522770689064131 79/16167053833464253010532287404800000000000000-3748619435624523040662 096435375495632409181/1405830768127326348741938035200000000000000*6^(1 /2), a[14,6] = 5966995986367380181027852718263477/68458296294293148072 9600000000000+2897291244884828193281565089/190492650558033107681280000 00*6^(1/2), a[15,14] = 38733614144315448443538220691259283298308750000 000/93806291160324220065540465392781147766752569954001, a[14,8] = 4198 2993267604400562166630899538221911/11663582231140195102930560000000000 0000*6^(1/2)+463264582311323448626935784428360393666199191685092746043 3/5096189922215990577140952099757378096638035200000000000000, a[15,8] \+ = -4220334940168717525563719481764309232553/52298626520074835198571070 84128791040000*6^(1/2)-10128486019754425336362855829794634260766582925 227304979/2051440458733049774272501566471676463052668060718080000, a[1 4,9] = 293880952873230803935166416296767553377/16544712239967276477054 5400000000000000*6^(1/2)-121354756771846880705598195017571549684439925 07349800863475059/3337534088557268857775113382843672473526974754800000 000000000\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "printed coefficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 283 "subs(e13,ma trix([seq([c[i],seq(a[i,j],j=1..i-1),``$(8-i)],i=2..8)]));\nfor ii fro m 9 to 15 do\n print(``);\n print(c[ii]=subs(e13,c[ii])); \n for jj to ii-1 do\n print(a[ii,jj]=subs(e13,a[ii,jj]));\n end do: \nend do:\n``;\nfor ii to 15 do\n print(b[ii]=subs(e13,b[ii]));\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7*#\"\"\"\"#DF(% !GF+F+F+F+F+7*,&#\"#[\"$N$F)*(\"#KF)\"%v;!\"\"\"\"'#F)\"\"#F4,&#\"&#z: \"'DA6F4*(\"%ObF)F;F4F5F6F),&#\"&s=$F;F)*(\"%O:F)\"&XC#F4F5F6F4F+F+F+F +F+7*,&#\"#sF0F)*(F/F)F3F4F5F6F4,&#\"#=F0F)*(\"#7F)F3F4F5F6F4\"\"!,&# \"#aF0F)*(\"#OF)F3F4F5F6F4F+F+F+F+7*#FG\"$D\",&#\"%9S\"%DJF)*(\"$_#F) \"$D'F4F5F6F)FN,&#\"&UZ\"FZF4*(\"$s*F)FgnF4F5F6F4,&#\"&GD\"FZF)*(\"$W \"F)FVF4F5F6F)F+F+F+7*,&#F/FVF)*(\"\")F)FVF4F5F6F4,&#\"%K7\"&vo\"F)*( \"$_\"F)FjoF4F5F6F4FNFN,&#\"&%oH\"'vo5F)*(\"&sL\"F)\"'D1KF4F5F6F4,&#\" %K@\"&DT'F)*(\"$%GF)\"&v8#F4F5F6F4F+F+7*,&FdoF)*(FfoF)FVF4F5F6F),&#\"% K?FjoF)*(F\\pF)FjoF4F5F6F)FNFN,&#\"%[t\"&D$)*F4*(\"&_O$F)\"'v\\HF4F5F6 F4,&#\"&K,\"FgpF)*(\"$;(F)FjpF4F5F6F4,&#\"%#f#\"&vV\"F)*(\"%7HF)FarF4F 5F6F)F+7*#\"#;F*#Ffr\"$D#FNFNFNFN,&#\"#kFhrF)*(\"\"%F)FhrF4F5F6F),&Fjr F)*(F]sF)FhrF4F5F6F4Q)pprint196\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% !G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"*#\"#7\"#D" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"\"#\"#d\"$+)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"#\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"\"*\"\"',&#\"$x\"\"$+)\"\"\"*(\"#pF-\"%+;!\"\"F(#F-\"\"#F-" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"(,&#\"$x\"\"$+)\"\" \"*(\"#pF-\"%+;!\"\"\"\"'#F-\"\"#F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"aG6$\"\"*\"\")#!#F\"$+)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#5#\"%xL\"&++&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"\"#\"7dE-ug/H3`WG\"8+++++++ +S#)*e" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"#\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"$\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"',&#\"7.UY)f_'fo:(G%\"8++++++++S#)*e\"\"\"*(\"7 fa-eOLiZ'))f\"F-\"9++++++++!['z6!\"\"F(#F-\"\"#F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"(,&#\"7.UY)f_'fo:(G%\"8++++++++S#)*e\" \"\"*(\"7fa-eOLiZ'))f\"F-\"9++++++++!['z6!\"\"\"\"'#F-\"\"#F-" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\")#!6VtL/'=i)Q.T\"\"7+ +++++++g`l" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"*#!5F>( \\b[[E49#\"6++++++++![?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#6#\"\"\"\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"\",&#\",r(*Q*=s\".++Cy\"f**!\" \"*(\"-*ysNmf%F(\"/++Wp]vfF-\"\"'#F(\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"#6\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\" \"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"'#\"\"\" \"#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"(,&#\"26pEE4C ,U\"\"3++[+XOprb!\"\"*(\"2H+mdBz!zJ\"\"\"F,F-\"\"'#F0\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"),&#\",TpV9C#\".++W(>j:\"\" \"*(\"-*ysNmf%F-\"/++%y5vi&!\"\"\"\"'#F-\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"*,&#\"-.\\g!=a\"\".++CaT`#\"\"\"*(\"-* ysNmf%F-\"/++3%p.9\"!\"\"\"\"'#F-\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"#5,&#\"5+++NCL([r=#\"6>Mez&zG&p`D\"\"\"\"*(\"5+++ g:\"Ha'Q=F-\"7r2D;i\"fdK)H6!\"\"\"\"'#F-\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#7#\" /y)*\\Gqhd\"/*pry#[4&)" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"# 7\"\"\",&#\"[qtF]iqW#)\\![t2(39R0auPHv&RmY\">h#)z(=#[herE'\\$=!3$R`8dW \"y\"\"\\q++Cy(HQT/`S?\"zQ?C8\\tj:(Ro0gZ\\8)f``U5k^XhuaS*H65!=xC\"!\" \"*(\"[qdL^QDcM5`G)3m/[ETJ\\%4l%[yWQ!))ol-(4B7*4=^c6<$pbd%>\"f_$F(\"\\ q++ceN*)*oyWj#G9_K2OpFk^\"e%p.%fr(yQ)zkxmce%*\\eNhWtlq;5\")F-\"\"'#F( \"\"#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"#\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"$\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"&\"\"!" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"',&#\"gpny8$e]=vX4@?l.\"fN9?.F7J=;XA]#)39v*))>$ ee)ozdkXD-:\"R\"gp![3znPMl&y(*zY**p/249J$=#)ob7.xn;VI[*e362nB5o#)zGaUQ !\"\"*(\"epD1kY))\\uy&R0,e\"=;Z`9-7Ha\"Ro=+uvH6'>L)fSXX(G'R+n#3\\\"\" \"\"gp#Rj62v8E9\"*>(y*z=GOcCL(GvA]73rms@$zNMWGo%4C2$>:;D$4o'Hk?i\"aq++C1\"H9S>DJxwKb\" p3Oyj86*eKRO3rc2OOXg)**fBElOw]!fqc.Z_It_,\"=T82M!\\\"=F-\"\"'#F0\"\"#F0" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"),&#\"\\q@:3B6RF4\\Sp4PWo#f!)y=+n jR&zM>-jmQ6I h&4Rh]]Qwy#e4'>^56:F-\"\\q++kGo=AANl81!=ni1>g1%3*4SR4]$=)p(RnemH1q[RqR Y_d@mLF$!\"\"\"\"'#F-\"\"#F1" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"a G6$\"#7\"\"*,&#\"jp\\/%)fgnlCO3cF)4;LISrU,i$z\\xVV+,zh*G4;.f_\\%y8qchY \"oj\"[q+]rH[Jcfq(HU`;8z5F,o2XWyggn,&[6p2yx(p0y_3\\$*=$\\jd!>\"\"\"\"* (\"[qdL^QDcM5`G)3m/[ETJ\\%4l%[yWQ!))ol-(4B7*4=^c6<$pbd%>\"f_$F-\"\\q+] H'y#4Ku!z)y@0$\\)**\\662u9'3\"Q:Y6a#\\xa\"!\"\"\" \"'#F-\"\"#F1" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"#5,&# \"fq+++SYyO)zDiz\")fNG3wHggcD1C&>pw?\"y`-)z#QkFc1$fjocqRKw[S$*y-\"\"fq (>g/\\VjPF\\6`5N7&z>qF?+lCV,te(3`vM3Y*z2^Z2;R&Qc/+leNL2mq?\"\"\"*(\"fq +++?vuS9cDSW`c[CqEQNc7=#*=lRm7^p3f&yN_UHe#pEpER/Mozi%zvCF-\"gqhD)fPlC* e0%\\!pj0mLd7gj-X?i=\\jQ,>=&3*HR,k<(*345I$f+XiiL&*e=p#!\"\"\"\"'#F-\" \"#F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"#6#\"dpcuy!)Gu !GeE#)4[WXn#H))RCj$)Q6ZG$\\4HeXcR%3Qm%4l4+]j>\"dpLH4u3-Nm=nTB:o\">g&G5 KDMyFAqt8ow#z^r8([xMKCJ$*4]\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#8#\"%B;\"%+?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"\",&#\"VZ#ovn(Q0q.si'H%=wj@ gm([4&>^)[D\"V+++++++K_hRxA;x.5GX\\kSVOV`#!\"\"*(\"IFNW<'QWik*3jQj&\\> C3c_#F(\"J+++++++#fy!3$[Ncx*y,;C\"F-\"\"'#F(\"\"#F(" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"#8\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\" \"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"',&#\"F8_ l/-M=(=uCpan\"3Yvh#\"E+++++S1O$p?%))4re#RJ%!\"\"*(\"?*)*f2k$RfeO'*4NqC b\"\"\"\"@++Sq+]7Mqj#GNz+CF-F(#F0\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"(,&#\"P,7B@a:U;eEENyvEKsS=YHpf%\"P+++++++K;'=#> 1%*e\"yLT:Jx=5!\"\"*(\"O*)opj1^Y:`@g!*QPI=GeON\\S:\"\"\"\"N+++++++oz)z U=*Hd]odl'*)e))F-\"\"'#F0\"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/& %\"aG6$\"#8\"\"),&#\"T>xR@f=#o`,b*[x2!))e,W,=M(Q,.#\"U+++++++/\"*>\"op t$4\")=JM,[SEQt\">_34wa7@-wn\"\"\"\"H+++++++C!HT5**QM9F9dj! \"\"\"\"'#F,\"\"#F.#\"V*Q#R6!>&\\%G_\"3ScT?#\\Ryj9>zs+Md\"V+++++++))=5 O@\"4NOG'=EJpR)QUu>F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8 \"#5,&#\"Y<2=ed()fI!oySedA#>R]r)y+T#\"X+?H'GRp8l?%*>RkU!3R&\\6=3@ ,qri'*\"\"\"*(\"F@Dc)oVPlrK^(H%f<(G+!))F-\"G+&[A&eT6R:wq>(*oui$\\/#!\" \"\"\"'#F-\"\"#F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"#6 #\"NPZ1X&*eS+6IZ:D#RM_Lg\"*eV7\"M+++++S/(y,^-][pc%*\\C\"f$*>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"#7#\"^qRWJ;tQbXyTU#=vh@<(Q&)R AB_%zN&oR2L'))H?f.hN#3vWn'[(HRhr%f\"^q+++++g&eq.K(f+g:J?eVH0%p'f&zV(3_ e'Q8$Q75pdsJ'f$y'>r%)>!=)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#9#\"$`%\"$+&" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#9\"\"\",&#\"gnrnfp$HzQAnY<$f%*=3#RUf uOvq9OzEf%=\"gn+++++++;KrX1b;t0U,kLBc1$=4<7*y7F(*(\"HxLbnnH;k;NR!3BtG& 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"6#/&%\"bG6#\"#9#\"B++++D \"GX]q<\"*=l[q6\"B,w\"Q]$[k)3S(y!e " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "RK9_15eqs := [op(RowSumConditions(15,'expanded')),op(OrderConditio ns(9,15,'expanded'))]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Check:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "e xpand(subs(e13,RK9_15eqs)):\nmap(u_->`if`(lhs(u_)=rhs(u_),0,1),%);\nno ps(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7`jl\"\"!F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$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 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 34 "Appendix: related order conditions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "#-------------------------------------------" }}{PARA 0 "" 0 " " {TEXT -1 145 "unrelated 129, 130, 134, 135, 136, 139, 140, 141, 143 , 151, 152, 156, 157, 166, 168, 173, 175, 180, 185, 186, 187, 212, 221 , 223\n\nrelated groups" }}{PARA 0 "" 0 "" {TEXT -1 19 "132, 137, 148, 153," }}{PARA 0 "" 0 "" {TEXT -1 19 "133, 138, 150, 163," }}{PARA 0 " " 0 "" {TEXT -1 59 "131, 147, 149, 192,\n142, 164, 170, 222,\n162, 184 , 188, 227," }}{PARA 0 "" 0 "" {TEXT -1 18 "181, 197, 202, 239" }} {PARA 0 "" 0 "" {TEXT -1 38 "144, 158, 167, 177, 193, 207, 213, 244" } }{PARA 0 "" 0 "" {TEXT -1 38 "145, 159, 169, 178, 194, 209, 215, 245" }}{PARA 0 "" 0 "" {TEXT -1 38 "154, 165, 172, 189, 206, 208, 224, 248 " }}{PARA 0 "" 0 "" {TEXT -1 38 "183, 200, 204, 205, 228, 231, 235, 25 1" }}{PARA 0 "" 0 "" {TEXT -1 78 "146, 160, 161, 174, 182, 196, 198, 2 01, 203, 211, 218, 230, 232, 238, 241, 253" }}{PARA 0 "" 0 "" {TEXT -1 78 "155, 171, 176, 179, 190, 195, 199, 210, 214, 216, 225, 229, 237 , 240, 246, 255" }}{PARA 0 "" 0 "" {TEXT -1 78 "191, 217, 219, 220, 22 6, 233, 234, 236, 242, 243, 247, 249, 250, 252, 254, 256" }}{PARA 0 " " 0 "" {TEXT -1 43 "#------------------------------------------" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 44 "#------- ------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 43 "construction of the order 8 embedded scheme" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "e13 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7419 "e13 := \{c [7] = 769/5000, c[6] = 923/2000, c[5] = 1923/5000, a[8,3] = 0, a[8,2] \+ = 0, a[7,3] = 0, a[7,2] = 0, a[6,3] = 0, a[6,2] = 0, a[5,2] = 0, a[4,2 ] = 0, c[8] = 8571/10000, a[8,7] = 9, c[2] = 139/2500, c[9] = 14280181 565693441800/15023510624554266963, a[5,3] = -1122077307242443/77847674 6531250, a[4,3] = 4261491/36928000, a[3,1] = 94226774499/1184694553600 0, a[6,1] = 3279216863/71027928000, a[12,7] = 948339742665210716931560 767459705347432692833888702670755668800833857753125000/541002030685238 16523920547203712158257383235486719126879290708372319555115953, a[2,1] = 139/2500, b[10] = 594562755257530592552224345703125000000/123802720 910327682301431417435953442122031, b[9] = 6399841965907450296097946702 7044380533513499562179716145788153981258354882170183557294261050806789 914954901252698438375102943237148428019253408419/140673838782241599806 7693599950032426772772394751630953646406168313006343761645295132681982 4531063499969576146304645311203502485409773697867038000, b[7] = 621973 1958882093270433753490048828125000000/25268792314562899182186819800834 272764054341, c[10] = 3611/5000, c[11] = 15/16, c[12] = 1, a[10,9] = - 3347565461896560762549026667823123036634583052726552531003001623087575 5239420324728600957368877132012320553021/56360718148650508277541923761 4873016043169125130359330269370345097328575740250457475506596993780051 575000000000, a[9,8] = -4738143867684122189593816244199450540483384372 163549951990525387550768038015218275414120082248510000000000/456273815 5611920982891652838532643427337613715822889250315879256708180176162834 4925618992749059885819540261, b[5] = 0, b[2] = 0, b[3] = 0, b[4] = 0, \+ b[11] = -1886691133979705639959153870454656/39794758588583538395156331 2487675, a[12,10] = 12459468894943123671827871807868857124596796164158 4151713065625000/24422661199951239145999264873446308735658803040031364 01896150897137, b[6] = 101652048214282205518610445783893750000000/2895 86658278060310247144250081091360673509, a[11,8] = -5476716391734935351 87249719091874257724281347519101322819710937500/5545897345777551056363 997101744668576293320106258184122878184564689, a[7,1] = 22628776819090 891/378192197956950000, b[1] = 55284231195707975647126708111723/125881 4250475014554517603612114000, a[11,7] = 271466662889835128614810796916 069488062485834784939879774912795880859375/170800578682088075849119195 72395871967414618088449127584294090356526117, a[10,4] = 35076261889213 578261995286390053983221937920015616/890366240305246889023432168040903 9895089390971875, b[12] = -50061468875139778913910254637881/1411866419 36036184819986782313781, c[4] = 1420497/9232000, a[8,6] = 503350581600 824913/125990342179812500, a[7,6] = 182198401/10649112500, a[8,5] = -2 9670244019740727/6363654238520000, a[7,5] = -21022506263989/6144360561 90000, a[7,4] = 116326878837670912/1047320101758291075, a[8,4] = -4704 8174572430533795112/6781573975400382234375, a[6,4] = 1418466643672/614 6756958375, a[5,4] = 1683359084698258/1167715119796875, c[3] = 473499/ 4616000, a[9,6] = 9941127982121041338714935249721178582954714935869695 2646781033905129048593757052549024957474512389085654050445280000000/10 2197500711543555291993605654791795484972275164188674551435468249918836 16697162903707552409044338358355837306818391433, a[6,5] = 725783021041 /3932390759616, a[5,1] = 3590880889914277/9341720958375000, a[8,1] = - 150702609045465151/280142368857000000, a[9,1] = -160459180355445265310 8566670820097679909295434905910255443941805885523703480313807623591448 5211856830024864166368323768200/98294831577286648441985440349216559753 7825783008095189369352011632206843937744670849326949363324414642152273 0587295739613, a[3,2] = 224201303001/2369389107200, b[8] = 47152588201 4932587321673707929687500000000/52372881739119879572881083964931404449 5553, a[4,1] = 1420497/36928000, a[9,2] = 0, a[10,2] = 0, a[11,2] = 0, a[12,2] = 0, a[9,3] = 0, a[10,3] = 0, a[11,3] = 0, a[12,3] = 0, a[12, 8] = -4693599865307842423678411639679979852809920520624795226940377187 50000/1262000068797740029604893636717124646814536106030908513651797310 900389, a[11,10] = -7641031089620713267943821483116886435546875/186511 5549729059107236140767953375513727732867, a[10,1] = 100509763879264306 824096153463041174636629364248095333923106653001873/229490324007644628 042361756217436155832461488260089524115475000000000, a[10,5] = 2987705 3472248545227782869189767925950557009/10443709158645362958089905740134 206606110000, a[11,5] = -304301954438407952266341438991435702455078125 /26817588459995310678515661957421952038871616, a[11,4] = -739282853758 412257967453242147288028248514875000/672441828756548290983239437133742 07709631980757, a[12,1] = -2173296165244568434534168496725754283210370 856714048295955495704392191998074219/826329709817197468253912814996276 721297719271873047121496939265708086637102000, a[12,4] = -117648588933 4472345397948024769479865991267657667189808793600000/12823827412574578 4857395023683055391720808636521055761250085619, a[10,6] = -72602025182 798889442893966553844286012776770019588838776297031451/409184393804050 07926071673834718276106178863851548244800289187500, a[12,6] = 44619197 7093458430386099958102902220927137343575035730212893991884410752778938 25000/3047226829867759667805678237083584916028456629069014496162891378 892137605913688013, a[9,5] = -9911366389726261686789033714164561000939 0040553516492468305812280242970735403338282694739815868376532443961828 2500000000/79848142008846002789298925227605775190331269194726743910364 273272231784282184770467794155269096224513726772081370189773, a[11,6] \+ = 79448939923998931164766316975203638555016544623086227984461001450043 71780765625/8870051616029286592528276040357686572984358849394897544622 73147402132735951354, a[10,7] = -1532291206386437051213014598849209860 5486502994107816190/31300694006456048761836695496055390763335792905249 19889, a[9,7] = 194327599672380134095898291719912961363678073793023525 007081328425098431574448809779310732532821200046895000000000/119960114 8822772264965667393113649089125646329262047360184187511517025904353198 7881275965965525693289412424400177, a[11,1] = -51010977601978416151375 71256611109219669965728737004453654940399435749927779/3460254790065218 498025394912113654440547987641021174968985010753183759952128, a[11,9] \+ = 27880505422945647305191478577013040105627752107138627482025385282602 3602554035252878888463451556710568650135086751734733498388408614837563 /570327897589938424716739897067258193159963418613722438239821039844076 4234613614101505591454120074773344256118618572821579031366188464747033 6, a[12,5] = -90164152627308866729367798396043430614573156600775937500 0/47006051321531726477650870675115597779536141098588395119, a[12,11] = 561064742310110985699639624014768981793940948761680464947352/67137939 1526329055316725343694451428305592235340194400714475, a[12,9] = -66412 7481006260866245417353167879111725438608160134129788715555378465573117 8130017572382606930519018125943416089220164619687237540672159347990548 52630446938063/9474053136757793278864620448618348468794738615395834799 9510116829111616884404043875158439278591322423989402984382146436322440 5440877235651675827150051218506000, a[9,4] = -235853914788187787303944 5811751103289867403691705393271280994482043263320271984518766662589229 245453919623760032009543680000000/217835526181955565558022800341429464 0863851721229066080313495435838630220158871171685663384701630900324981 96064101837445053273, a[10,8] = 66085154677219418645471125072555541174 985695924/310222736648235062495097951962638800384203417327\}:" }}} {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 61 "We can obtain an embedded 14 stage order 8 scheme as follows." }} {PARA 0 "" 0 "" {TEXT -1 94 "We remove stages 14 and 15 from the 15 st age order 9 scheme and introduce a new stage 14 with:" }}{PARA 256 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "c[14] = 1;" "6#/&%\"cG6#\"#9\" \"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[14,2] = 0;" "6#/&%\"aG6$\"#9 \"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[14,3] = 0;" "6#/&%\"aG 6$\"#9\"\"$\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[14,4]=0" "6#/&% \"aG6$\"#9\"\"%\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[14,5]=0" "6# /&%\"aG6$\"#9\"\"&\"\"!" }{TEXT -1 1 "," }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`b*`[2]=0" "6#/&%#b*G6#\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[3]=0" "6#/&%#b*G6#\"\"$\"\"!" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "`b*`[4]=0" "6#/&%#b*G6#\"\"%\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[5]=0" "6#/&%#b*G6#\"\"&\"\"!" }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "`b*`[6]=0" "6#/&%#b*G6#\"\"'\"\"!" }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "`b*`[7] = 0;" "6#/&%#b*G6#\"\"(\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 56 "where the weights of the order 8 sch eme are denoted by " }{XPPEDIT 18 0 "`b*`" "6#%#b*G" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 156 "We inc orporate the order 8 quadrature conditions, the row sum conditions for this stage and stage-order conditions so that this new stage has stag e-order 4." }}{PARA 0 "" 0 "" {TEXT -1 48 "We also incorporate the sim plifying conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(`b*`[i]*a[i,j],i = j+1 .. 14) = `b*`[j]*(1-c[j]);" "6#/-%$Su mG6$*&&%#b*G6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,\"#9*&&F)6#F0F ,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 5 "for " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 10 " ( with " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\"\"!" }{TEXT -1 5 " ), \+ " }{XPPEDIT 18 0 "j = 9;" "6#/%\"jG\"\"*" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "j = 11;" "6#/%\"jG\"#6" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "j = 13;" "6#/%\"jG\"#8" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 451 "`Qeqs* ` := subs(b=`b*`,QuadratureConditions(8,14,'expanded')):\nSO_eqs2 := [ add(a[14,j],j=1..13)=c[14],add(a[14,j]*c[j],j=2..13)=1/2*c[14]^2,\n \+ add(a[14,j]*c[j]^2,j=2..13)=1/3*c[14]^3,add(a[14,j]*c[j]^3,j=2..1 3)=1/4*c[14]^4,\n add(a[14,j]*c[j]^4,j=2..13)=1/5*c[14]^5]:\n`s imp_eqs*` := [add(`b*`[i]*a[i,1],i=2..14)=`b*`[1],seq(add(`b*`[i]*a[i, j],i=j+1..14)=`b*`[j]*(1-c[j]),j=[9,11,13])]:\n`cdns*` := [op(`simp_eq s*`),op(SO_eqs2),op(`Qeqs*`)]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "e14 := `union`(remove(u_->member(op(1,lhs(u_)),[14,15]) or op(0,l hs(u_))=b,e13),\n \{c[14]=1,seq(a[14,i]=0,i=2..5),seq(`b*`[i]= 0,i=2..7)\}):\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "We have 17 equations for the 17 unknown coefficients." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "eqns3 := subs(e14,`cdns*`): \nnops(%);\nindets(eqns3);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#\"#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<3&%#b*G6#\"\"\"&%\"aG6$\"#9 F'&F)6$F+\"\"*&F)6$F+\"#5&F)6$F+\"\"(&F)6$F+\"\")&F)6$F+\"\"'&F)6$F+\" #8&F)6$F+\"#7&F)6$F+\"#6&F%6#F+&F%6#F=&F%6#F@&F%6#FC&F%6#F1&F%6#F.&F%6 #F7" }}{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 74 "e15 := solve(\{op(e qns3)\}):\ninfolevel[solve] := 0:\ne16 := `union`(e14,e15):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "ev alb(e16=e_16);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e16" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9027 "e16 := \{a[14,2] = \+ 0, a[12,3] = 0, a[14,3] = 0, a[13,3] = 0, a[12,4] = 0, a[13,4] = 0, a[ 12,5] = 0, a[14,4] = 0, a[13,5] = 0, a[14,5] = 0, a[4,2] = 0, c[11] = \+ 1/4, a[2,1] = 1/25, a[10,1] = 2844530829046074022657/58982400000000000 000000, c[4] = 72/335-48/1675*6^(1/2), a[10,7] = 428715685965259846420 3/58982400000000000000000+1598864762333658025459/117964800000000000000 000*6^(1/2), a[11,9] = 154180604903/2534154240000+459663572789/1140369 4080000*6^(1/2), a[10,8] = -141033886218604337343/65536000000000000000 00, `b*`[1] = 34542436255316150799031/1697695107285568386175488, a[9,1 ] = 57/800, `b*`[8] = 11089416604965799654140625/103676454803990129602 54464, c[2] = 1/25, c[5] = 72/125, c[8] = 16/25, c[10] = 3377/50000, a [5,2] = 0, a[6,2] = 0, a[6,3] = 0, a[7,2] = 0, a[7,3] = 0, a[8,2] = 0, a[8,3] = 0, a[8,4] = 0, a[8,5] = 0, a[9,2] = 0, a[9,3] = 0, a[9,4] = \+ 0, a[9,5] = 0, a[10,2] = 0, a[10,3] = 0, a[10,4] = 0, a[10,5] = 0, a[1 1,2] = 0, a[11,3] = 0, a[11,4] = 0, a[11,5] = 0, a[11,6] = 1/30, `b*`[ 9] = 19661377435148646805703125/255979697183364488207083392, a[11,1] = -72189389771/9959178240000-459663572789/59755069440000*6^(1/2), a[6,1 ] = 1232/16875-152/16875*6^(1/2), a[5,3] = -14742/3125-972/625*6^(1/2) , c[13] = 1623/2000, a[11,7] = -14201240926266911/557169364500480000-3 1790792357660029/557169364500480000*6^(1/2), c[3] = 48/335-32/1675*6^( 1/2), a[4,3] = 54/335-36/1675*6^(1/2), a[14,1] = -15631765399556785018 4921063401246768027/212359240683067701291661898043594240000*6^(1/2)+18 092408213832965447840389945226814727838921316097789/456826090866320289 2681034746459555501706277524480000, c[7] = 48/125+8/125*6^(1/2), a[7,4 ] = -7348/98325-33652/294975*6^(1/2), a[6,4] = 29684/106875-13372/3206 25*6^(1/2), c[9] = 12/25, `b*`[14] = 16723862451391031122709/339047561 915509883967882, a[14,13] = 894887352566754892758772116806035181797324 80000000000/58814073298029244653337926984054935524805348324518339, a[1 4,7] = -38800907634871851557429961190168666395776983/61985241540425821 21698476759940763269120000*6^(1/2)+86929658981342392065135182481190067 69975359/535962614823230784958890847664050959360000, a[14,6] = 4198448 4090942118562371132909375/52494462812567108248894805557248*6^(1/2)+659 46837846509057750066287355594411/3018431611722608724311451319541760, a [9,8] = -27/800, c[12] = 57617028499878/85094827871699, a[8,1] = 16/22 5, `b*`[10] = 688790393936688343750000000000000000000000/6091295374506 475008386869675533556427693991, a[10,9] = -21409264848554971927/204800 000000000000000, a[6,5] = 2132/64125-284/21375*6^(1/2), a[11,10] = 218 71487332435000000/125536952879579583419+18386542911560000000/112983257 5916216250771*6^(1/2), a[7,1] = 2032/16875+152/16875*6^(1/2), a[4,1] = 18/335-12/1675*6^(1/2), a[11,8] = 22414436941/1563197440000+459663572 789/56275107840000*6^(1/2), a[8,6] = 64/225+4/225*6^(1/2), a[5,4] = 12 528/3125+144/125*6^(1/2), a[5,1] = 4014/3125+252/625*6^(1/2), a[3,2] = 31872/112225-1536/22445*6^(1/2), a[3,1] = -15792/112225+5536/112225*6 ^(1/2), a[7,5] = 10132/64125-716/21375*6^(1/2), c[6] = 48/125-8/125*6^ (1/2), a[9,7] = 177/800-69/1600*6^(1/2), a[7,6] = 2592/14375+2912/1437 5*6^(1/2), a[9,6] = 177/800+69/1600*6^(1/2), a[10,6] = 428715685965259 8464203/58982400000000000000000-1598864762333658025459/117964800000000 000000000*6^(1/2), a[8,7] = 64/225-4/225*6^(1/2), a[14,9] = -193422103 474923353694542105476136625982637168991506421/160483573437102114320728 33749116213173856151273040000+156317653995567850184921063401246768027/ 40526767661819895652001992553977680000*6^(1/2), a[12,2] = 0, a[13,2] = 0, `b*`[12] = -124286991333373091036558909908082266527720007186430450 82918121983471100560973332724469319414156511978021429/1264891923709564 3063555422745116719205173888728920704277722434371356124846214898200346 632677347950735102176, a[14,8] = 2641630262222426284468500071736583566 105608571199325883/235343729805469979809271071241826755571130067491840 000+156317653995567850184921063401246768027/19999205560725979054621914 6109360640000*6^(1/2), a[12,8] = 1203943546728385294644268186854769106 596033156989737459970836017592684437096940490927391123081521/654673243 1504927940789740125933173479539636700187880198168132038132534360122730 704444373657280000-151110511960958278763850506139095613011386663021919 350564690684891134877118069370443383945077153/327336621575246397039487 0062966586739769818350093940099084066019066267180061365352222186828640 000*6^(1/2), a[13,8] = 20301387341801440158880077748955015368218592139 7719/6003783517480134311881093736968119910400000000000000-396881295163 5028178485515838403541696971/18374628007211630439078297600000000000000 *6^(1/2), a[13,9] = -677602211254760908521917338264019314117/635714271 434389910412902400000000000000*6^(1/2)+5734007279191463783949220415640 0815228449519011392389/19744238839693126186283635091213610188800000000 000000, a[13,12] = 594716139297486674475082356103592029886330739685357 945223223985387172161751824241784555387316314439/818019847119678359631 7257691012383133865852087437955966940529435820311560005973203705856000 00000000, a[13,6] = -2617546081675469247418718340204655213/43139258710 9884206933606400000000000-552470350996365859393640759989/2400793528263 703412500070400000*6^(1/2), a[13,7] = -4596929461840723226757835262658 1642155421231201/10187731154133781589406192186163200000000000000+15404 93536582818303738906021531546510663696889/8858896655768505729918427987 96800000000000000*6^(1/2), a[12,6] = -39115022545645779688585831988975 140882502245161831122703201435591036520210945751850583137867/384254287 9826810236707110858948304316677703125568821833114090704699946799778565 3437677908480-49082670039628745454059833196112975740018683915429120214 5347161815801053957874498846640625/15370171519307240946828443435793217 266710812502275287332456362818799787199114261375071163392*6^(1/2), a[1 2,7] = -62206429668093251619352586547306026465246348663559566744785569 9920023546211607797417635623004817316459/90745170356705905076366526235 9998604536360756710836393258911136378360869155327677312519401429106240 000+643210041535328932923955834959360270277930780334485030265105750796 567439186095378077346484617022694073/181490340713411810152733052471999 7209072721513421672786517822272756721738310655354625038802858212480000 *6^(1/2), a[14,12] = -194853016061874333335484555436551917329750324944 158872786342511110263341294405932910123138589192221578993200454377/158 4060589458414075357115090086066650800046113494753774958834164204847903 0548925808450142813874420689113167632464, a[13,1] = -25488511950948766 602163761842966272037005387677568247/253433643406449452810037716227739 61523200000000000000+2525608241949563386308964624438617443527/12416017 897756354830807859200000000000000*6^(1/2), a[13,10] = 2410078871503919 2225758197856261740786803059875758180717/96627170012108181149539080426 43919942065136939286292000-8800028717594297513271653743688562521/20449 362746897197076153911415852248500*6^(1/2), a[12,1] = -1781445713533930 8018349626715861482187798261191466639575293774540539140870773480498244 7062502773/12477180101129940547461455164104253535981349476005683971563 73491324203879120405304413829778240000+3525911945755693171156511809912 23097026568880384478484650944931412648046608828531034562538513357/8110 1670657344613558499458566677647983878771594036945815164276936073252142 82634478689893558560000*6^(1/2), a[13,11] = 12435891603352343922515473 0110040589545064737/19935912449945669485002510178704400000000000, a[12 ,11] = 196350000965094663808439564558290949328471138836324398829267454 44809822658280742880787456/1500993312432347748713715179276681373702227 7834253210285601916815234167186635020874092933, c[14] = 1, a[12,10] = \+ 1027893404876323970566863593065627643827980253781207669195240625566060 2976082835598179622579836784640000000/20706607333558650004563853916074 7510779946083475530875873014324650020277019795123510531149273763434904 60197-2475794627968340439266926692582942523578559086951126639651892181 2563538267024485653444025561440747520000000/26918589533626245005933010 0908971764013929908518190138634918622045026360125733660563690494055892 465375982561*6^(1/2), a[12,9] = 63681466156701378449525903160928961790 100434377497936201427140303316098275608362465676059840449/119057634931 8934908527805697777807691148501676060784445076801271079131653422977059 56314829715000-3525911945755693171156511809912230970265688803844784846 50944931412648046608828531034562538513357/1547749254114615381086147407 111149998493052178879019778599841652402871149449870177432092786295000* 6^(1/2), `b*`[7] = 0, `b*`[6] = 0, `b*`[5] = 0, `b*`[4] = 0, `b*`[3] = 0, `b*`[2] = 0, a[14,10] = -19419587503350096681694941672242361995243 037384025897671049255680000000/215573892923309540745628400274055192560 7350553898010429805538450410571+16006927769146147858935916892287669045 96480000000/1027899053306897212338272667278244239283200723671*6^(1/2), `b*`[13] = 185286960316915979617280000000000000/465364379785423284294 027008596613217, a[14,11] = -27285760059113968837908932750404506400064 658240/1336728621411169422812695849204871581103756891, `b*`[11] = 2387 113868151976968347648/9351742108599933990994683\}:" }{TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "printed coe fficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 290 "subs(e16,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$( 9-i)],i=2..9)]));\nfor ii from 10 to 14 do\n print(``);\n print(c[ ii]=subs(e16,c[ii])); \n for jj to ii-1 do\n print(a[ii,jj]=sub s(e16,a[ii,jj]));\n end do:\nend do:\n``;\nfor ii to 14 do\n print (`b*`[ii]=subs(e16,`b*`[ii]));\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7*7+#\"\"\"\"#DF(%!GF+F+F+F+F+F+7+,&#\"#[\"$N$F)*(\" #KF)\"%v;!\"\"\"\"'#F)\"\"#F4,&#\"&#z:\"'DA6F4*(\"%ObF)F;F4F5F6F),&#\" &s=$F;F)*(\"%O:F)\"&XC#F4F5F6F4F+F+F+F+F+F+7+,&#\"#sF0F)*(F/F)F3F4F5F6 F4,&#\"#=F0F)*(\"#7F)F3F4F5F6F4\"\"!,&#\"#aF0F)*(\"#OF)F3F4F5F6F4F+F+F +F+F+7+#FG\"$D\",&#\"%9S\"%DJF)*(\"$_#F)\"$D'F4F5F6F)FN,&#\"&UZ\"FZF4* (\"$s*F)FgnF4F5F6F4,&#\"&GD\"FZF)*(\"$W\"F)FVF4F5F6F)F+F+F+F+7+,&#F/FV F)*(\"\")F)FVF4F5F6F4,&#\"%K7\"&vo\"F)*(\"$_\"F)FjoF4F5F6F4FNFN,&#\"&% oH\"'vo5F)*(\"&sL\"F)\"'D1KF4F5F6F4,&#\"%K@\"&DT'F)*(\"$%GF)\"&v8#F4F5 F6F4F+F+F+7+,&FdoF)*(FfoF)FVF4F5F6F),&#\"%K?FjoF)*(F\\pF)FjoF4F5F6F)FN FN,&#\"%[t\"&D$)*F4*(\"&_O$F)\"'v\\HF4F5F6F4,&#\"&K,\"FgpF)*(\"$;(F)Fj pF4F5F6F4,&#\"%#f#\"&vV\"F)*(\"%7HF)FarF4F5F6F)F+F+7+#\"#;F*#Ffr\"$D#F NFNFNFN,&#\"#kFhrF)*(\"\"%F)FhrF4F5F6F),&FjrF)*(F]sF)FhrF4F5F6F4F+7+#F MF*#\"#d\"$+)FNFNFNFN,&#\"$x\"FdsF)*(\"#pF)\"%+;F4F5F6F),&FfsF)*(FisF) FjsF4F5F6F4#!#FFdsQ(pprint26\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#5#\"%xL\"&++&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"\"#\"7dE-ug/H3`WG\"8+ +++++++S#)*e" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"#\"\" !" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"',&#\"7.UY)f_'fo:(G%\"8++++++++S#)*e\"\"\"*(\"7 fa-eOLiZ'))f\"F-\"9++++++++!['z6!\"\"F(#F-\"\"#F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"(,&#\"7.UY)f_'fo:(G%\"8++++++++S#)*e\" \"\"*(\"7fa-eOLiZ'))f\"F-\"9++++++++!['z6!\"\"\"\"'#F-\"\"#F-" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\")#!6VtL/'=i)Q.T\"\"7+ +++++++g`l" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"*#!5F>( \\b[[E49#\"6++++++++![?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#6#\"\"\"\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"\",&#\",r(*Q*=s\".++Cy\"f**!\" \"*(\"-*ysNmf%F(\"/++Wp]vfF-\"\"'#F(\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"#6\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\" \"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"'#\"\"\" \"#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"(,&#\"26pEE4C ,U\"\"3++[+XOprb!\"\"*(\"2H+mdBz!zJ\"\"\"F,F-\"\"'#F0\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"),&#\",TpV9C#\".++W(>j:\"\" \"*(\"-*ysNmf%F-\"/++%y5vi&!\"\"\"\"'#F-\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"*,&#\"-.\\g!=a\"\".++CaT`#\"\"\"*(\"-* ysNmf%F-\"/++3%p.9\"!\"\"\"\"'#F-\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"#5,&#\"5+++NCL([r=#\"6>Mez&zG&p`D\"\"\"\"*(\"5+++ g:\"Ha'Q=F-\"7r2D;i\"fdK)H6!\"\"\"\"'#F-\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#7#\" /y)*\\Gqhd\"/*pry#[4&)" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"# 7\"\"\",&#\"[qtF]iqW#)\\![t2(39R0auPHv&RmY\">h#)z(=#[herE'\\$=!3$R`8dW 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"6#/&%\"cG6#\"#8#\"%B;\"%+?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"\",&#\"VZ#ovn(Q0q.si'H%=wj@ gm([4&>^)[D\"V+++++++K_hRxA;x.5GX\\kSVOV`#!\"\"*(\"IFNW<'QWik*3jQj&\\> C3c_#F(\"J+++++++#fy!3$[Ncx*y,;C\"F-\"\"'#F(\"\"#F(" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"#8\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\" \"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"',&#\"F8_ l/-M=(=uCpan\"3Yvh#\"E+++++S1O$p?%))4re#RJ%!\"\"*(\"?*)*f2k$RfeO'*4NqC b\"\"\"\"@++Sq+]7Mqj#GNz+CF-F(#F0\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"(,&#\"P,7B@a:U;eEENyvEKsS=YHpf%\"P+++++++K;'=#> 1%*e\"yLT:Jx=5!\"\"*(\"O*)opj1^Y:`@g!*QPI=GeON\\S:\"\"\"\"N+++++++oz)z U=*Hd]odl'*)e))F-\"\"'#F0\"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/& %\"aG6$\"#8\"\"),&#\"T>xR@f=#o`,b*[x2!))e,W,=M(Q,.#\"U+++++++/\"*>\"op t$4\")=JM,[SEQt\">_34wa7@-wn\"\"\"\"H+++++++C!HT5**QM9F9dj! \"\"\"\"'#F,\"\"#F.#\"V*Q#R6!>&\\%G_\"3ScT?#\\Ryj9>zs+Md\"V+++++++))=5 O@\"4NOG'=EJpR)QUu>F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8 \"#5,&#\"Y<2=ed()fI!oySedA#>R]r)y+T#\"X+?H'GRp8l?%*>RkU!3R&\\6=3@ ,qri'*\"\"\"*(\"F@Dc)oVPlrK^(H%f<(G+!))F-\"G+&[A&eT6R:wq>(*oui$\\/#!\" \"\"\"'#F-\"\"#F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"#6 #\"NPZ1X&*eS+6IZ:D#RM_Lg\"*eV7\"M+++++S/(y,^-][pc%*\\C\"f$*>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"#7#\"^qRWJ;tQbXyTU#=vh@<(Q&)R AB_%zN&oR2L'))H?f.hN#3vWn'[(HRhr%f\"^q+++++g&eq.K(f+g:J?eVH0%p'f&zV(3_ e'Q8$Q75pdsJ'f$y'>r%)>!=)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#9\"\"\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"#9\"\"\",&*(\"HF!onC,M1@\\=]yc&*Rl%\"\"\"\"A[sb0[*)[#3rc7GY%\\_!\"\"F(#F,\"\"#F,#\"D6Wf btGm+vd!4l%y$o%f'\"Cg8X6Vs3Es6;V=IF," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#9\"\"(,&*(\"M$)px&Rm'o,>h*Hub^=([j24!)Q\"\"\"\"L++7p KwS*fnZ)p@@eUS:C&)>'!\"\"\"\"'#F,\"\"#F.#\"Lf`(*pn+>\"[#=N^1#RU8)*e'Hp )\"K++Of40kw%3*)e\\yIK#[hif`F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"#9\"\"),&#\"X$)eK*>r&3c5mNeO<2+&oWGECAi-jTE\"W++%=\\n+8rbvE=Cr 5F4)z*pa!)HPMN#\"\"\"*(\"HF!onC,M1@\\=]yc&*Rlia!zfsgb ?***>!\"\"\"\"'#F-\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"#9\"\"*,&#\"W@k]\"**orj#)fiOhZ0@a%p`L#\\Z.@U$>\"V++/t7:cQ<8i6\\P$G2K 9@5PMd$[g\"!\"\"*(\"HF!onC,M1@\\=]yc&*Rl+_c*)>=mn n_SF-\"\"'#F0\"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#9 \"#5,&#\"bo+++!ob#\\5n(*e-%QPIC&*>OUAnT\\p\"o'4]L](e>%>\"aor0T]%Qb!)H/ ,)*Qb]tgD>bSF+%GcuS&4L#H*Qd:#!\"\"*(\"R+++!['f/pwG#*o\"f$*ey9Y\"px#p+; \"\"\"\"RrOs+KGRUCysms#QB@(*oI`!**y-\"F-\"\"'#F0\"\"#F0" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#9\"#6#!PS#eY1+k]//vK*3z$)oR6f+w&GF\" O\"*ov.6er[?\\ep7GUp6T@'GnL\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"#9\"#7#!]rxVX+K**y:A#>*eQJ75H$fS%HTLE56^Ujys)eT%\\K](Ht\">bOab%[ NLLu=1;I&[>\"\\rkCjnJ6*o?W(Q\"G9]%3e#*[0.z%[?kT$)e\\x`Z\\8h/+3lmg3!4:r NvSTe%*egSe\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#9\"#8#\"V +++++[K(z\"=Ng!o6s(eF*[vmDN()[*)\"VR$=XK[`![_N\\0%)p#zL`YCH!)HtS\")e" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"\"#\"8J!*z]hJbiVUX$\":)[vhQobG2^p(p\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6# \"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"&\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"'\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%#b*G6#\"\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%#b*G6#\"\")#\";D19a'*zl\\g;%*36\";kWDgH,*R![XwO5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"*#\";DJq0ok[^Vx8m>\"<#R$32#)[kL=(pzfD " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#5#\"K+++++++++++vV$)oO RR!z)o\"L\"*RpFkbLbnpoQ3]Z1XP&H\"4'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%#b*G6#\"#6#\":[wMop(>:oQ6(Q#\":$o%*4*R$**f3@u^$*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#7#!fqH9-y>^cTT>$pWsKL(4c+6Z$)>7=H3XIk=2+ sFlE#33*4*el.\"4tLL\"*pGC\"\"fqw@5N2&zMxEjY.?)*[@Y[7c8PMCsxUq?*G())Q<0 #>n6XFUbN1Vc4P#>*[E\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#8 #\"E++++++!G<'zf\"pJgpG&=\"EcZ!R$" }}} {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 0 104 "`RK8_14eqs*` := subs(b=`b*`,[op(RowSumConditions(14, 'expanded')),op(OrderConditions(8,14,'expanded'))]):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Check:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "expand(subs(e16,`RK8_14eqs*`)):\nmap(u->l hs(u)-rhs(u),%);\nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7ax\"\"! F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$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#\"$8#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "We can calculate the prin cipal error norm, that is, the 2-norm of the principal error terms. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 185 "`errterms8_14*` := subs(b =`b*`,PrincipalErrorTerms(8,14,'expanded')):\nsm := 0:\nfor ct to nops (`errterms8_14*`) do\n sm := sm+(evalf(subs(e16,`errterms8_14*`[ct]) ))^2;\nend do:\nsqrt(sm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+*Rh5: %!#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 138 "We can include the new stage for the embedded scheme as an additional 16th stage added to the order 9 scheme along with the coefficients \+ " }{XPPEDIT 18 0 "a[16,14] = 0;" "6#/&%\"aG6$\"#;\"#9\"\"!" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[16,15] = 0;" "6#/&%\"aG6$\"#;\"#:\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 13 "The weights " } {XPPEDIT 18 0 "`b*`[i]" "6#&%#b*G6#%\"iG" }{TEXT -1 7 " for " } {XPPEDIT 18 0 "i=1" "6#/%\"iG\"\"\"" }{TEXT -1 89 " . . 13 of the 16 s tage combined scheme are those of the 14 stage scheme and the weight \+ " }{XPPEDIT 18 0 "`b*`[14];" "6#&%#b*G6#\"#9" }{TEXT -1 34 " in the 1 4 stage scheme becomes " }{XPPEDIT 18 0 "`b*`[16];" "6#&%#b*G6#\"#;" }{TEXT -1 25 " in the 16 stage scheme." }}{PARA 0 "" 0 "" {TEXT -1 8 "We set " }{XPPEDIT 18 0 "c[16] = 1;" "6#/&%\"cG6#\"#;\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[14] = 0;" "6#/&%#b*G6#\"#9\"\"!" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "`b*`[15] = 0;" "6#/&%#b*G6#\"#:\" \"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 66 "We can make the or der 9 scheme into a 16 stage scheme by setting " }{XPPEDIT 18 0 "b[16 ] = 0;" "6#/&%\"bG6#\"#;\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 201 "e17 := \{c[ 16]=1,seq(a[16,i]=subs(e16,a[14,i]),i=1..13),a[16,14]=0,a[16,15]=0,b[1 6]=0,\nseq(`b*`[i]=subs(e16,`b*`[i]),i=1..13),`b*`[14]=0,`b*`[15]=0,`b *`[16]=subs(e16,`b*`[14])\}:\ne18 := `union`(e13,e17):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e18" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13737 "e18 := \{a[14,2] = 0 , a[12,3] = 0, a[15,2] = 0, a[14,3] = 0, a[13,3] = 0, a[12,4] = 0, a[1 5,3] = 0, a[13,4] = 0, a[12,5] = 0, a[15,4] = 0, a[14,4] = 0, a[13,5] \+ = 0, a[15,5] = 0, a[14,5] = 0, c[16] = 1, a[4,2] = 0, a[16,14] = 0, a[ 16,15] = 0, c[11] = 1/4, b[16] = 0, a[2,1] = 1/25, a[10,1] = 284453082 9046074022657/58982400000000000000000, c[4] = 72/335-48/1675*6^(1/2), \+ a[10,7] = 4287156859652598464203/58982400000000000000000+1598864762333 658025459/117964800000000000000000*6^(1/2), a[11,9] = 154180604903/253 4154240000+459663572789/11403694080000*6^(1/2), a[10,8] = -14103388621 8604337343/6553600000000000000000, `b*`[1] = 34542436255316150799031/1 697695107285568386175488, a[9,1] = 57/800, `b*`[8] = 11089416604965799 654140625/10367645480399012960254464, c[2] = 1/25, c[5] = 72/125, c[8] = 16/25, c[10] = 3377/50000, a[5,2] = 0, a[6,2] = 0, a[6,3] = 0, a[7, 2] = 0, a[7,3] = 0, a[8,2] = 0, a[8,3] = 0, a[8,4] = 0, a[8,5] = 0, a[ 9,2] = 0, a[9,3] = 0, a[9,4] = 0, a[9,5] = 0, a[10,2] = 0, a[10,3] = 0 , a[10,4] = 0, a[10,5] = 0, a[11,2] = 0, a[11,3] = 0, a[11,4] = 0, a[1 1,5] = 0, a[11,6] = 1/30, b[14] = 117048651891177050452812500000000/90 3958175807874008864483503817601, `b*`[15] = 0, `b*`[14] = 0, a[16,5] = 0, a[16,4] = 0, a[16,3] = 0, a[16,2] = 0, a[16,1] = -1563176539955678 50184921063401246768027/212359240683067701291661898043594240000*6^(1/2 )+18092408213832965447840389945226814727838921316097789/45682609086632 02892681034746459555501706277524480000, a[16,7] = -3880090763487185155 7429961190168666395776983/6198524154042582121698476759940763269120000* 6^(1/2)+8692965898134239206513518248119006769975359/535962614823230784 958890847664050959360000, a[16,6] = 41984484090942118562371132909375/5 2494462812567108248894805557248*6^(1/2)+659468378465090577500662873555 94411/3018431611722608724311451319541760, a[16,10] = -1941958750335009 6681694941672242361995243037384025897671049255680000000/21557389292330 95407456284002740551925607350553898010429805538450410571+1600692776914 614785893591689228766904596480000000/102789905330689721233827266727824 4239283200723671*6^(1/2), a[16,9] = -193422103474923353694542105476136 625982637168991506421/160483573437102114320728337491162131738561512730 40000+156317653995567850184921063401246768027/405267676618198956520019 92553977680000*6^(1/2), a[16,8] = 264163026222242628446850007173658356 6105608571199325883/23534372980546997980927107124182675557113006749184 0000+156317653995567850184921063401246768027/1999920556072597905462191 46109360640000*6^(1/2), a[16,12] = -1948530160618743333354845554365519 1732975032494415887278634251111026334129440593291012313858919222157899 3200454377/15840605894584140753571150900860666508000461134947537749588 341642048479030548925808450142813874420689113167632464, a[16,11] = -27 285760059113968837908932750404506400064658240/133672862141116942281269 5849204871581103756891, `b*`[9] = 19661377435148646805703125/255979697 183364488207083392, a[16,13] = 894887352566754892758772116806035181797 32480000000000/58814073298029244653337926984054935524805348324518339, \+ `b*`[16] = 16723862451391031122709/339047561915509883967882, b[10] = 1 967337516701564001434375000000000000000000000000/154313648631198549430 71851131903908575429289017877, a[11,1] = -72189389771/9959178240000-45 9663572789/59755069440000*6^(1/2), a[6,1] = 1232/16875-152/16875*6^(1/ 2), a[5,3] = -14742/3125-972/625*6^(1/2), c[13] = 1623/2000, c[14] = 4 53/500, c[15] = 1, a[11,7] = -14201240926266911/557169364500480000-317 90792357660029/557169364500480000*6^(1/2), b[11] = 2323713252076974806 855457536/10352378514220126928031114081, b[2] = 0, b[3] = 0, b[4] = 0, b[5] = 0, c[3] = 48/335-32/1675*6^(1/2), a[4,3] = 54/335-36/1675*6^(1 /2), c[7] = 48/125+8/125*6^(1/2), a[7,4] = -7348/98325-33652/294975*6^ (1/2), a[6,4] = 29684/106875-13372/320625*6^(1/2), b[6] = 0, b[7] = 0, c[9] = 12/25, b[15] = 145778296653275182685983/4945417885871057962703 934, b[1] = 100976787617015984669475787/6921502952403262310437464576, \+ a[15,13] = -4759443892077050695292772766912275498897121280000000000/46 14026158486489304426198097242323507384603506591241137, b[9] = 96191657 2949681511747758515625/4416417715504587515036809762176, a[9,8] = -27/8 00, c[12] = 57617028499878/85094827871699, a[8,1] = 16/225, `b*`[10] = 688790393936688343750000000000000000000000/60912953745064750083868696 75533556427693991, b[13] = 296136352341197653422080000000000000/389943 2561650270968394778037550931439, a[10,9] = -21409264848554971927/20480 0000000000000000, b[8] = 8877148253451235588984375/4385514038208782482 187638272, b[12] = 283086002932414569543119391171389373911412640543251 5812108885979809005699950463699380263412267180656606126698463122636361 9/15839154654067008096806575700717389088587763113250985823298798221316 9380898029758612356596464561682203552186100842929847264, a[6,5] = 2132 /64125-284/21375*6^(1/2), a[11,10] = 21871487332435000000/125536952879 579583419+18386542911560000000/1129832575916216250771*6^(1/2), a[7,1] \+ = 2032/16875+152/16875*6^(1/2), a[4,1] = 18/335-12/1675*6^(1/2), a[11, 8] = 22414436941/1563197440000+459663572789/56275107840000*6^(1/2), a[ 8,6] = 64/225+4/225*6^(1/2), a[5,4] = 12528/3125+144/125*6^(1/2), a[5, 1] = 4014/3125+252/625*6^(1/2), a[3,2] = 31872/112225-1536/22445*6^(1/ 2), a[3,1] = -15792/112225+5536/112225*6^(1/2), a[7,5] = 10132/64125-7 16/21375*6^(1/2), c[6] = 48/125-8/125*6^(1/2), a[9,7] = 177/800-69/160 0*6^(1/2), a[7,6] = 2592/14375+2912/14375*6^(1/2), a[9,6] = 177/800+69 /1600*6^(1/2), a[10,6] = 4287156859652598464203/5898240000000000000000 0-1598864762333658025459/117964800000000000000000*6^(1/2), a[8,7] = 64 /225-4/225*6^(1/2), a[12,2] = 0, a[13,2] = 0, `b*`[12] = -124286991333 3730910365589099080822665277200071864304508291812198347110056097333272 4469319414156511978021429/12648919237095643063555422745116719205173888 728920704277722434371356124846214898200346632677347950735102176, a[12, 8] = 12039435467283852946442681868547691065960331569897374599708360175 92684437096940490927391123081521/6546732431504927940789740125933173479 539636700187880198168132038132534360122730704444373657280000-151110511 9609582787638505061390956130113866630219193505646906848911348771180693 70443383945077153/3273366215752463970394870062966586739769818350093940 099084066019066267180061365352222186828640000*6^(1/2), a[15,10] = 8222 2042178435392241610634610511219031021036676513878210889246011776000000 0/11253996143693043613068245539207658293443583785731438071890740125300 8522853-43216229787327667461772487493266526541342720000000/26879922067 906624005297106621487120800647611685231*6^(1/2), a[15,9] = 90164396775 224825317874747969895939573106537263617688583/993219458219557282041149 2715910530662743409752296080000-42203349401687175255637194817643092325 53/1059789239915401287502749181781142480000*6^(1/2), a[15,8] = -422033 4940168717525563719481764309232553/52298626520074835198571070841287910 40000*6^(1/2)-10128486019754425336362855829794634260766582925227304979 /2051440458733049774272501566471676463052668060718080000, a[14,10] = - 8031428603597180964147750321204281053557209400385686643427859804352/25 13586440178391306900178164909088113749933981949109368563389778875+2149 5292553013453087829315020563569618432/29973653446264566614372595657785 433238875*6^(1/2), a[13,8] = 20301387341801440158880077748955015368218 5921397719/6003783517480134311881093736968119910400000000000000-396881 2951635028178485515838403541696971/18374628007211630439078297600000000 000000*6^(1/2), a[14,8] = 46326458231132344862693578442836039366619919 16850927460433/5096189922215990577140952099757378096638035200000000000 000+41982993267604400562166630899538221911/116635822311401951029305600 000000000000*6^(1/2), a[13,9] = -6776022112547609085219173382640193141 17/635714271434389910412902400000000000000*6^(1/2)+5734007279191463783 9492204156400815228449519011392389/19744238839693126186283635091213610 188800000000000000, a[14,12] = -35998853393475254381336822594505813796 1504440675167771570563740579175484411998973149563531328527853237660249 760109643/234128137445668240736067711662442379892483470931740696615716 602345729933343308008758732590194294958617150200000000000, a[13,12] = \+ 5947161392974866744750823561035920298863307396853579452232239853871721 61751824241784555387316314439/8180198471196783596317257691012383133865 85208743795596694052943582031156000597320370585600000000000, a[15,11] \+ = 209814213871916216679569640811090009729840605120/1165197466096073809 6987230622554824075229159217, a[13,6] = -26175460816754692474187183402 04655213/431392587109884206933606400000000000-552470350996365859393640 759989/2400793528263703412500070400000*6^(1/2), a[13,7] = -45969294618 407232267578352626581642155421231201/101877311541337815894061921861632 00000000000000+1540493536582818303738906021531546510663696889/88588966 5576850572991842798796800000000000000*6^(1/2), a[12,6] = -391150225456 4577968858583198897514088250224516183112270320143559103652021094575185 0583137867/38425428798268102367071108589483043166777031255688218331140 907046999467997785653437677908480-490826700396287454540598331961129757 400186839154291202145347161815801053957874498846640625/153701715193072 4094682844343579321726671081250227528733245636281879978719911426137507 1163392*6^(1/2), a[14,7] = 9900924061444672761159422545227706890641317 9/16167053833464253010532287404800000000000000-37486194356245230406620 96435375495632409181/1405830768127326348741938035200000000000000*6^(1/ 2), a[12,7] = -6220642966809325161935258654730602646524634866355956674 47855699920023546211607797417635623004817316459/9074517035670590507636 6526235999860453636075671083639325891113637836086915532767731251940142 9106240000+64321004153532893292395583495936027027793078033448503026510 5750796567439186095378077346484617022694073/18149034071341181015273305 2471999720907272151342167278651782227275672173831065535462503880285821 2480000*6^(1/2), a[15,6] = -492170504431044248358505817476691843/26311 016381549020957251979880325120-65372649360291914372644744384375/457582 893592156886213077910962176*6^(1/2), a[14,6] = 59669959863673801810278 52718263477/684582962942931480729600000000000+289729124488482819328156 5089/19049265055803310768128000000*6^(1/2), a[13,1] = -254885119509487 66602163761842966272037005387677568247/2534336434064494528100377162277 3961523200000000000000+2525608241949563386308964624438617443527/124160 17897756354830807859200000000000000*6^(1/2), a[13,10] = 24100788715039 192225758197856261740786803059875758180717/966271700121081811495390804 2643919942065136939286292000-8800028717594297513271653743688562521/204 49362746897197076153911415852248500*6^(1/2), a[12,1] = -17814457135339 3080183496267158614821877982611914666395752937745405391408707734804982 447062502773/124771801011299405474614551641042535359813494760056839715 6373491324203879120405304413829778240000+35259119457556931711565118099 1223097026568880384478484650944931412648046608828531034562538513357/81 1016706573446135584994585666776479838787715940369458151642769360732521 4282634478689893558560000*6^(1/2), a[13,11] = 124358916033523439225154 730110040589545064737/19935912449945669485002510178704400000000000, a[ 12,11] = 1963500009650946638084395645582909493284711388363243988292674 5444809822658280742880787456/15009933124323477487137151792766813737022 277834253210285601916815234167186635020874092933, a[14,11] = -44866632 128961269158599595825347949153517725172080973/542960717582130814123228 3339099249860726338769531250, a[15,1] = 422033494016871752556371948176 4309232553/5553268895030690235789996003491120640000*6^(1/2)-4311793849 4612449223384237106139955697243762787772847383/14030104862294690648579 566521727488445420965260989440000, a[15,14] = 387336141443154484435382 20691259283298308750000000/9380629116032422006554046539278114776675256 9954001, a[12,10] = 10278934048763239705668635930656276438279802537812 076691952406255660602976082835598179622579836784640000000/207066073335 5865000456385391607475107799460834755308758730143246500202770197951235 1053114927376343490460197-24757946279683404392669266925829425235785590 869511266396518921812563538267024485653444025561440747520000000/269185 8953362624500593301009089717640139299085181901386349186220450263601257 33660563690494055892465375982561*6^(1/2), a[12,9] = 636814661567013784 4952590316092896179010043437749793620142714030331609827560836246567605 9840449/11905763493189349085278056977778076911485016760607844450768012 7107913165342297705956314829715000-35259119457556931711565118099122309 7026568880384478484650944931412648046608828531034562538513357/15477492 5411461538108614740711114999849305217887901977859984165240287114944987 0177432092786295000*6^(1/2), a[14,1] = 1845926793614707536745942392081 89459317466722387929369596771/1278912170918306562333640142057316550645 71321600000000000000-293880952873230803935166416296767553377/866938749 696854853908947200000000000000*6^(1/2), a[15,7] = -6024202572220608364 7640080390153021797730167/4671870346091448695506240355969515576320000+ 312292306982904450161582808948983992497619279/540311961765358849132460 84116864832317440000*6^(1/2), a[15,12] = 39415508928639522811841227154 8581315021451985814824457854376666199907962545343216942004611160316807 1904019820197053706993310801913/57634880694574783556174939816096694539 0916610765210182493857593996732757363976976471505530337022074730630638 475548489174498085584, a[14,13] = 1011136807359189181222571916914688/2 927578889427871359661497484969729, a[14,9] = -121354756771846880705598 19501757154968443992507349800863475059/3337534088557268857775113382843 672473526974754800000000000000+293880952873230803935166416296767553377 /165447122399672764770545400000000000000*6^(1/2), `b*`[7] = 0, `b*`[6] = 0, `b*`[5] = 0, `b*`[4] = 0, `b*`[3] = 0, `b*`[2] = 0, `b*`[13] = 1 85286960316915979617280000000000000/4653643797854232842940270085966132 17, `b*`[11] = 2387113868151976968347648/9351742108599933990994683\}: " }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "printed coefficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 363 "subs(e18,matrix([seq([c[i],seq(a[i ,j],j=1..i-1),``$(9-i)],i=2..9)]));\nfor ii from 10 to 16 do\n print (``);\n print(c[ii]=subs(e18,c[ii])); \n for jj to ii-1 do\n \+ print(a[ii,jj]=subs(e18,a[ii,jj]));\n end do:\nend do:print(``);\nfo r ii to 16 do\n print(b[ii]=subs(e18,b[ii]));\nend do:print(``);\nfo r ii to 16 do\n print(`b*`[ii]=subs(e18,`b*`[ii]));\nend do:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7*7+#\"\"\"\"#DF(%!GF+F+F+ F+F+F+7+,&#\"#[\"$N$F)*(\"#KF)\"%v;!\"\"\"\"'#F)\"\"#F4,&#\"&#z:\"'DA6 F4*(\"%ObF)F;F4F5F6F),&#\"&s=$F;F)*(\"%O:F)\"&XC#F4F5F6F4F+F+F+F+F+F+7 +,&#\"#sF0F)*(F/F)F3F4F5F6F4,&#\"#=F0F)*(\"#7F)F3F4F5F6F4\"\"!,&#\"#aF 0F)*(\"#OF)F3F4F5F6F4F+F+F+F+F+7+#FG\"$D\",&#\"%9S\"%DJF)*(\"$_#F)\"$D 'F4F5F6F)FN,&#\"&UZ\"FZF4*(\"$s*F)FgnF4F5F6F4,&#\"&GD\"FZF)*(\"$W\"F)F VF4F5F6F)F+F+F+F+7+,&#F/FVF)*(\"\")F)FVF4F5F6F4,&#\"%K7\"&vo\"F)*(\"$_ \"F)FjoF4F5F6F4FNFN,&#\"&%oH\"'vo5F)*(\"&sL\"F)\"'D1KF4F5F6F4,&#\"%K@ \"&DT'F)*(\"$%GF)\"&v8#F4F5F6F4F+F+F+7+,&FdoF)*(FfoF)FVF4F5F6F),&#\"%K ?FjoF)*(F\\pF)FjoF4F5F6F)FNFN,&#\"%[t\"&D$)*F4*(\"&_O$F)\"'v\\HF4F5F6F 4,&#\"&K,\"FgpF)*(\"$;(F)FjpF4F5F6F4,&#\"%#f#\"&vV\"F)*(\"%7HF)FarF4F5 F6F)F+F+7+#\"#;F*#Ffr\"$D#FNFNFNFN,&#\"#kFhrF)*(\"\"%F)FhrF4F5F6F),&Fj rF)*(F]sF)FhrF4F5F6F4F+7+#FMF*#\"#d\"$+)FNFNFNFN,&#\"$x\"FdsF)*(\"#pF) \"%+;F4F5F6F),&FfsF)*(FisF)FjsF4F5F6F4#!#FFdsQ(pprint36\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6 #\"#5#\"%xL\"&++&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\" \"#\"7dE-ug/H3`WG\"8++++++++S#)*e" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ 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" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"',&#\"7.UY)f_'fo:(G%\"8++++ ++++S#)*e\"\"\"*(\"7fa-eOLiZ'))f\"F-\"9++++++++!['z6!\"\"F(#F-\"\"#F1 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"(,&#\"7.UY)f_'fo: (G%\"8++++++++S#)*e\"\"\"*(\"7fa-eOLiZ'))f\"F-\"9++++++++!['z6!\"\"\" \"'#F-\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\")#!6 VtL/'=i)Q.T\"\"7++++++++g`l" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG 6$\"#5\"\"*#!5F>(\\b[[E49#\"6++++++++![?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#6#\"\"\"\"\"% " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"\",&#\",r(*Q*=s\" .++Cy\"f**!\"\"*(\"-*ysNmf%F(\"/++Wp]vfF-\"\"'#F(\"\"#F-" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"#6\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\" \"'#\"\"\"\"#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"(,& #\"26pEE4C,U\"\"3++[+XOprb!\"\"*(\"2H+mdBz!zJ\"\"\"F,F-\"\"'#F0\"\"#F- " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"),&#\",TpV9C#\".+ +W(>j:\"\"\"*(\"-*ysNmf%F-\"/++%y5vi&!\"\"\"\"'#F-\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"*,&#\"-.\\g!=a\"\".++CaT`#\"\" \"*(\"-*ysNmf%F-\"/++3%p.9\"!\"\"\"\"'#F-\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"#5,&#\"5+++NCL([r=#\"6>Mez&zG&p`D\"\"\" \"*(\"5+++g:\"Ha'Q=F-\"7r2D;i\"fdK)H6!\"\"\"\"'#F-\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6 #\"#7#\"/y)*\\Gqhd\"/*pry#[4&)" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"#7\"\"\",&#\"[qtF]iqW#)\\![t2(39R0auPHv&RmY\">h#)z(=#[herE'\\$ =!3$R`8dW\"y\"\"\\q++Cy(HQT/`S?\"zQ?C8\\tj:(Ro0gZ\\8)f``U5k^XhuaS*H65! =xC\"!\"\"*(\"[qdL^QDcM5`G)3m/[ETJ\\%4l%[yWQ!))ol-(4B7*4=^c6<$pbd%>\"f _$F(\"\\q++ceN*)*oyWj#G9_K2OpFk^\"e%p.%fr(yQ)zkxmce%*\\eNhWtlq;5\")F- \"\"'#F(\"\"#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"# \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"$\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"%\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"&\"\"!" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"',&#\"gpny8$e]=vX4@?l.\"fN9?.F7J=;XA]# )39v*))>$ee)ozdkXD-:\"R\"gp![3znPMl&y(*zY**p/249J$=#)ob7.xn;VI[*e362nB 5o#)zGaUQ!\"\"*(\"epD1kY))\\uy&R0,e\"=;Z`9-7Ha\"Ro=+uvH6'>L)fSXX(G'R+n #3\\\"\"\"\"gp#Rj62v8E9\"*>(y*z=GOcCL(GvA]73rms@$zNMWGo%4C2$>:;D$4o'Hk?i\"aq++C1\"H9S> DJxwKb\"p3Oyj86*eKRO3rc2OOXg)**fBElOw]!fqc.Z_It_,\"=T82M!\\\"=F-\"\"'#F0\"\"#F0" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"),&#\"\\q@:3B6RF4\\Sp 4PWo#f!)y=+njR&zM>-jmQ6Ih&4Rh]]Qwy#e4'>^56:F-\"\\q++kGo=AANl81!=ni1>g1%3*4SR4]$=)p (RnemH1q[RqRY_d@mLF$!\"\"\"\"'#F-\"\"#F1" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"*,&#\"jp\\/%)fgnlCO3cF)4;LISrU,i$z\\xVV+,zh*G4; .f_\\%y8qchY\"oj\"[q+]rH[Jcfq(HU`;8z5F,o2XWyggn,&[6p2yx(p0y_3\\$*=$\\j d!>\"\"\"\"*(\"[qdL^QDcM5`G)3m/[ETJ\\%4l%[yWQ!))ol-(4B7*4=^c6<$pbd%>\" f_$F-\"\\q+]H'y#4Ku!z)y@0$\\)**\\662u9'3\"Q:Y6a# \\xa\"!\"\"\"\"'#F-\"\"#F1" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6 $\"#7\"#5,&#\"fq+++SYyO)zDiz\")fNG3wHggcD1C&>pw?\"y`-)z#QkFc1$fjocqRKw [S$*y-\"\"fq(>g/\\VjPF\\6`5N7&z>qF?+lCV,te(3`vM3Y*z2^Z2;R&Qc/+leNL2mq? 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82031156000597320370585600000000000,\na[14,1]=184592679361470753674594 239208189459317466722387929369596771/\n1278912170918306562333640142057 31655064571321600000000000000-293880952873230803935166416296767553377/ 866938749696854853908947200000000000000*6^(1/2),\na[14,2]=0,\na[14,3]= 0,\na[14,4]=0,\na[14,5]=0,\na[14,6]=5966995986367380181027852718263477 /684582962942931480729600000000000+2897291244884828193281565089/190492 65055803310768128000000*6^(1/2),\na[14,7]=9900924061444672761159422545 2277068906413179/16167053833464253010532287404800000000000000-37486194 35624523040662096435375495632409181/1405830768127326348741938035200000 000000000*6^(1/2),\na[14,8]=463264582311323448626935784428360393666199 1916850927460433/50961899222159905771409520997573780966380352000000000 00000+41982993267604400562166630899538221911/1166358223114019510293056 00000000000000*6^(1/2),\na[14,9]=-121354756771846880705598195017571549 68443992507349800863475059/\n33375340885572688577751133828436724735269 74754800000000000000+293880952873230803935166416296767553377/165447122 399672764770545400000000000000*6^(1/2),\na[14,10]=-8031428603597180964 147750321204281053557209400385686643427859804352/251358644017839130690 0178164909088113749933981949109368563389778875+21495292553013453087829 315020563569618432/29973653446264566614372595657785433238875*6^(1/2), \na[14,11]=-44866632128961269158599595825347949153517725172080973/5429 607175821308141232283339099249860726338769531250,\na[14,12]=-359988533 9347525438133682259450581379615044406751677715705637405791754844119989 73149563531328527853237660249760109643/2341281374456682407360677116624 4237989248347093174069661571660234572993334330800875873259019429495861 7150200000000000,\na[14,13]=1011136807359189181222571916914688/2927578 889427871359661497484969729,\na[15,1]=-4311793849461244922338423710613 9955697243762787772847383/14030104862294690648579566521727488445420965 260989440000+4220334940168717525563719481764309232553/\n55532688950306 90235789996003491120640000*6^(1/2),\na[15,2]=0,\na[15,3]=0,\na[15,4]=0 ,\na[15,5]=0,\na[15,6]=-492170504431044248358505817476691843/263110163 81549020957251979880325120-65372649360291914372644744384375/4575828935 92156886213077910962176*6^(1/2),\na[15,7]=-602420257222060836476400803 90153021797730167/4671870346091448695506240355969515576320000+\n312292 306982904450161582808948983992497619279/540311961765358849132460841168 64832317440000*6^(1/2),\na[15,8]=-101284860197544253363628558297946342 60766582925227304979/2051440458733049774272501566471676463052668060718 080000-4220334940168717525563719481764309232553/5229862652007483519857 107084128791040000*6^(1/2),\na[15,9]=901643967752248253178747479698959 39573106537263617688583/9932194582195572820411492715910530662743409752 296080000-4220334940168717525563719481764309232553/1059789239915401287 502749181781142480000*6^(1/2),\na[15,10]=82222042178435392241610634610 5112190310210366765138782108892460117760000000/11253996143693043613068 2455392076582934435837857314380718907401253008522853-43216229787327667 461772487493266526541342720000000/268799220679066240052971066214871208 00647611685231*6^(1/2),\na[15,11]=209814213871916216679569640811090009 729840605120/11651974660960738096987230622554824075229159217,\na[15,12 ]=39415508928639522811841227154858131502145198581482445785437666619990 79625453432169420046111603168071904019820197053706993310801913/5763488 0694574783556174939816096694539091661076521018249385759399673275736397 6976471505530337022074730630638475548489174498085584,\na[15,13]=-47594 43892077050695292772766912275498897121280000000000/4614026158486489304 426198097242323507384603506591241137,\na[15,14]=3873361414431544844353 8220691259283298308750000000/93806291160324220065540465392781147766752 569954001,\n\na[16,1]=180924082138329654478403899452268147278389213160 97789/4568260908663202892681034746459555501706277524480000-15631765399 5567850184921063401246768027/212359240683067701291661898043594240000*6 ^(1/2),\na[16,2]=0,\na[16,3]=0,\na[16,4]=0,\na[16,5]=0,\na[16,6]=65946 837846509057750066287355594411/3018431611722608724311451319541760+4198 4484090942118562371132909375/52494462812567108248894805557248*6^(1/2), \na[16,7]=8692965898134239206513518248119006769975359/5359626148232307 84958890847664050959360000-3880090763487185155742996119016866639577698 3/6198524154042582121698476759940763269120000*6^(1/2),\na[16,8]=264163 0262222426284468500071736583566105608571199325883/23534372980546997980 9271071241826755571130067491840000+15631765399556785018492106340124676 8027/199992055607259790546219146109360640000*6^(1/2),\na[16,9]=-193422 103474923353694542105476136625982637168991506421/160483573437102114320 72833749116213173856151273040000+1563176539955678501849210634012467680 27/40526767661819895652001992553977680000*6^(1/2),\na[16,10]=-19419587 503350096681694941672242361995243037384025897671049255680000000/215573 8929233095407456284002740551925607350553898010429805538450410571+16006 92776914614785893591689228766904596480000000/1027899053306897212338272 667278244239283200723671*6^(1/2),\na[16,11]=-2728576005911396883790893 2750404506400064658240/1336728621411169422812695849204871581103756891, \na[16,12]=-1948530160618743333354845554365519173297503249441588727863 42511110263341294405932910123138589192221578993200454377/1584060589458 4140753571150900860666508000461134947537749588341642048479030548925808 450142813874420689113167632464,\na[16,13]=8948873525667548927587721168 0603518179732480000000000/58814073298029244653337926984054935524805348 324518339,\na[16,14]=0,\na[16,15]=0,\n \nb[1]=100976787617015984669475 787/6921502952403262310437464576,\nseq(b[i]=0,i=2..7),\nb[8]=887714825 3451235588984375/4385514038208782482187638272,\nb[9]=96191657294968151 1747758515625/4416417715504587515036809762176,\nb[10]=1967337516701564 001434375000000000000000000000000/154313648631198549430718511319039085 75429289017877,\nb[11]=2323713252076974806855457536/103523785142201269 28031114081,\nb[12]=28308600293241456954311939117138937391141264054325 1581210888597980900569995046369938026341226718065660612669846312263636 19/1583915465406700809680657570071738908858776311325098582329879822131 69380898029758612356596464561682203552186100842929847264,\nb[13]=29613 6352341197653422080000000000000/3899432561650270968394778037550931439, \nb[14]=117048651891177050452812500000000/9039581758078740088644835038 17601,\nb[15]=145778296653275182685983/4945417885871057962703934,\nb[1 6]=0,\n\n`b*`[1]=34542436255316150799031/1697695107285568386175488,\n` b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=0,\n`b*`[5]=0,\n`b*`[6]=0,\n`b*`[7]=0, \n`b*`[8]=11089416604965799654140625/10367645480399012960254464,\n`b*` [9]=19661377435148646805703125/255979697183364488207083392,\n`b*`[10]= 688790393936688343750000000000000000000000/609129537450647500838686967 5533556427693991,\n`b*`[11]=2387113868151976968347648/9351742108599933 990994683,\n`b*`[12]=-124286991333373091036558909908082266527720007186 43045082918121983471100560973332724469319414156511978021429/1264891923 7095643063555422745116719205173888728920704277722434371356124846214898 200346632677347950735102176,\n`b*`[13]=1852869603169159796172800000000 00000/465364379785423284294027008596613217,\n`b*`[14]=0,\n`b*`[15]=0, \n`b*`[16]=16723862451391031122709/339047561915509883967882\}:" }}} {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[9,16];" "6#&%\"TG6$\"\"*\"#;" }{TEXT -1 129 " denote the vector whose components are the principal error terms of \+ the 16 stage, order 9 scheme (the error terms of order 10)." }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "`T*`[8,16];" "6#&%#T*G6$ \"\")\"#;" }{TEXT -1 146 " denote the vector whose components are the principal error terms of the embedded 16 stage, order 8 scheme (the e rror terms of order 9) and let " }{XPPEDIT 18 0 "`T*`[9, 16];" "6#&%# T*G6$\"\"*\"#;" }{TEXT -1 100 " denote the vector whose components ar e the error terms of order 10 of the embedded order 8 scheme." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Denote th e 2-norm of these vectors by " }{XPPEDIT 18 0 "abs(abs(T[9,16]));" "6 #-%$absG6#-F$6#&%\"TG6$\"\"*\"#;" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "ab s(abs(`T*`[8,16]));" "6#-%$absG6#-F$6#&%#T*G6$\"\")\"#;" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "abs(abs(`T*`[9,16]));" "6#-%$absG6#-F$6#&%#T* G6$\"\"*\"#;" }{TEXT -1 15 " respectively." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Define: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A[10] = abs(abs(T[9, 16]));" "6#/&%\"AG 6#\"#5-%$absG6#-F)6#&%\"TG6$\"\"*\"#;" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "B[10] = abs(abs(`T*`[9,16]))/abs(abs(`T*`[8,16]));" "6#/&%\"BG6# \"#5*&-%$absG6#-F*6#&%#T*G6$\"\"*\"#;\"\"\"-F*6#-F*6#&F/6$\"\")F2!\"\" " }{TEXT -1 6 ", " }{XPPEDIT 18 0 "C[10] = abs(abs(`T*`[9,16]-T[9, 16]))/abs(abs(`T*`[8,16]));" "6#/&%\"CG6#\"#5*&-%$absG6#-F*6#,&&%#T*G6 $\"\"*\"#;\"\"\"&%\"TG6$F2F3!\"\"F4-F*6#-F*6#&F06$\"\")F3F8" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Dormand and Prince have suggested that as well as attempting to en sure that " }{XPPEDIT 18 0 "A[10];" "6#&%\"AG6#\"#5" }{TEXT -1 73 " \+ is a minimum, if the embedded scheme is to be used for error control, \+ " }{XPPEDIT 18 0 "B[10];" "6#&%\"BG6#\"#5" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "C[10];" "6#&%\"CG6#\"#5" }{TEXT -1 27 " should be chos en so that " }}{PARA 0 "" 0 "" {TEXT -1 68 "they are \"similar in magn itude\" and also not differ too much from 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 168 "errterms9_16 := P rincipalErrorTerms(9,16,'expanded'):\n`errterms9_16*` :=subs(b=`b*`,er rterms9_16):\n`errterms8_16*` := subs(b=`b*`,PrincipalErrorTerms(8,16, 'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 510 "nmB := 0: \nfor ct to nops(`errterms9_16*`) do \n print(ct);\n nmB := nmB+evalf(subs(ee,`errterms9_16*`[ct]))^2; \nend do:\nsnmB := sqrt(nmB):\ndnB := 0:\nfor ct to nops(`errterms8_16 *`) do\n print(ct);\n dnB := dnB+evalf(subs(ee,`errterms8_16*`[ct] ))^2;\nend do:\nsdnB := sqrt(dnB):\nnmC := 0:\nfor ct to nops(errterms 9_16) do\n print(ct);\n nmC := nmC+(evalf(subs(ee,`errterms9_16*`[ ct]))-evalf(subs(ee,errterms9_16[ct])))^2;\nend do:\nsnmC := sqrt(nmC) :\n'B[10]'= evalf[8](snmB/sdnB);\n'C[10]'= evalf[8](snmC/sdnB);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"BG6#\"#5$\")v`CM!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"#5$\")L#fT$!\"(" }}}{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 regio ns" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "coefficient s of the combined scheme " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13385 "ee := \{c[2]=1/25,\nc[3]=48/335- 32/1675*6^(1/2),\nc[4]=72/335-48/1675*6^(1/2),\nc[5]=72/125,\nc[6]=48/ 125-8/125*6^(1/2),\nc[7]=48/125+8/125*6^(1/2),\nc[8]=16/25,\nc[9]=12/2 5,\nc[10]=3377/50000,\nc[11]=1/4,\nc[12]=57617028499878/85094827871699 ,\nc[13]=1623/2000,\nc[14]=453/500,\nc[15]=1,\nc[16]=1,\n\na[2,1]=1/25 ,\na[3,1]=-15792/112225+5536/112225*6^(1/2),\na[3,2]=31872/112225-1536 /22445*6^(1/2),\na[4,1]=18/335-12/1675*6^(1/2),\na[4,2]=0,\na[4,3]=54/ 335-36/1675*6^(1/2),\na[5,1]=4014/3125+252/625*6^(1/2),\na[5,2]=0,\na[ 5,3]=-14742/3125-972/625*6^(1/2),\na[5,4]=12528/3125+144/125*6^(1/2), \na[6,1]=1232/16875-152/16875*6^(1/2),\na[6,2]=0,\na[6,3]=0,\na[6,4]=2 9684/106875-13372/320625*6^(1/2),\na[6,5]=2132/64125-284/21375*6^(1/2) ,\na[7,1]=2032/16875+152/16875*6^(1/2),\na[7,2]=0,\na[7,3]=0,\na[7,4]= -7348/98325-33652/294975*6^(1/2),\na[7,5]=10132/64125-716/21375*6^(1/2 ),\na[7,6]=2592/14375+2912/14375*6^(1/2),\na[8,1]=16/225,\na[8,2]=0,\n a[8,3]=0,\na[8,4]=0,\na[8,5]=0,\na[8,6]=64/225+4/225*6^(1/2),\na[8,7]= 64/225-4/225*6^(1/2),\na[9,1]=57/800,\na[9,2]=0,\na[9,3]=0,\na[9,4]=0, \na[9,5]=0,\na[9,6]=177/800+69/1600*6^(1/2),\na[9,7]=177/800-69/1600*6 ^(1/2),\na[9,8]=-27/800,\na[10,1]=2844530829046074022657/5898240000000 0000000000,\na[10,2]=0,\na[10,3]=0,\na[10,4]=0,\na[10,5]=0,\na[10,6]=4 287156859652598464203/58982400000000000000000-1598864762333658025459/1 17964800000000000000000*6^(1/2),\na[10,7]=4287156859652598464203/58982 400000000000000000+1598864762333658025459/117964800000000000000000*6^( 1/2),\na[10,8]=-141033886218604337343/6553600000000000000000,\na[10,9] =-21409264848554971927/204800000000000000000,\na[11,1]=-72189389771/99 59178240000-459663572789/59755069440000*6^(1/2),\na[11,2]=0,\na[11,3]= 0,\na[11,4]=0,\na[11,5]=0,\na[11,6]=1/30,\na[11,7]=-14201240926266911/ 557169364500480000-31790792357660029/557169364500480000*6^(1/2),\na[11 ,8]=22414436941/1563197440000+459663572789/56275107840000*6^(1/2),\na[ 11,9]=154180604903/2534154240000+459663572789/11403694080000*6^(1/2), \na[11,10]=21871487332435000000/125536952879579583419+1838654291156000 0000/1129832575916216250771*6^(1/2),\na[12,1]=-17814457135339308018349 6267158614821877982611914666395752937745405391408707734804982447062502 773/124771801011299405474614551641042535359813494760056839715637349132 4203879120405304413829778240000+35259119457556931711565118099122309702 6568880384478484650944931412648046608828531034562538513357/81101670657 3446135584994585666776479838787715940369458151642769360732521428263447 8689893558560000*6^(1/2),\na[12,2]=0,\na[12,3]=0,\na[12,4]=0,\na[12,5] =0,\na[12,6]=-39115022545645779688585831988975140882502245161831122703 201435591036520210945751850583137867/\n3842542879826810236707110858948 3043166777031255688218331140907046999467997785653437677908480-49082670 0396287454540598331961129757400186839154291202145347161815801053957874 498846640625/\n1537017151930724094682844343579321726671081250227528733 2456362818799787199114261375071163392*6^(1/2),\na[12,7]=-6220642966809 3251619352586547306026465246348663559566744785569992002354621160779741 7635623004817316459/90745170356705905076366526235999860453636075671083 6393258911136378360869155327677312519401429106240000+64321004153532893 2923955834959360270277930780334485030265105750796567439186095378077346 484617022694073/181490340713411810152733052471999720907272151342167278 6517822272756721738310655354625038802858212480000*6^(1/2),\na[12,8]=12 0394354672838529464426818685476910659603315698973745997083601759268443 7096940490927391123081521/65467324315049279407897401259331734795396367 00187880198168132038132534360122730704444373657280000-1511105119609582 7876385050613909561301138666302191935056469068489113487711806937044338 3945077153/\n327336621575246397039487006296658673976981835009394009908 4066019066267180061365352222186828640000*6^(1/2),\na[12,9]=63681466156 7013784495259031609289617901004343774979362014271403033160982756083624 65676059840449/1190576349318934908527805697777807691148501676060784445 07680127107913165342297705956314829715000-3525911945755693171156511809 91223097026568880384478484650944931412648046608828531034562538513357/1 5477492541146153810861474071111499984930521788790197785998416524028711 49449870177432092786295000*6^(1/2),\na[12,10]=102789340487632397056686 3593065627643827980253781207669195240625566060297608283559817962257983 6784640000000/20706607333558650004563853916074751077994608347553087587 301432465002027701979512351053114927376343490460197-247579462796834043 9266926692582942523578559086951126639651892181256353826702448565344402 5561440747520000000/26918589533626245005933010090897176401392990851819 0138634918622045026360125733660563690494055892465375982561*6^(1/2),\na [12,11]=19635000096509466380843956455829094932847113883632439882926745 444809822658280742880787456/150099331243234774871371517927668137370222 77834253210285601916815234167186635020874092933,\na[13,1]=-25488511950 948766602163761842966272037005387677568247/253433643406449452810037716 22773961523200000000000000+2525608241949563386308964624438617443527/12 416017897756354830807859200000000000000*6^(1/2),\na[13,2]=0,\na[13,3]= 0,\na[13,4]=0,\na[13,5]=0,\na[13,6]=-261754608167546924741871834020465 5213/431392587109884206933606400000000000-5524703509963658593936407599 89/2400793528263703412500070400000*6^(1/2),\na[13,7]=-4596929461840723 2267578352626581642155421231201/10187731154133781589406192186163200000 000000000+1540493536582818303738906021531546510663696889/8858896655768 50572991842798796800000000000000*6^(1/2),\na[13,8]=2030138734180144015 88800777489550153682185921397719/6003783517480134311881093736968119910 400000000000000-3968812951635028178485515838403541696971/1837462800721 1630439078297600000000000000*6^(1/2),\na[13,9]=57340072791914637839492 204156400815228449519011392389/197442388396931261862836350912136101888 00000000000000-677602211254760908521917338264019314117/635714271434389 910412902400000000000000*6^(1/2),\na[13,10]=\n241007887150391922257581 97856261740786803059875758180717/9662717001210818114953908042643919942 065136939286292000-8800028717594297513271653743688562521/2044936274689 7197076153911415852248500*6^(1/2),\na[13,11]=1243589160335234392251547 30110040589545064737/19935912449945669485002510178704400000000000,\na[ 13,12]=594716139297486674475082356103592029886330739685357945223223985 387172161751824241784555387316314439/818019847119678359631725769101238 313386585208743795596694052943582031156000597320370585600000000000,\na [14,1]=184592679361470753674594239208189459317466722387929369596771/\n 127891217091830656233364014205731655064571321600000000000000-293880952 873230803935166416296767553377/866938749696854853908947200000000000000 *6^(1/2),\na[14,2]=0,\na[14,3]=0,\na[14,4]=0,\na[14,5]=0,\na[14,6]=596 6995986367380181027852718263477/684582962942931480729600000000000+2897 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54031196176535884913246084116864832317440000*6^(1/2),\na[15,8]=-101284 86019754425336362855829794634260766582925227304979/2051440458733049774 272501566471676463052668060718080000-422033494016871752556371948176430 9232553/5229862652007483519857107084128791040000*6^(1/2),\na[15,9]=901 64396775224825317874747969895939573106537263617688583/9932194582195572 820411492715910530662743409752296080000-422033494016871752556371948176 4309232553/1059789239915401287502749181781142480000*6^(1/2),\na[15,10] =822220421784353922416106346105112190310210366765138782108892460117760 000000/112539961436930436130682455392076582934435837857314380718907401 253008522853-43216229787327667461772487493266526541342720000000/268799 22067906624005297106621487120800647611685231*6^(1/2),\na[15,11]=209814 213871916216679569640811090009729840605120/116519746609607380969872306 22554824075229159217,\na[15,12]=39415508928639522811841227154858131502 1451985814824457854376666199907962545343216942004611160316807190401982 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20000*6^(1/2),\na[16,8]=2641630262222426284468500071736583566105608571 199325883/235343729805469979809271071241826755571130067491840000+15631 7653995567850184921063401246768027/19999205560725979054621914610936064 0000*6^(1/2),\na[16,9]=-1934221034749233536945421054761366259826371689 91506421/16048357343710211432072833749116213173856151273040000+1563176 53995567850184921063401246768027/4052676766181989565200199255397768000 0*6^(1/2),\na[16,10]=-194195875033500966816949416722423619952430373840 25897671049255680000000/2155738929233095407456284002740551925607350553 898010429805538450410571+160069277691461478589359168922876690459648000 0000/1027899053306897212338272667278244239283200723671*6^(1/2),\na[16, 11]=-27285760059113968837908932750404506400064658240/13367286214111694 22812695849204871581103756891,\na[16,12]=-1948530160618743333354845554 3655191732975032494415887278634251111026334129440593291012313858919222 1578993200454377/15840605894584140753571150900860666508000461134947537 749588341642048479030548925808450142813874420689113167632464,\na[16,13 ]=89488735256675489275877211680603518179732480000000000/58814073298029 244653337926984054935524805348324518339,\na[16,14]=0,\na[16,15]=0,\n \+ \nb[1]=100976787617015984669475787/6921502952403262310437464576,\nseq( b[i]=0,i=2..7),\nb[8]=8877148253451235588984375/4385514038208782482187 638272,\nb[9]=961916572949681511747758515625/4416417715504587515036809 762176,\nb[10]=1967337516701564001434375000000000000000000000000/15431 364863119854943071851131903908575429289017877,\nb[11]=2323713252076974 806855457536/10352378514220126928031114081,\nb[12]=2830860029324145695 4311939117138937391141264054325158121088859798090056999504636993802634 122671806566061266984631226363619/158391546540670080968065757007173890 8858776311325098582329879822131693808980297586123565964645616822035521 86100842929847264,\nb[13]=296136352341197653422080000000000000/3899432 561650270968394778037550931439,\nb[14]=1170486518911770504528125000000 00/903958175807874008864483503817601,\nb[15]=145778296653275182685983/ 4945417885871057962703934,\nb[16]=0,\n\n`b*`[1]=3454243625531615079903 1/1697695107285568386175488,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=0,\n`b*` [5]=0,\n`b*`[6]=0,\n`b*`[7]=0,\n`b*`[8]=11089416604965799654140625/103 67645480399012960254464,\n`b*`[9]=19661377435148646805703125/255979697 183364488207083392,\n`b*`[10]=6887903939366883437500000000000000000000 00/6091295374506475008386869675533556427693991,\n`b*`[11]=238711386815 1976968347648/9351742108599933990994683,\n`b*`[12]=-124286991333373091 0365589099080822665277200071864304508291812198347110056097333272446931 9414156511978021429/12648919237095643063555422745116719205173888728920 704277722434371356124846214898200346632677347950735102176,\n`b*`[13]=1 85286960316915979617280000000000000/4653643797854232842940270085966132 17,\n`b*`[14]=0,\n`b*`[15]=0,\n`b*`[16]=16723862451391031122709/339047 561915509883967882\}:" }}}{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 stability function R for the 16 s tage, order 9 scheme is given (approximately) as follows." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "expa nd(subs(ee,StabilityFunction(9,16,'expanded'))):\nmap(convert,evalf[28 ](%),rational,24):\nR := unapply(%,z):\n'R(z)'=R(z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"RG6#%\"zG,B\"\"\"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)*$)F'\"\")F)F)F)*&#F)\"'!)GOF)*$)F'\"\"*F)F)F)*&#\"*KE)z7\" 0p*fyH**pXF)*$)F'\"#5F)F)F)*&#\")&)RSY\"1CBL%eJ,[\"F)*$)F'\"#6F)F)F)*& #\"(^(Qz\"2'p%z15'y[9F)*$)F'\"#7F)F)!\"\"*&#\"(p%3!)\"1%\\Vtjhy\"HF)*$ )F'\"#8F)F)Fgo*&#\"(UgC&\"2rI(GX0FwBF)*$)F'\"#9F)F)F)*&#\"'*p`)\"38MBD 2]McrF)*$)F'\"#:F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can find the point where the boundary of the sta bility region intersects the negative real axis by solving the equatio n: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R(z) = 1;" "6# /-%\"RG6#%\"zG\"\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "z0 := newton(R(z)=1,z=-4.5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$!+^xW@X!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 304 "z0 := newton(R(z)=1,z=-4.5) :\np1 := plot([R(z),1],z=-5.19..0.49,color=[red,blue]):\np2 := plot([[ [z0,1]]$3],style=point,symbol=[circle,cross,diamond],color=black):\np3 := plot([[z0,0],[z0,1]],linestyle=3,color=COLOR(RGB,0,.5,0)):\nplots[ display]([p1,p2,p3],view=[-5.19..0.49,-0.07..1.47],font=[HELVETICA,9]) ;" }}{PARA 13 "" 1 "" {GLPLOT2D 360 253 253 {PLOTDATA 2 "6+-%'CURVESG6 $7Z7$$!3Q++++++!>&!#<$\"3+!oaJMK#ykF*7$$!3QML3T![!f^F*$\"3E5l=Ye*H(fF* 7$$!3Ynm;#3'4G^F*$\"3Bj^I>'eW]&F*7$$!3a++DBT9(4&F*$\"3yZgvq%f,2&F*7$$! 3kLLLk@>m]F*$\"3c`_`C')ynYF*7$$!3E+]U'*)HB,&F*$\"3gM\"RK)\\MPSF*7$$!3! pm;&GwYe\\F*$\"3Cw5$yiQiFho7$$!3$QL$)f7eWC%F*$\"3\\bC77RH4VFho7$$!3A++lN ]MCTF*$\"3G:/VrU;sHFho7$$!3ummYeRz+SF*$\"3>Wa#QeSQ.#Fho7$$!3_LLV-,(>*Q F*$\"3[\"\\FOX)\\q9Fho7$$!35++S:-YpPF*$\"3s2H_,*Gb/\"Fho7$$!3K+++\"HZk k$F*$\"3[UOYmnMgx!#>7$$!3;++gW:!z_$F*$\"35&y-sn6d>'F\\r7$$!3hLL)*\\1D? 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%(DEFAULTG-%%VIEWG6$;$!$>&!\"#$\"#\\Fidl;$!\"(Fidl$\"$Z\"Fidl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "The following picture shows the st ability region." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1453 "R := z ->add(z^j/j!,j=0..9)+\n 127982632 /456999297859969*z^10+46403985/1480131584332324*z^11-\n 7938751/14 487861006794696*z^12-8008469/2917861637343494*z^13+\n 5246042/2376 2705452873071*z^14+853699/715634500725233413*z^15:\npts := []: z0 := 0 :\nfor ct from 0 to 320 do\n zz := newton(R(z)=exp(ct*Pi/20*I),z=z0) :\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np1 := \+ plot(pts,color=COLOR(RGB,.22,0,.45)):\np2 := plots[polygonplot]([seq([ pts[i-1],pts[i],[-2.3,0]],i=2..nops(pts))],\n style=patchnogr id,color=COLOR(RGB,.45,.33,.95)):\npts := []: z0 := 1.1+4.8*I:\nfor ct from 0 to 60 do\n zz := newton(R(z)=exp(ct*Pi/30*I),z=z0):\n z0 : = zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np3 := plot(pts,c olor=COLOR(RGB,.22,0,.45)):\np4 := plots[polygonplot]([seq([pts[i-1],p ts[i],[0.97,4.7]],i=2..nops(pts))],\n style=patchnogrid,color =COLOR(RGB,.45,.33,.95)):\npts := []: z0 := 1.1-4.8*I:\nfor ct from 0 \+ to 60 do\n zz := newton(R(z)=exp(ct*Pi/30*I),z=z0):\n z0 := zz:\n \+ pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np5 := plot(pts,color=COL OR(RGB,.22,0,.45)):\np6 := plots[polygonplot]([seq([pts[i-1],pts[i],[0 .97,-4.7]],i=2..nops(pts))],\n style=patchnogrid,color=COLOR( RGB,.45,.33,.95)):\np7 := plot([[[-5.29,0],[1.89,0]],[[0,-5.29],[0,5.2 9]]],color=black,linestyle=3):\nplots[display]([p||(1..7)],view=[-5.29 ..1.89,-5.29..5.29],font=[HELVETICA,9],\n labels=[`Re(z)` ,`Im(z)`],axes=boxed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 501 551 551 {PLOTDATA 2 "6/-%'CURVESG6$7]_l7$$\"\"!F)F(7$F($ \"3++++Fjzq:!#=7$F($\"3)******Rl#fTJF-7$F($\"3<+++!)*)Q7ZF-7$$\"3.+++ 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}{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 288 "We can distort the boundary curve horizo ntally by taking the 11th root of the real part of points along the cu rve. In this way we see that the largest interval on the nonnegative i maginary axis that contains the origin and lies inside the stability r egion is [ 0, 2.7 ] approximately. " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 483 "R := z ->add(z^j/j!,j=0 ..9)+\n 127982632/456999297859969*z^10+46403985/1480131584332324*z ^11-\n 7938751/14487861006794696*z^12-8008469/2917861637343494*z^1 3+\n 5246042/23762705452873071*z^14+853699/715634500725233413*z^15 :\nDigits := 25:\npts := []: z0 := 0:\nfor ct from 0 to 90 do\n zz : = newton(R(z)=exp(ct*Pi/100*I),z=z0):\n z0 := zz:\n pts := [op(pts ),[surd(Re(zz),11),Im(zz)]]:\nend do:\nplot(pts,color=COLOR(RGB,.5,.2, .95),thickness=2,font=[HELVETICA,9]);\nDigits := 10:" }}{PARA 13 "" 1 "" {GLPLOT2D 287 308 308 {PLOTDATA 2 "6(-%'CURVESG6#7gp7$$\"\"!F)F(7$$ \":,LP,[q9Oclf\\(!#F$\":\\EYQKz*e`EfTJ!#E7$$\":@N>&o&[fu#R?39F0$\":;i$ pZ'ezrI&=$G'F07$$\":]tOcOGbk')=s.#F0$\":\"zFjrz$p2'zxC%*F07$$\":]i'QKI HR7Ti[EF0$\":i>f(H#F@7$$\":<(y[b.W\\y4U/]F0$\":s4pX)Q=(G7uK^#F@7$$\":u*o .%)HrWB5r\"e&F0$\":.%*\\1\"H3B)QLu#GF@7$$\":q$4mi;Cx%4>p:'F0$\":Frgv.% )*e`EfTJF@7$$\":rxG0.,!f(=(*3t'F0$\":&GVr'H!*[*=>vbMF@7$$\":E&fM9j:O,$ [VI(F0$\":!H$oN'z!3V=6*pPF@7$$\":/I'4DZ:+!Hdy(yF0$\":fGC,,\\n'\\/2%3%F @7$$\":#RTMzfYA0D\">X)F0$\":[3aglMF]rH#)R%F@7$$\":n1j.lgYuk@p-*F0$\":4 $[:%***zQ!)*)Q7ZF@7$$\":JzZuL%)=,#QA.'*F0$\":rT;@=**\\dC[l-&F@7$$\":yM AR.1O***)4\"=5F@$\":*=#>9y496^22M&F@7$$\":))QhU?Ulycxg2\"F@$\":vh4bEL \"[wn'[l&F@7$$\":$z]`U3m'yrVU8\"F@$\":0CIw@'G&=/E!pfF@7$$\":!zYsC=!*o% pAE>\"F@$\":\\\">\\0H(HsI&=$G'F@7$$\":X5vT-p5oID7D\"F@$\":H>))\\DC7EdW tf'F@7$$\":kXUV0=MS@f+J\"F@$\":\"fSJ!QQ.JmDsF@7$$\":g\\=H4h*RxxVG9F@$\":ou(R2WTvoB#)RvF@7$$\": #z*o'G6!oP\"[)z[\"F@$\":d&)>j&*e$3M;)R&yF@7$$\":^sytamE.Qoxa\"F@$\":F@$\":tG4x5@Zyk4`+\"!#C7$$ \":%e%)\\)R0*)*QqNs>F@$\"::Q!\\vVR\\tbsO5Fgu7$$\":g*pjKi%R/ZoP.#F@$\": 'eV?$3&[k)\\T\"o5Fgu7$$\":SA<\\<^*\\A4L&4#F@$\":`)ekT+l1Bub*4\"Fgu7$$ \":\"z>$*R&)*p76Hq:#F@$\":by#QLg:VYL(48\"Fgu7$$\":X'y\\$*Qs'eyY)=AF@$ \":7&=g$>['Go#*Qi6Fgu7$$\":VshGR_ncum2G#F@$\":70sV1t4!)=0Q>\"Fgu7$$\": N$G&eh\"yhZ3xUBF@$\":t$3x0\\#eZ5@_A\"Fgu7$$\":R%p9H&RJQ+S[S#F@$\":1=Y& yy$*R 'e8Fgu7$$\":)GK%*f'oH]UK7f#F@$\":S6,HF@$\": #RAudSJ)p%Q'z]\"Fgu7$$\":h\">%3\")[KBuRF'HF@$\":^6NhG4R#f&z$R:Fgu7$$\" :$4:55![K*>)zT-$F@$\":LOTH#[?.1_zq:Fgu7$$\":![[$*3VoW[:S&3$F@$\":![d$4 $\\'4'p2@-;Fgu7$$\":6*)ohL[_5Lqj9$F@$\":y\\aC3)4*zAEOj\"Fgu7$$\":x1gVd 6\\4[\\q?$F@$\":T&p]qZm4a:/l;Fgu7$$\":guKPM/2G***RnKF@$\":u!p6)ftGZrck p\"Fgu7$$\":[GHM%H#4tF!QFLF@$\":c=N_p=!Qp;(ys\"Fgu7$$\":]]!)*f\\g7_e%p Q$F@$\":>1ps/E^)ojGfFgu7$$\":A\"oE.i0KZ#pEt$F@$\":V4+wes0' )*[wZ>Fgu7$$\":E_M%z?qSR)[xy$F@$\":&Hl!>\"H[h+`Fgu7$$\":8!*=NB$zzwL *=%QF@$\":\"ydBE>\"e/\\%e5?Fgu7$$\":vc?h:\"[I9\\+&*QF@$\":JTP@]$**HLA* >/#Fgu7$$\":bIGuI9wrltp%RF@$\":JVsBV6^]D)Rt?Fgu7$$\":%>`(e(on]ucn(*RF@ $\":`P!GAV2&pB-[5#Fgu7$$\":ghRU\\._C-qp/%F@$\":gTuLtM%f4Q?O@Fgu7$$\":U Y.G!pw&=3&p%4%F@$\":hbr'QKVQYDgn@Fgu7$$\":2z`/!3[8MQmSTF@$\":ban2)*e$Q cz**)>#Fgu7$$\":;pcpz*Hixwl%=%F@$\":4&Q04&yflZ*QIAFgu7$$\":^xxqw'>W*>= kA%F@$\":ZfTsD6;MYw-OcE%F@$\":H8[(4(Q9S=eJH#Fgu7$ $\":Sg4hXH$3i`$>I%F@$\":@+j!4!Q3k!Q`CBFgu7$$\":sk'*Q(*QP>8_[L%F@$\":;` /_&y$>&*Q-fN#Fgu7$$\":AQ#e(p42#**>!QO%F@$\":h\"42$p1-D(GE(Q#Fgu7$$\":j ;C;O$*)GZA.)Q%F@$\":qSS4h#pmjSh=CFgu7$$\":#4Q0by->Fja1WF@$\":nMr\">2_/ HY&*\\CFgu7$$\":(*peNB\"*z4()zzT%F@$\":;%)p'3/Qq!3$G\"[#Fgu7$$\"::\\,7 'zuf2YQ?WF@$\":>)\\7N[%R\\w(f7DFgu7$$\":>0kPk\"[CHC#3T%F@$\":sy+piw.0& o*Qa#Fgu7$$\":myR!Rp:+_H_%Q%F@$\":9E+xiwJqJy^d#Fgu7$$\":sm%yJ\\(H'GH(G L%F@$\":1b:b2$['Q%*Rkg#Fgu7$$\":2)oChOLY!>ruB%F@$\":\\$\\sUxMZ*Hzwj#Fg u7$$\":mBaU1(3G.j5H1O7t#Fgu7$$!:m!yX#\\ %R_]TBQWF@$\":&4#3Q(Hf-#[dBw#Fgu7$$!:qO1Tll&4-&p+JW&[F@$\":26U(4@g()yF[CGFgu-%*THICKNESSG6#\"\"#-%&CO LORG6&%$RGBG$\"\"&!\"\"$F]hlFdhl$\"#&*!\"#-%%FONTG6$%*HELVETICAG\"\"*- %+AXESLABELSG6$Q!6\"Fail-%%VIEWG6$%(DEFAULTGFfil" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 123 "The relevant intersectio n point of the boundary curve with the imaginary axis can be determine d more accurately as follows." }}{PARA 0 "" 0 "" {TEXT -1 86 "First we look for points on the boundary curve either side of the intersection point. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "Digits := 15:\nz0 := 2.7*I:\nfor ct from 85 to 88 do \n print(newton(R(z)=exp(ct*Pi/100*I),z=z0));\nend do;\nDigits := 10 :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"003)=(QHt%!#>$\"0(GpJP*)oE!# 9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0Z8Q>DBY$!#?$\"0s#)*e.3+F!#9 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!04IK3&)e^&!#>$\"0>5H1O7t#!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0FN\\+rhJ\"!#=$\"0$f-#[dBw#!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "Then \+ we apply the bisection method to calculate the parameter value associa ted with the intersection point." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 " real_part := proc(u)\n Re(newton(R(z)=exp(u*Pi*I),z=2.7*I ))\nend proc:\nDigits := 15:\nu0 := bisect('real_part'(u),u=0.85..0.88 );\nnewton(R(z)=exp(u0*Pi*I),z=2.7*I);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"0X[+e`ng)!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"0Q=W\\&=-F!#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 insid e the stability region" }{TEXT -1 5 " is " }{XPPEDIT 18 0 "[0, 2.7022 ];" "6#7$\"\"!-%&FloatG6$\"&Aq#!\"%" }{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 97 "The stability function R* for the 16 st age, order 8 scheme is given (approximately) as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "expand(subs(ee,subs(b=`b*`,Stabili tyFunction(8,16,'expanded')))):\nmap(convert,evalf[28](%),rational,24) :\n`R*` := unapply(%,z):\n'`R*`(z)'=`R*`(z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%#R*G6#%\"zG,@\"\"\"F)F'F)*&#F)\"\"#F)*$)F'F,F)F)F)*& #F)\"\"'F)*$)F'\"\"$F)F)F)*&#F)\"#CF)*$)F'\"\"%F)F)F)*&#F)\"$?\"F)*$)F '\"\"&F)F)F)*&#F)\"$?(F)*$)F'F1F)F)F)*&#F)\"%S]F)*$)F'\"\"(F)F)F)*&#F) \"&?.%F)*$)F'\"\")F)F)F)*&#\"*\"eg*[&\"0Z[hOip(=F)*$)F'\"\"*F)F)F)*&# \"**)G=h\"\"0nE4]LbC#F)*$)F'\"#5F)F)F)*&#\")**4\"\\)\"0v$\\lO2DiF)*$)F '\"#6F)F)F)*&#\"*=s*3K\"1rn-:[:S_F)*$)F'\"#7F)F)!\"\"*&#\")I$)=a\"2x*e \\*=W#)R\"F)*$)F'\"#8F)F)F)*&#\"(H08\"\"2b4LcKi#3`F)*$)F'\"#9F)F)F)" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can find the point where the boundary of the stability region intersects \+ the negative real axis by solving the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`R*`(z) = -1;" "6#/-%#R*G6#%\"zG,$\"\" \"!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "z_0 := newton(`R*`(z)=-1,z=-4);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$z_0G$!+fHS4R!\"*" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 322 "z_0 := ne wton(`R*`(z)=-1,z=-4):\np_1 := plot([`R*`(z),-1],z=-4.59..0.49,color=[ red,blue]):\np_2 := plot([[[z_0,-1]]$3],style=point,symbol=[circle,cro ss,diamond],color=black):\np_3 := plot([[z_0,0],[z_0,-1]],linestyle=3, color=COLOR(RGB,0,.5,0)):\nplots[display]([p_1,p_2,p_3],view=[-4.59..0 .49,-1.47..1.47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 369 264 264 {PLOTDATA 2 "6+-%'CURVESG6$7Y7$$!3')*************e%!#<$!3i 4$\\%[TGMuF*7$$!3EL$3Uh%F*7$$! 3n*\\(yb,vGVF*$!3CI@E=4/'f$F*7$$!3SLL=Q]duUF*$!3I7a%Hp?X2$F*7$$!34LLt7 3/?UF*$!3)>JNj'p#)>EF*7$$!3yKLG(e1b;%F*$!3]l'=%*pIpA#F*7$$!3/n;p>l&p0% F*$!3cLI8oD(*)f\"F*7$$!3KL$e;r;j&RF*$!3IURDhm*R;\"F*7$$!3++]s>+6_QF*$! 3ESu&3B%Gw#)!#=7$$!3IL$e)4$RVu$F*$!3Vx[EomuBdFho7$$!39+]-AU\"pj$F*$!3# 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$$!3E'**\\A^nfv'Fho$\"3Q%e'\\Kn_)3&Fho7$$!3%zmm;Qn5r&Fho$\"3e-]el*)**[ cFho7$$!3J'**\\7y`rh%Fho$\"3'=ixg$o,-jFho7$$!3!)GLLp\\ejNFho$\"3[cQ`za @-qFho7$$!39LL$))oeh[#Fho$\"35+mL=\\z)z(Fho7$$!3yim\">/awT\"Fho$\"35Ii s1#[#y')Fho7$$!3Mf****>C4eVFbq$\"3K3Pix2bt&*Fho7$$\"3wMLL)*p&\\*oFbq$ \"3yp['z<#Qr5F*7$$\"31mmma=)fp\"Fho$\"35Q&yUnG[=\"F*7$$\"370+v?g5pFFho $\"3#HusbW[!>8F*7$$\"3g1+DvFA'z$Fho$\"3yU#RgNK " 0 "" {MPLTEXT 1 0 1465 "`R*` := z -> add(z^j/j!,j=0..8)+\n 548960581/187696236614847 *z^9+161182889/224553350092667*z^10+\n 84910999/622507366549375*z^ 11-320897218/5240154815026771*z^12+\n 54188330/13982441894958977*z ^13+1130529/53082623256330955*z^14:\npts := []: z0 := 0:\nfor ct from \+ 0 to 280 do\n zz := newton(`R*`(z)=exp(ct*Pi/20*I),z=z0):\n z0 := \+ zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np_1 := plot(pts,co lor=COLOR(RGB,.17,0,.35)):\np_2 := plots[polygonplot]([seq([pts[i-1],p ts[i],[-2,0]],i=2..nops(pts))],\n style=patchnogrid,color=COL OR(RGB,.38,0,.75)):\npts := []: z0 := 0.9+4.2*I:\nfor ct from 0 to 60 \+ do\n zz := newton(`R*`(z)=exp(ct*Pi/30*I),z=z0):\n z0 := zz:\n p ts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np_3 := plot(pts,color=COLOR (RGB,.17,0,.35)):\np_4 := plots[polygonplot]([seq([pts[i-1],pts[i],[.8 1,4.05]],i=2..nops(pts))],\n style=patchnogrid,color=COLOR(RG B,.38,0,.75)):\npts := []: z0 := 0.9-4.2*I:\nfor ct from 0 to 60 do\n \+ zz := newton(`R*`(z)=exp(ct*Pi/30*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np_5 := plot(pts,color=COLOR(RGB, .17,0,.35)):\np_6 := plots[polygonplot]([seq([pts[i-1],pts[i],[.81,-4. 05]],i=2..nops(pts))],\n style=patchnogrid,color=COLOR(RGB,.3 8,0,.75)):\np_7 := plot([[[-4.49,0],[1.49,0]],[[0,-4.99],[0,4.99]]],co lor=black,linestyle=3):\nplots[display]([p_||(1..7)],view=[-4.49..1.49 ,-4.99..4.99],font=[HELVETICA,9],\n labels=[`Re(z)`,`Im(z)` ],axes=boxed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 396 515 515 {PLOTDATA 2 "6/-%'CURVESG6$7e\\l7$$\"\"!F)F(7$F($\"3++++Fjzq:! 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"Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "C urve 11" "Curve 12" "Curve 13" "Curve 14" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#--------------------------------------------------- " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 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 36 "coefficient s of the combined scheme " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13385 "ee := \{c[2]=1/25,\nc[3]=48/335- 32/1675*6^(1/2),\nc[4]=72/335-48/1675*6^(1/2),\nc[5]=72/125,\nc[6]=48/ 125-8/125*6^(1/2),\nc[7]=48/125+8/125*6^(1/2),\nc[8]=16/25,\nc[9]=12/2 5,\nc[10]=3377/50000,\nc[11]=1/4,\nc[12]=57617028499878/85094827871699 ,\nc[13]=1623/2000,\nc[14]=453/500,\nc[15]=1,\nc[16]=1,\n\na[2,1]=1/25 ,\na[3,1]=-15792/112225+5536/112225*6^(1/2),\na[3,2]=31872/112225-1536 /22445*6^(1/2),\na[4,1]=18/335-12/1675*6^(1/2),\na[4,2]=0,\na[4,3]=54/ 335-36/1675*6^(1/2),\na[5,1]=4014/3125+252/625*6^(1/2),\na[5,2]=0,\na[ 5,3]=-14742/3125-972/625*6^(1/2),\na[5,4]=12528/3125+144/125*6^(1/2), \na[6,1]=1232/16875-152/16875*6^(1/2),\na[6,2]=0,\na[6,3]=0,\na[6,4]=2 9684/106875-13372/320625*6^(1/2),\na[6,5]=2132/64125-284/21375*6^(1/2) ,\na[7,1]=2032/16875+152/16875*6^(1/2),\na[7,2]=0,\na[7,3]=0,\na[7,4]= -7348/98325-33652/294975*6^(1/2),\na[7,5]=10132/64125-716/21375*6^(1/2 ),\na[7,6]=2592/14375+2912/14375*6^(1/2),\na[8,1]=16/225,\na[8,2]=0,\n a[8,3]=0,\na[8,4]=0,\na[8,5]=0,\na[8,6]=64/225+4/225*6^(1/2),\na[8,7]= 64/225-4/225*6^(1/2),\na[9,1]=57/800,\na[9,2]=0,\na[9,3]=0,\na[9,4]=0, \na[9,5]=0,\na[9,6]=177/800+69/1600*6^(1/2),\na[9,7]=177/800-69/1600*6 ^(1/2),\na[9,8]=-27/800,\na[10,1]=2844530829046074022657/5898240000000 0000000000,\na[10,2]=0,\na[10,3]=0,\na[10,4]=0,\na[10,5]=0,\na[10,6]=4 287156859652598464203/58982400000000000000000-1598864762333658025459/1 17964800000000000000000*6^(1/2),\na[10,7]=4287156859652598464203/58982 400000000000000000+1598864762333658025459/117964800000000000000000*6^( 1/2),\na[10,8]=-141033886218604337343/6553600000000000000000,\na[10,9] =-21409264848554971927/204800000000000000000,\na[11,1]=-72189389771/99 59178240000-459663572789/59755069440000*6^(1/2),\na[11,2]=0,\na[11,3]= 0,\na[11,4]=0,\na[11,5]=0,\na[11,6]=1/30,\na[11,7]=-14201240926266911/ 557169364500480000-31790792357660029/557169364500480000*6^(1/2),\na[11 ,8]=22414436941/1563197440000+459663572789/56275107840000*6^(1/2),\na[ 11,9]=154180604903/2534154240000+459663572789/11403694080000*6^(1/2), \na[11,10]=21871487332435000000/125536952879579583419+1838654291156000 0000/1129832575916216250771*6^(1/2),\na[12,1]=-17814457135339308018349 6267158614821877982611914666395752937745405391408707734804982447062502 773/124771801011299405474614551641042535359813494760056839715637349132 4203879120405304413829778240000+35259119457556931711565118099122309702 6568880384478484650944931412648046608828531034562538513357/81101670657 3446135584994585666776479838787715940369458151642769360732521428263447 8689893558560000*6^(1/2),\na[12,2]=0,\na[12,3]=0,\na[12,4]=0,\na[12,5] =0,\na[12,6]=-39115022545645779688585831988975140882502245161831122703 201435591036520210945751850583137867/\n3842542879826810236707110858948 3043166777031255688218331140907046999467997785653437677908480-49082670 0396287454540598331961129757400186839154291202145347161815801053957874 498846640625/\n1537017151930724094682844343579321726671081250227528733 2456362818799787199114261375071163392*6^(1/2),\na[12,7]=-6220642966809 3251619352586547306026465246348663559566744785569992002354621160779741 7635623004817316459/90745170356705905076366526235999860453636075671083 6393258911136378360869155327677312519401429106240000+64321004153532893 2923955834959360270277930780334485030265105750796567439186095378077346 484617022694073/181490340713411810152733052471999720907272151342167278 6517822272756721738310655354625038802858212480000*6^(1/2),\na[12,8]=12 0394354672838529464426818685476910659603315698973745997083601759268443 7096940490927391123081521/65467324315049279407897401259331734795396367 00187880198168132038132534360122730704444373657280000-1511105119609582 7876385050613909561301138666302191935056469068489113487711806937044338 3945077153/\n327336621575246397039487006296658673976981835009394009908 4066019066267180061365352222186828640000*6^(1/2),\na[12,9]=63681466156 7013784495259031609289617901004343774979362014271403033160982756083624 65676059840449/1190576349318934908527805697777807691148501676060784445 07680127107913165342297705956314829715000-3525911945755693171156511809 91223097026568880384478484650944931412648046608828531034562538513357/1 5477492541146153810861474071111499984930521788790197785998416524028711 49449870177432092786295000*6^(1/2),\na[12,10]=102789340487632397056686 3593065627643827980253781207669195240625566060297608283559817962257983 6784640000000/20706607333558650004563853916074751077994608347553087587 301432465002027701979512351053114927376343490460197-247579462796834043 9266926692582942523578559086951126639651892181256353826702448565344402 5561440747520000000/26918589533626245005933010090897176401392990851819 0138634918622045026360125733660563690494055892465375982561*6^(1/2),\na [12,11]=19635000096509466380843956455829094932847113883632439882926745 444809822658280742880787456/150099331243234774871371517927668137370222 77834253210285601916815234167186635020874092933,\na[13,1]=-25488511950 948766602163761842966272037005387677568247/253433643406449452810037716 22773961523200000000000000+2525608241949563386308964624438617443527/12 416017897756354830807859200000000000000*6^(1/2),\na[13,2]=0,\na[13,3]= 0,\na[13,4]=0,\na[13,5]=0,\na[13,6]=-261754608167546924741871834020465 5213/431392587109884206933606400000000000-5524703509963658593936407599 89/2400793528263703412500070400000*6^(1/2),\na[13,7]=-4596929461840723 2267578352626581642155421231201/10187731154133781589406192186163200000 000000000+1540493536582818303738906021531546510663696889/8858896655768 50572991842798796800000000000000*6^(1/2),\na[13,8]=2030138734180144015 88800777489550153682185921397719/6003783517480134311881093736968119910 400000000000000-3968812951635028178485515838403541696971/1837462800721 1630439078297600000000000000*6^(1/2),\na[13,9]=57340072791914637839492 204156400815228449519011392389/197442388396931261862836350912136101888 00000000000000-677602211254760908521917338264019314117/635714271434389 910412902400000000000000*6^(1/2),\na[13,10]=\n241007887150391922257581 97856261740786803059875758180717/9662717001210818114953908042643919942 065136939286292000-8800028717594297513271653743688562521/2044936274689 7197076153911415852248500*6^(1/2),\na[13,11]=1243589160335234392251547 30110040589545064737/19935912449945669485002510178704400000000000,\na[ 13,12]=594716139297486674475082356103592029886330739685357945223223985 387172161751824241784555387316314439/818019847119678359631725769101238 313386585208743795596694052943582031156000597320370585600000000000,\na [14,1]=184592679361470753674594239208189459317466722387929369596771/\n 127891217091830656233364014205731655064571321600000000000000-293880952 873230803935166416296767553377/866938749696854853908947200000000000000 *6^(1/2),\na[14,2]=0,\na[14,3]=0,\na[14,4]=0,\na[14,5]=0,\na[14,6]=596 6995986367380181027852718263477/684582962942931480729600000000000+2897 291244884828193281565089/19049265055803310768128000000*6^(1/2),\na[14, 7]=99009240614446727611594225452277068906413179/1616705383346425301053 2287404800000000000000-3748619435624523040662096435375495632409181/140 5830768127326348741938035200000000000000*6^(1/2),\na[14,8]=46326458231 13234486269357844283603936661991916850927460433/5096189922215990577140 952099757378096638035200000000000000+419829932676044005621666308995382 21911/116635822311401951029305600000000000000*6^(1/2),\na[14,9]=-12135 475677184688070559819501757154968443992507349800863475059/\n3337534088 557268857775113382843672473526974754800000000000000+293880952873230803 935166416296767553377/165447122399672764770545400000000000000*6^(1/2), \na[14,10]=-8031428603597180964147750321204281053557209400385686643427 859804352/251358644017839130690017816490908811374993398194910936856338 9778875+21495292553013453087829315020563569618432/29973653446264566614 372595657785433238875*6^(1/2),\na[14,11]=-4486663212896126915859959582 5347949153517725172080973/54296071758213081412322833390992498607263387 69531250,\na[14,12]=-3599885339347525438133682259450581379615044406751 67771570563740579175484411998973149563531328527853237660249760109643/2 3412813744566824073606771166244237989248347093174069661571660234572993 3343308008758732590194294958617150200000000000,\na[14,13]=101113680735 9189181222571916914688/2927578889427871359661497484969729,\na[15,1]=-4 3117938494612449223384237106139955697243762787772847383/14030104862294 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\\s%e&3)\\u\"*[[bZQ1jIZ2AqLIb]rk(p(ROdFt'*RfdQ\\#=5_w5m\"4RXp'4;)R\\EWI*['[eh-9Y/&F%6$Fc`mFhjl#\"S+++](3$)H$Gf7p?#QNW[aJWThL(Q \"S,S&*pDvmx9\"y#Rl/al+AC.;\"H1Q*/&F%6$FbuF(,&*(\"HF!onC,M1@\\=]yc&*Rl %F(\"A[sb0[*)[ #3rc7GY%\\_F3F6F7F(#\"D6WfbtGm+vd!4l%y$o%f'\"Cg8X6Vs3Es6;V=IF(/&F%6 $FbuFgr,&*(\"M$)px&Rm'o,>h*Hub^=([j24!)QF(\"L++7pKwS*fnZ)p@@eUS:C&)>'F 3F6F7F3#\"Lf`(*pn+>\"[#=N^1#RU8)*e'Hp)\"K++Of40kw%3*)e\\yIK#[hif`F(/&F %6$FbuF`u,&#\"X$)eK*>r&3c5mNeO<2+&oWGECAi-jTE\"W++%=\\n+8rbvE=Cr5F4)z* pa!)HPMN#F(*(\"HF!onC,M1@\\=]yc&*Rlia!zfsgb?***>F3F6F 7F(/&F%6$FbuF_w,&#\"W@k]\"**orj#)fiOhZ0@a%p`L#\\Z.@U$>\"V++/t7:cQ<8i6 \\P$G2K9@5PMd$[g\"F3*(FcimF(\"G++oxRb#*>+_c*)>=mnn_SF3F6F7F(/&F%6$FbuF ey,&#\"bo+++!ob#\\5n(*e-%QPIC&*>OUAnT\\p\"o'4]L](e>%>\"aor0T]%Qb!)H/,) *Qb]tgD>bSF+%GcuS&4L#H*Qd:#F3*(\"R+++!['f/pwG#*o\"f$*ey9Y\"px#p+;F(\"R rOs+KGRUCysms#QB@(*oI`!**y-\"F3F6F7F(/&F%6$FbuFc\\l#!PS#eY1+k]//vK*3z$ )oR6f+w&GF\"O\"*ov.6er[?\\ep7GUp6T@'GnL\"/&F%6$FbuFJ#!]rxVX+K**y:A#>*e QJ75H$fS%HTLE56^Ujys)eT%\\K](Ht\">bOab%[NLLu=1;I&[>\"\\rkCjnJ6*o?W(Q\" G9]%3e#*[0.z%[?kT$)e\\x`Z\\8h/+3lmg3!4:rNvSTe%*egSe\"/&F%6$FbuFcel#\"V +++++[K(z\"=Ng!o6s(eF*[vmDN()[*)\"VR$=XK[`![_N\\0%)p#zL`YCH!)HtS\")e/& F%6$FbuFhjlFO/&F%6$FbuFc`mFO" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 259 39 "weights for the 16 stage order 9 scheme " }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "seq(b[i] =subs(ee,b[i]),i=1..16);" }}{PARA 12 "" 1 "" {XPPMATH 20 "62/&%\"bG6# \"\"\"#\"<(yv%pY)f,'*\"@w @w4o.:ve/br\"Sxy,*GHad3R!>8 ^=2V\\&)>J'[OJa\"/&F%6#\"#6#\"=OvXbo![(p2_KrBB\">\"396.Gp7?U^yBN5/&F%6 #\"#7#\"er>OOE7j%)pEhgc1=nATj-Q*pj/&**p0!4)zf))3@\"e^KaSET6RP*Qr6R>Jap XTKH+'3$G\"frks%)HH%35'=_N?#ohXY'fcBhe(H!)*3QpJ@#)z)HBe)4D8Jw(e)3*Q<2q vl!o43q1aY:Re\"/&F%6#\"#8#\"E++++++!3AMl(>TBNOhH\"FR9$4bP!yZRo4F];cK%* *Q/&F%6#\"#9#\"B++++D\"GX]q<\"*=l[q6\"B,w\"Q]$[k)3S(y!e " 0 "" {MPLTEXT 1 0 38 "seq(`b*`[i]=subs(ee,`b*`[i]),i=1..16);" }}{PARA 12 "" 1 "" {XPPMATH 20 "62/&%#b*G6#\"\"\"#\"8J!*z]hJbiVUX$\":)[vhQobG2^p(p\"/&F%6 #\"\"#\"\"!/&F%6#\"\"$F//&F%6#\"\"%F//&F%6#\"\"&F//&F%6#\"\"'F//&F%6# \"\"(F//&F%6#\"\")#\";D19a'*zl\\g;%*36\";kWDgH,*R![XwO5/&F%6#\"\"*#\"; DJq0ok[^Vx8m>\"<#R$32#)[kL=(pzfD/&F%6#\"#5#\"K+++++++++++vV$)oORR!z)o \"L\"*RpFkbLbnpoQ3]Z1XP&H\"4'/&F%6#\"#6#\":[wMop(>:oQ6(Q#\":$o%*4*R$** f3@u^$*/&F%6#\"#7#!fqH9-y>^cTT>$pWsKL(4c+6Z$)>7=H3XIk=2+sFlE#33*4*el. \"4tLL\"*pGC\"\"fqw@5N2&zMxEjY.?)*[@Y[7c8PMCsxUq?*G())Q<0#>n6XFUbN1Vc4 P#>*[E\"/&F%6#\"#8#\"E++++++!G<'zf\"pJgpG&=\"EcZ!R$" }} }{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 30 "#=============================" }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 32 "Verner's \"most efficient\" schem e" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 63 "The following ( \"most robust\" ) scheme comes from his w ebsite: " }{URLLINK 17 "http://www.math.sfu.ca/~jverner/" 4 "http://ww w.math.sfu.ca/~jverner/" "" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 144 "See: J.H. Verner, SIAM Journal of Numerical Analysis 1978, 772 -790, \"Explicit Runge-Kutta methods with estimates of the Local Trunc ation Error.\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "The nodes are: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "c[2] = 1731/50000" "6#/&%\"cG6#\"\"#*&\"%J<\"\"\"\"&++& !\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3] = 7630049/53810000-98353 9/53810000*sqrt(6)" "6#/&%\"cG6#\"\"$,&*&\"(\\+j(\"\"\"\")++\"Q&!\"\"F +*(\"'RN)*F+F,F--%%sqrtG6#\"\"'F+F-" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 22890147/107620000-2950617/107620000*sqrt(6)" "6#/&%\"cG6#\"\" %,&*&\")Z,*G#\"\"\"\"*++i2\"!\"\"F+*(\"(<1&HF+F,F--%%sqrtG6#\"\"'F+F- " }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5] = 561/1000" "6#/&%\"cG6#\"\"& *&\"$h&\"\"\"\"%+5!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6] = 387/ 1000-129/2000*sqrt(6)" "6#/&%\"cG6#\"\"',&*&\"$(Q\"\"\"\"%+5!\"\"F+*( \"$H\"F+\"%+?F--%%sqrtG6#F'F+F-" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7 ] = 387/1000+129/2000*sqrt(6)" "6#/&%\"cG6#\"\"(,&*&\"$(Q\"\"\"\"%+5! \"\"F+*(\"$H\"F+\"%+?F--%%sqrtG6#\"\"'F+F+" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[8] = 129/200" "6#/&%\"cG6 #\"\")*&\"$H\"\"\"\"\"$+#!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[9] = 387/800" "6#/&%\"cG6#\"\"**&\"$(Q\"\"\"\"$+)!\"\"" }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "c[10] = 6757/100000" "6#/&%\"cG6#\"#5*&\"%dn\"\"\"\" '++5!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[11] = 1/4" "6#/&%\"cG6# \"#6*&\"\"\"F)\"\"%!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[12] = 14 27971650951258372/2166662646162554701" "6#/&%\"cG6#\"#7*&\"4s$e7&4lrzU \"\"\"\"\"4,Zbihkim;#!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[13] = \+ 4103/5000" "6#/&%\"cG6#\"#8*&\"%.T\"\"\"\"%+]!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[14] = 2253/2500" "6#/&%\"cG6#\"#9*&\"%`A\"\"\"\"%+D! \"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[15] = 1" "6#/&%\"cG6#\"#:\" \"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[16] = 1" "6#/&%\"cG6#\"#;\" \"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "#---------------------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "checking the combined scheme" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "coefficients of the co mbined scheme " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18354 "ee := \{c[2]=1731/50000,\nc[3]=7630049/538100 00-983539/53810000*6^(1/2),\nc[4]=22890147/107620000-2950617/107620000 *6^(1/2),\nc[5]=561/1000,\nc[6]=387/1000-129/2000*6^(1/2),\nc[7]=387/1 000+129/2000*6^(1/2),\nc[8]=129/200,\nc[9]=387/800,\nc[10]=6757/100000 ,\nc[11]=1/4,\nc[12]=1427971650951258372/2166662646162554701,\nc[13]=4 103/5000,\nc[14]=2253/2500,\nc[15]=1,\nc[16]=1,\n\na[2,1]=1731/50000, \na[3,1]=-177968356965557/1002427673820000+14180534491313/250606918455 000*6^(1/2),\na[3,2]=64021741529527/200485534764000-7504450763411/1002 42767382000*6^(1/2),\na[4,1]=22890147/430480000-2950617/430480000*6^(1 /2),\na[4,2]=0,\na[4,3]=68670441/430480000-8851851/430480000*6^(1/2), \na[5,1]=592203994261020339/513126355505556250+730386990293623641/2052 505422022225000*6^(1/2),\na[5,2]=0,\na[5,3]=-8712153884182794903/20525 05422022225000-2843421359195851533/2052505422022225000*6^(1/2),\na[5,4 ]=1873698362223295443/513126355505556250+528258592225556973/5131263555 05556250*6^(1/2),\na[6,1]=11380823631/157617812000-339148869/394044530 00*6^(1/2),\na[6,2]=0,\na[6,3]=0,\na[6,4]=16193232887091831/5886434180 8507450-2355345717024309/58864341808507450*6^(1/2),\na[6,5]=1659122826 16977/4179075230308000-33181894472511/2089537615154000*6^(1/2),\na[7,1 ]=26523528363/231790900000+863255358/123138915625*6^(1/2),\na[7,2]=0, \na[7,3]=0,\na[7,4]=-38208748178016484817787/842517966262441068418750- \n 86118788556282369822807/842517966262441068418750*6^(1/2), \na[7,5]=92362336407446913/290322814529044000-232039320950012997/24677 43923496874000*6^(1/2),\na[7,6]=-362925891/1690350537500+857800423623/ 3380701075000*6^(1/2),\na[8,1]=43/600,\na[8,2]=0,\na[8,3]=0,\na[8,4]=0 ,\na[8,5]=0,\na[8,6]=43/150+43/2400*6^(1/2),\na[8,7]=43/150-43/2400*6^ (1/2),\na[9,1]=7353/102400,\na[9,2]=0,\na[9,3]=0,\na[9,4]=0,\na[9,5]=0 ,\na[9,6]=22833/102400+8901/204800*6^(1/2),\na[9,7]=22833/102400-8901/ 204800*6^(1/2),\na[9,8]=-3483/102400,\na[10,1]=37670874247221498870085 3/7788456028125000000000000,\na[10,2]=0,\na[10,3]=0,\na[10,4]=0,\na[10 ,5]=0,\na[10,6]=187914666753956840195279/2596152009375000000000000-\n \+ 210440846556290693268911/15576912056250000000000000*6^(1/2), \na[10,7]=187914666753956840195279/2596152009375000000000000+\n \+ 210440846556290693268911/15576912056250000000000000*6^(1/2),\na[10,8] =-18552667221896744226647/865384003125000000000000,\na[10,9]=-31677998 60072183913409/30423656359863281250000,\na[11,1]=-426968570497/5439441 5898750-92754382349/12087647977500*6^(1/2),\na[11,2]=0,\na[11,3]=0,\na [11,4]=0,\na[11,5]=0,\na[11,6]=1/30,\na[11,7]=-2865012129681958/114898 584332330625-12962517687655099/229797168664661250*6^(1/2),\na[11,8]=43 89715333607/309890657317500+92754382349/11477431752500*6^(1/2),\na[11, 9]=4990058173976/83757096376875+371017529396/9306344041875*6^(1/2),\na [11,10]=1099523524595993125000/6257667909869756018891+\n 10095 7348037989687500/6257667909869756018891*6^(1/2),\na[12,1]=\n1838203110 4798403869938539009154656587521498573595595063164077882800315372787284 683238439478955141517997198007108623761931447163756/139742569444997243 4491896099389093361416102532297045004793268899809500852862082123960473 4608111291769444706187497807869179550841329375+\n407885778185158609210 7938925175825953058964707564676126367962596114914082608964134468834508 91351622914818800693274034252252905536/2808492638860122607362409616917 5002956970191576455110633226765141161372294098693275117181239385312198 137508846535933127837167926875*6^(1/2),\na[12,2]=0,\na[12,3]=0,\na[12, 4]=0,\na[12,5]=0,\na[12,6]=\n-3338813117898494119715734728681282814382 0221072172312325174214536773458288757739554777822876017406875808613438 9952015563403904/22708720046081030371276898486040396230866390354413729 3405018059381649379612940534991414898146071420223298872773877849455772 7635+\n481927289247776817137330866672068912142109195362579297027804407 1549950640195056472955523769829034800621890424847009130000000/23162894 4470026509787024364557612041554837181615020039273118420569282367205199 34569124319610899284862776485022935540644488821877*6^(1/2),\na[12,7]= \n-1366666074964636222701356088637720764436254687981394803904267409930 2480394698176320934836471610872131282261984572615169366759843769996441 6/37192864653424042747885853272541808281952824273420576501948556349178 2111356343287068137204351252040188714143706710610568394480233242236937 5+\n169845085565361336805556009296394374527636952379388961026066628725 1555218327620868756323669964775679286575359121913961555667654578261399 04/1593979913718173260623679425966077497797978183146596135797795272107 6376200986140874348737329339373150944891873144740452931192009996095868 75*6^(1/2),\na[12,8]=5610987899273278525411960528081442902198567594809 7643797561956736732657005510768128839255833702537657025532355947644271 73637673766208/9288159819814403301827880474062633413542335679163959810 9358867770361609232846012626732332450844264293840456574956036349633197 336361875-\n5587476413495323413846491678323049250765705078855720721052 003556321800113162964567765526724539063327600257543743479921263738432/ 3653030893622016645164135969252861614944735753371152962505117528597281 08868696929614024803255122785403232359817965288739565550625*6^(1/2),\n a[12,9]=\n545985398180836152335661486022032448966969589107343397540652 7098543350794516270773775946921467448080727221064814847749923878327625 9328/30124791909229885263488687512995931079466293201418449982714507585 1637298698312074030567479239502011693447423026416040794479934024058125 -\n6526172450962537747372702280281321524894343532103481802188740153783 862532174342615150135214261625966637100811092384548036046488576/864909 3284303728183602838792132050266857965317662489228456648746817034128576 2869374265713247057712228954184044334206372230816544375*6^(1/2),\na[12 ,10]=\n939166734840458401095542221032870712500612066161106190888975080 5619418785820948002455890360939221912190524731087070645107486913457760 000000/581572669687730206124190285037387083035152858549707256623268015 3129538726578484984317222364519327722935843448874220309127298193173915 2584783-\n810882514508508810434472104816632522517372949568936469642672 0161112012414227752328969720658987315654179873760357725235734000399440 000000/265558296661064021061274102756797754810572081529546692522040189 6406182066930815061332065006629830010472987876198274113756757165832838 01757*6^(1/2),\na[12,11]=\n1234617126598879151772713393966068608104790 2877786934801487045060626091401956028566128821249812840047601569596034 1952/\n281629106670320674754245209358840703704235147307838896741075511 220826056829047205614324978253226176275078922716132461,\na[13,1]=-5604 2772675322042139227629978042586330633622706053363946766144416933631/\n 5880854077232319052559012261322343050735211853455734266601562 5000000000+\n 281404579734699232141455524604487724159024972527 /\n 1478009944832743180452316204077188415527343750000*6^(1/2), \na[13,2]=0,\na[13,3]=0,\na[13,4]=0,\na[13,5]=0,\na[13,6]=-10271639002 29750356561238237947225332675621517/1792618944311326640787476982928674 31640625000-\n 2745292391641202525373103979336813513372321/117 02216468464340311060649744558385937500000*6^(1/2),\na[13,7]=-157229999 853748227305165773364426925282378072238332930121/\n 3669990736 7985458573273204094330716033963413238525390625+\n 575760644280 2795095318986067317837904184278650664590252101/\n 352319110732 6604023034227593055748739260487670898437500000*6^(1/2),\na[13,8]=-9311 448168593934146015965019904013602133802943325818346622781285907057/\n \+ 42559708490101242171931354496687399854013133630055761593627929 68750000-\n 844213739204097696424366573813463172477074917581/ \n 4210188359946578336976868164966163024902343750000*6^(1/2), \na[13,9]=\n8857742338566725902229518676953278164573401303916391530705 21335485617578/\n30109854138029501101546924846546529011250565614375779 9934635162353515625-\\\n2814045797346992321414555246044877241590249725 27/\n284481916364737983221402322504830303192138671875*6^(1/2),\na[13,1 0]=3154791167297801539564121240521996850977442393866390237873591079592 54802182/\n 13448185050650584801258784221551557438021254320089 4932329128471154748828125-\n 294039645364787227664606877659229 2229737651937934623/\n 734546505878198371079583742953078477724 5286520703125*6^(1/2),\na[13,11]=2250996163406545378616532039018846586 217631599453822541/\n 3824913037970959935633041482042756364335 04028320312500,\na[13,12]=\n268934095730769185329490238833445400395937 8146957529866233529251986359392336044151708949720958809747970514366293 458424272174024493/\n9595163860195788085005691147808717084668947522804 8283510540802781519489531905544384278222710212049396080564957556179687 5000000000,\na[14,1]=4734200384802439149870797684768889301308307444115 9779465719863625051668939887702630319/\n 4480254687392605073040 1222636656855760802419993852060264615320801485392456054687500000-\n \+ 866369530987077991125562402829092187100493209601/\n 332552 2375873672156017711459173673934936523437500*6^(1/2),\na[14,2]=0,\na[14 ,3]=0,\na[14,4]=0,\na[14,5]=0,\na[14,6]=871779321807802447463310035318 238762878527157/\n 13444642082334949805906077371965057373046875 0+\n 107641268480999396081848975271849857994818/\n 10970 82793918531904161935913552348681640625*6^(1/2),\na[14,7]=4961037863518 62292800034805114190705484800743513354117014/\n 110099722103956 375719819612282992148101890239715576171875-\n 13299384126061974 85769312599390307351191540891599374831099/\n 660598332623738254 318917673697952888611341438293457031250*6^(1/2),\na[14,8]=407740772777 4763635459845170889116549412313138377723522953861198939217519328599426 6471/\n 1526429054624816210105898594158807951825674125537703173 6357946125713524703979492187500+\n 1237670758552968558750803432 61298883871499029943/\n 451091609994276250390378731960660324096 679687500*6^(1/2),\na[14,9]=-10522038608500556459828649038302068473735 749030796372764961618751973793724796364606986664/\n 3899417425 0054222540345740003973828622358928296533758351973409182715560555076599 12109375+\n 3465478123948311964502249611316368748401972838404/ \n 2560337247282641848992620902543472728729248046875*6^(1/2), \na[14,10]=-2784376447126269318936520113562067049032847532328282021947 4851621693895769527094334687108984/\n 12257041066285164222002 594300605593929434139193022166317802121412999357024704596261133984375+ \n 574774300271998598683873114105472016699241495055292/\n \+ 1049352151254569101542262489932969253892183788671875*6^(1/2),\na [14,11]=-3424113435184824562423280943767688900943193050352985303257641 7589898516/\n 56133478243586519811009850090242810076032300624 39942682713165283203125,\na[14,12]=\n-34320443758939323781023685680522 8650103385091051699920208853270521163343279392054770280096153243800840 1883737341854688972639605334600163938610268855705742764072609/\n114317 4106341682260971647690410567292143926198650927778920823267461111371275 9075998017148701658133941475190682109317668444949946165802584355181814 34575195312500000,\na[14,13]=47469308760239193350794516127267176492182 64199984/\n 18592065538407049755200144388134089346432755594877 ,\na[15,1]=-2518832924925882544374852703814240987992301213373898531326 5430932280250855708601/\n 113706413255744693120569618740772985 50827642308774647316995717036347558064286250+\n 12342730589818 60170179592598535508631343082535549881956/\n 21056337714696287 44518390642968552144069898845895808125*6^(1/2),\na[15,2]=0,\na[15,3]=0 ,\na[15,4]=0,\na[15,5]=0,\na[15,6]=-5482114211968505556247721620542861 3949905430396088/\n 395943983700946128908558774674809794739310 1278095-\n 1511276753825982856072891469504471256664975925000/ \n 40386286337496505148672995016830599063409633036569*6^(1/2), \na[15,7]=-60922424274061599918603524049390657305431262635197540405697 952/\n 6484861747489032169774584624759953148531564032417461909 516875+\n 8455857575163597873310996189398423878692955046261537 5699341616/\n 194545852424670965093237538742798594455946920972 52385728550625*6^(1/2),\na[15,8]=-116118147575045169733222875835719955 334334798191459879782123534889390467935109772/\n 8810626901954 835245672275131295870892503713957512170681453300814988417642493125-\n \+ 176324722711694310025656085505072661620440362221411708/\n \+ 285619406719829107485771207042040133465420149964555625*6^(1/2),\na [15,9]=177694487225138983422768374906650972869276072470733356185669871 43467294900183033216/\n 255121700813788961505634214608456186712 2485163596619283719957742418751029506356875-\n 1974836894370976 2722873481576568138101489320568798111296/\n 6484554262322259071 286545935997129135111813687175650625*6^(1/2),\na[15,10]=97659266139124 0748181932648019295477816599265437863815101909541842185707462150338239 93530000000/\n 18560076654469706205963482908787056850812308205 603127326855360961727608242796551101182080033599-\n 8529708461 1782122474911131363078900058888025224607913745000000/\n 692106 59450201393843166746722954036326338355649915383851733911*6^(1/2),\na[1 5,11]=4733897490497529632561146492313538224929122595096495198708697505 25/\n 35412440882360341799798842428365422941216508121322622479 260846291,\na[15,12]=\n33351439245158438248073494056784144097872912773 4159045364007283876903345639683941147024141088075051581063851164687328 53458202899966748488718531545706559142895903144848764637/\n23166110253 2728742771480201132225288609079390498990062159236562764909757810216357 2190502232425490606773312310665593424982745744299371285598588298606088 543376742054644818966,\na[15,13]=-387149926569584133897432527260168975 99283911682945255636643554687500000/\n 4854049492697158749929 4589382572212036169135429877901702347521300421767,\na[15,14]=148002502 00940323717124616175641261235119295795768814717803955078125/\n \+ 33565577125141877760287380588632421223433194078156948298488471160489, \na[16,1]=230578569608639756108085818693989717364564133108504131394438 9849986584101287/\n 6175082443452822658190873700782751226712461 64669900462139876057008239440000-\n 854046233055897126321659052 33974183137607899140719/\n 124822287169084833758410283469525117 460541643292500*6^(1/2),\na[16,2]=0,\na[16,3]=0,\na[16,4]=0,\na[16,5]= 0,\na[16,6]=102903996961580448264190625267026062654799259083/\n \+ 5046398084890004857481629999673320438819484730+\n 413209254873 04219313300272052128374567081128125/\n 514732604658780495463126 25996667868475958744246*6^(1/2),\na[16,7]=6279844334987645750671892084 3975661399949564598018488144466/\n 4132553498782573324058263582 553715220777051359780141380625-\n 72308807081932961554425711089 716771013571419950657300729103/\n 12397660496347719972174790747 661145662331154079340424141875*6^(1/2),\na[16,8]=179490914212648256439 0848522924225553221469019751470544959297614654661293377/\n 5259 6481193994264435601626109752988674679691644275456716633975785978672500 +\n 12200660472227101804595129319139169019658271305817/\n \+ 16931561456559959115207709344056578263397760602500*6^(1/2),\na[16,9] =-27752447327801096673428456123947393191156626363714773004557470224232 70475907256/\n 22841715367558402972501804542270695582799632820 8181619436454383447149337555625+\n 341618493222358850528663620 935896732550431596562876/\n 9610133837877335746924521195491150 5447551097205625*6^(1/2),\na[16,10]=-276805546597690166235309791767274 48251292244310769996015342190819068970556083063125000/\n 3299 5577774296489605765613822566068446772584387970729553415813540513750365 22231471437+\n 4426552127579895373479670356100179759944766558 141730312500/\n 307711373866732070774887719980463674649497700 0658967987677*6^(1/2),\na[16,11]=-292603171929706291053929402159930330 736639136252680853622275/\n 154736228262791611502270768872902 62443510550964275858143964,\na[16,12]=-9815717129569106988569302193220 9993438249320845820936475960869317546660986625941530952589885163051657 94739744873539829069617203523509136682216933020431/2864769911709341530 7614664109440217180193725006859654293102867866950176225328769329439768 9327797388113854588113430063939405071979092547998950955940992,\na[16,1 3]=2729491144709837905799148766650782532906050298971406518524169921875 /\n 2158115888622139473142775812109447802920656149243127309253 686951469,\na[16,14]=0,\na[16,15]=0, \n\nb[1]=819816036620317341111994 3711500331/561057579384085860167277847128765528,\nb[2]=0,\nb[3]=0,\nb[ 4]=0,\nb[5]=0,\nb[6]=0,\nb[7]=0,\nb[8]=-455655493073428838813281446213 740000000/1163808011150910561240464225837312497869,\nb[9]=199651636487 06008081135075746915614720000000/8639440419053708686839468620578243251 6544599,\nb[10]=892311079199814187055669708043437500000000000000000000 00/\n 699979870988335674445594679856445060562597693583175985391, \nb[11]=47104273954945906713184913871143492/20968463912233960193463111 3492763467,\nb[12]=208450044214045004640105847407967506508321767983703 8308422635129473073119667364731106233097274073473727950311938762714638 1678677156136042524139311907482802844083/36670849891136373020238225328 2651002506051447185019263051409665867580548476046814663361031692847559 87753542321202462371554120593858149755539878561976786592389608,\nb[13] =6053037282142306509795911286909179687500000000/1038992573505180634552 90077573775162739725126989,\nb[14]=91740110492099349836035840609672546 3867187500/6724249815911346653315790737453607382989551463,\nb[15]=2585 449557665268951371699596493957/84574345160764140163208606048427531,\nb [16]=0,\n\n`b*`[1]=552562031208180939317806684253/27669654257734667858 523344041464,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=0,\n`b*`[5]=0,\n`b*`[6] =0,\n`b*`[7]=0,\n`b*`[8]=221223388631423597589898601690000000/10094613 6798587090054685074667127461,\n`b*`[9]=1018354087913052979846578125619 20000000/1149763833200743759976506650241312100139,\n`b*`[10]=131372030 9077630014453239843750000000000000000000/\n 115182019232155109 89126466531107437037395719117133,\n`b*`[11]=48336112327014405045080861 51728/19081321241454145230196661524503,\n`b*`[12]=-2129662374582324648 1069197957033736453531182730667422307241727310258139647124736471440105 99206669825382719359113196238857709025512340589957/1035543739272367080 8851905462010972188912687281182073325925959875548518829722926708817941 78380097716583123063485287435793657425889233080568,\n`b*`[13]=10847615 91753640855844358063964843750000000/3182895486031249071938549691320502 488733423,\n`b*`[14]=0,\n`b*`[15]=0,\n`b*`[16]=18391900710606498871278 95100784/38045139523510634351420875415397\}:" }}}{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 0 357 "s ubs(ee,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(4-i)],i=2..4)]));\nf or ii from 5 to 16 do\n print(``);\n print(c[ii]=subs(ee,c[ii])); \+ \n for jj to ii-1 do\n print(a[ii,jj]=subs(ee,a[ii,jj]));\n e nd do:\nend do:print(``);\nfor ii to 16 do\n print(b[ii]=subs(ee,b[i i]));\nend do:print(``);\nfor ii to 16 do\n print(`b*`[ii]=subs(ee,` b*`[ii]));\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7%7 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#:#\"CdR\\'f*pr8&*o_md&\\ae#\"DJvU[gg3K;STwg^MuX)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#;\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% !G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"\"#\"?`Uo1yJR4=37.i Db\"Ak9/WL_eymMxDa'pw#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\" \"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"$\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"%\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%#b*G6#\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%#b*G6#\"\"'\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\" (\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\")#\"E+++!p,')*) *e(fB9j)QB7A\"Ehu7nY2&oa+4(e)zOh%45" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%#b*G6#\"\"*#\"H+++?>c7yl%)zH08z3a$=5\"IR,578C]m]w*fPu+K$Qw\\6" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#5#\"R+++++++++]P%)RKX9+jx! 4.s88\"SLr6>dRPqV26`mk7*)4^:K#>?=:\"" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%#b*G6#\"#6#\"@G<:'33X]S9qK7hL[\"A.X_hm>I_9a9C@83>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#7#!fsd**eSB^D!4x&)Qi>8\"f$>FQD)pm?*f5S 9ZOZ7Z'R\"e-JFp5[YK#euBmH@\"fso03L#*)eUdOzNuG&[jI7$ e;x4!QyTz\")3n#HsH)=&[b()ff#fKt?=\"G(o7*)=s4,ia!>&)33nBFRPaN5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#8#\"L+++]P%['R1eV%e&3k`2\\7.'[&*G=$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%#b*G6#\"#9\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#:\" \"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#;#\"@%y+^*y7())\\11 r+>R=\"A(R:a(3U^Vj5N_R^/Q" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "RK9_16eqs := [op(RowSum Conditions(16,'expanded')),op(OrderConditions(9,16,'expanded'))]:\n`RK 8_16eqs*` := subs(b=`b*`,OrderConditions(8,16,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Check:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "expand(subs(ee,RK9_16eqs)): \nmap(u->lhs(u)-rhs(u),%);\nnops(%);\nexpand(subs(ee,`RK8_16eqs*`)):\n map(u->lhs(u)-rhs(u),%);\nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# 7ajl\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$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 12 "" 1 "" {XPPMATH 20 "6#7dw\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$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 257 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 "Next we set-up stage-order conditions to check for stage-orders from 2 to 6 i nclusive. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "for ct from 2 \+ to 6 do\n so||ct||_16 := StageOrderConditions(ct,16,'expanded');\nen d do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 138 "Stage 3 has stage-order 2, stages 4 and 5 have stage-order 3, sta ges 6 and 7 have stage-order 4, while stages 8 to 16 have stage-order \+ 5. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 175 "[seq([seq(expand(sub s(ee,so||i||_16[j])),i=2..6)],j=1..14)]:\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#70\"\"#\"\"$F%\" \"%F&\"\"&F'F'F'F'F'F'F'F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The simplifying conditions:" }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j+1 .. 16) = b[j] *(1-c[j]);" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F 0F,F,F,\"#;*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 3 ", " } {XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 9 " . . 15, " }}{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 0 151 "[Sum(b[i]*a[i,1], i=1+1..16)=b[1],seq(Sum(b[i]*a[i,j],i=j+1..16)=b[j]*(1-c[j]),j=2..15)] :\nmap(u->lhs(u)-rhs(u),subs(ee,eval(subs(Sum=add,%))));\nnops(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#71\"\"!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 27 "The simplifying conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(`b*`[i]*a[i,j],i = j+1 \+ .. 16) = `b*`[j]*(1-c[j]);" "6#/-%$SumG6$*&&%#b*G6#%\"iG\"\"\"&%\"aG6$ F+%\"jGF,/F+;,&F0F,F,F,\"#;*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 9 " . . 15, " }}{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 0 163 "[Sum(`b* `[i]*a[i,1],i=1+1..16)=`b*`[1],seq(Sum(`b*`[i]*a[i,j],i=j+1..16)=`b*`[ j]*(1-c[j]),j=2..15)]:\nmap(u->lhs(u)-rhs(u),subs(ee,eval(subs(Sum=add ,%))));\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#71\"\"!F$F$F$F$F$ F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "We can calculat e the principal error norm, that is, the 2-norm of the principal error terms. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "errterms9_16 := PrincipalErrorTerms(9,16,'expanded'):\nsm := 0:\nfor ct to nops(errte rms9_16) do\n sm := sm+(evalf(subs(ee,errterms9_16[ct])))^2;\nend do :\nsqrt(sm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+]q_!\\$!#;" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "We can al so calculate the principal error norm of the order 8 embedded scheme. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "`errterms8_16*` := subs (b=`b*`,PrincipalErrorTerms(8,16,'expanded')):\nsm := 0:\nfor ct to no ps(`errterms8_16*`) do\n sm := sm+(evalf(subs(ee,`errterms8_16*`[ct] )))^2;\nend do:\nsqrt(sm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+q!3h C%!#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#----------------- ----------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 34 "construction of the order 9 scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 94 "First we construct a 15 stage orde r 9 scheme starting with a consideration of stages 2 to 11. " }}{PARA 0 "" 0 "" {TEXT -1 21 "We specify the nodes:" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "c[2] = 1731/50000;" "6#/&%\"cG6#\"\"#* &\"%J<\"\"\"\"&++&!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5] = 561/ 1000;" "6#/&%\"cG6#\"\"&*&\"$h&\"\"\"\"%+5!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[8] = 129/200;" "6#/&%\"cG6#\"\")*&\"$H\"\"\"\"\"$+#! \"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[10] = 6757/100000;" "6#/&%\" cG6#\"#5*&\"%dn\"\"\"\"'++5!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[ 11] = 1/4;" "6#/&%\"cG6#\"#6*&\"\"\"F)\"\"%!\"\"" }{TEXT -1 2 " " }} {PARA 0 "" 0 "" {TEXT -1 35 "and the zero linking coefficients: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[4,2]=0" "6#/&%\"aG6 $\"\"%\"\"#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[5,2]=0" "6#/&%\"a G6$\"\"&\"\"#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[6,2]=0" "6#/&% \"aG6$\"\"'\"\"#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[6,3]=0" "6#/ &%\"aG6$\"\"'\"\"$\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[7,2]=0" "6 #/&%\"aG6$\"\"(\"\"#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[7,3]=0" "6#/&%\"aG6$\"\"(\"\"$\"\"!" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[8,2]=0" "6#/&%\"aG6$\"\")\"\"#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[8,3]=0" "6#/&%\"aG6$\"\")\"\"$\"\"! " }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[8,4]=0" "6#/&%\"aG6$\"\")\"\"%\" \"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[8,5]=0" "6#/&%\"aG6$\"\")\"\"& \"\"!" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[9,2]=0" "6#/&%\"aG6$\"\"*\"\"#\"\"!" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "a[9,3]=0" "6#/&%\"aG6$\"\"*\"\"$\"\"!" }{TEXT -1 2 ", \+ " }{XPPEDIT 18 0 "a[9,4]=0" "6#/&%\"aG6$\"\"*\"\"%\"\"!" }{TEXT -1 2 " , " }{XPPEDIT 18 0 "a[9,5]=0" "6#/&%\"aG6$\"\"*\"\"&\"\"!" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[10,2]=0" " 6#/&%\"aG6$\"#5\"\"#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[10,3]=0 " "6#/&%\"aG6$\"#5\"\"$\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[10,4] =0" "6#/&%\"aG6$\"#5\"\"%\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[10, 5]=0" "6#/&%\"aG6$\"#5\"\"&\"\"!" }{TEXT -1 2 ", " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "a[11,2]=0" "6#/&%\"aG6$\"#6\"\"#\"\"! " }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[11,3]=0" "6#/&%\"aG6$\"#6\"\"$\" \"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[11,4]=0" "6#/&%\"aG6$\"#6\"\"% \"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[11,5]=0" "6#/&%\"aG6$\"#6\" \"&\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 20 " in stages 2 \+ to 11. " }}{PARA 0 "" 0 "" {TEXT -1 17 "We also specify " }{XPPEDIT 18 0 "a[11,6] = 1/30;" "6#/&%\"aG6$\"#6\"\"'*&\"\"\"F*\"#I!\"\"" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "The stage-order equations relating to these stages, such \+ that stages " }{XPPEDIT 18 0 "3,4,5,6,7,8,9,10,11;" "6+\"\"$\"\"%\"\"& \"\"'\"\"(\"\")\"\"*\"#5\"#6" }{TEXT -1 19 " have stage orders " } {XPPEDIT 18 0 "2,3,3,4,4,5,5,5,5;" "6+\"\"#\"\"$F$\"\"%F%\"\"&F&F&F&" }{TEXT -1 96 " respectively taken together with the row-sum conditions can then be solved to obtain the nodes " }{XPPEDIT 18 0 "c[3],c[4],c[ 6],c[7],c[9],c[11];" "6(&%\"cG6#\"\"$&F$6#\"\"%&F$6#\"\"'&F$6#\"\"(&F$ 6#\"\"*&F$6#\"#6" }{TEXT -1 67 " and the remaining non-zero linking co efficients for these stages. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 42 "The computation is made more efficient by " }{TEXT 260 48 "including explicitly relations between the nodes" } {TEXT -1 42 " arising from the stage-order conditions: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " c[3]=2/3" "6#/&%\"cG6#\"\"$*&\"\"#\"\"\"F'!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 8 ", " } {XPPEDIT 18 0 "c[4] = (6-sqrt(6))*(9*c[8]-24*c[5]-4*c[5]*sqrt(6))*c[8] /(60*(2*c[8]-6*c[5]-c[5]*sqrt(6)));" "6#/&%\"cG6#\"\"%**,&\"\"'\"\"\"- %%sqrtG6#F*!\"\"F+,(*&\"\"*F+&F%6#\"\")F+F+*&\"#CF+&F%6#\"\"&F+F/*(F'F +&F%6#F:F+-F-6#F*F+F/F+&F%6#F5F+*&\"#gF+,(*&\"\"#F+&F%6#F5F+F+*&F*F+&F %6#F:F+F/*&&F%6#F:F+-F-6#F*F+F/F+F/" }{TEXT -1 2 ", " }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " c[6]=((6-sqrt(6))*c[8])/10" "6#/&%\"cG6#\"\"'*(,&F'\"\"\"-%%sqrtG6#F'! \"\"F*&F%6#\"\")F*\"#5F." }{TEXT -1 8 ", " }{XPPEDIT 18 0 "c[7]= ((6+sqrt(6))*c[8])/10" "6#/&%\"cG6#\"\"(*(,&\"\"'\"\"\"-%%sqrtG6#F*F+F +&F%6#\"\")F+\"#5!\"\"" }{TEXT -1 8 ", " }{XPPEDIT 18 0 "c[9]=3/ 4" "6#/&%\"cG6#\"\"**&\"\"$\"\"\"\"\"%!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "c[8]" "6#&%\"cG6#\"\")" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "The equations that l ead to " }{XPPEDIT 18 0 "c[6] < c[7]" "6#2&%\"cG6#\"\"'&F%6#\"\"(" } {TEXT -1 19 ", have been chosen." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 826 "RSeqs := RowSumConditions(1 1,'expanded'):\nSOeqs := [op(StageOrderConditions(2,11,'expanded')),\n op(StageOrderConditions(3,4..11,'expanded')),\n \+ op(StageOrderConditions(4,6..11,'expanded')),\n o p(StageOrderConditions(5,8..11,'expanded'))]:\nnode_eqsA := [c[3]=2/3* c[4],c[9]=3/4*c[8],c[6]=1/10*(6-6^(1/2))*c[8],c[7]=1/10*(6+6^(1/2))*c[ 8],\n c[4]=1/60*(6-6^(1/2))*(9*c[8]-24*c[5]-4*c[5]*6^(1/2 ))*c[8]/(2*c[8]-6*c[5]-c[5]*6^(1/2))]:\n\ne1 := \{c[2]=1731/50000,c[5] =561/1000,c[8]=129/200,c[10]=6757/100000,c[11]=1/4,\n seq(a[i ,2]=0,i=4..11),seq(a[i,3]=0,i=6..11),\n seq(a[i,4]=0,i=8..11) ,seq(a[i,5]=0,i=8..11),a[11,6]=1/30\}:\neqns := expand(rationalize(sub s(e1,[op(RSeqs),op(SOeqs),op(node_eqsA)]))):\nconvert(ListTools[Enumer ate](%),matrix);\n``;\nindets(eqns);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7L7$\"\"\"/&%\"aG6$\"\"#F(#\"%J<\"&++&7$F-/ ,&&F+6$\"\"$F(F(&F+6$F6F-F(&%\"cG6#F67$F6/,&&F+6$\"\"%F(F(&F+6$FAF6F(& F:6#FA7$FA/,(&F+6$\"\"&F(F(&F+6$FKF6F(&F+6$FKFAF(#\"$h&\"%+57$FK/,(&F+ 6$\"\"'F(F(&F+6$FXFAF(&F+6$FXFKF(&F:6#FX7$FX/,*&F+6$\"\"(F(F(&F+6$F^oF AF(&F+6$F^oFKF(&F+6$F^oFXF(&F:6#F^o7$F^o/,(&F+6$\"\")F(F(&F+6$F\\pFXF( &F+6$F\\pF^oF(#\"$H\"\"$+#7$F\\p/,*&F+6$\"\"*F(F(&F+6$FipFXF(&F+6$FipF ^oF(&F+6$FipF\\pF(&F:6#Fip7$Fip/,,&F+6$\"#5F(F(&F+6$FgqFXF(&F+6$FgqF^o F(&F+6$FgqF\\pF(&F+6$FgqFipF(#\"%dn\"'++57$Fgq/,.&F+6$\"#6F(F(#F(\"#IF (&F+6$FhrF^oF(&F+6$FhrF\\pF(&F+6$FhrFipF(&F+6$FhrFgqF(#F(FA7$Fhr/,$*&F .F(F7F(F(,$*&#F(F-F(*$)F9F-F(F(F(7$\"#7/*&FBF(F9F(,$*&FjsF(*$)FDF-F(F( F(7$\"#8/,&*&FLF(F9F(F(*&FNF(FDF(F(#\"'@ZJ\"(+++#7$\"#9/,&*&FYF(FDF(F( *&FPF(FenF(F(,$*&FjsF(*$)FgnF-F(F(F(7$\"#:/,(*&F_oF(FDF(F(*&FPF(FaoF(F (*&FcoF(FgnF(F(,$*&FjsF(*$)FeoF-F(F(F(7$\"#;/,&*&F]pF(FgnF(F(*&F_pF(Fe oF(F(#\"&Tm\"\"&++)7$\"#/,,*&FirF(FgnF(F(*&F[sF(F eoF(F(*&FapF(F]sF(F(*&F_sF(F`qF(F(*&F`rF(FasF(F(#F(\"#K7$\"#?/*&FBF(F \\tF(,$*&#F(F6F(*$)FDF6F(F(F(7$\"#@/,&*&FLF(F\\tF(F(*&FNF(FdtF(F(#\")F G&)e\"+++++57$\"#A/,&*&FYF(FdtF(F(*&#F\\u\"(+++\"F(FenF(F(,$*&FcyF(*$) FgnF6F(F(F(7$\"#B/,(*&F_oF(FdtF(F(*&FezF(FaoF(F(*&FcoF(FguF(F(,$*&FcyF (*$)FeoF6F(F(F(7$\"#C/,&*&F]pF(FguF(F(*&F_pF(FbvF(F(#\"'jbr\"(+++)7$\" #D/,(*&FjpF(FguF(F(*&F\\qF(FbvF(F(*&#Fjv\"&++%F(F^qF(F(,$*&FcyF(*$)F`q F6F(F(F(7$\"#E/,**&FhqF(FguF(F(*&FjqF(FbvF(F(*&Ff\\lF(F\\rF(F(*&F^rF(F fwF(F(#\"-$4!o/&3$\"1+++++++I7$\"#F/,,*&FirF(FfuF(F(*&F[sF(FbvF(F(*&Ff \\lF(F]sF(F(*&F_sF(FfwF(F(*&#F`x\",+++++\"F(FasF(F(#F(\"$#>7$\"#G/,&*& FYF(FeyF(F(*&#\"*\"[el " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "e2 := expand(rationalize(sol ve(\{op(eqns)\}))):\ne3 := `union`(e1,e2):\ninfolevel[solve] := 0:" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" } {TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 106 " If the equations givi ng the relations between the nodes are omitted we need to select the s olution with " }{XPPEDIT 18 0 "c[6] < c[7]" "6#2&%\"cG6#\"\"'&F%6#\" \"(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 23 " Thus we require t hat " }{XPPEDIT 18 0 "c[6] = 387/1000-129/2000*sqrt(6);" "6#/&%\"cG6# \"\"',&*&\"$(Q\"\"\"\"%+5!\"\"F+*(\"$H\"F+\"%+?F--%%sqrtG6#F'F+F-" } {TEXT -1 1 " " }{TEXT 268 1 "~" }{TEXT -1 20 " 0.22900791 and " } {XPPEDIT 18 0 "c[7] = 387/1000+129/2000*sqrt(6);" "6#/&%\"cG6#\"\"(,&* &\"$(Q\"\"\"\"%+5!\"\"F+*(\"$H\"F+\"%+?F--%%sqrtG6#\"\"'F+F+" }{TEXT -1 1 " " }{TEXT 269 1 "~" }{TEXT -1 47 " 0.54499209, rather than the \+ other way round. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "The following commands achieve this." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[so lve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 185 "eqns := subs (e1,[op(RSeqs),op(SOeqs)]):\nsol := solve(\{op(eqns)\}):\ne2 := op(sel ect(u_->evalf(subs(u_,c[6]) " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e3 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2803 "e3 := \{a[4,2] = 0, c[11] = 1/4, \+ a[10,9] = -3167799860072183913409/30423656359863281250000, c[8] = 129/ 200, a[5,2] = 0, a[6,2] = 0, a[6,3] = 0, a[7,2] = 0, a[7,3] = 0, a[8,2 ] = 0, a[8,3] = 0, a[8,4] = 0, a[8,5] = 0, a[9,2] = 0, a[9,3] = 0, a[9 ,4] = 0, a[9,5] = 0, a[10,2] = 0, a[10,3] = 0, a[10,4] = 0, a[10,5] = \+ 0, a[11,2] = 0, a[11,3] = 0, a[11,4] = 0, a[11,5] = 0, a[11,6] = 1/30, a[9,8] = -3483/102400, a[7,4] = -38208748178016484817787/842517966262 441068418750-86118788556282369822807/842517966262441068418750*6^(1/2), a[8,6] = 43/150+43/2400*6^(1/2), a[9,6] = 22833/102400+8901/204800*6^ (1/2), a[9,7] = 22833/102400-8901/204800*6^(1/2), a[3,2] = 64021741529 527/200485534764000-7504450763411/100242767382000*6^(1/2), a[11,8] = 4 389715333607/309890657317500+92754382349/11477431752500*6^(1/2), a[10, 6] = 187914666753956840195279/2596152009375000000000000-21044084655629 0693268911/15576912056250000000000000*6^(1/2), a[8,7] = 43/150-43/2400 *6^(1/2), c[7] = 387/1000+129/2000*6^(1/2), c[3] = 7630049/53810000-98 3539/53810000*6^(1/2), a[7,1] = 26523528363/231790900000+863255358/123 138915625*6^(1/2), a[7,6] = -362925891/1690350537500+857800423623/3380 701075000*6^(1/2), a[10,7] = 187914666753956840195279/2596152009375000 000000000+210440846556290693268911/15576912056250000000000000*6^(1/2), a[5,4] = 1873698362223295443/513126355505556250+528258592225556973/51 3126355505556250*6^(1/2), a[4,1] = 22890147/430480000-2950617/43048000 0*6^(1/2), c[4] = 22890147/107620000-2950617/107620000*6^(1/2), a[5,3] = -8712153884182794903/2052505422022225000-2843421359195851533/205250 5422022225000*6^(1/2), a[5,1] = 592203994261020339/513126355505556250+ 730386990293623641/2052505422022225000*6^(1/2), a[11,7] = -28650121296 81958/114898584332330625-12962517687655099/229797168664661250*6^(1/2), c[6] = 387/1000-129/2000*6^(1/2), a[3,1] = -177968356965557/100242767 3820000+14180534491313/250606918455000*6^(1/2), a[7,5] = 9236233640744 6913/290322814529044000-232039320950012997/2467743923496874000*6^(1/2) , a[4,3] = 68670441/430480000-8851851/430480000*6^(1/2), a[11,1] = -42 6968570497/54394415898750-92754382349/12087647977500*6^(1/2), a[11,9] \+ = 4990058173976/83757096376875+371017529396/9306344041875*6^(1/2), a[6 ,5] = 165912282616977/4179075230308000-33181894472511/2089537615154000 *6^(1/2), a[6,4] = 16193232887091831/58864341808507450-235534571702430 9/58864341808507450*6^(1/2), a[11,10] = 1099523524595993125000/6257667 909869756018891+100957348037989687500/6257667909869756018891*6^(1/2), \+ a[9,1] = 7353/102400, a[10,1] = 376708742472214988700853/7788456028125 000000000000, a[6,1] = 11380823631/157617812000-339148869/39404453000* 6^(1/2), c[2] = 1731/50000, c[10] = 6757/100000, c[5] = 561/1000, c[9] = 387/800, a[2,1] = 1731/50000, a[10,8] = -18552667221896744226647/86 5384003125000000000000, a[8,1] = 43/600\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "printed coefficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 219 "subs(e3,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(4-i)],i=2..4 )]));\nfor ii from 5 to 11 do\n print(``);\n print(c[ii]=subs(e3,c [ii])); \n for jj to ii-1 do\n print(a[ii,jj]=subs(e3,a[ii,jj]) );\n end do:\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6 #7%7&#\"%J<\"&++&F(%!GF+7&,&#\"(\\+j(\")++\"Q&\"\"\"*(\"'RN)*F1F0!\"\" \"\"'#F1\"\"#F4,&#\"0db'pNoz<\"1++#QnFC+\"F4*(\"/88\\M0=9F1\"0+]X=pg]# F4F5F6F1,&#\"/F&H:u@S'\"0+SwMb[+#F1*(\".6Mw]W](F1\"0+?QnFC+\"F4F5F6F4F +7&,&#\")Z,*G#\"*++i2\"F1*(\"(<1&HF1FJF4F5F6F4,&#FI\"*++[I%F1*(FLF1FOF 4F5F6F4\"\"!,&#\")T/noFOF1*(\"(^=&))F1FOF4F5F6F4Q)pprint106\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"cG6#\"\"&#\"$h&\"%+5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6 $\"\"&\"\"\",&#\"3R.-hU*R?#f\"3]ib0bNEJ^F(*(\"3TOi$H!*pQI(F(\"4+]AA?U0 D0#!\"\"\"\"'#F(\"\"#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\" \"&\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"$,& #\"4.\\z#=%)Q:7()\"4+]AA?U0D0#!\"\"*(\"4L:&e>f8UVG\"\"\"F,F-\"\"'#F0\" \"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"%,&#\"4VaHB AO)pt=\"3]ib0bNEJ^\"\"\"*(\"3tpbDAfe#G&F-F,!\"\"\"\"'#F-\"\"#F-" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"cG6#\"\"',&#\"$(Q\"%+5\"\"\"*(\"$H\"F,\"%+?!\"\"F'#F,\"\"#F0" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"\",&#\",JO#3Q6\"-+? \"yhd\"F(*(\"*p)[\"R$F(\",+IX/%R!\"\"F'#F(\"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"\"'\"\"%,&#\"2J=4()GB$>;\"2]u]3=Mk)e\"\"\"*(\"14V-So&RvmY\"z=\":+++ +++v$4?:'f#\"\"\"*(\"96*oKp!Hcl%3W5#F-\";++++++]i07pd:!\"\"F(#F-\"\"#F 1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"(,&#\"9z_>So&Rvm Y\"z=\":++++++v$4?:'f#\"\"\"*(\"96*oKp!Hcl%3W5#F-\";++++++]i07pd:!\"\" \"\"'#F-\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\")# !8ZmAWn*=AnEb=\"9++++++DJ+%Ql)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"#5\"\"*#!74M\"R=s+')*znJ\"8++D\"Gj)fjlB/$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#6#\" \"\"\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"\",&#\"- (\\q&opU\"/]()*eT%Ra!\"\"*(\",\\BQaF*F(\"/+v(zk(37F-\"\"'#F(\"\"#F-" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"#\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"#6\"\"'#\"\"\"\"#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"#6\"\"(,&#\"1e>oH@,lG\"3D1LKVe)*[6!\"\"*(\"2*4bwo " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 27 "We do not need to specify " }{XPPEDIT 18 0 "c[12]" "6# &%\"cG6#\"#7" }{TEXT -1 43 ", because, according to Verner, the nodes \+ " }{XPPEDIT 18 0 "c[8]" "6#&%\"cG6#\"\")" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "c[9]" "6#&%\"cG6#\"\"*" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "c[10]" "6#&%\"cG6#\"#5" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "c[11]" "6#&%\"cG6#\" #6" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "c[12]" "6#&%\"cG6#\"#7" }{TEXT -1 30 " 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) ,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-F- F,F3F-/F,;F:F-F-*&-F&6$*&-F*6#F,F--F/6#*&,&F-F-F,F3\"\"#-F66#FNF3F-/F, ;F:F-F--F&6$*&-F?6#F,F--F/6#*&,&F-F-F,F3FN-F66#FNF3F-/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[8])*(x-c[9])*(x-c[10] )*(x-c[11])" "6#/-%\"pG6#%\"xG*,F'\"\"\",&F'F)&%\"cG6#\"\")!\"\"F),&F' F)&F,6#\"\"*F/F),&F'F)&F,6#\"#5F/F),&F'F)&F,6#\"#6F/F)" }{TEXT -1 7 " \+ and " }{XPPEDIT 18 0 "q(x)=(x-c[12])*p(x)" "6#/-%\"qG6#%\"xG*&,&F'\" \"\"&%\"cG6#\"#7!\"\"F*-%\"pG6#F'F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 155 "See: J.H. Verner, SIA M Journal of Numerical Analysis 1978, 772-790, \"Explicit Runge-Kutta \+ methods with estimates of the Local Truncation Error.\" (page 780)" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "p := x -> x*(x-c[8])*(x-c[9])*(x-c[10])*(x-c[11]):\n'p(x)'=p(x); \nq := x -> (x-c[12])*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),&F'F)&F,6#\"#5F/F),&F'F)&F,6#\"#6F/F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"qG6#%\"xG*.,&F'\"\"\"&%\"cG6#\"#7!\"\"F*F'F*, &F'F*&F,6#\"\")F/F*,&F'F*&F,6#\"\"*F/F*,&F'F*&F,6#\"#5F/F*,&F'F*&F,6# \"#6F/F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "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);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$IeqG/*&-%$IntG6$,$*&#\"\"\" \"\"'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-F8F--F(6$,$*&FE F-*&F>F-FHF-F-F-F8F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "e4 := solve(subs(e3,value(Ieq)),\{c[12]\} ):\nc[12]=subs(e4,c[12]);\ne5 := `union`(e3,e4):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#7#\"4s$e7&4lrzU\"\"4,Zbihkim;#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Thus " }{XPPEDIT 18 0 "c[12] = 1427971650951258372/2166662646162554701" "6#/&%\"cG6#\"# 7*&\"4s$e7&4lrzU\"\"\"\"\"4,Zbihkim;#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 99 "Now we can use the quadrature equations to find the weights once the remaining nodes once the nodes" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "c[13] = 4103/5000;" "6#/&%\"cG6#\"#8*&\"%.T\"\"\" \"%+]!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[14] = 2253/2500;" "6#/ &%\"cG6#\"#9*&\"%`A\"\"\"\"%+D!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[15]=1" "6#/&%\"cG6#\"#:\"\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 38 "are specified along with the weights " }{XPPEDIT 18 0 "b [2]=0" "6#/&%\"bG6#\"\"#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "b[3]=0 " "6#/&%\"bG6#\"\"$\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "b[4]=0" "6# /&%\"bG6#\"\"%\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "b[5]=0" "6#/&%\" bG6#\"\"&\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "b[6]=0" "6#/&%\"bG6# \"\"'\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "b[7]=0" "6#/&%\"bG6#\"\"( \"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 296 "Qeqs := QuadratureConditions(9,15, 'expanded'):\ne6 := \{seq(b[i]=0,i=2..7),c[13]=4103/5000,c[14]=2253/25 00,c[15]=1\}:\ne7 := `union`(e5,e6):\nquadeqns := subs(e7,Qeqs):\nfor \+ ct to nops(quadeqns) do\n print(`equation `||ct); print(``);print( quadeqns[ct]);print(``);\nend do:\nindets(quadeqns);\nnops(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,4&%\"bG6#\"\" \"F(&F&6#\"\")F(&F&6#\"\"*F(&F&6#\"#5F(&F&6#\"#6F(&F&6#\"#7F(&F&6#\"#8 F(&F&6#\"#9F(&F&6#\"#:F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~2G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,2*&#\"$H\"\"$+ #\"\"\"&%\"bG6#\"\")F)F)*&#\"$(Q\"$+)F)&F+6#\"\"*F)F)*&#\"%dn\"'++5F)& F+6#\"#5F)F)*&#F)\"\"%F)&F+6#\"#6F)F)*&#\"4s$e7&4lrzU\"\"4,Zbihkim;#F) &F+6#\"#7F)F)*&#\"%.T\"%+]F)&F+6#\"#8F)F)*&#\"%`A\"%+DF)&F+6#\"#9F)F)& F+6#\"#:F)#F)\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~3G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,2*&#\"&Tm\"\"&++%\"\"\" &%\"bG6#\"\")F)F)*&#\"'p(\\\"\"'++kF)&F+6#\"\"*F)F)*&#\")\\qlX\",+++++ \"F)&F+6#\"#5F)F)*&#F)\"#;F)&F+6#\"#6F)F)*&#\"F%Q!4+.4+s)\\Zi/#f..\"R? \"F,%*>Tz:%\"eo,)eZ*\\!>*\\k\\Cyb.b5w?6Z/%4r(3'GastiZn]kP?#F)&F+6#\"#7F)F )*&#\"0\")G=gSS$G\"0++++++D'F)&F+6#\"#8F)F)*&#\"/\"3ot'ewD\"/++++D1RF) &F+6#\"#9F)F)&F+6#\"#:F)#F)\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%! G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~6G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,2*&#\",\\;0 Bd$\"-+++++K\"\"\"&%\"bG6#\"\")F)F)*&#\".22b,2o)\"0+++++oF$F)&F+6#\"\" *F)F)*&#\"5dbUNdH8a39\";++++++++++++5F)&F+6#\"#5F)F)*&#F)\"%C5F)&F+6# \"#6F)F)*&#\"fpK;9F%4g*\\ItHwb?j4G>XLRqG()zx+2R._.erSbCcP8s9Au$f\"gp,N nj%p3Hq^$RhZ7e4doD.O#G6sbeW#[j3TYd^%od9$*3xB9[x%F)&F+6#\"#7F)F)*&#\"4V 2OI*eo!G;\"\"4+++++++]7$F)&F+6#\"#8F)F)*&#\"2$\\'G!=*\\]!e\"2+++++Dcw* F)&F+6#\"#9F)F)&F+6#\"#:F)#F)\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~7G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,2*&#\".@F mt#3Y\"/++++++k\"\"\"&%\"bG6#\"\")F)F)*&#\"14O7+:VfL\"3++++++W@EF)&F+6 #\"\"*F)F)*&#\"8\\')[qUGRw8v^*\"@+++++++++++++++\"F)&F+6#\"#5F)F)*&#F) \"%'4%F)&F+6#\"#6F)F)*&#\"hq/Jup9?#*[Ua43HS?DXphoIq*yK0&[%*eao\"\\*))H a'z!>^?9)>*H=%\\3&f]q%y%)\"jq,#ymJA6WD\\WVo,mf2:4L5'4]>>quI\"ou9]/A&3b 3F1[)>)o!o5\\247l6aM5F)&F+6#\"#7F)F)*&#\"7H&G,F\">Ul*4x%\"8+++++++++Dc \"F)&F+6#\"#8F)F)*&#\"6H(oa=`Yx(yI\"\"6++++++D19W#F)&F+6#\"#9F)F)&F+6# \"#:F)#F)\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~8G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,2*&#\"045\\-tY%f\"2+++++++G \"\"\"\"&%\"bG6#\"\")F)F)*&#\"4$oOya!***4+8\"6+++++++_r4#F)&F+6#\"\"*F )F)*&#\"<$H,=p:QkG]S)4V'\"E+++++++++++++++++5F)&F+6#\"#5F)F)*&#F)\"&%Q ;F)&F+6#\"#6F)F)*&#\"[s)om#pBV019P1Jnt]f8P&ec*fJ*)3oy]\\F/B:4'[[]ToXBn M<(Rb@]U\\n5H8`=Cf_:q57\"\\s,Hxw#)*=ZS)*oG&\\>ol!)**G8Z*G$*>W&*o#H#Hd` w\"y:In@$y(>'4aSP'4L\"Q,T=(Gky&\\Q)p,:C#F)&F+6#\"#7F)F)*&#\";([atL\"y5 E\"))Rv&>\";++++++++++]7yF)&F+6#\"#8F)F)*&#\"9Pk/hoi$Hc[m%H\"9+++++++D c^.hF)&F+6#\"#9F)F)&F+6#\"#:F)#F)F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~9G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,2*&#\"2 h,M@?G'ow\"4++++++++c#\"\"\"&%\"bG6#\"\")F)F)*&#\"6@jz7?M'pQJ]\"9+++++ +++;sx;F)&F+6#\"\"*F)F)*&#\"@,oL,&HEEY&yF#fTXV\"J++++++++++++++++++++ \"F)&F+6#\"#5F)F)*&#F)\"&Ob'F)&F+6#\"#6F)F)*&#\"]tO>rgC$*RLW?-'R'Rar^) p=C&paLoN#f'o8E<$H(R@_[N&et1C#y^UQx#H&*RGQi;Md'*oP&pblZY'[\\Z)G<\"^t,w &p;!GR]9h_$pF%*oL!4mM\"zmP&*45^i8wf#)4TIUFA[E`G46-YrgJ%*zk!=j!f3[^<#Qp ?rLh+7+yl&[F)&F+6#\"#7F)F)*&#\">h,Ylr#e.U\"G8'yJ!)\"?++++++++++++D1RF) &F+6#\"#8F)F)*&#\" " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 73 "e8 := solve(\{op(quadeqns)\}):\ne9 := `union`(e7,e8 ):\ninfolevel[solve] := 0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "for ii to 15 do\n wt_val := subs (e9,b[ii]);\n if wt_val<>0 then print(b[ii]=wt_val) end if;\nend do: \n``;\nevalf[8](subs(e9,[seq(b[i],i=1..15)]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"\"#\"CJ.]6P%*>6TtJ?m.;)>)\"EGbwGr%yFn,'e3% Qzv0h&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\")#!H+++SP@Y9G8) Q)GM2$\\bcX\"Ipy\\7t$eAk/Ch0\"4:6!3Q;\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"*#\"M+++?Zh:puv]8\"33gq[O;l*>\"M*fWl^KCy0io%Roo3P0> /WR')" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#5#\"Y+++++++++++v VV!3(pc0(=9)*>z5J#*)\"Z\"R&)f<$e$p(fi01Xk&)zYfXWnN$))4()z**p" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#6#\"D#\\V6(Q\"\\=8n!f%\\&RF/r%\" EnMw#\\86jM>gRB7RYo4#" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#7 #\"\\u$3WG!G[2>JRT_Ug8crny;QYri(Q>J]zstM2us4Li5JZOn'>J2t%H^jA%3$Qq$)zw @$3l]nzSZe5SY+XS@W+X3#\"\\u3'*Q#f'yw>cy)Rbv\\\"eQf?TbrBY-7KUNv()fv%GpJ 5OjY\"o/w%[0enem490j#>]=Z901D+^EG`AQ--tj8\"*)\\3nO" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"bG6#\"#8#\"O++++voz\"4pG6fz4lIU@GPI0'\"Q*)p7D(RF ;vPdx+HbM1=0Nd#**Q5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#9# \"N+v=nQYDn41%e.O)\\$*4#\\5,u\"*\"Oj9b*)HQ2OXP2z:LlY8\"f\")\\Us'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#:#\"CdR\\'f*pr8&*o_md&\\ae #\"DJvU[gg3K;STwg^MuX)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#71$\")x>h9!\"*$\"\"!F(F'F'F'F'F'$!)>@: R!\")$\")D$4J#F+$\")owu7F+$\")UVYAF+$\")FN%o&F+$\");(e#eF&$\")uJk8F+$ \")S,dIF&" }}}{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 4055 "e9 := \{b[10] = 892311079199814187055669708043437500000000000000 00000000/699979870988335674445594679856445060562597693583175985391, b[ 12] = 2084500442140450046401058474079675065083217679837038308422635129 4730731196673647311062330972740734737279503119387627146381678677156136 042524139311907482802844083/366708498911363730202382253282651002506051 4471850192630514096658675805484760468146633610316928475598775354232120 2462371554120593858149755539878561976786592389608, a[4,2] = 0, c[11] = 1/4, a[10,9] = -3167799860072183913409/30423656359863281250000, c[8] \+ = 129/200, a[5,2] = 0, a[6,2] = 0, a[6,3] = 0, a[7,2] = 0, a[7,3] = 0, a[8,2] = 0, a[8,3] = 0, a[8,4] = 0, a[8,5] = 0, a[9,2] = 0, a[9,3] = \+ 0, a[9,4] = 0, a[9,5] = 0, a[10,2] = 0, a[10,3] = 0, a[10,4] = 0, a[10 ,5] = 0, a[11,2] = 0, a[11,3] = 0, a[11,4] = 0, a[11,5] = 0, a[11,6] = 1/30, c[15] = 1, b[2] = 0, b[3] = 0, b[4] = 0, b[5] = 0, b[6] = 0, b[ 7] = 0, a[9,8] = -3483/102400, a[7,4] = -38208748178016484817787/84251 7966262441068418750-86118788556282369822807/842517966262441068418750*6 ^(1/2), a[8,6] = 43/150+43/2400*6^(1/2), b[8] = -455655493073428838813 281446213740000000/1163808011150910561240464225837312497869, a[9,6] = \+ 22833/102400+8901/204800*6^(1/2), a[9,7] = 22833/102400-8901/204800*6^ (1/2), a[3,2] = 64021741529527/200485534764000-7504450763411/100242767 382000*6^(1/2), a[11,8] = 4389715333607/309890657317500+92754382349/11 477431752500*6^(1/2), a[10,6] = 187914666753956840195279/2596152009375 000000000000-210440846556290693268911/15576912056250000000000000*6^(1/ 2), a[8,7] = 43/150-43/2400*6^(1/2), c[7] = 387/1000+129/2000*6^(1/2), c[3] = 7630049/53810000-983539/53810000*6^(1/2), a[7,1] = 26523528363 /231790900000+863255358/123138915625*6^(1/2), a[7,6] = -362925891/1690 350537500+857800423623/3380701075000*6^(1/2), a[10,7] = 18791466675395 6840195279/2596152009375000000000000+210440846556290693268911/15576912 056250000000000000*6^(1/2), a[5,4] = 1873698362223295443/5131263555055 56250+528258592225556973/513126355505556250*6^(1/2), a[4,1] = 22890147 /430480000-2950617/430480000*6^(1/2), c[4] = 22890147/107620000-295061 7/107620000*6^(1/2), a[5,3] = -8712153884182794903/2052505422022225000 -2843421359195851533/2052505422022225000*6^(1/2), a[5,1] = 59220399426 1020339/513126355505556250+730386990293623641/2052505422022225000*6^(1 /2), a[11,7] = -2865012129681958/114898584332330625-12962517687655099/ 229797168664661250*6^(1/2), c[6] = 387/1000-129/2000*6^(1/2), a[3,1] = -177968356965557/1002427673820000+14180534491313/250606918455000*6^(1 /2), a[7,5] = 92362336407446913/290322814529044000-232039320950012997/ 2467743923496874000*6^(1/2), a[4,3] = 68670441/430480000-8851851/43048 0000*6^(1/2), a[11,1] = -426968570497/54394415898750-92754382349/12087 647977500*6^(1/2), a[11,9] = 4990058173976/83757096376875+371017529396 /9306344041875*6^(1/2), a[6,5] = 165912282616977/4179075230308000-3318 1894472511/2089537615154000*6^(1/2), a[6,4] = 16193232887091831/588643 41808507450-2355345717024309/58864341808507450*6^(1/2), a[11,10] = 109 9523524595993125000/6257667909869756018891+100957348037989687500/62576 67909869756018891*6^(1/2), a[9,1] = 7353/102400, b[9] = 19965163648706 008081135075746915614720000000/863944041905370868683946862057824325165 44599, a[10,1] = 376708742472214988700853/7788456028125000000000000, a [6,1] = 11380823631/157617812000-339148869/39404453000*6^(1/2), c[12] \+ = 1427971650951258372/2166662646162554701, c[2] = 1731/50000, c[10] = \+ 6757/100000, c[5] = 561/1000, c[9] = 387/800, a[2,1] = 1731/50000, b[1 ] = 8198160366203173411119943711500331/5610575793840858601672778471287 65528, c[13] = 4103/5000, c[14] = 2253/2500, b[11] = 47104273954945906 713184913871143492/209684639122339601934631113492763467, b[15] = 25854 49557665268951371699596493957/84574345160764140163208606048427531, b[1 4] = 917401104920993498360358406096725463867187500/6724249815911346653 315790737453607382989551463, a[10,8] = -18552667221896744226647/865384 003125000000000000, b[13] = 605303728214230650979591128690917968750000 0000/103899257350518063455290077573775162739725126989, a[8,1] = 43/600 \}:" }{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 -1 109 "It remains to determine the linking coef ficients in stages 12 to 15. We have the following zero coefficients. " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "a[12,2]=0" "6#/& %\"aG6$\"#7\"\"#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[12,3]=0" "6# /&%\"aG6$\"#7\"\"$\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[12,4]=0" " 6#/&%\"aG6$\"#7\"\"%\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[12,5]=0 " "6#/&%\"aG6$\"#7\"\"&\"\"!" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "a[13,2]=0" "6#/&%\"aG6$\"#8\"\"#\"\"! " }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[13,3]=0" "6#/&%\"aG6$\"#8\"\"$\" \"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[13,4]=0" "6#/&%\"aG6$\"#8\"\"% \"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[13,5]=0" "6#/&%\"aG6$\"#8\" \"&\"\"!" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "a[14,2]=0" "6#/&%\"aG6$\"#9\"\"#\"\"!" }{TEXT -1 2 ", \+ " }{XPPEDIT 18 0 "a[14,3]=0" "6#/&%\"aG6$\"#9\"\"$\"\"!" }{TEXT -1 2 " , " }{XPPEDIT 18 0 "a[14,4]=0" "6#/&%\"aG6$\"#9\"\"%\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[14,5]=0" "6#/&%\"aG6$\"#9\"\"&\"\"!" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[15,2]=0 " "6#/&%\"aG6$\"#:\"\"#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[15,3] =0" "6#/&%\"aG6$\"#:\"\"$\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[15, 4]=0" "6#/&%\"aG6$\"#:\"\"%\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[1 5,5]=0" "6#/&%\"aG6$\"#:\"\"&\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 154 "We make use of the stage -order conditions for stages 12 to 15 so that all these stages all hav e stage-order 5 and incorporate the simplifying conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j+1 .. \+ 15) = b[j]*(1-c[j]);" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+%\"j GF,/F+;,&F0F,F,F,\"#:*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "for " }{XPPEDIT 18 0 "j=1" "6#/%\"jG \"\"\"" }{TEXT -1 10 " ( with " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6# \"\"\"\"\"!" }{TEXT -1 5 " ), " }{XPPEDIT 18 0 "j=7" "6#/%\"jG\"\"(" }{TEXT -1 8 " . . 13." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 185 "Then it turns out that the following collection of \" simple\" order conditions (given in abreviated form) is sufficient to \+ determine the remaining linking coefficients in stages 11 to 15." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 194 "SO9 := SimpleOrderConditions(9):\n[seq([i,SO9[i]],i=[102,106,125, 212,223,239,245,251,253])]:\nlinalg[augment](linalg[delcols](%,2..2),m atrix([[` `]$(linalg[rowdim](%))]),linalg[delcols](%,1..1));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7+7%\"$-\"%#~~G/*(%\"bG\" \"\"%\"cGF--%!G6#*&%\"aGF--F06#*&F3F--F06#*&)F.\"\"$F-F3F-F-F-F-#F-\"$ g*7%\"$1\"F)/*&F,F--F06#*&F3F--F06#*&F3F--F06#*&)F.\"\"%F-F3F-F-F-F-#F -\"%!o\"7%\"$D\"F)/*(F,F-F.F--F06#*&)F.\"\"&F-F3F-F-#F-\"#[7%\"$7#F)/* (F,F-)F.\"\"#F-F/F-#F-\"%!3\"7%\"$B#F)/*(F,F-F.F-FBF-#F-\"%!*=7%\"$R#F )/*(F,F-FhnF-FEF-#F-\"$q#7%\"$X#F)/*(F,F-F.F--F06#*&F3F-FSF-F-#F-\"$y$ 7%\"$^#F)/*(F,F-FhnF-FSF-#F-\"#a7%\"$`#F)/*(F,F-F.F--F06#*&)F.\"\"'F-F 3F-F-#F-\"#jQ(pprint16\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "The associated trees" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 456 "AST9 := AllSimpleTrees(9):\nwhch : = [102,106,125,212,223,239,245,251,253]:\nm := 3: ## number of trees p er row\nnn := nops(whch): q := iquo(nn,m,'r'):\nfor i to nn do \n p|| i := DrawTree(AST9[whch[i]],height=4,width=2,show_ordercondition=true, \n color=COLOR(RGB,.5,0,.9),font_color=black);\nend do:\npp := plot ([[1,1]],style=line,axes=none):\nplots[display](convert([seq([p||((k-1 )*m+1..m*k)],k=1..q),\n `if`(r>0,[p||(m*q+1..nn),pp$(m*(q+1)-nn)],NUL L)],array));" }}{PARA 13 "" 1 "" {GLPLOT2D 782 948 948 {PLOTDATA 2 "6f u-%%TEXTG6&7$$\"#g\"\"!$!+E!G3\"R!\")Q5b~c~~(a~(a~c~))~=~~~6\"-%'COLOU RG6&%$RGBGF)F)F)-%%FONTG6$%(COURIERG\"#5-F$6&7$F'$!+uu)G%QF,Q7~~~~~~~~ ~~~~~~~~~~~~~1F.F/F3-F$6&7$F'$!+(*)=T'QF,Q9~~~~~~~~~~~~~~~~~~~~~___F.F /F3-F$6&7$F'$!+,+++SF,Q9~~~~~~~~~~~~~~~~~~~~~270F.F/F3-F$6&7$F'$!+0\\R MQF,Q7~~~~2~~~~~~~~4~~~~~~~~F.F/F3-%'CURVESG6&7+7$F'$!+,Q^)p$F,7$$\"++ +++bF,$!+^`)QF$F,7$F'FZ7$$\"+++++lF,FZ7$Fhn$!+,pD\\GF,7$$\"+LLLLjF,$!+ ]%GYU#F,7$$\"+WWWWkF,F`o7$$\"+cbbblF,F`o7$$\"+nmmmmF,F`o-%'SYMBOLG6#%' CIRCLEG-%&COLORG6&F2$F)F)FbpFbp-%&STYLEG6#%&POINTG-FQ6&FS-F\\p6#%(DIAM ONDGF_pFcp-FQ6&FS-F\\p6#%&CROSSGF_pFcp-FQ6%7$FTFW-F`p6&F2$\"\"&!\"\"Fb p$\"\"*Fhq-%*THICKNESSG6#\"\"#-FQ6%7$FTFfnFdqF[r-FQ6%7$FTFgnFdqF[r-FQ6 %7$FgnFjnFdqF[r-FQ6%7$FjnF]oFdqF[r-FQ6%7$FjnFboFdqF[r-FQ6%7$FjnFeoFdqF [r-FQ6%7$FjnFhoFdqF[r-FQ6%7#7$F'$!+++++?F,-F06&F2$F7FhqFbpFbp-Fdp6#%%L INEG-FQ6%7#7$$\"+++++]F,FUFjsF]t-FQ6%7#7$$\"+++++qF,FUFjsF]t-F$6&7$$\" #IF)$!+d-G3\"*!\"*Q7b~(a~(a~(a~c~)))~=~~~~F.F/F3-F$6&7$F_u$!+PZ()G%)Fc uQ9~~~~~~~~~~~~~~~~~~~~~~~1F.F/F3-F$6&7$F_u$!+i*)=T')FcuQ<~~~~~~~~~~~~ ~~~~~~~~~~~____F.F/F3-F$6&7$F_u$!+++++5F,Q<~~~~~~~~~~~~~~~~~~~~~~~1680 F.F/F3-F$6&7$F_u$!+Z!\\RM)FcuQ9~~~~~~~~~~~~~4~~~~~~~~~~F.F/F3-FQ6&7*7$ F_u$!+1!Q^)pFcu7$F_u$!+/N&)QFFcu7$F_u$\"+)*4V2:Fcu7$F_u$\"+/br`dFcu7$$ \"++++DEF,$\"+++++5F,7$$\"++++vGF,F_x7$$\"++++DJF,F_x7$$\"++++vLF,F_xF [pF_pFcp-FQ6&F_wFipF_pFcp-FQ6&F_wF^qF_pFcp-FQ6%7$F`wFcwFdqF[r-FQ6%7$Fc wFfwFdqF[r-FQ6%7$FfwFiwFdqF[r-FQ6%7$FiwF\\xFdqF[r-FQ6%7$FiwFaxFdqF[r-F Q6%7$FiwFdxFdqF[r-FQ6%7$FiwFgxFdqF[r-FQ6%7#7$F_uF_xFjsF]t-FQ6%7#7$$\"+ ++++?F,FawFjsF]t-FQ6%7#7$$\"+++++SF,FawFjsF]t-FQ6#-%'LEGENDG6#QB__neve r_display_this_legend_entryF.-F$6&7$FbpF*Q:b~c~~(a~(a~(a~c~)))~=~~~~F. 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66" "Curve 67" "Curve 68" "Curve 69" "Curve 70" "Curve 71 " "Curve 72" "Curve 73" "Curve 74" "Curve 75" "Curve 76" "Curve 77" "C urve 78" "Curve 79" "Curve 80" "Curve 81" "Curve 82" "Curve 83" "Curve 84" "Curve 85" "Curve 86" "Curve 87" "Curve 88" "Curve 89" "Curve 90 " "Curve 91" "Curve 92" "Curve 93" "Curve 94" "Curve 95" "Curve 96" "C urve 97" "Curve 98" "Curve 99" "Curve 100" "Curve 101" "Curve 102" "Cu rve 103" "Curve 104" "Curve 105" "Curve 106" "Curve 107" "Curve 108" " Curve 109" "Curve 110" "Curve 111" "Curve 112" "Curve 113" "Curve 114 " "Curve 115" "Curve 116" "Curve 117" "Curve 118" "Curve 119" "Curve 1 20" "Curve 121" "Curve 122" "Curve 123" "Curve 124" "Curve 125" "Curve 126" "Curve 127" "Curve 128" "Curve 129" "Curve 130" "Curve 131" "Cur ve 132" "Curve 133" "Curve 134" "Curve 135" "Curve 136" "Curve 137" "C urve 138" "Curve 139" "Curve 140" "Curve 141" "Curve 142" "Curve 143" "Curve 144" "Curve 145" "Curve 146" "Curve 147" "Curve 148" "Curve 149 " "Curve 150" "Curve 151" "Curve 152" "Curve 153" "Curve 154" "Curve 1 55" "Curve 156" "Curve 157" "Curve 158" "Curve 159" "Curve 160" "Curve 161" "Curve 162" "Curve 163" "Curve 164" "Curve 165" "Curve 166" "Cur ve 167" "Curve 168" "Curve 169" }}}}{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 49 "SO9_15 := SimpleOr derConditions(9,15,'expanded'):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 485 "SOeqs2 := [seq(add(a[i,j],j= 1..i-1)=c[i],i=12..15),op(StageO rderConditions(2,12..15,'expanded')),\n op(StageOrderCondition s(3,12..15,'expanded')),op(StageOrderConditions(4,12..15,'expanded')), \n op(StageOrderConditions(5,12..15,'expanded'))]:\nord_ cdns := [seq(SO9_15[i],i=[102,106,125,212,223,239,245,251,253])]:\nsim p_eqs := [add(b[i]*a[i,1],i=1+1..15)=b[1],seq(add(b[i]*a[i,j],i=j+1..1 5)=b[j]*(1-c[j]),j=[$7..13])]:\ncdns := [op(SOeqs2),op(ord_cdns),op(si mp_eqs)]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 127 "It is po ssible to manage with fewer equations, but the computation may be less efficient if the number of equations is reduced." }}{PARA 0 "" 0 "" {TEXT -1 50 "For example, the simplifying conditions given by " } {XPPEDIT 18 0 "j=8" "6#/%\"jG\"\")" }{TEXT -1 3 ", " }{XPPEDIT 18 0 " j=10" "6#/%\"jG\"#5" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "j=12" "6#/% \"jG\"#7" }{TEXT -1 17 " may be omitted." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "e10 :=\{seq(seq(a[i,j ]=0,i=12..15),j=2..5)\}:\ne11 := `union`(e9,e10):\neqns2 := subs(e11,c dns):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "There are 37 equations and 34 unknowns." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "nops(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 74 "e12 := solve(\{op(eqns2)\}): \ninfolevel[solve] := 0:\ne13 := `union`(e11,e12):" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "e13 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14542 "e13 := \{a[14,2] = 0, a[12,3] = 0, a[15,2] = 0, a[14,3] = 0, a[13,3] = 0, a[12,4] = 0, a [15,3] = 0, a[13,4] = 0, a[12,5] = 0, a[15,4] = 0, a[14,4] = 0, a[13,5 ] = 0, a[15,5] = 0, a[14,5] = 0, b[10] = 89231107919981418705566970804 343750000000000000000000000/699979870988335674445594679856445060562597 693583175985391, b[12] = 208450044214045004640105847407967506508321767 9837038308422635129473073119667364731106233097274073473727950311938762 7146381678677156136042524139311907482802844083/36670849891136373020238 2253282651002506051447185019263051409665867580548476046814663361031692 84755987753542321202462371554120593858149755539878561976786592389608, \+ a[4,2] = 0, c[11] = 1/4, a[10,9] = -3167799860072183913409/30423656359 863281250000, c[8] = 129/200, a[5,2] = 0, a[6,2] = 0, a[6,3] = 0, a[7, 2] = 0, a[7,3] = 0, a[8,2] = 0, a[8,3] = 0, a[8,4] = 0, a[8,5] = 0, a[ 9,2] = 0, a[9,3] = 0, a[9,4] = 0, a[9,5] = 0, a[10,2] = 0, a[10,3] = 0 , a[10,4] = 0, a[10,5] = 0, a[11,2] = 0, a[11,3] = 0, a[11,4] = 0, a[1 1,5] = 0, a[11,6] = 1/30, c[15] = 1, b[2] = 0, b[3] = 0, b[4] = 0, b[5 ] = 0, b[6] = 0, b[7] = 0, a[9,8] = -3483/102400, a[7,4] = -3820874817 8016484817787/842517966262441068418750-86118788556282369822807/8425179 66262441068418750*6^(1/2), a[8,6] = 43/150+43/2400*6^(1/2), b[8] = -45 5655493073428838813281446213740000000/11638080111509105612404642258373 12497869, a[9,6] = 22833/102400+8901/204800*6^(1/2), a[9,7] = 22833/10 2400-8901/204800*6^(1/2), a[3,2] = 64021741529527/200485534764000-7504 450763411/100242767382000*6^(1/2), a[11,8] = 4389715333607/30989065731 7500+92754382349/11477431752500*6^(1/2), a[10,6] = 1879146667539568401 95279/2596152009375000000000000-210440846556290693268911/1557691205625 0000000000000*6^(1/2), a[8,7] = 43/150-43/2400*6^(1/2), c[7] = 387/100 0+129/2000*6^(1/2), c[3] = 7630049/53810000-983539/53810000*6^(1/2), a [7,1] = 26523528363/231790900000+863255358/123138915625*6^(1/2), a[7,6 ] = -362925891/1690350537500+857800423623/3380701075000*6^(1/2), a[10, 7] = 187914666753956840195279/2596152009375000000000000+21044084655629 0693268911/15576912056250000000000000*6^(1/2), a[5,4] = 18736983622232 95443/513126355505556250+528258592225556973/513126355505556250*6^(1/2) , a[4,1] = 22890147/430480000-2950617/430480000*6^(1/2), c[4] = 228901 47/107620000-2950617/107620000*6^(1/2), a[5,3] = -8712153884182794903/ 2052505422022225000-2843421359195851533/2052505422022225000*6^(1/2), a [5,1] = 592203994261020339/513126355505556250+730386990293623641/20525 05422022225000*6^(1/2), a[11,7] = -2865012129681958/114898584332330625 -12962517687655099/229797168664661250*6^(1/2), c[6] = 387/1000-129/200 0*6^(1/2), a[3,1] = -177968356965557/1002427673820000+14180534491313/2 50606918455000*6^(1/2), a[7,5] = 92362336407446913/290322814529044000- 232039320950012997/2467743923496874000*6^(1/2), a[4,3] = 68670441/4304 80000-8851851/430480000*6^(1/2), a[11,1] = -426968570497/5439441589875 0-92754382349/12087647977500*6^(1/2), a[11,9] = 4990058173976/83757096 376875+371017529396/9306344041875*6^(1/2), a[6,5] = 165912282616977/41 79075230308000-33181894472511/2089537615154000*6^(1/2), a[6,4] = 16193 232887091831/58864341808507450-2355345717024309/58864341808507450*6^(1 /2), a[11,10] = 1099523524595993125000/6257667909869756018891+10095734 8037989687500/6257667909869756018891*6^(1/2), a[9,1] = 7353/102400, b[ 9] = 19965163648706008081135075746915614720000000/86394404190537086868 394686205782432516544599, a[10,1] = 376708742472214988700853/778845602 8125000000000000, a[6,1] = 11380823631/157617812000-339148869/39404453 000*6^(1/2), c[12] = 1427971650951258372/2166662646162554701, c[2] = 1 731/50000, c[10] = 6757/100000, c[5] = 561/1000, c[9] = 387/800, a[14, 13] = 4746930876023919335079451612726717649218264199984/18592065538407 049755200144388134089346432755594877, a[15,14] = 148002502009403237171 24616175641261235119295795768814717803955078125/3356557712514187776028 7380588632421223433194078156948298488471160489, a[2,1] = 1731/50000, a [12,2] = 0, b[1] = 8198160366203173411119943711500331/5610575793840858 60167277847128765528, c[13] = 4103/5000, c[14] = 2253/2500, a[13,2] = \+ 0, b[11] = 47104273954945906713184913871143492/20968463912233960193463 1113492763467, a[14,10] = 57477430027199859868387311410547201669924149 5055292/1049352151254569101542262489932969253892183788671875*6^(1/2)-2 7843764471262693189365201135620670490328475323282820219474851621693895 769527094334687108984/122570410662851642220025943006055939294341391930 22166317802121412999357024704596261133984375, b[15] = 2585449557665268 951371699596493957/84574345160764140163208606048427531, b[14] = 917401 104920993498360358406096725463867187500/672424981591134665331579073745 3607382989551463, a[14,11] = -3424113435184824562423280943767688900943 1930503529853032576417589898516/56133478243586519811009850090242810076 03230062439942682713165283203125, a[12,8] = -5587476413495323413846491 6783230492507657050788557207210520035563218001131629645677655267245390 63327600257543743479921263738432/3653030893622016645164135969252861614 9447357533711529625051175285972810886869692961402480325512278540323235 9817965288739565550625*6^(1/2)+561098789927327852541196052808144290219 8567594809764379756195673673265700551076812883925583370253765702553235 594764427173637673766208/928815981981440330182788047406263341354233567 9163959810935886777036160923284601262673233245084426429384045657495603 6349633197336361875, a[14,6] = 871779321807802447463310035318238762878 527157/134446420823349498059060773719650573730468750+10764126848099939 6081848975271849857994818/1097082793918531904161935913552348681640625* 6^(1/2), a[12,7] = -13666660749646362227013560886377207644362546879813 9480390426740993024803946981763209348364716108721312822619845726151693 667598437699964416/371928646534240427478858532725418082819528242734205 7650194855634917821113563432870681372043512520401887141437067106105683 944802332422369375+169845085565361336805556009296394374527636952379388 9610260666287251555218327620868756323669964775679286575359121913961555 66765457826139904/1593979913718173260623679425966077497797978183146596 1357977952721076376200986140874348737329339373150944891873144740452931 19200999609586875*6^(1/2), a[13,7] = -15722999985374822730516577336442 6925282378072238332930121/36699907367985458573273204094330716033963413 238525390625+575760644280279509531898606731783790418427865066459025210 1/3523191107326604023034227593055748739260487670898437500000*6^(1/2), \+ a[14,7] = 496103786351862292800034805114190705484800743513354117014/11 0099722103956375719819612282992148101890239715576171875-13299384126061 97485769312599390307351191540891599374831099/6605983326237382543189176 73697952888611341438293457031250*6^(1/2), a[12,6] = 481927289247776817 1373308666720689121421091953625792970278044071549950640195056472955523 769829034800621890424847009130000000/231628944470026509787024364557612 0415548371816150200392731184205692823672051993456912431961089928486277 6485022935540644488821877*6^(1/2)-333881311789849411971573472868128281 4382022107217231232517421453677345828875773955477782287601740687580861 34389952015563403904/2270872004608103037127689848604039623086639035441 3729340501805938164937961294053499141489814607142022329887277387784945 57727635, a[15,6] = -5482114211968505556247721620542861394990543039608 8/3959439837009461289085587746748097947393101278095-151127675382598285 6072891469504471256664975925000/40386286337496505148672995016830599063 409633036569*6^(1/2), a[13,6] = -2745292391641202525373103979336813513 372321/11702216468464340311060649744558385937500000*6^(1/2)-1027163900 229750356561238237947225332675621517/179261894431132664078747698292867 431640625000, a[10,8] = -18552667221896744226647/865384003125000000000 000, a[12,11] = 123461712659887915177271339396606860810479028777869348 014870450606260914019560285661288212498128400476015695960341952/281629 1066703206747542452093588407037042351473078388967410755112208260568290 47205614324978253226176275078922716132461, a[12,1] = 40788577818515860 9210793892517582595305896470756467612636796259611491408260896413446883 450891351622914818800693274034252252905536/280849263886012260736240961 6917500295697019157645511063322676514116137229409869327511718123938531 2198137508846535933127837167926875*6^(1/2)+183820311047984038699385390 0915465658752149857359559506316407788280031537278728468323843947895514 1517997198007108623761931447163756/13974256944499724344918960993890933 6141610253229704500479326889980950085286208212396047346081112917694447 06187497807869179550841329375, a[15,8] = -1763247227116943100256560855 05072661620440362221411708/2856194067198291074857712070420401334654201 49964555625*6^(1/2)-11611814757504516973322287583571995533433479819145 9879782123534889390467935109772/88106269019548352456722751312958708925 03713957512170681453300814988417642493125, a[13,9] = -2814045797346992 32141455524604487724159024972527/2844819163647379832214023225048303031 92138671875*6^(1/2)+88577423385667259022295186769532781645734013039163 9153070521335485617578/30109854138029501101546924846546529011250565614 3757799934635162353515625, a[15,12] = 33351439245158438248073494056784 1440978729127734159045364007283876903345639683941147024141088075051581 0638511646873285345820289996674848871853154570655914289590314484876463 7/23166110253272874277148020113222528860907939049899006215923656276490 9757810216357219050223242549060677331231066559342498274574429937128559 8588298606088543376742054644818966, a[14,12] = -3432044375893932378102 3685680522865010338509105169992020885327052116334327939205477028009615 3243800840188373734185468897263960533460016393861026885570574276407260 9/11431741063416822609716476904105672921439261986509277789208232674611 1137127590759980171487016581339414751906821093176684449499461658025843 5518181434575195312500000, a[13,1] = 281404579734699232141455524604487 724159024972527/1478009944832743180452316204077188415527343750000*6^(1 /2)-560427726753220421392276299780425863306336227060533639467661444169 33631/5880854077232319052559012261322343050735211853455734266601562500 0000000, a[14,8] = 123767075855296855875080343261298883871499029943/45 1091609994276250390378731960660324096679687500*6^(1/2)+407740772777476 3635459845170889116549412313138377723522953861198939217519328599426647 1/15264290546248162101058985941588079518256741255377031736357946125713 524703979492187500, a[12,9] = -652617245096253774737270228028132152489 4343532103481802188740153783862532174342615150135214261625966637100811 092384548036046488576/864909328430372818360283879213205026685796531766 2489228456648746817034128576286937426571324705771222895418404433420637 2230816544375*6^(1/2)+545985398180836152335661486022032448966969589107 3433975406527098543350794516270773775946921467448080727221064814847749 9238783276259328/30124791909229885263488687512995931079466293201418449 9827145075851637298698312074030567479239502011693447423026416040794479 934024058125, a[15,9] = -197483689437097627228734815765681381014893205 68798111296/6484554262322259071286545935997129135111813687175650625*6^ (1/2)+1776944872251389834227683749066509728692760724707333561856698714 3467294900183033216/25512170081378896150563421460845618671224851635966 19283719957742418751029506356875, a[13,8] = -8442137392040976964243665 73813463172477074917581/4210188359946578336976868164966163024902343750 000*6^(1/2)-9311448168593934146015965019904013602133802943325818346622 781285907057/425597084901012421719313544966873998540131336300557615936 2792968750000, a[14,1] = 473420038480243914987079768476888930130830744 41159779465719863625051668939887702630319/4480254687392605073040122263 6656855760802419993852060264615320801485392456054687500000-86636953098 7077991125562402829092187100493209601/33255223758736721560177114591736 73934936523437500*6^(1/2), a[15,11] = 47338974904975296325611464923135 3822492912259509649519870869750525/35412440882360341799798842428365422 941216508121322622479260846291, a[13,10] = -29403964536478722766460687 76592292229737651937934623/7345465058781983710795837429530784777245286 520703125*6^(1/2)+3154791167297801539564121240521996850977442393866390 23787359107959254802182/1344818505065058480125878422155155743802125432 00894932329128471154748828125, b[13] = 6053037282142306509795911286909 179687500000000/103899257350518063455290077573775162739725126989, a[8, 1] = 43/600, a[13,11] = 2250996163406545378616532039018846586217631599 453822541/382491303797095993563304148204275636433504028320312500, a[15 ,1] = 1234273058981860170179592598535508631343082535549881956/21056337 71469628744518390642968552144069898845895808125*6^(1/2)-25188329249258 825443748527038142409879923012133738985313265430932280250855708601/113 7064132557446931205696187407729855082764230877464731699571703634755806 4286250, a[12,10] = -8108825145085088104344721048166325225173729495689 3646964267201611120124142277523289697206589873156541798737603577252357 34000399440000000/2655582966610640210612741027567977548105720815295466 9252204018964061820669308150613320650066298300104729878761982741137567 5716583283801757*6^(1/2)+939166734840458401095542221032870712500612066 1611061908889750805619418785820948002455890360939221912190524731087070 645107486913457760000000/581572669687730206124190285037387083035152858 5497072566232680153129538726578484984317222364519327722935843448874220 3091272981931739152584783, a[13,12] = 26893409573076918532949023883344 5400395937814695752986623352925198635939233604415170894972095880974797 0514366293458424272174024493/95951638601957880850056911478087170846689 4752280482835105408027815194895319055443842782227102120493960805649575 561796875000000000, a[15,13] = -38714992656958413389743252726016897599 283911682945255636643554687500000/485404949269715874992945893825722120 36169135429877901702347521300421767, a[14,9] = 34654781239483119645022 49611316368748401972838404/2560337247282641848992620902543472728729248 046875*6^(1/2)-1052203860850055645982864903830206847373574903079637276 4961618751973793724796364606986664/38994174250054222540345740003973828 62235892829653375835197340918271556055507659912109375, a[15,10] = -852 97084611782122474911131363078900058888025224607913745000000/6921065945 0201393843166746722954036326338355649915383851733911*6^(1/2)+976592661 3912407481819326480192954778165992654378638151019095418421857074621503 3823993530000000/18560076654469706205963482908787056850812308205603127 326855360961727608242796551101182080033599, a[15,7] = -609224242740615 99918603524049390657305431262635197540405697952/6484861747489032169774 584624759953148531564032417461909516875+845585757516359787331099618939 84238786929550462615375699341616/1945458524246709650932375387427985944 5594692097252385728550625*6^(1/2)\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "printed coefficients" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 283 "subs(e13,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(4-i)],i=2..4) ]));\nfor ii from 5 to 15 do\n print(``);\n print(c[ii]=subs(e13,c [ii])); \n for jj to ii-1 do\n print(a[ii,jj]=subs(e13,a[ii,jj] ));\n end do:\nend do:\n``;\nfor ii to 15 do\n print(b[ii]=subs(e1 3,b[ii]));\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7%7 &#\"%J<\"&++&F(%!GF+7&,&#\"(\\+j(\")++\"Q&\"\"\"*(\"'RN)*F1F0!\"\"\"\" '#F1\"\"#F4,&#\"0db'pNoz<\"1++#QnFC+\"F4*(\"/88\\M0=9F1\"0+]X=pg]#F4F5 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20 "6#/&%\"aG6$\"#6\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"#6\"\"'#\"\"\"\"#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"#6\"\"(,&#\"1e>oH@,lG\"3D1LKVe)*[6!\"\"*(\"2*4bwo7`QR7=<^F$p)4%Hs8;T^wEKj5^Xw:>qp&H+v\"ph4CO2E7g)QE\\3G!\"\"\"\"'#F(\" \"#F(#\"^scP;Z9$>wB'3r+)>(*z^T^&*y%R%QKo%G(ys`J+G)y2kJ1&f&ft&)\\@vecY: 4!R&Q*pQS)z/6.#Q=\"asv$H8%3bz\"py!y\\(=1ZWpS1&*\\:2W!y-(HzDO&>4@97*o?nm3LPr\"oxZ#*GF>[\"\"\"\"[s x=#))[W1aNH-&[wF'[G**3h>V7pX$*>0sO#Gp0U=JFR+-:;=P[bT?hdXOCqy4l-qW%*G;B !\"\"F(#F,\"\"#F,#\"ir/RSjb,_**QMh3e(oST\\)*y68)QL\"jrNwsdX\\y(Qxs))HB-Urg9)*[T\"*\\`SHhz$\\;Qf!= ]S$HPTa.Rm3B'RSg[)*oFr..\"3Y+s3F#F." }}{PARA 12 "" 1 "" {XPPMATH 20 "6 #/&%\"aG6$\"#7\"\"(,&#\"es;W'**pP%)fnOp^hsX)>E#GJ@(3hrk$[$4Kw\")p%R![- $*4uE/R![R\")zoaiVk2sP')3c8qAijk\\2mmO\"\"fsv$pBUKB![%Ro0h5nqVTr)=S?D^ V?P\"oqGVjN6@y\"\\jb[>]w0UtU#G&>G3=asK&e)yuUSU`Y'G>P!\"\"*(\"es/*Rh#yX 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"6#/&%\"bG6#\"#6#\"D#\\V6(Q\"\\=8n!f%\\&RF/r%\"EnMw#\\86 jM>gRB7RYo4#" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#7#\"\\u$3W G!G[2>JRT_Ug8crny;QYri(Q>J]zstM2us4Li5JZOn'>J2t%H^jA%3$Qq$)zw@$3l]nzSZ e5SY+XS@W+X3#\"\\u3'*Q#f'yw>cy)Rbv\\\"eQf?TbrBY-7KUNv()fv%GpJ5OjY\"o/w %[0enem490j#>]=Z901D+^EG`AQ--tj8\"*)\\3nO" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#8#\"O++++voz\"4pG6fz4lIU@GPI0'\"Q*)p7D(RF;v Pdx+HbM1=0Nd#**Q5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#9#\"N +v=nQYDn41%e.O)\\$*4#\\5,u\"*\"Oj9b*)HQ2OXP2z:LlY8\"f\")\\Us'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#:#\"CdR\\'f*pr8&*o_md&\\ae #\"DJvU[gg3K;STwg^MuX)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "RK9_15eqs := [op(RowSumC onditions(15,'expanded')),op(OrderConditions(9,15,'expanded'))]:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Check:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "expand(subs(e13,RK9_15eqs)): \nmap(u_->`if`(lhs(u_)=rhs(u_),0,1),%);\nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7`jl\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$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 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 34 "Appendix: related order conditions" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "#-------- -----------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 145 "u nrelated 129, 130, 134, 135, 136, 139, 140, 141, 143, 151, 152, 156, \+ 157, 166, 168, 173, 175, 180, 185, 186, 187, 212, 221, 223\n\nrelated \+ groups" }}{PARA 0 "" 0 "" {TEXT -1 19 "132, 137, 148, 153," }}{PARA 0 "" 0 "" {TEXT -1 19 "133, 138, 150, 163," }}{PARA 0 "" 0 "" {TEXT -1 59 "131, 147, 149, 192,\n142, 164, 170, 222,\n162, 184, 188, 227," }} {PARA 0 "" 0 "" {TEXT -1 18 "181, 197, 202, 239" }}{PARA 0 "" 0 "" {TEXT -1 38 "144, 158, 167, 177, 193, 207, 213, 244" }}{PARA 0 "" 0 " " {TEXT -1 38 "145, 159, 169, 178, 194, 209, 215, 245" }}{PARA 0 "" 0 "" {TEXT -1 38 "154, 165, 172, 189, 206, 208, 224, 248" }}{PARA 0 "" 0 "" {TEXT -1 38 "183, 200, 204, 205, 228, 231, 235, 251" }}{PARA 0 " " 0 "" {TEXT -1 78 "146, 160, 161, 174, 182, 196, 198, 201, 203, 211, \+ 218, 230, 232, 238, 241, 253" }}{PARA 0 "" 0 "" {TEXT -1 78 "155, 171, 176, 179, 190, 195, 199, 210, 214, 216, 225, 229, 237, 240, 246, 255 " }}{PARA 0 "" 0 "" {TEXT -1 78 "191, 217, 219, 220, 226, 233, 234, 23 6, 242, 243, 247, 249, 250, 252, 254, 256" }}{PARA 0 "" 0 "" {TEXT -1 43 "#------------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 44 "#----------------------- --------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 43 "construction of the order 8 embedded scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "e13 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14542 "e13 := \{a[14,2] = 0, a[1 2,3] = 0, a[15,2] = 0, a[14,3] = 0, a[13,3] = 0, a[12,4] = 0, a[15,3] \+ = 0, a[13,4] = 0, a[12,5] = 0, a[15,4] = 0, a[14,4] = 0, a[13,5] = 0, \+ a[15,5] = 0, a[14,5] = 0, b[10] = 892311079199814187055669708043437500 00000000000000000000/6999798709883356744455946798564450605625976935831 75985391, b[12] = 2084500442140450046401058474079675065083217679837038 3084226351294730731196673647311062330972740734737279503119387627146381 678677156136042524139311907482802844083/366708498911363730202382253282 6510025060514471850192630514096658675805484760468146633610316928475598 7753542321202462371554120593858149755539878561976786592389608, a[4,2] \+ = 0, c[11] = 1/4, a[10,9] = -3167799860072183913409/304236563598632812 50000, c[8] = 129/200, a[5,2] = 0, a[6,2] = 0, a[6,3] = 0, a[7,2] = 0, a[7,3] = 0, a[8,2] = 0, a[8,3] = 0, a[8,4] = 0, a[8,5] = 0, a[9,2] = \+ 0, a[9,3] = 0, a[9,4] = 0, a[9,5] = 0, a[10,2] = 0, a[10,3] = 0, a[10, 4] = 0, a[10,5] = 0, a[11,2] = 0, a[11,3] = 0, a[11,4] = 0, a[11,5] = \+ 0, a[11,6] = 1/30, c[15] = 1, b[2] = 0, b[3] = 0, b[4] = 0, b[5] = 0, \+ b[6] = 0, b[7] = 0, a[9,8] = -3483/102400, a[7,4] = -38208748178016484 817787/842517966262441068418750-86118788556282369822807/84251796626244 1068418750*6^(1/2), a[8,6] = 43/150+43/2400*6^(1/2), b[8] = -455655493 073428838813281446213740000000/116380801115091056124046422583731249786 9, a[9,6] = 22833/102400+8901/204800*6^(1/2), a[9,7] = 22833/102400-89 01/204800*6^(1/2), a[3,2] = 64021741529527/200485534764000-75044507634 11/100242767382000*6^(1/2), a[11,8] = 4389715333607/309890657317500+92 754382349/11477431752500*6^(1/2), a[10,6] = 187914666753956840195279/2 596152009375000000000000-210440846556290693268911/15576912056250000000 000000*6^(1/2), a[8,7] = 43/150-43/2400*6^(1/2), c[7] = 387/1000+129/2 000*6^(1/2), c[3] = 7630049/53810000-983539/53810000*6^(1/2), a[7,1] = 26523528363/231790900000+863255358/123138915625*6^(1/2), a[7,6] = -36 2925891/1690350537500+857800423623/3380701075000*6^(1/2), a[10,7] = 18 7914666753956840195279/2596152009375000000000000+210440846556290693268 911/15576912056250000000000000*6^(1/2), a[5,4] = 1873698362223295443/5 13126355505556250+528258592225556973/513126355505556250*6^(1/2), a[4,1 ] = 22890147/430480000-2950617/430480000*6^(1/2), c[4] = 22890147/1076 20000-2950617/107620000*6^(1/2), a[5,3] = -8712153884182794903/2052505 422022225000-2843421359195851533/2052505422022225000*6^(1/2), a[5,1] = 592203994261020339/513126355505556250+730386990293623641/205250542202 2225000*6^(1/2), a[11,7] = -2865012129681958/114898584332330625-129625 17687655099/229797168664661250*6^(1/2), c[6] = 387/1000-129/2000*6^(1/ 2), a[3,1] = -177968356965557/1002427673820000+14180534491313/25060691 8455000*6^(1/2), a[7,5] = 92362336407446913/290322814529044000-2320393 20950012997/2467743923496874000*6^(1/2), a[4,3] = 68670441/430480000-8 851851/430480000*6^(1/2), a[11,1] = -426968570497/54394415898750-92754 382349/12087647977500*6^(1/2), a[11,9] = 4990058173976/83757096376875+ 371017529396/9306344041875*6^(1/2), a[6,5] = 165912282616977/417907523 0308000-33181894472511/2089537615154000*6^(1/2), a[6,4] = 161932328870 91831/58864341808507450-2355345717024309/58864341808507450*6^(1/2), a[ 11,10] = 1099523524595993125000/6257667909869756018891+100957348037989 687500/6257667909869756018891*6^(1/2), a[9,1] = 7353/102400, b[9] = 19 965163648706008081135075746915614720000000/863944041905370868683946862 05782432516544599, a[10,1] = 376708742472214988700853/7788456028125000 000000000, a[6,1] = 11380823631/157617812000-339148869/39404453000*6^( 1/2), c[12] = 1427971650951258372/2166662646162554701, c[2] = 1731/500 00, c[10] = 6757/100000, c[5] = 561/1000, c[9] = 387/800, a[14,13] = 4 746930876023919335079451612726717649218264199984/185920655384070497552 00144388134089346432755594877, a[15,14] = 1480025020094032371712461617 5641261235119295795768814717803955078125/33565577125141877760287380588 632421223433194078156948298488471160489, a[2,1] = 1731/50000, a[12,2] \+ = 0, b[1] = 8198160366203173411119943711500331/56105757938408586016727 7847128765528, c[13] = 4103/5000, c[14] = 2253/2500, a[13,2] = 0, b[11 ] = 47104273954945906713184913871143492/209684639122339601934631113492 763467, a[14,10] = 574774300271998598683873114105472016699241495055292 /1049352151254569101542262489932969253892183788671875*6^(1/2)-27843764 4712626931893652011356206704903284753232828202194748516216938957695270 94334687108984/1225704106628516422200259430060559392943413919302216631 7802121412999357024704596261133984375, b[15] = 25854495576652689513716 99596493957/84574345160764140163208606048427531, b[14] = 9174011049209 93498360358406096725463867187500/6724249815911346653315790737453607382 989551463, a[14,11] = -34241134351848245624232809437676889009431930503 529853032576417589898516/561334782435865198110098500902428100760323006 2439942682713165283203125, a[12,8] = -55874764134953234138464916783230 4925076570507885572072105200355632180011316296456776552672453906332760 0257543743479921263738432/36530308936220166451641359692528616149447357 5337115296250511752859728108868696929614024803255122785403232359817965 288739565550625*6^(1/2)+5610987899273278525411960528081442902198567594 8097643797561956736732657005510768128839255833702537657025532355947644 27173637673766208/9288159819814403301827880474062633413542335679163959 8109358867770361609232846012626732332450844264293840456574956036349633 197336361875, a[14,6] = 871779321807802447463310035318238762878527157/ 134446420823349498059060773719650573730468750+107641268480999396081848 975271849857994818/1097082793918531904161935913552348681640625*6^(1/2) , a[12,7] = -136666607496463622270135608863772076443625468798139480390 4267409930248039469817632093483647161087213128226198457261516936675984 37699964416/3719286465342404274788585327254180828195282427342057650194 8556349178211135634328706813720435125204018871414370671061056839448023 32422369375+1698450855653613368055560092963943745276369523793889610260 6662872515552183276208687563236699647756792865753591219139615556676545 7826139904/15939799137181732606236794259660774977979781831465961357977 9527210763762009861408743487373293393731509448918731447404529311920099 9609586875*6^(1/2), a[13,7] = -157229999853748227305165773364426925282 378072238332930121/366999073679854585732732040943307160339634132385253 90625+5757606442802795095318986067317837904184278650664590252101/35231 91107326604023034227593055748739260487670898437500000*6^(1/2), a[14,7] = 496103786351862292800034805114190705484800743513354117014/110099722 103956375719819612282992148101890239715576171875-132993841260619748576 9312599390307351191540891599374831099/66059833262373825431891767369795 2888611341438293457031250*6^(1/2), a[12,6] = 4819272892477768171373308 6667206891214210919536257929702780440715499506401950564729555237698290 34800621890424847009130000000/2316289444700265097870243645576120415548 3718161502003927311842056928236720519934569124319610899284862776485022 935540644488821877*6^(1/2)-3338813117898494119715734728681282814382022 1072172312325174214536773458288757739554777822876017406875808613438995 2015563403904/22708720046081030371276898486040396230866390354413729340 5018059381649379612940534991414898146071420223298872773877849455772763 5, a[15,6] = -54821142119685055562477216205428613949905430396088/39594 39837009461289085587746748097947393101278095-1511276753825982856072891 469504471256664975925000/403862863374965051486729950168305990634096330 36569*6^(1/2), a[13,6] = -2745292391641202525373103979336813513372321/ 11702216468464340311060649744558385937500000*6^(1/2)-10271639002297503 56561238237947225332675621517/1792618944311326640787476982928674316406 25000, a[10,8] = -18552667221896744226647/865384003125000000000000, a[ 12,11] = 1234617126598879151772713393966068608104790287778693480148704 50606260914019560285661288212498128400476015695960341952/2816291066703 2067475424520935884070370423514730783889674107551122082605682904720561 4324978253226176275078922716132461, a[12,1] = 407885778185158609210793 8925175825953058964707564676126367962596114914082608964134468834508913 51622914818800693274034252252905536/2808492638860122607362409616917500 2956970191576455110633226765141161372294098693275117181239385312198137 508846535933127837167926875*6^(1/2)+1838203110479840386993853900915465 6587521498573595595063164077882800315372787284683238439478955141517997 198007108623761931447163756/139742569444997243449189609938909336141610 2532297045004793268899809500852862082123960473460811129176944470618749 7807869179550841329375, a[15,8] = -17632472271169431002565608550507266 1620440362221411708/28561940671982910748577120704204013346542014996455 5625*6^(1/2)-116118147575045169733222875835719955334334798191459879782 123534889390467935109772/881062690195483524567227513129587089250371395 7512170681453300814988417642493125, a[13,9] = -28140457973469923214145 5524604487724159024972527/28448191636473798322140232250483030319213867 1875*6^(1/2)+885774233856672590222951867695327816457340130391639153070 521335485617578/301098541380295011015469248465465290112505656143757799 934635162353515625, a[15,12] = 333514392451584382480734940567841440978 7291277341590453640072838769033456396839411470241410880750515810638511 6468732853458202899966748488718531545706559142895903144848764637/23166 1102532728742771480201132225288609079390498990062159236562764909757810 2163572190502232425490606773312310665593424982745744299371285598588298 606088543376742054644818966, a[14,12] = -34320443758939323781023685680 5228650103385091051699920208853270521163343279392054770280096153243800 8401883737341854688972639605334600163938610268855705742764072609/11431 7410634168226097164769041056729214392619865092777892082326746111137127 5907599801714870165813394147519068210931766844494994616580258435518181 434575195312500000, a[13,1] = 2814045797346992321414555246044877241590 24972527/1478009944832743180452316204077188415527343750000*6^(1/2)-560 42772675322042139227629978042586330633622706053363946766144416933631/5 8808540772323190525590122613223430507352118534557342666015625000000000 , a[14,8] = 123767075855296855875080343261298883871499029943/451091609 994276250390378731960660324096679687500*6^(1/2)+4077407727774763635459 8451708891165494123131383777235229538611989392175193285994266471/15264 2905462481621010589859415880795182567412553770317363579461257135247039 79492187500, a[12,9] = -6526172450962537747372702280281321524894343532 1034818021887401537838625321743426151501352142616259666371008110923845 48036046488576/8649093284303728183602838792132050266857965317662489228 4566487468170341285762869374265713247057712228954184044334206372230816 544375*6^(1/2)+5459853981808361523356614860220324489669695891073433975 4065270985433507945162707737759469214674480807272210648148477499238783 276259328/301247919092298852634886875129959310794662932014184499827145 0758516372986983120740305674792395020116934474230264160407944799340240 58125, a[15,9] = -1974836894370976272287348157656813810148932056879811 1296/6484554262322259071286545935997129135111813687175650625*6^(1/2)+1 7769448722513898342276837490665097286927607247073335618566987143467294 900183033216/255121700813788961505634214608456186712248516359661928371 9957742418751029506356875, a[13,8] = -84421373920409769642436657381346 3172477074917581/4210188359946578336976868164966163024902343750000*6^( 1/2)-93114481685939341460159650199040136021338029433258183466227812859 07057/4255970849010124217193135449668739985401313363005576159362792968 750000, a[14,1] = 4734200384802439149870797684768889301308307444115977 9465719863625051668939887702630319/44802546873926050730401222636656855 760802419993852060264615320801485392456054687500000-866369530987077991 125562402829092187100493209601/332552237587367215601771145917367393493 6523437500*6^(1/2), a[15,11] = 473389749049752963256114649231353822492 912259509649519870869750525/354124408823603417997988424283654229412165 08121322622479260846291, a[13,10] = -294039645364787227664606877659229 2229737651937934623/73454650587819837107958374295307847772452865207031 25*6^(1/2)+31547911672978015395641212405219968509774423938663902378735 9107959254802182/13448185050650584801258784221551557438021254320089493 2329128471154748828125, b[13] = 60530372821423065097959112869091796875 00000000/103899257350518063455290077573775162739725126989, a[8,1] = 43 /600, a[13,11] = 22509961634065453786165320390188465862176315994538225 41/382491303797095993563304148204275636433504028320312500, a[15,1] = 1 234273058981860170179592598535508631343082535549881956/210563377146962 8744518390642968552144069898845895808125*6^(1/2)-251883292492588254437 48527038142409879923012133738985313265430932280250855708601/1137064132 5574469312056961874077298550827642308774647316995717036347558064286250 , a[12,10] = -81088251450850881043447210481663252251737294956893646964 2672016111201241422775232896972065898731565417987376035772523573400039 9440000000/26555829666106402106127410275679775481057208152954669252204 0189640618206693081506133206500662983001047298787619827411375675716583 283801757*6^(1/2)+9391667348404584010955422210328707125006120661611061 9088897508056194187858209480024558903609392219121905247310870706451074 86913457760000000/5815726696877302061241902850373870830351528585497072 5662326801531295387265784849843172223645193277229358434488742203091272 981931739152584783, a[13,12] = 268934095730769185329490238833445400395 9378146957529866233529251986359392336044151708949720958809747970514366 293458424272174024493/959516386019578808500569114780871708466894752280 4828351054080278151948953190554438427822271021204939608056495755617968 75000000000, a[15,13] = -387149926569584133897432527260168975992839116 82945255636643554687500000/4854049492697158749929458938257221203616913 5429877901702347521300421767, a[14,9] = 346547812394831196450224961131 6368748401972838404/2560337247282641848992620902543472728729248046875* 6^(1/2)-10522038608500556459828649038302068473735749030796372764961618 751973793724796364606986664/389941742500542225403457400039738286223589 2829653375835197340918271556055507659912109375, a[15,10] = -8529708461 1782122474911131363078900058888025224607913745000000/69210659450201393 843166746722954036326338355649915383851733911*6^(1/2)+9765926613912407 4818193264801929547781659926543786381510190954184218570746215033823993 530000000/185600766544697062059634829087870568508123082056031273268553 60961727608242796551101182080033599, a[15,7] = -6092242427406159991860 3524049390657305431262635197540405697952/64848617474890321697745846247 59953148531564032417461909516875+8455857575163597873310996189398423878 6929550462615375699341616/19454585242467096509323753874279859445594692 097252385728550625*6^(1/2)\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "We can obtain a n embedded 14 stage order 8 scheme as follows." }}{PARA 0 "" 0 "" {TEXT -1 94 "We remove stages 14 and 15 from the 15 stage order 9 sche me and introduce a new stage 14 with:" }}{PARA 256 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "c[14] = 1;" "6#/&%\"cG6#\"#9\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[14,2] = 0;" "6#/&%\"aG6$\"#9\"\"#\"\"!" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "a[14,3] = 0;" "6#/&%\"aG6$\"#9\"\"$\" \"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[14,4]=0" "6#/&%\"aG6$\"#9\"\" %\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[14,5]=0" "6#/&%\"aG6$\"#9 \"\"&\"\"!" }{TEXT -1 1 "," }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "`b*`[2]=0" "6#/&%#b*G6#\"\"#\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "`b*`[3]=0" "6#/&%#b*G6#\"\"$\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "`b*`[4]=0" "6#/&%#b*G6#\"\"%\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "`b*`[5]=0" "6#/&%#b*G6#\"\"&\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "`b*`[6]=0" "6#/&%#b*G6#\"\"'\"\"!" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "`b*`[7] = 0;" "6#/&%#b*G6#\"\"(\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 56 "where the weights of the order 8 scheme are denoted by " }{XPPEDIT 18 0 "`b*`" "6#%#b*G" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 156 "We incor porate the order 8 quadrature conditions, the row sum conditions for t his stage and stage-order conditions so that this new stage has stage- order 4." }}{PARA 0 "" 0 "" {TEXT -1 48 "We also incorporate the simpl ifying conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(`b*`[i]*a[i,j],i = j+1 .. 14) = `b*`[j]*(1-c[j]);" "6#/-%$SumG6 $*&&%#b*G6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,\"#9*&&F)6#F0F,,& F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 5 "f or " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 10 " ( with " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\"\"!" }{TEXT -1 5 " ), " } {XPPEDIT 18 0 "j = 9;" "6#/%\"jG\"\"*" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j = 11;" "6#/%\"jG\"#6" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "j = 1 3;" "6#/%\"jG\"#8" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 451 "`Qeqs*` := subs(b=`b*`,Quad ratureConditions(8,14,'expanded')):\nSO_eqs2 := [add(a[14,j],j=1..13)= c[14],add(a[14,j]*c[j],j=2..13)=1/2*c[14]^2,\n add(a[14,j]*c[j] ^2,j=2..13)=1/3*c[14]^3,add(a[14,j]*c[j]^3,j=2..13)=1/4*c[14]^4,\n \+ add(a[14,j]*c[j]^4,j=2..13)=1/5*c[14]^5]:\n`simp_eqs*` := [add(`b* `[i]*a[i,1],i=2..14)=`b*`[1],seq(add(`b*`[i]*a[i,j],i=j+1..14)=`b*`[j] *(1-c[j]),j=[9,11,13])]:\n`cdns*` := [op(`simp_eqs*`),op(SO_eqs2),op(` Qeqs*`)]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 146 "e14 := `union`(remove(u_->member(op(1,lhs(u_)),[14 ,15]) or op(0,lhs(u_))=b,e13),\n \{c[14]=1,seq(a[14,i]=0,i=2.. 5),seq(`b*`[i]=0,i=2..7)\}):\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 53 "We have 17 equations for the 17 unknown c oefficients." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "eqns3 := sub s(e14,`cdns*`):\nnops(%);\nindets(eqns3);\nnops(%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#\"#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<3&%#b*G6#\" \"\"&%\"aG6$\"#9F'&F)6$F+\"\"*&F)6$F+\"#5&F)6$F+\"\"(&F)6$F+\"\")&F)6$ F+\"\"'&F)6$F+\"#8&F)6$F+\"#7&F)6$F+\"#6&F%6#F+&F%6#F=&F%6#F@&F%6#FC&F %6#F1&F%6#F.&F%6#F7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#<" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "in folevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "e15 := solve(\{op(eqns3)\}):\ninfolevel[solve] := 0:\ne16 := `union`(e14, e15):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e16" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11736 "e16 := \{a[14,2] = 0, a[12,3] = 0, a[14,3] = 0, a[13,3] = 0, a[ 12,4] = 0, a[13,4] = 0, a[12,5] = 0, a[14,4] = 0, a[13,5] = 0, a[14,5] = 0, a[4,2] = 0, c[11] = 1/4, a[10,9] = -3167799860072183913409/30423 656359863281250000, c[8] = 129/200, a[5,2] = 0, a[6,2] = 0, a[6,3] = 0 , a[7,2] = 0, a[7,3] = 0, a[8,2] = 0, a[8,3] = 0, a[8,4] = 0, a[8,5] = 0, a[9,2] = 0, a[9,3] = 0, a[9,4] = 0, a[9,5] = 0, a[10,2] = 0, a[10, 3] = 0, a[10,4] = 0, a[10,5] = 0, a[11,2] = 0, a[11,3] = 0, a[11,4] = \+ 0, a[11,5] = 0, a[11,6] = 1/30, a[14,12] = -98157171295691069885693021 9322099934382493208458209364759608693175466609866259415309525898851630 5165794739744873539829069617203523509136682216933020431/28647699117093 4153076146641094402171801937250068596542931028678669501762253287693294 397689327797388113854588113430063939405071979092547998950955940992, a[ 14,9] = -2775244732780109667342845612394739319115662636371477300455747 022423270475907256/228417153675584029725018045422706955827996328208181 619436454383447149337555625+341618493222358850528663620935896732550431 596562876/96101338378773357469245211954911505447551097205625*6^(1/2), \+ a[9,8] = -3483/102400, a[7,4] = -38208748178016484817787/8425179662624 41068418750-86118788556282369822807/842517966262441068418750*6^(1/2), \+ a[8,6] = 43/150+43/2400*6^(1/2), a[9,6] = 22833/102400+8901/204800*6^( 1/2), a[9,7] = 22833/102400-8901/204800*6^(1/2), a[3,2] = 640217415295 27/200485534764000-7504450763411/100242767382000*6^(1/2), a[11,8] = 43 89715333607/309890657317500+92754382349/11477431752500*6^(1/2), a[10,6 ] = 187914666753956840195279/2596152009375000000000000-210440846556290 693268911/15576912056250000000000000*6^(1/2), a[8,7] = 43/150-43/2400* 6^(1/2), c[7] = 387/1000+129/2000*6^(1/2), c[3] = 7630049/53810000-983 539/53810000*6^(1/2), a[7,1] = 26523528363/231790900000+863255358/1231 38915625*6^(1/2), a[7,6] = -362925891/1690350537500+857800423623/33807 01075000*6^(1/2), a[10,7] = 187914666753956840195279/25961520093750000 00000000+210440846556290693268911/15576912056250000000000000*6^(1/2), \+ a[5,4] = 1873698362223295443/513126355505556250+528258592225556973/513 126355505556250*6^(1/2), a[4,1] = 22890147/430480000-2950617/430480000 *6^(1/2), c[4] = 22890147/107620000-2950617/107620000*6^(1/2), a[5,3] \+ = -8712153884182794903/2052505422022225000-2843421359195851533/2052505 422022225000*6^(1/2), a[5,1] = 592203994261020339/513126355505556250+7 30386990293623641/2052505422022225000*6^(1/2), a[11,7] = -286501212968 1958/114898584332330625-12962517687655099/229797168664661250*6^(1/2), \+ c[6] = 387/1000-129/2000*6^(1/2), a[3,1] = -177968356965557/1002427673 820000+14180534491313/250606918455000*6^(1/2), a[7,5] = 92362336407446 913/290322814529044000-232039320950012997/2467743923496874000*6^(1/2), a[4,3] = 68670441/430480000-8851851/430480000*6^(1/2), a[11,1] = -426 968570497/54394415898750-92754382349/12087647977500*6^(1/2), a[11,9] = 4990058173976/83757096376875+371017529396/9306344041875*6^(1/2), a[6, 5] = 165912282616977/4179075230308000-33181894472511/2089537615154000* 6^(1/2), a[6,4] = 16193232887091831/58864341808507450-2355345717024309 /58864341808507450*6^(1/2), a[11,10] = 1099523524595993125000/62576679 09869756018891+100957348037989687500/6257667909869756018891*6^(1/2), a [9,1] = 7353/102400, a[14,8] = 179490914212648256439084852292422555322 1469019751470544959297614654661293377/52596481193994264435601626109752 988674679691644275456716633975785978672500+122006604722271018045951293 19139169019658271305817/1693156145655995911520770934405657826339776060 2500*6^(1/2), `b*`[1] = 552562031208180939317806684253/276696542577346 67858523344041464, a[14,10] = -276805546597690166235309791767274482512 92244310769996015342190819068970556083063125000/3299557777429648960576 561382256606844677258438797072955341581354051375036522231471437+442655 2127579895373479670356100179759944766558141730312500/30771137386673207 07748877199804636746494977000658967987677*6^(1/2), a[10,1] = 376708742 472214988700853/7788456028125000000000000, a[6,1] = 11380823631/157617 812000-339148869/39404453000*6^(1/2), c[12] = 1427971650951258372/2166 662646162554701, `b*`[10] = 131372030907763001445323984375000000000000 0000000/11518201923215510989126466531107437037395719117133, `b*`[11] = 4833611232701440504508086151728/19081321241454145230196661524503, c[2 ] = 1731/50000, c[10] = 6757/100000, `b*`[14] = 1839190071060649887127 895100784/38045139523510634351420875415397, c[5] = 561/1000, c[9] = 38 7/800, a[2,1] = 1731/50000, a[12,2] = 0, c[13] = 4103/5000, a[13,2] = \+ 0, `b*`[8] = 221223388631423597589898601690000000/10094613679858709005 4685074667127461, a[14,11] = -2926031719297062910539294021599303307366 39136252680853622275/1547362282627916115022707688729026244351055096427 5858143964, a[12,8] = -55874764134953234138464916783230492507657050788 5572072105200355632180011316296456776552672453906332760025754374347992 1263738432/36530308936220166451641359692528616149447357533711529625051 1752859728108868696929614024803255122785403232359817965288739565550625 *6^(1/2)+5610987899273278525411960528081442902198567594809764379756195 6736732657005510768128839255833702537657025532355947644271736376737662 08/9288159819814403301827880474062633413542335679163959810935886777036 1609232846012626732332450844264293840456574956036349633197336361875, a [12,7] = -136666607496463622270135608863772076443625468798139480390426 7409930248039469817632093483647161087213128226198457261516936675984376 99964416/3719286465342404274788585327254180828195282427342057650194855 6349178211135634328706813720435125204018871414370671061056839448023324 22369375+1698450855653613368055560092963943745276369523793889610260666 2872515552183276208687563236699647756792865753591219139615556676545782 6139904/15939799137181732606236794259660774977979781831465961357977952 7210763762009861408743487373293393731509448918731447404529311920099960 9586875*6^(1/2), a[13,7] = -157229999853748227305165773364426925282378 072238332930121/366999073679854585732732040943307160339634132385253906 25+5757606442802795095318986067317837904184278650664590252101/35231911 07326604023034227593055748739260487670898437500000*6^(1/2), a[12,6] = \+ 4819272892477768171373308666720689121421091953625792970278044071549950 640195056472955523769829034800621890424847009130000000/231628944470026 5097870243645576120415548371816150200392731184205692823672051993456912 4319610899284862776485022935540644488821877*6^(1/2)-333881311789849411 9715734728681282814382022107217231232517421453677345828875773955477782 28760174068758086134389952015563403904/2270872004608103037127689848604 0396230866390354413729340501805938164937961294053499141489814607142022 32988727738778494557727635, a[13,6] = -2745292391641202525373103979336 813513372321/11702216468464340311060649744558385937500000*6^(1/2)-1027 163900229750356561238237947225332675621517/179261894431132664078747698 292867431640625000, c[14] = 1, `b*`[7] = 0, `b*`[6] = 0, `b*`[5] = 0, \+ `b*`[4] = 0, `b*`[3] = 0, `b*`[2] = 0, a[10,8] = -18552667221896744226 647/865384003125000000000000, a[14,1] = 230578569608639756108085818693 9897173645641331085041313944389849986584101287/61750824434528226581908 7370078275122671246164669900462139876057008239440000-85404623305589712 632165905233974183137607899140719/124822287169084833758410283469525117 460541643292500*6^(1/2), a[12,11] = 1234617126598879151772713393966068 6081047902877786934801487045060626091401956028566128821249812840047601 5695960341952/28162910667032067475424520935884070370423514730783889674 1075511220826056829047205614324978253226176275078922716132461, a[12,1] = 4078857781851586092107938925175825953058964707564676126367962596114 91408260896413446883450891351622914818800693274034252252905536/2808492 6388601226073624096169175002956970191576455110633226765141161372294098 693275117181239385312198137508846535933127837167926875*6^(1/2)+1838203 1104798403869938539009154656587521498573595595063164077882800315372787 284683238439478955141517997198007108623761931447163756/139742569444997 2434491896099389093361416102532297045004793268899809500852862082123960 4734608111291769444706187497807869179550841329375, a[14,13] = 27294911 44709837905799148766650782532906050298971406518524169921875/2158115888 622139473142775812109447802920656149243127309253686951469, a[14,6] = 1 02903996961580448264190625267026062654799259083/5046398084890004857481 629999673320438819484730+413209254873042193133002720521283745670811281 25/51473260465878049546312625996667868475958744246*6^(1/2), a[14,7] = \+ 62798443349876457506718920843975661399949564598018488144466/4132553498 782573324058263582553715220777051359780141380625-723088070819329615544 25711089716771013571419950657300729103/1239766049634771997217479074766 1145662331154079340424141875*6^(1/2), a[13,9] = -281404579734699232141 455524604487724159024972527/284481916364737983221402322504830303192138 671875*6^(1/2)+8857742338566725902229518676953278164573401303916391530 70521335485617578/3010985413802950110154692484654652901125056561437577 99934635162353515625, `b*`[13] = 1084761591753640855844358063964843750 000000/3182895486031249071938549691320502488733423, a[13,1] = 28140457 9734699232141455524604487724159024972527/14780099448327431804523162040 77188415527343750000*6^(1/2)-56042772675322042139227629978042586330633 622706053363946766144416933631/588085407723231905255901226132234305073 52118534557342666015625000000000, a[12,9] = -6526172450962537747372702 2802813215248943435321034818021887401537838625321743426151501352142616 25966637100811092384548036046488576/8649093284303728183602838792132050 2668579653176624892284566487468170341285762869374265713247057712228954 184044334206372230816544375*6^(1/2)+5459853981808361523356614860220324 4896696958910734339754065270985433507945162707737759469214674480807272 210648148477499238783276259328/301247919092298852634886875129959310794 6629320141844998271450758516372986983120740305674792395020116934474230 26416040794479934024058125, a[13,8] = -8442137392040976964243665738134 63172477074917581/4210188359946578336976868164966163024902343750000*6^ (1/2)-9311448168593934146015965019904013602133802943325818346622781285 907057/425597084901012421719313544966873998540131336300557615936279296 8750000, a[13,10] = -2940396453647872276646068776592292229737651937934 623/7345465058781983710795837429530784777245286520703125*6^(1/2)+31547 9116729780153956412124052199685097744239386639023787359107959254802182 /134481850506505848012587842215515574380212543200894932329128471154748 828125, a[8,1] = 43/600, `b*`[12] = -212966237458232464810691979570337 3645353118273066742230724172731025813964712473647144010599206669825382 719359113196238857709025512340589957/103554373927236708088519054620109 7218891268728118207332592595987554851882972292670881794178380097716583 123063485287435793657425889233080568, a[13,11] = 225099616340654537861 6532039018846586217631599453822541/38249130379709599356330414820427563 6433504028320312500, a[12,10] = -8108825145085088104344721048166325225 1737294956893646964267201611120124142277523289697206589873156541798737 60357725235734000399440000000/2655582966610640210612741027567977548105 7208152954669252204018964061820669308150613320650066298300104729878761 9827411375675716583283801757*6^(1/2)+939166734840458401095542221032870 7125006120661611061908889750805619418785820948002455890360939221912190 524731087070645107486913457760000000/581572669687730206124190285037387 0830351528585497072566232680153129538726578484984317222364519327722935 8434488742203091272981931739152584783, a[13,12] = 26893409573076918532 9490238833445400395937814695752986623352925198635939233604415170894972 0958809747970514366293458424272174024493/95951638601957880850056911478 0871708466894752280482835105408027815194895319055443842782227102120493 960805649575561796875000000000, `b*`[9] = 1018354087913052979846578125 61920000000/1149763833200743759976506650241312100139\}:" }{TEXT -1 0 " " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "p rinted coefficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 289 "subs(e16,matrix([seq([c[i],seq(a[i,j],j=1. .i-1),``$(4-i)],i=2..4)]));\nfor ii from 5 to 14 do\n print(``);\n \+ print(c[ii]=subs(e16,c[ii])); \n for jj to ii-1 do\n print(a[i i,jj]=subs(e16,a[ii,jj]));\n end do:\nend do:\n``;\nfor ii to 14 do \n print(`b*`[ii]=subs(e16,`b*`[ii]));\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7%7&#\"%J<\"&++&F(%!GF+7&,&#\"(\\+j(\")++\" Q&\"\"\"*(\"'RN)*F1F0!\"\"\"\"'#F1\"\"#F4,&#\"0db'pNoz<\"1++#QnFC+\"F4 *(\"/88\\M0=9F1\"0+]X=pg]#F4F5F6F1,&#\"/F&H:u@S'\"0+SwMb[+#F1*(\".6Mw] W](F1\"0+?QnFC+\"F4F5F6F4F+7&,&#\")Z,*G#\"*++i2\"F1*(\"(<1&HF1FJF4F5F6 F4,&#FI\"*++[I%F1*(FLF1FOF4F5F6F4\"\"!,&#\")T/noFOF1*(\"(^=&))F1FOF4F5 F6F4Q)pprint156\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"&#\"$h&\"%+5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"\",&#\"3R.-hU*R?#f\"3]ib0bNEJ^F(*(\"3 TOi$H!*pQI(F(\"4+]AA?U0D0#!\"\"\"\"'#F(\"\"#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"$,&#\"4.\\z#=%)Q:7()\"4+]AA?U0D0#!\"\"*(\"4L:& e>f8UVG\"\"\"F,F-\"\"'#F0\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/& %\"aG6$\"\"&\"\"%,&#\"4VaHBAO)pt=\"3]ib0bNEJ^\"\"\"*(\"3tpbDAfe#G&F-F, !\"\"\"\"'#F-\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"',&#\"$(Q\"%+5\"\"\"*(\"$H\"F, \"%+?!\"\"F'#F,\"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\" \"'\"\"\",&#\",JO#3Q6\"-+?\"yhd\"F(*(\"*p)[\"R$F(\",+IX/%R!\"\"F'#F(\" \"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"#\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"$\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"%,&#\"2J=4()GB$>;\"2]u]3=Mk)e \"\"\"*(\"14V-So&RvmY\"z =\":++++++v$4?:'f#\"\"\"*(\"96*oKp!Hcl%3W5#F-\";++++++]i07pd:!\"\"F(#F -\"\"#F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"(,&#\"9z_ >So&RvmY\"z=\":++++++v$4?:'f#\"\"\"*(\"96*oKp!Hcl%3W5#F-\";++++++]i07p 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" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 185 "`errterms8_14*` := subs(b=`b*`,Pri ncipalErrorTerms(8,14,'expanded')):\nsm := 0:\nfor ct to nops(`errterm s8_14*`) do\n sm := sm+(evalf(subs(e16,`errterms8_14*`[ct])))^2;\nen d do:\nsqrt(sm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+kg5YU!#:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 138 "We can i nclude the new stage for the embedded scheme as an additional 16th sta ge added to the order 9 scheme along with the coefficients " } {XPPEDIT 18 0 "a[16,14] = 0;" "6#/&%\"aG6$\"#;\"#9\"\"!" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[16,15] = 0;" "6#/&%\"aG6$\"#;\"#:\"\"!" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 13 "The weights " } {XPPEDIT 18 0 "`b*`[i]" "6#&%#b*G6#%\"iG" }{TEXT -1 7 " for " } {XPPEDIT 18 0 "i=1" "6#/%\"iG\"\"\"" }{TEXT -1 89 " . . 13 of the 16 s tage combined scheme are those of the 14 stage scheme and the weight \+ " }{XPPEDIT 18 0 "`b*`[14];" "6#&%#b*G6#\"#9" }{TEXT -1 34 " in the 1 4 stage scheme becomes " }{XPPEDIT 18 0 "`b*`[16];" "6#&%#b*G6#\"#;" }{TEXT -1 25 " in the 16 stage scheme." }}{PARA 0 "" 0 "" {TEXT -1 8 "We set " }{XPPEDIT 18 0 "c[16] = 1;" "6#/&%\"cG6#\"#;\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[14] = 0;" "6#/&%#b*G6#\"#9\"\"!" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "`b*`[15] = 0;" "6#/&%#b*G6#\"#:\" \"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 66 "We can make the or der 9 scheme into a 16 stage scheme by setting " }{XPPEDIT 18 0 "b[16 ] = 0;" "6#/&%\"bG6#\"#;\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 201 "e17 := \{c[ 16]=1,seq(a[16,i]=subs(e16,a[14,i]),i=1..13),a[16,14]=0,a[16,15]=0,b[1 6]=0,\nseq(`b*`[i]=subs(e16,`b*`[i]),i=1..13),`b*`[14]=0,`b*`[15]=0,`b *`[16]=subs(e16,`b*`[14])\}:\ne18 := `union`(e13,e17):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e18" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17871 "e18 := \{a[14,2] = 0 , a[12,3] = 0, a[15,2] = 0, a[14,3] = 0, a[13,3] = 0, a[12,4] = 0, a[1 5,3] = 0, a[13,4] = 0, a[12,5] = 0, a[15,4] = 0, a[14,4] = 0, a[13,5] \+ = 0, a[15,5] = 0, a[14,5] = 0, b[10] = 8923110791998141870556697080434 3750000000000000000000000/69997987098833567444559467985644506056259769 3583175985391, b[12] = 20845004421404500464010584740796750650832176798 3703830842263512947307311966736473110623309727407347372795031193876271 46381678677156136042524139311907482802844083/3667084989113637302023822 5328265100250605144718501926305140966586758054847604681466336103169284 755987753542321202462371554120593858149755539878561976786592389608, c[ 16] = 1, a[4,2] = 0, a[16,14] = 0, a[16,15] = 0, c[11] = 1/4, b[16] = \+ 0, a[10,9] = -3167799860072183913409/30423656359863281250000, c[8] = 1 29/200, a[5,2] = 0, a[6,2] = 0, a[6,3] = 0, a[7,2] = 0, a[7,3] = 0, a[ 8,2] = 0, a[8,3] = 0, a[8,4] = 0, a[8,5] = 0, a[9,2] = 0, a[9,3] = 0, \+ a[9,4] = 0, a[9,5] = 0, a[10,2] = 0, a[10,3] = 0, a[10,4] = 0, a[10,5] = 0, a[11,2] = 0, a[11,3] = 0, a[11,4] = 0, a[11,5] = 0, a[11,6] = 1/ 30, `b*`[15] = 0, `b*`[14] = 0, a[16,5] = 0, a[16,4] = 0, a[16,3] = 0, a[16,2] = 0, c[15] = 1, b[2] = 0, b[3] = 0, b[4] = 0, b[5] = 0, b[6] \+ = 0, b[7] = 0, a[9,8] = -3483/102400, a[7,4] = -3820874817801648481778 7/842517966262441068418750-86118788556282369822807/8425179662624410684 18750*6^(1/2), a[8,6] = 43/150+43/2400*6^(1/2), b[8] = -45565549307342 8838813281446213740000000/1163808011150910561240464225837312497869, a[ 9,6] = 22833/102400+8901/204800*6^(1/2), a[9,7] = 22833/102400-8901/20 4800*6^(1/2), a[3,2] = 64021741529527/200485534764000-7504450763411/10 0242767382000*6^(1/2), a[11,8] = 4389715333607/309890657317500+9275438 2349/11477431752500*6^(1/2), a[10,6] = 187914666753956840195279/259615 2009375000000000000-210440846556290693268911/1557691205625000000000000 0*6^(1/2), a[8,7] = 43/150-43/2400*6^(1/2), c[7] = 387/1000+129/2000*6 ^(1/2), c[3] = 7630049/53810000-983539/53810000*6^(1/2), a[7,1] = 2652 3528363/231790900000+863255358/123138915625*6^(1/2), a[7,6] = -3629258 91/1690350537500+857800423623/3380701075000*6^(1/2), a[10,7] = 1879146 66753956840195279/2596152009375000000000000+210440846556290693268911/1 5576912056250000000000000*6^(1/2), a[5,4] = 1873698362223295443/513126 355505556250+528258592225556973/513126355505556250*6^(1/2), a[4,1] = 2 2890147/430480000-2950617/430480000*6^(1/2), c[4] = 22890147/107620000 -2950617/107620000*6^(1/2), a[5,3] = -8712153884182794903/205250542202 2225000-2843421359195851533/2052505422022225000*6^(1/2), a[5,1] = 5922 03994261020339/513126355505556250+730386990293623641/20525054220222250 00*6^(1/2), a[11,7] = -2865012129681958/114898584332330625-12962517687 655099/229797168664661250*6^(1/2), c[6] = 387/1000-129/2000*6^(1/2), a [3,1] = -177968356965557/1002427673820000+14180534491313/2506069184550 00*6^(1/2), a[7,5] = 92362336407446913/290322814529044000-232039320950 012997/2467743923496874000*6^(1/2), a[4,3] = 68670441/430480000-885185 1/430480000*6^(1/2), a[11,1] = -426968570497/54394415898750-9275438234 9/12087647977500*6^(1/2), a[11,9] = 4990058173976/83757096376875+37101 7529396/9306344041875*6^(1/2), a[6,5] = 165912282616977/41790752303080 00-33181894472511/2089537615154000*6^(1/2), a[6,4] = 16193232887091831 /58864341808507450-2355345717024309/58864341808507450*6^(1/2), a[11,10 ] = 1099523524595993125000/6257667909869756018891+10095734803798968750 0/6257667909869756018891*6^(1/2), a[9,1] = 7353/102400, b[9] = 1996516 3648706008081135075746915614720000000/86394404190537086868394686205782 432516544599, `b*`[1] = 552562031208180939317806684253/276696542577346 67858523344041464, a[10,1] = 376708742472214988700853/7788456028125000 000000000, a[6,1] = 11380823631/157617812000-339148869/39404453000*6^( 1/2), c[12] = 1427971650951258372/2166662646162554701, `b*`[10] = 1313 720309077630014453239843750000000000000000000/115182019232155109891264 66531107437037395719117133, `b*`[11] = 4833611232701440504508086151728 /19081321241454145230196661524503, c[2] = 1731/50000, c[10] = 6757/100 000, c[5] = 561/1000, c[9] = 387/800, a[14,13] = 474693087602391933507 9451612726717649218264199984/18592065538407049755200144388134089346432 755594877, a[15,14] = 148002502009403237171246161756412612351192957957 68814717803955078125/3356557712514187776028738058863242122343319407815 6948298488471160489, a[2,1] = 1731/50000, a[12,2] = 0, b[1] = 81981603 66203173411119943711500331/561057579384085860167277847128765528, c[13] = 4103/5000, c[14] = 2253/2500, a[13,2] = 0, b[11] = 4710427395494590 6713184913871143492/209684639122339601934631113492763467, a[16,1] = 23 0578569608639756108085818693989717364564133108504131394438984998658410 1287/61750824434528226581908737007827512267124616466990046213987605700 8239440000-85404623305589712632165905233974183137607899140719/12482228 7169084833758410283469525117460541643292500*6^(1/2), a[14,10] = 574774 300271998598683873114105472016699241495055292/104935215125456910154226 2489932969253892183788671875*6^(1/2)-278437644712626931893652011356206 70490328475323282820219474851621693895769527094334687108984/1225704106 6285164222002594300605593929434139193022166317802121412999357024704596 261133984375, b[15] = 2585449557665268951371699596493957/8457434516076 4140163208606048427531, b[14] = 91740110492099349836035840609672546386 7187500/6724249815911346653315790737453607382989551463, `b*`[8] = 2212 23388631423597589898601690000000/100946136798587090054685074667127461, a[14,11] = -342411343518482456242328094376768890094319305035298530325 76417589898516/5613347824358651981100985009024281007603230062439942682 713165283203125, a[16,6] = 1029039969615804482641906252670260626547992 59083/5046398084890004857481629999673320438819484730+41320925487304219 313300272052128374567081128125/514732604658780495463126259966678684759 58744246*6^(1/2), a[12,8] = -55874764134953234138464916783230492507657 0507885572072105200355632180011316296456776552672453906332760025754374 3479921263738432/36530308936220166451641359692528616149447357533711529 6250511752859728108868696929614024803255122785403232359817965288739565 550625*6^(1/2)+5610987899273278525411960528081442902198567594809764379 7561956736732657005510768128839255833702537657025532355947644271736376 73766208/9288159819814403301827880474062633413542335679163959810935886 7770361609232846012626732332450844264293840456574956036349633197336361 875, a[14,6] = 871779321807802447463310035318238762878527157/134446420 823349498059060773719650573730468750+107641268480999396081848975271849 857994818/1097082793918531904161935913552348681640625*6^(1/2), a[12,7] = -136666607496463622270135608863772076443625468798139480390426740993 0248039469817632093483647161087213128226198457261516936675984376999644 16/3719286465342404274788585327254180828195282427342057650194855634917 8211135634328706813720435125204018871414370671061056839448023324223693 75+1698450855653613368055560092963943745276369523793889610260666287251 5552183276208687563236699647756792865753591219139615556676545782613990 4/15939799137181732606236794259660774977979781831465961357977952721076 3762009861408743487373293393731509448918731447404529311920099960958687 5*6^(1/2), a[13,7] = -157229999853748227305165773364426925282378072238 332930121/36699907367985458573273204094330716033963413238525390625+575 7606442802795095318986067317837904184278650664590252101/35231911073266 04023034227593055748739260487670898437500000*6^(1/2), a[14,7] = 496103 786351862292800034805114190705484800743513354117014/110099722103956375 719819612282992148101890239715576171875-132993841260619748576931259939 0307351191540891599374831099/66059833262373825431891767369795288861134 1438293457031250*6^(1/2), a[12,6] = 4819272892477768171373308666720689 1214210919536257929702780440715499506401950564729555237698290348006218 90424847009130000000/2316289444700265097870243645576120415548371816150 2003927311842056928236720519934569124319610899284862776485022935540644 488821877*6^(1/2)-3338813117898494119715734728681282814382022107217231 2325174214536773458288757739554777822876017406875808613438995201556340 3904/22708720046081030371276898486040396230866390354413729340501805938 16493796129405349914148981460714202232988727738778494557727635, a[15,6 ] = -54821142119685055562477216205428613949905430396088/39594398370094 61289085587746748097947393101278095-1511276753825982856072891469504471 256664975925000/40386286337496505148672995016830599063409633036569*6^( 1/2), a[13,6] = -2745292391641202525373103979336813513372321/117022164 68464340311060649744558385937500000*6^(1/2)-10271639002297503565612382 37947225332675621517/179261894431132664078747698292867431640625000, `b *`[7] = 0, `b*`[6] = 0, `b*`[5] = 0, `b*`[4] = 0, `b*`[3] = 0, `b*`[2] = 0, a[16,7] = 627984433498764575067189208439756613999495645980184881 44466/4132553498782573324058263582553715220777051359780141380625-72308 807081932961554425711089716771013571419950657300729103/123976604963477 19972174790747661145662331154079340424141875*6^(1/2), a[16,8] = 179490 9142126482564390848522924225553221469019751470544959297614654661293377 /525964811939942644356016261097529886746796916442754567166339757859786 72500+12200660472227101804595129319139169019658271305817/1693156145655 9959115207709344056578263397760602500*6^(1/2), a[16,9] = -277524473278 0109667342845612394739319115662636371477300455747022423270475907256/22 8417153675584029725018045422706955827996328208181619436454383447149337 555625+341618493222358850528663620935896732550431596562876/96101338378 773357469245211954911505447551097205625*6^(1/2), a[16,10] = -276805546 5976901662353097917672744825129224431076999601534219081906897055608306 3125000/32995577774296489605765613822566068446772584387970729553415813 54051375036522231471437+4426552127579895373479670356100179759944766558 141730312500/307711373866732070774887719980463674649497700065896798767 7*6^(1/2), a[16,11] = -29260317192970629105392940215993033073663913625 2680853622275/15473622826279161150227076887290262443510550964275858143 964, a[16,12] = -98157171295691069885693021932209993438249320845820936 4759608693175466609866259415309525898851630516579473974487353982906961 7203523509136682216933020431/28647699117093415307614664109440217180193 7250068596542931028678669501762253287693294397689327797388113854588113 430063939405071979092547998950955940992, a[10,8] = -185526672218967442 26647/865384003125000000000000, a[12,11] = 123461712659887915177271339 3966068608104790287778693480148704506062609140195602856612882124981284 00476015695960341952/2816291066703206747542452093588407037042351473078 38896741075511220826056829047205614324978253226176275078922716132461, \+ a[12,1] = 407885778185158609210793892517582595305896470756467612636796 259611491408260896413446883450891351622914818800693274034252252905536/ 2808492638860122607362409616917500295697019157645511063322676514116137 2294098693275117181239385312198137508846535933127837167926875*6^(1/2)+ 1838203110479840386993853900915465658752149857359559506316407788280031 5372787284683238439478955141517997198007108623761931447163756/13974256 9444997243449189609938909336141610253229704500479326889980950085286208 21239604734608111291769444706187497807869179550841329375, a[16,13] = 2 729491144709837905799148766650782532906050298971406518524169921875/215 8115888622139473142775812109447802920656149243127309253686951469, `b*` [16] = 1839190071060649887127895100784/3804513952351063435142087541539 7, a[15,8] = -176324722711694310025656085505072661620440362221411708/2 85619406719829107485771207042040133465420149964555625*6^(1/2)-11611814 7575045169733222875835719955334334798191459879782123534889390467935109 772/881062690195483524567227513129587089250371395751217068145330081498 8417642493125, a[13,9] = -28140457973469923214145552460448772415902497 2527/284481916364737983221402322504830303192138671875*6^(1/2)+88577423 3856672590222951867695327816457340130391639153070521335485617578/30109 8541380295011015469248465465290112505656143757799934635162353515625, a [15,12] = 333514392451584382480734940567841440978729127734159045364007 2838769033456396839411470241410880750515810638511646873285345820289996 6748488718531545706559142895903144848764637/23166110253272874277148020 1132225288609079390498990062159236562764909757810216357219050223242549 0606773312310665593424982745744299371285598588298606088543376742054644 818966, `b*`[13] = 1084761591753640855844358063964843750000000/3182895 486031249071938549691320502488733423, a[14,12] = -34320443758939323781 0236856805228650103385091051699920208853270521163343279392054770280096 1532438008401883737341854688972639605334600163938610268855705742764072 609/114317410634168226097164769041056729214392619865092777892082326746 1111371275907599801714870165813394147519068210931766844494994616580258 435518181434575195312500000, a[13,1] = 2814045797346992321414555246044 87724159024972527/1478009944832743180452316204077188415527343750000*6^ (1/2)-5604277267532204213922762997804258633063362270605336394676614441 6933631/58808540772323190525590122613223430507352118534557342666015625 000000000, a[14,8] = 123767075855296855875080343261298883871499029943/ 451091609994276250390378731960660324096679687500*6^(1/2)+4077407727774 7636354598451708891165494123131383777235229538611989392175193285994266 471/152642905462481621010589859415880795182567412553770317363579461257 13524703979492187500, a[12,9] = -6526172450962537747372702280281321524 8943435321034818021887401537838625321743426151501352142616259666371008 11092384548036046488576/8649093284303728183602838792132050266857965317 6624892284566487468170341285762869374265713247057712228954184044334206 372230816544375*6^(1/2)+5459853981808361523356614860220324489669695891 0734339754065270985433507945162707737759469214674480807272210648148477 499238783276259328/301247919092298852634886875129959310794662932014184 4998271450758516372986983120740305674792395020116934474230264160407944 79934024058125, a[15,9] = -1974836894370976272287348157656813810148932 0568798111296/6484554262322259071286545935997129135111813687175650625* 6^(1/2)+17769448722513898342276837490665097286927607247073335618566987 143467294900183033216/255121700813788961505634214608456186712248516359 6619283719957742418751029506356875, a[13,8] = -84421373920409769642436 6573813463172477074917581/42101883599465783369768681649661630249023437 50000*6^(1/2)-93114481685939341460159650199040136021338029433258183466 22781285907057/4255970849010124217193135449668739985401313363005576159 362792968750000, a[14,1] = 4734200384802439149870797684768889301308307 4441159779465719863625051668939887702630319/44802546873926050730401222 636656855760802419993852060264615320801485392456054687500000-866369530 987077991125562402829092187100493209601/332552237587367215601771145917 3673934936523437500*6^(1/2), a[15,11] = 473389749049752963256114649231 353822492912259509649519870869750525/354124408823603417997988424283654 22941216508121322622479260846291, a[13,10] = -294039645364787227664606 8776592292229737651937934623/73454650587819837107958374295307847772452 86520703125*6^(1/2)+31547911672978015395641212405219968509774423938663 9023787359107959254802182/13448185050650584801258784221551557438021254 3200894932329128471154748828125, b[13] = 60530372821423065097959112869 09179687500000000/103899257350518063455290077573775162739725126989, a[ 8,1] = 43/600, `b*`[12] = -2129662374582324648106919795703373645353118 2730667422307241727310258139647124736471440105992066698253827193591131 96238857709025512340589957/1035543739272367080885190546201097218891268 7281182073325925959875548518829722926708817941783800977165831230634852 87435793657425889233080568, a[13,11] = 2250996163406545378616532039018 846586217631599453822541/382491303797095993563304148204275636433504028 320312500, a[15,1] = 1234273058981860170179592598535508631343082535549 881956/2105633771469628744518390642968552144069898845895808125*6^(1/2) -251883292492588254437485270381424098799230121337389853132654309322802 50855708601/1137064132557446931205696187407729855082764230877464731699 5717036347558064286250, a[12,10] = -8108825145085088104344721048166325 2251737294956893646964267201611120124142277523289697206589873156541798 73760357725235734000399440000000/2655582966610640210612741027567977548 1057208152954669252204018964061820669308150613320650066298300104729878 7619827411375675716583283801757*6^(1/2)+939166734840458401095542221032 8707125006120661611061908889750805619418785820948002455890360939221912 190524731087070645107486913457760000000/581572669687730206124190285037 3870830351528585497072566232680153129538726578484984317222364519327722 9358434488742203091272981931739152584783, a[13,12] = 26893409573076918 5329490238833445400395937814695752986623352925198635939233604415170894 9720958809747970514366293458424272174024493/95951638601957880850056911 4780871708466894752280482835105408027815194895319055443842782227102120 493960805649575561796875000000000, `b*`[9] = 1018354087913052979846578 12561920000000/1149763833200743759976506650241312100139, a[15,13] = -3 8714992656958413389743252726016897599283911682945255636643554687500000 /485404949269715874992945893825722120361691354298779017023475213004217 67, a[14,9] = 3465478123948311964502249611316368748401972838404/256033 7247282641848992620902543472728729248046875*6^(1/2)-105220386085005564 5982864903830206847373574903079637276496161875197379372479636460698666 4/38994174250054222540345740003973828622358928296533758351973409182715 56055507659912109375, a[15,10] = -852970846117821224749111313630789000 58888025224607913745000000/6921065945020139384316674672295403632633835 5649915383851733911*6^(1/2)+976592661391240748181932648019295477816599 26543786381510190954184218570746215033823993530000000/1856007665446970 6205963482908787056850812308205603127326855360961727608242796551101182 080033599, a[15,7] = -609224242740615999186035240493906573054312626351 97540405697952/6484861747489032169774584624759953148531564032417461909 516875+84558575751635978733109961893984238786929550462615375699341616/ 19454585242467096509323753874279859445594692097252385728550625*6^(1/2) \}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 20 "printed coefficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 362 "subs(e18,matrix([seq([c[i] ,seq(a[i,j],j=1..i-1),``$(4-i)],i=2..4)]));\nfor ii from 5 to 16 do\n \+ print(``);\n print(c[ii]=subs(e18,c[ii])); \n for jj to ii-1 do \n print(a[ii,jj]=subs(e18,a[ii,jj]));\n end do:\nend do:print( ``);\nfor ii to 16 do\n print(b[ii]=subs(e18,b[ii]));\nend do:print( ``);\nfor ii to 16 do\n print(`b*`[ii]=subs(e18,`b*`[ii]));\nend do: " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7%7&#\"%J<\"&++&F(%!GF +7&,&#\"(\\+j(\")++\"Q&\"\"\"*(\"'RN)*F1F0!\"\"\"\"'#F1\"\"#F4,&#\"0db 'pNoz<\"1++#QnFC+\"F4*(\"/88\\M0=9F1\"0+]X=pg]#F4F5F6F1,&#\"/F&H:u@S' \"0+SwMb[+#F1*(\".6Mw]W](F1\"0+?QnFC+\"F4F5F6F4F+7&,&#\")Z,*G#\"*++i2 \"F1*(\"(<1&HF1FJF4F5F6F4,&#FI\"*++[I%F1*(FLF1FOF4F5F6F4\"\"!,&#\")T/n oFOF1*(\"(^=&))F1FOF4F5F6F4Q)pprint166\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"&#\"$h&\"%+5 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"\",&#\"3R.-hU*R? #f\"3]ib0bNEJ^F(*(\"3TOi$H!*pQI(F(\"4+]AA?U0D0#!\"\"\"\"'#F(\"\"#F(" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"$,&#\"4.\\z#=%)Q:7()\"4+]AA?U 0D0#!\"\"*(\"4L:&e>f8UVG\"\"\"F,F-\"\"'#F0\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"%,&#\"4VaHBAO)pt=\"3]ib0bNEJ^\"\"\"*( \"3tpbDAfe#G&F-F,!\"\"\"\"'#F-\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"',&#\"$(Q\"%+5 \"\"\"*(\"$H\"F,\"%+?!\"\"F'#F,\"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"\",&#\",JO#3Q6\"-+?\"yhd\"F(*(\"*p)[\"R$F(\",+ IX/%R!\"\"F'#F(\"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\" \"'\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"$\" \"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"%,&#\"2J=4()G B$>;\"2]u]3=Mk)e\"\"\"*(\"14V-So&RvmY\"z=\":++++++v$4?:'f#\"\"\"*( \"96*oKp!Hcl%3W5#F-\";++++++]i07pd:!\"\"F(#F-\"\"#F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"(,&#\"9z_>So&RvmY\"z=\":++++++v$4?: 'f#\"\"\"*(\"96*oKp!Hcl%3W5#F-\";++++++]i07pd:!\"\"\"\"'#F-\"\"#F-" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\")#!8ZmAWn*=AnEb=\"9++ ++++DJ+%Ql)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"*#!74M \"R=s+')*znJ\"8++D\"Gj)fjlB/$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#6#\"\"\"\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"\",&#\"-(\\q&opU\"/]()*eT%R a!\"\"*(\",\\BQaF*F(\"/+v(zk(37F-\"\"'#F(\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"#6\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\" \"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"'#\"\"\" \"#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"(,&#\"1e>oH@, lG\"3D1LKVe)*[6!\"\"*(\"2*4bwo7`QR7=<^F$p)4%Hs8 ;T^wEKj5^Xw:>qp&H+v\"ph4CO2E7g)QE\\3G!\"\"\"\"'#F(\"\"#F(#\"^scP;Z9$>w B'3r+)>(*z^T^&*y%R%QKo%G(ys`J+G)y2kJ1&f&ft&)\\@vecY:4!R&Q*pQS)z/6.#Q= \"asv$H8%3bz\"py!y\\(=1ZWp S1&*\\:2W!y-(HzDO&>4@97*o?nm3LPr\"oxZ#*GF>[\"\"\"\"[sx=#))[W1aNH-&[wF' [G**3h>V7pX$*>0sO#Gp0U=JFR+-:;=P[bT?hdXOCqy4l-qW%*G;B!\"\"F(#F,\"\"#F, #\"ir/RSjb,_**QMh3e(oST\\)* y68)QL\"jrNwsdX\\y(Qxs))HB-Urg9)*[T\"*\\`SHhz$\\;Qf!=]S$HPTa.Rm3B'RSg[ )*oFr..\"3Y+s3F#F." }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\" \"(,&#\"es;W'**pP%)fnOp^hsX)>E#GJ@(3hrk$[$4Kw\")p%R![-$*4uE/R![R\")zoa iVk2sP')3c8qAijk\\2mmO\"\"fsv$pBUKB![%Ro0h5nqVTr)=S?D^V?P\"oqGVjN6@y\" \\jb[>]w0UtU#G&>G3=asK&e)yuUSU`Y'G>P!\"\"*(\"es/*Rh#yXlncbhR\">7f`d'Gz cxk*pOKc(o3iF$=_b^sGm1E5'*)QzB&pjFXP%R'H4gb0oLh`c&3X)p\"\"\"\"\"fsvoe4 '**4?>JHXSZ9t=*[%4:t$R$Htt[V(39')4?wj2@F&z(zNhfYJ=yzz(\\xg'fUzOigK<=P \"*zRf\"F-\"\"'#F0\"\"#F0" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"#7\"\"),&*(\"jrK%QPE@*zMuVvD+wKj!RXsEbwnX'H;8,!=KcN+_5s?d&)y]ql2D\\I Ky;\\YQTB`\\8kZ(e&\"\"\"\"\\sD1bl&R()Glz\")fBB.ayA^D.[-9'Hppo)3\"G(fGv 60D'H:rLvNZ%\\hhGDpf8k^k;?i$*3.`O!\"\"\"\"'#F,\"\"#F.#\"`s3iwtwjtrUkZf NKb-dw`-P$eDR)G\"o2^0qlKntc>c(zVw4[fn&)>-HW\"3G0'>TD&yKF**y)4h&\"asv=O Ot>L'\\j.c\\dc/%QHkU%3XKBtEE,YGB4;Oqx')e$4\")fR;zcLUNTLE1u/)y#=I.W\")> )f\")G*F," }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"*,&*(\"] sw&)[Yg.[XQ#4635Pm'fihU@N,::EMu@`iQy`,u)=-=[.@`VV*[_@8G!G-FPZx`i4Xsh_' \"\"\"\"^svVa;3Bsj?MV/%=a*GArdqC8dEu$pGw>.zC,$F," }} {PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"#5,&*(\"cs+++S%*R+StN_s d.wt)zTl:t)*e1sp*GBvFUT7?6h,sEkpk$*o&\\HPwy)HZ5+$)Hm+l?Lh]\"3$p1#=1k*=S?_#pY&H:3s0\"[v(zc F5u71@S1hmHebE!\"\"\"\"'#F,\"\"#F.#\"fs+++gxX8p[2^kqq3JZ_!>7>AR4O!*eX- ![4#ey=%>c!3v*))3>16;m?h+DrqG.@Aa&4,%e/%[tm\"R*\"gs$y%e_\"R<$>)HF\"4.A u)[M%e$HsF$>XOAsJ%)\\[ylsQ&HJ:!oKicsq\\&eG:NI3(QP]G!>Ch?IxopEd\"eF," } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"#6#\"`r_>Mgfp:gZ+%G\") \\7#)GhcGg&>S\"4E11Xq[,[$pyxG!z/\"3'og'RR8Fx^\"z))fErhM7\"`rhC8;F#*y]F whA`#y\\K9c?Z!Ho0E3A6b2Tn*)QyIZ^B/PqS)e$4_CaZn?.n1\"H;G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6# \"#8#\"%.T\"%+]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"\" ,&*(\"QFD(\\-fTs([/Y_b99K#*pM(zXS\"GF(\"R++vVt_:%)=xS?;BX!=VF$[%*4!y9! \"\"\"\"'#F(\"\"#F(#\"boJO$pTWhwYRO`gqAOjIjeU!y*HwAR@/A`nsF/c\"bo++++] i:gmUtbM&=@N20VBKhA,fD0>BBxS&3)eF-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%\"aG6$\"#8\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\" #8\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"%\"\" !" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"',&*(\"L@BP8N\"oLzR5t`_-7k \"R#HXF\"\"\"\"M++]PfQeXu\\116.Mk%ok@-<\"!\"\"F(#F,\"\"#F.#\"O<:ivELDs %zBQ7cc.vH-!R;F5\"N+]iS;VnGH)pZ(ySmK6V%*=Ez\"F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"(,&#\"Z@,$HLQA2yBGDpUkLxl^IF#[P&)***Hs :\"YD1RD&QKTjR.;2L%4/KFt&ea)zO2**pO!\"\"*(\"en,@D!fk1lyU=/z$yJng)*=`4& z-GW1wv&\"\"\"\"en++]P%)*3n([g#R([d0$fFU.BSgEt5\">BNF-\"\"'#F0\"\"#F0 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"),&*(\"Q\"e<\\2xC ]'f,YT$Rfo\"[9J*\"ao++voHzi$fh d0IO88S&)*R(o'\\a8$>. .$[]ABS@K)ztkj\">[%G!\"\"\"\"'#F,\"\"#F.#\"coyvh&[N8_qI:R;RI,Mdk\"yK&p n=&HA!fsm&QBud))\"coDc^`B;NY$**zdP9cc]7,HlaY[#pa,6]H!QT&)4,$F," }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"#5,&*(\"UBY$z$>lP(HAH#f w(ogkwA(yk`kRSH\"\"\"\"UDJq?lGXsx%yI&Hu$ez5P)>ye]YXt!\"\"\"\"'#F,\"\"# 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$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$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 12 "" 1 " " {XPPMATH 20 "6#7dw\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$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#\"$+#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "seq(c[i]=sub s(e18,c[i]),i=2..16);" }}{PARA 12 "" 1 "" {XPPMATH 20 "61/&%\"cG6#\"\" ##\"%J<\"&++&/&F%6#\"\"$,&#\"(\\+j(\")++\"Q&\"\"\"*(\"'RN)*F3F2!\"\"\" \"'#F3F'F6/&F%6#\"\"%,&#\")Z,*G#\"*++i2\"F3*(\"(<1&HF3F@F6F7F8F6/&F%6# \"\"&#\"$h&\"%+5/&F%6#F7,&#\"$(QFIF3*(\"$H\"F3\"%+?F6F7F8F6/&F%6#\"\"( ,&FNF3*(FQF3FRF6F7F8F3/&F%6#\"\")#FQ\"$+#/&F%6#\"\"*#FO\"$+)/&F%6#\"#5 #\"%dn\"'++5/&F%6#\"#6#F3F " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 52 "#---------------------------------------- -----------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "characteristics of the embedded scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "coefficients of the combined scheme " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18354 "ee := \{c[2]=1731/500 00,\nc[3]=7630049/53810000-983539/53810000*6^(1/2),\nc[4]=22890147/107 620000-2950617/107620000*6^(1/2),\nc[5]=561/1000,\nc[6]=387/1000-129/2 000*6^(1/2),\nc[7]=387/1000+129/2000*6^(1/2),\nc[8]=129/200,\nc[9]=387 /800,\nc[10]=6757/100000,\nc[11]=1/4,\nc[12]=1427971650951258372/21666 62646162554701,\nc[13]=4103/5000,\nc[14]=2253/2500,\nc[15]=1,\nc[16]=1 ,\n\na[2,1]=1731/50000,\na[3,1]=-177968356965557/1002427673820000+1418 0534491313/250606918455000*6^(1/2),\na[3,2]=64021741529527/20048553476 4000-7504450763411/100242767382000*6^(1/2),\na[4,1]=22890147/430480000 -2950617/430480000*6^(1/2),\na[4,2]=0,\na[4,3]=68670441/430480000-8851 851/430480000*6^(1/2),\na[5,1]=592203994261020339/513126355505556250+7 30386990293623641/2052505422022225000*6^(1/2),\na[5,2]=0,\na[5,3]=-871 2153884182794903/2052505422022225000-2843421359195851533/2052505422022 225000*6^(1/2),\na[5,4]=1873698362223295443/513126355505556250+5282585 92225556973/513126355505556250*6^(1/2),\na[6,1]=11380823631/1576178120 00-339148869/39404453000*6^(1/2),\na[6,2]=0,\na[6,3]=0,\na[6,4]=161932 32887091831/58864341808507450-2355345717024309/58864341808507450*6^(1/ 2),\na[6,5]=165912282616977/4179075230308000-33181894472511/2089537615 154000*6^(1/2),\na[7,1]=26523528363/231790900000+863255358/12313891562 5*6^(1/2),\na[7,2]=0,\na[7,3]=0,\na[7,4]=-38208748178016484817787/8425 17966262441068418750-\n 86118788556282369822807/842517966262 441068418750*6^(1/2),\na[7,5]=92362336407446913/290322814529044000-232 039320950012997/2467743923496874000*6^(1/2),\na[7,6]=-362925891/169035 0537500+857800423623/3380701075000*6^(1/2),\na[8,1]=43/600,\na[8,2]=0, \na[8,3]=0,\na[8,4]=0,\na[8,5]=0,\na[8,6]=43/150+43/2400*6^(1/2),\na[8 ,7]=43/150-43/2400*6^(1/2),\na[9,1]=7353/102400,\na[9,2]=0,\na[9,3]=0, \na[9,4]=0,\na[9,5]=0,\na[9,6]=22833/102400+8901/204800*6^(1/2),\na[9, 7]=22833/102400-8901/204800*6^(1/2),\na[9,8]=-3483/102400,\na[10,1]=37 6708742472214988700853/7788456028125000000000000,\na[10,2]=0,\na[10,3] =0,\na[10,4]=0,\na[10,5]=0,\na[10,6]=187914666753956840195279/25961520 09375000000000000-\n 210440846556290693268911/15576912056250 000000000000*6^(1/2),\na[10,7]=187914666753956840195279/25961520093750 00000000000+\n 210440846556290693268911/15576912056250000000000 000*6^(1/2),\na[10,8]=-18552667221896744226647/86538400312500000000000 0,\na[10,9]=-3167799860072183913409/30423656359863281250000,\na[11,1]= -426968570497/54394415898750-92754382349/12087647977500*6^(1/2),\na[11 ,2]=0,\na[11,3]=0,\na[11,4]=0,\na[11,5]=0,\na[11,6]=1/30,\na[11,7]=-28 65012129681958/114898584332330625-12962517687655099/229797168664661250 *6^(1/2),\na[11,8]=4389715333607/309890657317500+92754382349/114774317 52500*6^(1/2),\na[11,9]=4990058173976/83757096376875+371017529396/9306 344041875*6^(1/2),\na[11,10]=1099523524595993125000/625766790986975601 8891+\n 100957348037989687500/6257667909869756018891*6^(1/2), \na[12,1]=\n1838203110479840386993853900915465658752149857359559506316 4077882800315372787284683238439478955141517997198007108623761931447163 756/139742569444997243449189609938909336141610253229704500479326889980 95008528620821239604734608111291769444706187497807869179550841329375+ \n40788577818515860921079389251758259530589647075646761263679625961149 1408260896413446883450891351622914818800693274034252252905536/28084926 3886012260736240961691750029569701915764551106332267651411613722940986 93275117181239385312198137508846535933127837167926875*6^(1/2),\na[12,2 ]=0,\na[12,3]=0,\na[12,4]=0,\na[12,5]=0,\na[12,6]=\n-33388131178984941 1971573472868128281438202210721723123251742145367734582887577395547778 228760174068758086134389952015563403904/227087200460810303712768984860 4039623086639035441372934050180593816493796129405349914148981460714202 232988727738778494557727635+\n4819272892477768171373308666720689121421 0919536257929702780440715499506401950564729555237698290348006218904248 47009130000000/2316289444700265097870243645576120415548371816150200392 7311842056928236720519934569124319610899284862776485022935540644488821 877*6^(1/2),\na[12,7]=\n-136666607496463622270135608863772076443625468 7981394803904267409930248039469817632093483647161087213128226198457261 51693667598437699964416/3719286465342404274788585327254180828195282427 3420576501948556349178211135634328706813720435125204018871414370671061 05683944802332422369375+\n16984508556536133680555600929639437452763695 2379388961026066628725155521832762086875632366996477567928657535912191 396155566765457826139904/159397991371817326062367942596607749779797818 3146596135797795272107637620098614087434873732933937315094489187314474 045293119200999609586875*6^(1/2),\na[12,8]=561098789927327852541196052 8081442902198567594809764379756195673673265700551076812883925583370253 765702553235594764427173637673766208/928815981981440330182788047406263 3413542335679163959810935886777036160923284601262673233245084426429384 0456574956036349633197336361875-\n558747641349532341384649167832304925 0765705078855720721052003556321800113162964567765526724539063327600257 543743479921263738432/365303089362201664516413596925286161494473575337 1152962505117528597281088686969296140248032551227854032323598179652887 39565550625*6^(1/2),\na[12,9]=\n54598539818083615233566148602203244896 6969589107343397540652709854335079451627077377594692146744808072722106 48148477499238783276259328/3012479190922988526348868751299593107946629 3201418449982714507585163729869831207403056747923950201169344742302641 6040794479934024058125-\n652617245096253774737270228028132152489434353 2103481802188740153783862532174342615150135214261625966637100811092384 548036046488576/864909328430372818360283879213205026685796531766248922 8456648746817034128576286937426571324705771222895418404433420637223081 6544375*6^(1/2),\na[12,10]=\n93916673484045840109554222103287071250061 2066161106190888975080561941878582094800245589036093922191219052473108 7070645107486913457760000000/58157266968773020612419028503738708303515 2858549707256623268015312953872657848498431722236451932772293584344887 42203091272981931739152584783-\n81088251450850881043447210481663252251 7372949568936469642672016111201241422775232896972065898731565417987376 0357725235734000399440000000/26555829666106402106127410275679775481057 2081529546692522040189640618206693081506133206500662983001047298787619 827411375675716583283801757*6^(1/2),\na[12,11]=\n123461712659887915177 2713393966068608104790287778693480148704506062609140195602856612882124 98128400476015695960341952/\n28162910667032067475424520935884070370423 5147307838896741075511220826056829047205614324978253226176275078922716 132461,\na[13,1]=-5604277267532204213922762997804258633063362270605336 3946766144416933631/\n 588085407723231905255901226132234305073 52118534557342666015625000000000+\n 28140457973469923214145552 4604487724159024972527/\n 147800994483274318045231620407718841 5527343750000*6^(1/2),\na[13,2]=0,\na[13,3]=0,\na[13,4]=0,\na[13,5]=0, \na[13,6]=-1027163900229750356561238237947225332675621517/179261894431 132664078747698292867431640625000-\n 2745292391641202525373103 979336813513372321/11702216468464340311060649744558385937500000*6^(1/2 ),\na[13,7]=-157229999853748227305165773364426925282378072238332930121 /\n 36699907367985458573273204094330716033963413238525390625+ \n 5757606442802795095318986067317837904184278650664590252101/ \n 3523191107326604023034227593055748739260487670898437500000* 6^(1/2),\na[13,8]=-931144816859393414601596501990401360213380294332581 8346622781285907057/\n 425597084901012421719313544966873998540 1313363005576159362792968750000-\n 844213739204097696424366573 813463172477074917581/\n 4210188359946578336976868164966163024 902343750000*6^(1/2),\na[13,9]=\n8857742338566725902229518676953278164 57340130391639153070521335485617578/\n30109854138029501101546924846546 5290112505656143757799934635162353515625-\\\n2814045797346992321414555 24604487724159024972527/\n28448191636473798322140232250483030319213867 1875*6^(1/2),\na[13,10]=3154791167297801539564121240521996850977442393 86639023787359107959254802182/\n 13448185050650584801258784221 5515574380212543200894932329128471154748828125-\n 294039645364 7872276646068776592292229737651937934623/\n 734546505878198371 0795837429530784777245286520703125*6^(1/2),\na[13,11]=2250996163406545 378616532039018846586217631599453822541/\n 3824913037970959935 63304148204275636433504028320312500,\na[13,12]=\n268934095730769185329 4902388334454003959378146957529866233529251986359392336044151708949720 958809747970514366293458424272174024493/\n9595163860195788085005691147 8087170846689475228048283510540802781519489531905544384278222710212049 3960805649575561796875000000000,\na[14,1]=4734200384802439149870797684 7688893013083074441159779465719863625051668939887702630319/\n 4 4802546873926050730401222636656855760802419993852060264615320801485392 456054687500000-\n 86636953098707799112556240282909218710049320 9601/\n 3325522375873672156017711459173673934936523437500*6^(1/ 2),\na[14,2]=0,\na[14,3]=0,\na[14,4]=0,\na[14,5]=0,\na[14,6]=871779321 807802447463310035318238762878527157/\n 13444642082334949805906 0773719650573730468750+\n 1076412684809993960818489752718498579 94818/\n 1097082793918531904161935913552348681640625*6^(1/2),\n a[14,7]=496103786351862292800034805114190705484800743513354117014/\n \+ 110099722103956375719819612282992148101890239715576171875-\n \+ 1329938412606197485769312599390307351191540891599374831099/\n \+ 660598332623738254318917673697952888611341438293457031250*6^(1/2), \na[14,8]=407740772777476363545984517088911654941231313837772352295386 11989392175193285994266471/\n 152642905462481621010589859415880 79518256741255377031736357946125713524703979492187500+\n 123767 075855296855875080343261298883871499029943/\n 45109160999427625 0390378731960660324096679687500*6^(1/2),\na[14,9]=-1052203860850055645 9828649038302068473735749030796372764961618751973793724796364606986664 /\n 3899417425005422254034574000397382862235892829653375835197 340918271556055507659912109375+\n 3465478123948311964502249611 316368748401972838404/\n 2560337247282641848992620902543472728 729248046875*6^(1/2),\na[14,10]=-2784376447126269318936520113562067049 0328475323282820219474851621693895769527094334687108984/\n 12 2570410662851642220025943006055939294341391930221663178021214129993570 24704596261133984375+\n 5747743002719985986838731141054720166 99241495055292/\n 1049352151254569101542262489932969253892183 788671875*6^(1/2),\na[14,11]=-3424113435184824562423280943767688900943 1930503529853032576417589898516/\n 56133478243586519811009850 09024281007603230062439942682713165283203125,\na[14,12]=\n-34320443758 9393237810236856805228650103385091051699920208853270521163343279392054 7702800961532438008401883737341854688972639605334600163938610268855705 742764072609/\n1143174106341682260971647690410567292143926198650927778 9208232674611113712759075998017148701658133941475190682109317668444949 94616580258435518181434575195312500000,\na[14,13]=47469308760239193350 79451612726717649218264199984/\n 18592065538407049755200144388 134089346432755594877,\na[15,1]=-2518832924925882544374852703814240987 9923012133738985313265430932280250855708601/\n 113706413255744 69312056961874077298550827642308774647316995717036347558064286250+\n \+ 1234273058981860170179592598535508631343082535549881956/\n \+ 2105633771469628744518390642968552144069898845895808125*6^(1/2),\n a[15,2]=0,\na[15,3]=0,\na[15,4]=0,\na[15,5]=0,\na[15,6]=-5482114211968 5055562477216205428613949905430396088/\n 395943983700946128908 5587746748097947393101278095-\n 151127675382598285607289146950 4471256664975925000/\n 403862863374965051486729950168305990634 09633036569*6^(1/2),\na[15,7]=-609224242740615999186035240493906573054 31262635197540405697952/\n 64848617474890321697745846247599531 48531564032417461909516875+\n 84558575751635978733109961893984 238786929550462615375699341616/\n 1945458524246709650932375387 4279859445594692097252385728550625*6^(1/2),\na[15,8]=-1161181475750451 69733222875835719955334334798191459879782123534889390467935109772/\n \+ 881062690195483524567227513129587089250371395751217068145330081 4988417642493125-\n 176324722711694310025656085505072661620440 362221411708/\n 2856194067198291074857712070420401334654201499 64555625*6^(1/2),\na[15,9]=1776944872251389834227683749066509728692760 7247073335618566987143467294900183033216/\n 2551217008137889615 056342146084561867122485163596619283719957742418751029506356875-\n \+ 19748368943709762722873481576568138101489320568798111296/\n \+ 6484554262322259071286545935997129135111813687175650625*6^(1/2),\na[1 5,10]=9765926613912407481819326480192954778165992654378638151019095418 4218570746215033823993530000000/\n 185600766544697062059634829 08787056850812308205603127326855360961727608242796551101182080033599- \n 85297084611782122474911131363078900058888025224607913745000 000/\n 6921065945020139384316674672295403632633835564991538385 1733911*6^(1/2),\na[15,11]=4733897490497529632561146492313538224929122 59509649519870869750525/\n 35412440882360341799798842428365422 941216508121322622479260846291,\na[15,12]=\n33351439245158438248073494 0567841440978729127734159045364007283876903345639683941147024141088075 0515810638511646873285345820289996674848871853154570655914289590314484 8764637/\n231661102532728742771480201132225288609079390498990062159236 5627649097578102163572190502232425490606773312310665593424982745744299 371285598588298606088543376742054644818966,\na[15,13]=-387149926569584 13389743252726016897599283911682945255636643554687500000/\n 4 8540494926971587499294589382572212036169135429877901702347521300421767 ,\na[15,14]=1480025020094032371712461617564126123511929579576881471780 3955078125/\n 335655771251418777602873805886324212234331940781 56948298488471160489,\na[16,1]=230578569608639756108085818693989717364 5641331085041313944389849986584101287/\n 6175082443452822658190 87370078275122671246164669900462139876057008239440000-\n 854046 23305589712632165905233974183137607899140719/\n 124822287169084 833758410283469525117460541643292500*6^(1/2),\na[16,2]=0,\na[16,3]=0, \na[16,4]=0,\na[16,5]=0,\na[16,6]=102903996961580448264190625267026062 654799259083/\n 5046398084890004857481629999673320438819484730+ \n 41320925487304219313300272052128374567081128125/\n 51 473260465878049546312625996667868475958744246*6^(1/2),\na[16,7]=627984 43349876457506718920843975661399949564598018488144466/\n 413255 3498782573324058263582553715220777051359780141380625-\n 7230880 7081932961554425711089716771013571419950657300729103/\n 1239766 0496347719972174790747661145662331154079340424141875*6^(1/2),\na[16,8] =179490914212648256439084852292422555322146901975147054495929761465466 1293377/\n 5259648119399426443560162610975298867467969164427545 6716633975785978672500+\n 1220066047222710180459512931913916901 9658271305817/\n 1693156145655995911520770934405657826339776060 2500*6^(1/2),\na[16,9]=-2775244732780109667342845612394739319115662636 371477300455747022423270475907256/\n 2284171536755840297250180 45422706955827996328208181619436454383447149337555625+\n 34161 8493222358850528663620935896732550431596562876/\n 961013383787 73357469245211954911505447551097205625*6^(1/2),\na[16,10]=-27680554659 7690166235309791767274482512922443107699960153421908190689705560830631 25000/\n 3299557777429648960576561382256606844677258438797072 955341581354051375036522231471437+\n 442655212757989537347967 0356100179759944766558141730312500/\n 30771137386673207077488 77199804636746494977000658967987677*6^(1/2),\na[16,11]=-29260317192970 6291053929402159930330736639136252680853622275/\n 15473622826 279161150227076887290262443510550964275858143964,\na[16,12]=-981571712 9569106988569302193220999343824932084582093647596086931754666098662594 1530952589885163051657947397448735398290696172035235091366822169330204 31/2864769911709341530761466410944021718019372500685965429310286786695 0176225328769329439768932779738811385458811343006393940507197909254799 8950955940992,\na[16,13]=272949114470983790579914876665078253290605029 8971406518524169921875/\n 215811588862213947314277581210944780 2920656149243127309253686951469,\na[16,14]=0,\na[16,15]=0, \n\nb[1]=81 98160366203173411119943711500331/561057579384085860167277847128765528, \nb[2]=0,\nb[3]=0,\nb[4]=0,\nb[5]=0,\nb[6]=0,\nb[7]=0,\nb[8]=-45565549 3073428838813281446213740000000/11638080111509105612404642258373124978 69,\nb[9]=19965163648706008081135075746915614720000000/863944041905370 86868394686205782432516544599,\nb[10]=89231107919981418705566970804343 750000000000000000000000/\n 6999798709883356744455946798564450605 62597693583175985391,\nb[11]=47104273954945906713184913871143492/20968 4639122339601934631113492763467,\nb[12]=208450044214045004640105847407 9675065083217679837038308422635129473073119667364731106233097274073473 7279503119387627146381678677156136042524139311907482802844083/36670849 8911363730202382253282651002506051447185019263051409665867580548476046 8146633610316928475598775354232120246237155412059385814975553987856197 6786592389608,\nb[13]=6053037282142306509795911286909179687500000000/1 03899257350518063455290077573775162739725126989,\nb[14]=91740110492099 3498360358406096725463867187500/67242498159113466533157907374536073829 89551463,\nb[15]=2585449557665268951371699596493957/845743451607641401 63208606048427531,\nb[16]=0,\n\n`b*`[1]=552562031208180939317806684253 /27669654257734667858523344041464,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=0, \n`b*`[5]=0,\n`b*`[6]=0,\n`b*`[7]=0,\n`b*`[8]=221223388631423597589898 601690000000/100946136798587090054685074667127461,\n`b*`[9]=1018354087 91305297984657812561920000000/1149763833200743759976506650241312100139 ,\n`b*`[10]=1313720309077630014453239843750000000000000000000/\n \+ 11518201923215510989126466531107437037395719117133,\n`b*`[11]=48336 11232701440504508086151728/19081321241454145230196661524503,\n`b*`[12] =-21296623745823246481069197957033736453531182730667422307241727310258 1396471247364714401059920666982538271935911319623885770902551234058995 7/10355437392723670808851905462010972188912687281182073325925959875548 5188297229267088179417838009771658312306348528743579365742588923308056 8,\n`b*`[13]=1084761591753640855844358063964843750000000/3182895486031 249071938549691320502488733423,\n`b*`[14]=0,\n`b*`[15]=0,\n`b*`[16]=18 39190071060649887127895100784/38045139523510634351420875415397\}:" }}} {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[9,16];" "6#&%\"TG6$\"\"*\"#;" }{TEXT -1 129 " denote the vector whose components are the principal error terms of \+ the 16 stage, order 9 scheme (the error terms of order 10)." }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "`T*`[8,16];" "6#&%#T*G6$ \"\")\"#;" }{TEXT -1 146 " denote the vector whose components are the principal error terms of the embedded 16 stage, order 8 scheme (the e rror terms of order 9) and let " }{XPPEDIT 18 0 "`T*`[9, 16];" "6#&%# T*G6$\"\"*\"#;" }{TEXT -1 100 " denote the vector whose components ar e the error terms of order 10 of the embedded order 8 scheme." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Denote th e 2-norm of these vectors by " }{XPPEDIT 18 0 "abs(abs(T[9,16]));" "6 #-%$absG6#-F$6#&%\"TG6$\"\"*\"#;" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "ab s(abs(`T*`[8,16]));" "6#-%$absG6#-F$6#&%#T*G6$\"\")\"#;" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "abs(abs(`T*`[9,16]));" "6#-%$absG6#-F$6#&%#T* G6$\"\"*\"#;" }{TEXT -1 15 " respectively." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Define: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A[10] = abs(abs(T[9, 16]));" "6#/&%\"AG 6#\"#5-%$absG6#-F)6#&%\"TG6$\"\"*\"#;" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "B[10] = abs(abs(`T*`[9,16]))/abs(abs(`T*`[8,16]));" "6#/&%\"BG6# \"#5*&-%$absG6#-F*6#&%#T*G6$\"\"*\"#;\"\"\"-F*6#-F*6#&F/6$\"\")F2!\"\" " }{TEXT -1 6 ", " }{XPPEDIT 18 0 "C[10] = abs(abs(`T*`[9,16]-T[9, 16]))/abs(abs(`T*`[8,16]));" "6#/&%\"CG6#\"#5*&-%$absG6#-F*6#,&&%#T*G6 $\"\"*\"#;\"\"\"&%\"TG6$F2F3!\"\"F4-F*6#-F*6#&F06$\"\")F3F8" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Dormand and Prince have suggested that as well as attempting to en sure that " }{XPPEDIT 18 0 "A[10];" "6#&%\"AG6#\"#5" }{TEXT -1 73 " \+ is a minimum, if the embedded scheme is to be used for error control, \+ " }{XPPEDIT 18 0 "B[10];" "6#&%\"BG6#\"#5" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "C[10];" "6#&%\"CG6#\"#5" }{TEXT -1 27 " should be chos en so that " }}{PARA 0 "" 0 "" {TEXT -1 68 "they are \"similar in magn itude\" and also not differ too much from 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 168 "errterms9_16 := P rincipalErrorTerms(9,16,'expanded'):\n`errterms9_16*` :=subs(b=`b*`,er rterms9_16):\n`errterms8_16*` := subs(b=`b*`,PrincipalErrorTerms(8,16, 'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 468 "nmB := 0: \nfor ct to nops(`errterms9_16*`) do \n nmB := nmB+evalf(subs(ee,`errterms9_16*`[ct]))^2;\nend do:\nsnmB \+ := sqrt(nmB):\ndnB := 0:\nfor ct to nops(`errterms8_16*`) do\n dnB : = dnB+evalf(subs(ee,`errterms8_16*`[ct]))^2;\nend do:\nsdnB := sqrt(dn B):\nnmC := 0:\nfor ct to nops(errterms9_16) do\n nmC := nmC+(evalf( subs(ee,`errterms9_16*`[ct]))-evalf(subs(ee,errterms9_16[ct])))^2;\nen d do:\nsnmC := sqrt(nmC):\n'B[10]'= evalf[8](snmB/sdnB);\n'C[10]'= eva lf[8](snmC/sdnB);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"BG6#\"#5$\") XVJL!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"#5$\")&oLK$!\"( " }}}{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 36 "coefficients of the combined scheme " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18354 "ee := \+ \{c[2]=1731/50000,\nc[3]=7630049/53810000-983539/53810000*6^(1/2),\nc[ 4]=22890147/107620000-2950617/107620000*6^(1/2),\nc[5]=561/1000,\nc[6] =387/1000-129/2000*6^(1/2),\nc[7]=387/1000+129/2000*6^(1/2),\nc[8]=129 /200,\nc[9]=387/800,\nc[10]=6757/100000,\nc[11]=1/4,\nc[12]=1427971650 951258372/2166662646162554701,\nc[13]=4103/5000,\nc[14]=2253/2500,\nc[ 15]=1,\nc[16]=1,\n\na[2,1]=1731/50000,\na[3,1]=-177968356965557/100242 7673820000+14180534491313/250606918455000*6^(1/2),\na[3,2]=64021741529 527/200485534764000-7504450763411/100242767382000*6^(1/2),\na[4,1]=228 90147/430480000-2950617/430480000*6^(1/2),\na[4,2]=0,\na[4,3]=68670441 /430480000-8851851/430480000*6^(1/2),\na[5,1]=592203994261020339/51312 6355505556250+730386990293623641/2052505422022225000*6^(1/2),\na[5,2]= 0,\na[5,3]=-8712153884182794903/2052505422022225000-284342135919585153 3/2052505422022225000*6^(1/2),\na[5,4]=1873698362223295443/51312635550 5556250+528258592225556973/513126355505556250*6^(1/2),\na[6,1]=1138082 3631/157617812000-339148869/39404453000*6^(1/2),\na[6,2]=0,\na[6,3]=0, \na[6,4]=16193232887091831/58864341808507450-2355345717024309/58864341 808507450*6^(1/2),\na[6,5]=165912282616977/4179075230308000-3318189447 2511/2089537615154000*6^(1/2),\na[7,1]=26523528363/231790900000+863255 358/123138915625*6^(1/2),\na[7,2]=0,\na[7,3]=0,\na[7,4]=-3820874817801 6484817787/842517966262441068418750-\n 861187885562823698228 07/842517966262441068418750*6^(1/2),\na[7,5]=92362336407446913/2903228 14529044000-232039320950012997/2467743923496874000*6^(1/2),\na[7,6]=-3 62925891/1690350537500+857800423623/3380701075000*6^(1/2),\na[8,1]=43/ 600,\na[8,2]=0,\na[8,3]=0,\na[8,4]=0,\na[8,5]=0,\na[8,6]=43/150+43/240 0*6^(1/2),\na[8,7]=43/150-43/2400*6^(1/2),\na[9,1]=7353/102400,\na[9,2 ]=0,\na[9,3]=0,\na[9,4]=0,\na[9,5]=0,\na[9,6]=22833/102400+8901/204800 *6^(1/2),\na[9,7]=22833/102400-8901/204800*6^(1/2),\na[9,8]=-3483/1024 00,\na[10,1]=376708742472214988700853/7788456028125000000000000,\na[10 ,2]=0,\na[10,3]=0,\na[10,4]=0,\na[10,5]=0,\na[10,6]=187914666753956840 195279/2596152009375000000000000-\n 210440846556290693268911 /15576912056250000000000000*6^(1/2),\na[10,7]=187914666753956840195279 /2596152009375000000000000+\n 210440846556290693268911/15576912 056250000000000000*6^(1/2),\na[10,8]=-18552667221896744226647/86538400 3125000000000000,\na[10,9]=-3167799860072183913409/3042365635986328125 0000,\na[11,1]=-426968570497/54394415898750-92754382349/12087647977500 *6^(1/2),\na[11,2]=0,\na[11,3]=0,\na[11,4]=0,\na[11,5]=0,\na[11,6]=1/3 0,\na[11,7]=-2865012129681958/114898584332330625-12962517687655099/229 797168664661250*6^(1/2),\na[11,8]=4389715333607/309890657317500+927543 82349/11477431752500*6^(1/2),\na[11,9]=4990058173976/83757096376875+37 1017529396/9306344041875*6^(1/2),\na[11,10]=1099523524595993125000/625 7667909869756018891+\n 100957348037989687500/62576679098697560 18891*6^(1/2),\na[12,1]=\n18382031104798403869938539009154656587521498 5735955950631640778828003153727872846832384394789551415179971980071086 23761931447163756/1397425694449972434491896099389093361416102532297045 0047932688998095008528620821239604734608111291769444706187497807869179 550841329375+\n4078857781851586092107938925175825953058964707564676126 3679625961149140826089641344688345089135162291481880069327403425225290 5536/28084926388601226073624096169175002956970191576455110633226765141 161372294098693275117181239385312198137508846535933127837167926875*6^( 1/2),\na[12,2]=0,\na[12,3]=0,\na[12,4]=0,\na[12,5]=0,\na[12,6]=\n-3338 8131178984941197157347286812828143820221072172312325174214536773458288 7577395547778228760174068758086134389952015563403904/22708720046081030 3712768984860403962308663903544137293405018059381649379612940534991414 8981460714202232988727738778494557727635+\n481927289247776817137330866 6720689121421091953625792970278044071549950640195056472955523769829034 800621890424847009130000000/231628944470026509787024364557612041554837 1816150200392731184205692823672051993456912431961089928486277648502293 5540644488821877*6^(1/2),\na[12,7]=\n-13666660749646362227013560886377 2076443625468798139480390426740993024803946981763209348364716108721312 822619845726151693667598437699964416/371928646534240427478858532725418 0828195282427342057650194855634917821113563432870681372043512520401887 141437067106105683944802332422369375+\n1698450855653613368055560092963 9437452763695237938896102606662872515552183276208687563236699647756792 8657535912191396155566765457826139904/15939799137181732606236794259660 7749779797818314659613579779527210763762009861408743487373293393731509 4489187314474045293119200999609586875*6^(1/2),\na[12,8]=56109878992732 7852541196052808144290219856759480976437975619567367326570055107681288 3925583370253765702553235594764427173637673766208/92881598198144033018 2788047406263341354233567916395981093588677703616092328460126267323324 50844264293840456574956036349633197336361875-\n55874764134953234138464 9167832304925076570507885572072105200355632180011316296456776552672453 9063327600257543743479921263738432/36530308936220166451641359692528616 1494473575337115296250511752859728108868696929614024803255122785403232 359817965288739565550625*6^(1/2),\na[12,9]=\n5459853981808361523356614 8602203244896696958910734339754065270985433507945162707737759469214674 480807272210648148477499238783276259328/301247919092298852634886875129 9593107946629320141844998271450758516372986983120740305674792395020116 93447423026416040794479934024058125-\n65261724509625377473727022802813 2152489434353210348180218874015378386253217434261515013521426162596663 7100811092384548036046488576/86490932843037281836028387921320502668579 6531766248922845664874681703412857628693742657132470577122289541840443 34206372230816544375*6^(1/2),\na[12,10]=\n9391667348404584010955422210 3287071250061206616110619088897508056194187858209480024558903609392219 12190524731087070645107486913457760000000/5815726696877302061241902850 3738708303515285854970725662326801531295387265784849843172223645193277 229358434488742203091272981931739152584783-\n8108825145085088104344721 0481663252251737294956893646964267201611120124142277523289697206589873 15654179873760357725235734000399440000000/2655582966610640210612741027 5679775481057208152954669252204018964061820669308150613320650066298300 1047298787619827411375675716583283801757*6^(1/2),\na[12,11]=\n12346171 2659887915177271339396606860810479028777869348014870450606260914019560 285661288212498128400476015695960341952/\n2816291066703206747542452093 5884070370423514730783889674107551122082605682904720561432497825322617 6275078922716132461,\na[13,1]=-560427726753220421392276299780425863306 33622706053363946766144416933631/\n 58808540772323190525590122 613223430507352118534557342666015625000000000+\n 2814045797346 99232141455524604487724159024972527/\n 14780099448327431804523 16204077188415527343750000*6^(1/2),\na[13,2]=0,\na[13,3]=0,\na[13,4]=0 ,\na[13,5]=0,\na[13,6]=-1027163900229750356561238237947225332675621517 /179261894431132664078747698292867431640625000-\n 274529239164 1202525373103979336813513372321/11702216468464340311060649744558385937 500000*6^(1/2),\na[13,7]=-15722999985374822730516577336442692528237807 2238332930121/\n 366999073679854585732732040943307160339634132 38525390625+\n 57576064428027950953189860673178379041842786506 64590252101/\n 35231911073266040230342275930557487392604876708 98437500000*6^(1/2),\na[13,8]=-931144816859393414601596501990401360213 3802943325818346622781285907057/\n 425597084901012421719313544 9668739985401313363005576159362792968750000-\n 844213739204097 696424366573813463172477074917581/\n 4210188359946578336976868 164966163024902343750000*6^(1/2),\na[13,9]=\n8857742338566725902229518 67695327816457340130391639153070521335485617578/\n30109854138029501101 5469248465465290112505656143757799934635162353515625-\\\n2814045797346 99232141455524604487724159024972527/\n28448191636473798322140232250483 0303192138671875*6^(1/2),\na[13,10]=3154791167297801539564121240521996 85097744239386639023787359107959254802182/\n 13448185050650584 8012587842215515574380212543200894932329128471154748828125-\n \+ 2940396453647872276646068776592292229737651937934623/\n 734546 5058781983710795837429530784777245286520703125*6^(1/2),\na[13,11]=2250 996163406545378616532039018846586217631599453822541/\n 3824913 03797095993563304148204275636433504028320312500,\na[13,12]=\n268934095 7307691853294902388334454003959378146957529866233529251986359392336044 151708949720958809747970514366293458424272174024493/\n9595163860195788 0850056911478087170846689475228048283510540802781519489531905544384278 2227102120493960805649575561796875000000000,\na[14,1]=4734200384802439 1498707976847688893013083074441159779465719863625051668939887702630319 /\n 44802546873926050730401222636656855760802419993852060264615 320801485392456054687500000-\n 86636953098707799112556240282909 2187100493209601/\n 3325522375873672156017711459173673934936523 437500*6^(1/2),\na[14,2]=0,\na[14,3]=0,\na[14,4]=0,\na[14,5]=0,\na[14, 6]=871779321807802447463310035318238762878527157/\n 13444642082 3349498059060773719650573730468750+\n 1076412684809993960818489 75271849857994818/\n 109708279391853190416193591355234868164062 5*6^(1/2),\na[14,7]=49610378635186229280003480511419070548480074351335 4117014/\n 1100997221039563757198196122829921481018902397155761 71875-\n 132993841260619748576931259939030735119154089159937483 1099/\n 6605983326237382543189176736979528886113414382934570312 50*6^(1/2),\na[14,8]=4077407727774763635459845170889116549412313138377 7235229538611989392175193285994266471/\n 1526429054624816210105 8985941588079518256741255377031736357946125713524703979492187500+\n \+ 123767075855296855875080343261298883871499029943/\n 451091 609994276250390378731960660324096679687500*6^(1/2),\na[14,9]=-10522038 6085005564598286490383020684737357490307963727649616187519737937247963 64606986664/\n 38994174250054222540345740003973828622358928296 53375835197340918271556055507659912109375+\n 34654781239483119 64502249611316368748401972838404/\n 25603372472826418489926209 02543472728729248046875*6^(1/2),\na[14,10]=-27843764471262693189365201 135620670490328475323282820219474851621693895769527094334687108984/\n \+ 1225704106628516422200259430060559392943413919302216631780212 1412999357024704596261133984375+\n 57477430027199859868387311 4105472016699241495055292/\n 10493521512545691015422624899329 69253892183788671875*6^(1/2),\na[14,11]=-34241134351848245624232809437 676889009431930503529853032576417589898516/\n 561334782435865 1981100985009024281007603230062439942682713165283203125,\na[14,12]=\n- 3432044375893932378102368568052286501033850910516999202088532705211633 4327939205477028009615324380084018837373418546889726396053346001639386 10268855705742764072609/\n11431741063416822609716476904105672921439261 9865092777892082326746111137127590759980171487016581339414751906821093 1766844494994616580258435518181434575195312500000,\na[14,13]=474693087 6023919335079451612726717649218264199984/\n 185920655384070497 55200144388134089346432755594877,\na[15,1]=-25188329249258825443748527 038142409879923012133738985313265430932280250855708601/\n 1137 0641325574469312056961874077298550827642308774647316995717036347558064 286250+\n 1234273058981860170179592598535508631343082535549881 956/\n 2105633771469628744518390642968552144069898845895808125 *6^(1/2),\na[15,2]=0,\na[15,3]=0,\na[15,4]=0,\na[15,5]=0,\na[15,6]=-54 821142119685055562477216205428613949905430396088/\n 3959439837 009461289085587746748097947393101278095-\n 1511276753825982856 072891469504471256664975925000/\n 4038628633749650514867299501 6830599063409633036569*6^(1/2),\na[15,7]=-6092242427406159991860352404 9390657305431262635197540405697952/\n 648486174748903216977458 4624759953148531564032417461909516875+\n 845585757516359787331 09961893984238786929550462615375699341616/\n 19454585242467096 509323753874279859445594692097252385728550625*6^(1/2),\na[15,8]=-11611 8147575045169733222875835719955334334798191459879782123534889390467935 109772/\n 8810626901954835245672275131295870892503713957512170 681453300814988417642493125-\n 1763247227116943100256560855050 72661620440362221411708/\n 28561940671982910748577120704204013 3465420149964555625*6^(1/2),\na[15,9]=17769448722513898342276837490665 097286927607247073335618566987143467294900183033216/\n 25512170 0813788961505634214608456186712248516359661928371995774241875102950635 6875-\n 1974836894370976272287348157656813810148932056879811129 6/\n 6484554262322259071286545935997129135111813687175650625*6^ (1/2),\na[15,10]=97659266139124074818193264801929547781659926543786381 510190954184218570746215033823993530000000/\n 1856007665446970 6205963482908787056850812308205603127326855360961727608242796551101182 080033599-\n 8529708461178212247491113136307890005888802522460 7913745000000/\n 692106594502013938431667467229540363263383556 49915383851733911*6^(1/2),\na[15,11]=473389749049752963256114649231353 822492912259509649519870869750525/\n 3541244088236034179979884 2428365422941216508121322622479260846291,\na[15,12]=\n3335143924515843 8248073494056784144097872912773415904536400728387690334563968394114702 4141088075051581063851164687328534582028999667484887185315457065591428 95903144848764637/\n23166110253272874277148020113222528860907939049899 0062159236562764909757810216357219050223242549060677331231066559342498 2745744299371285598588298606088543376742054644818966,\na[15,13]=-38714 992656958413389743252726016897599283911682945255636643554687500000/\n \+ 4854049492697158749929458938257221203616913542987790170234752 1300421767,\na[15,14]=148002502009403237171246161756412612351192957957 68814717803955078125/\n 33565577125141877760287380588632421223 433194078156948298488471160489,\na[16,1]=23057856960863975610808581869 39897173645641331085041313944389849986584101287/\n 617508244345 282265819087370078275122671246164669900462139876057008239440000-\n \+ 85404623305589712632165905233974183137607899140719/\n 12482 2287169084833758410283469525117460541643292500*6^(1/2),\na[16,2]=0,\na [16,3]=0,\na[16,4]=0,\na[16,5]=0,\na[16,6]=102903996961580448264190625 267026062654799259083/\n 50463980848900048574816299996733204388 19484730+\n 41320925487304219313300272052128374567081128125/\n \+ 51473260465878049546312625996667868475958744246*6^(1/2),\na[16, 7]=62798443349876457506718920843975661399949564598018488144466/\n \+ 4132553498782573324058263582553715220777051359780141380625-\n \+ 72308807081932961554425711089716771013571419950657300729103/\n \+ 12397660496347719972174790747661145662331154079340424141875*6^(1/2), \na[16,8]=179490914212648256439084852292422555322146901975147054495929 7614654661293377/\n 5259648119399426443560162610975298867467969 1644275456716633975785978672500+\n 1220066047222710180459512931 9139169019658271305817/\n 1693156145655995911520770934405657826 3397760602500*6^(1/2),\na[16,9]=-2775244732780109667342845612394739319 115662636371477300455747022423270475907256/\n 2284171536755840 29725018045422706955827996328208181619436454383447149337555625+\n \+ 341618493222358850528663620935896732550431596562876/\n 961 01338378773357469245211954911505447551097205625*6^(1/2),\na[16,10]=-27 6805546597690166235309791767274482512922443107699960153421908190689705 56083063125000/\n 3299557777429648960576561382256606844677258 438797072955341581354051375036522231471437+\n 442655212757989 5373479670356100179759944766558141730312500/\n 30771137386673 20707748877199804636746494977000658967987677*6^(1/2),\na[16,11]=-29260 3171929706291053929402159930330736639136252680853622275/\n 15 473622826279161150227076887290262443510550964275858143964,\na[16,12]=- 9815717129569106988569302193220999343824932084582093647596086931754666 0986625941530952589885163051657947397448735398290696172035235091366822 16933020431/2864769911709341530761466410944021718019372500685965429310 2867866950176225328769329439768932779738811385458811343006393940507197 9092547998950955940992,\na[16,13]=272949114470983790579914876665078253 2906050298971406518524169921875/\n 215811588862213947314277581 2109447802920656149243127309253686951469,\na[16,14]=0,\na[16,15]=0, \n \nb[1]=8198160366203173411119943711500331/5610575793840858601672778471 28765528,\nb[2]=0,\nb[3]=0,\nb[4]=0,\nb[5]=0,\nb[6]=0,\nb[7]=0,\nb[8]= -455655493073428838813281446213740000000/11638080111509105612404642258 37312497869,\nb[9]=19965163648706008081135075746915614720000000/863944 04190537086868394686205782432516544599,\nb[10]=89231107919981418705566 970804343750000000000000000000000/\n 6999798709883356744455946798 56445060562597693583175985391,\nb[11]=47104273954945906713184913871143 492/209684639122339601934631113492763467,\nb[12]=208450044214045004640 1058474079675065083217679837038308422635129473073119667364731106233097 2740734737279503119387627146381678677156136042524139311907482802844083 /366708498911363730202382253282651002506051447185019263051409665867580 5484760468146633610316928475598775354232120246237155412059385814975553 9878561976786592389608,\nb[13]=605303728214230650979591128690917968750 0000000/103899257350518063455290077573775162739725126989,\nb[14]=91740 1104920993498360358406096725463867187500/67242498159113466533157907374 53607382989551463,\nb[15]=2585449557665268951371699596493957/845743451 60764140163208606048427531,\nb[16]=0,\n\n`b*`[1]=552562031208180939317 806684253/27669654257734667858523344041464,\n`b*`[2]=0,\n`b*`[3]=0,\n` b*`[4]=0,\n`b*`[5]=0,\n`b*`[6]=0,\n`b*`[7]=0,\n`b*`[8]=221223388631423 597589898601690000000/100946136798587090054685074667127461,\n`b*`[9]=1 01835408791305297984657812561920000000/1149763833200743759976506650241 312100139,\n`b*`[10]=1313720309077630014453239843750000000000000000000 /\n 11518201923215510989126466531107437037395719117133,\n`b*`[ 11]=4833611232701440504508086151728/19081321241454145230196661524503, \n`b*`[12]=-2129662374582324648106919795703373645353118273066742230724 1727310258139647124736471440105992066698253827193591131962388577090255 12340589957/1035543739272367080885190546201097218891268728118207332592 5959875548518829722926708817941783800977165831230634852874357936574258 89233080568,\n`b*`[13]=1084761591753640855844358063964843750000000/318 2895486031249071938549691320502488733423,\n`b*`[14]=0,\n`b*`[15]=0,\n` b*`[16]=1839190071060649887127895100784/380451395235106343514208754153 97\}:" }}}{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 stability function R for the 16 stage, order 9 sche me is given (approximately) as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "expand(subs(ee,Stabili tyFunction(9,16,'expanded'))):\nmap(convert,evalf[28](%),rational,24): \nR := unapply(%,z):\n'R(z)'=R(z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# /-%\"RG6#%\"zG,B\"\"\"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)*&#F)\"'!)GOF)*$)F'\"\"*F)F)F)*&#\"*(eQ1J\"1z-RT6b/6F)*$)F'\" #5F)F)F)*&#\")>q-f\"1tJX!o\\hv\"F)*$)F'\"#6F)F)F)*&#\")xdc:\"2ML5_=K*y mF)*$)F'\"#7F)F)!\"\"*&#\")d?b5\"1%p,[>ZoT$F)*$)F'\"#8F)F)Fgo*&#\")uMz 6\"2 " 0 "" {MPLTEXT 1 0 28 "z0 := new ton(R(z)=1,z=-4.5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$!+BF " 0 "" {MPLTEXT 1 0 304 "z0 := newton(R(z)=1,z=-4.5):\np1 := plot([R(z),1],z= -5.19..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= [-5.19..0.49,-0.07..1.47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 253 253 {PLOTDATA 2 "6+-%'CURVESG6$7Z7$$!3Q++++++!>&!#<$ \"3F@A3Ren6uF*7$$!3QML3T![!f^F*$\"3?wOo+2;MoF*7$$!3Ynm;#3'4G^F*$\"3')) R#[xfa)H'F*7$$!3a++DBT9(4&F*$\"3VQjuF[--eF*7$$!3kLLLk@>m]F*$\"3X&yCQ)G '>M&F*7$$!3E+]U'*)HB,&F*$\"369\"=,:L5i%F*7$$!3!pm;&GwYe\\F*$\"3-h>'G%y ;\"*RF*7$$!3s+](\\(Q*y*[F*$\"3yW%[T%zQyLF*7$$!3nLLV@,KP[F*$\"3%3$oi+\" ))Q&GF*7$$!3'RLLd%[MwZF*$\"3DrT@M'HJS#F*7$$!3NLL.q&p`r%F*$\"34$[M\\xA$ >?F*7$$!3E+]<*4%oaYF*$\"3a8&Rd*>n%p\"F*7$$!3;nmJG')*Rf%F*$\"3`/&*)ycl# >9F*7$$!3uLLyGAZ\"[%F*$\"3!)pE8'GNg,\"F*7$$!3%3+])fw&\\O%F*$\"3`rQ%f7Z &Qr!#=7$$!3$QL$)f7eWC%F*$\"3]lfUXzzC\\F]p7$$!3A++lN]MCTF*$\"3)e]*=lk$) )Q$F]p7$$!3ummYeRz+SF*$\"3m\"HN)omB4BF]p7$$!3_LLV-,(>*QF*$\"3+L!*zc)\\ 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{TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 49 "The following picture shows the stabilit y region." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1456 "R := z ->add(z^j/j!,j=0..9)+\n 310638587/110455 1141390279*z^10+59027019/1756149680453173*z^11-\n 15565777/6678932 1852103334*z^12-10552057/3416847194801694*z^13+\n 11793474/4748367 3297105517*z^14+387677/313167245538822961*z^15:\npts := []: z0 := 0:\n for ct from 0 to 320 do\n zz := newton(R(z)=exp(ct*Pi/20*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np1 := plo t(pts,color=COLOR(RGB,.4,.2,.05)):\np2 := plots[polygonplot]([seq([pts [i-1],pts[i],[-2.3,0]],i=2..nops(pts))],\n style=patchnogrid, color=COLOR(RGB,.95,.45,.1)):\npts := []: z0 := 1.1+4.7*I:\nfor ct fro m 0 to 60 do\n zz := newton(R(z)=exp(ct*Pi/30*I),z=z0):\n z0 := zz :\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np3 := plot(pts,color =COLOR(RGB,.4,.2,.05)):\np4 := plots[polygonplot]([seq([pts[i-1],pts[i ],[0.99,4.67]],i=2..nops(pts))],\n style=patchnogrid,color=CO LOR(RGB,.95,.45,.1)):\npts := []: z0 := 1.1-4.7*I:\nfor ct from 0 to 6 0 do\n zz := newton(R(z)=exp(ct*Pi/30*I),z=z0):\n z0 := zz:\n pt s := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np5 := plot(pts,color=COLOR(R GB,.4,.2,.05)):\np6 := plots[polygonplot]([seq([pts[i-1],pts[i],[0.99, -4.67]],i=2..nops(pts))],\n style=patchnogrid,color=COLOR(RGB ,.95,.45,.1)):\np7 := plot([[[-5.29,0],[1.89,0]],[[0,-5.29],[0,5.29]]] ,color=black,linestyle=3):\nplots[display]([p||(1..7)],view=[-5.29..1. 89,-5.29..5.29],font=[HELVETICA,9],\n labels=[`Re(z)`,`Im (z)`],axes=boxed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 501 551 551 {PLOTDATA 2 "6/-%'CURVESG6$7]_l7$$\"\"!F)F(7$F($\"3++++Fjz q:!#=7$F($\"3)******Rl#fTJF-7$F($\"3<+++!)*)Q7ZF-7$$\"3R+++C*>xh)!#G$ \"3))*****pI&=$G'F-7$$\"3]+++u%)3w(*!#F$\"3C+++N;)R&yF-7$$\"3))*****\\ 8uqR(!#E$\"3M+++gzxC%*F-7$$\"35+++`yO%>%!#D$\"3)******\\Ud&*4\"!#<7$$ 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hz-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6%%&Re(z)G%&Im(z)G-Fihz6#%( DEFAULTG-%*AXESSTYLEG6#%$BOXG-%(SCALINGG6#%,CONSTRAINEDG-%%VIEWG6$;$!$ H&F\\]p$\"$*=F\\]p;Fajz$\"$H&F\\]p" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 30 "interval of absolute \+ stability" }{TEXT -1 89 " (or stability interval) is the intersection of the stability region with the real line." }}{PARA 0 "" 0 "" {TEXT -1 59 "For this scheme the stability interval is (approximately) " } {XPPEDIT 18 0 "[-4.5214, 0];" "6#7$,$-%&FloatG6$\"&9_%!\"%!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 289 "We can distort the boundary curve horizontally by taking the 11th root of the real part of points along the curve. In this way we see that the largest interval on the nonnegative imaginary axis th at contains the origin and lies inside the stability region is [ 0, 2 .75 ] approximately. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 485 "R := z ->add(z^j/j!,j=0..9)+\n \+ 310638587/1104551141390279*z^10+59027019/1756149680453173*z^11-\n \+ 15565777/66789321852103334*z^12-10552057/3416847194801694*z^13+\n \+ 11793474/47483673297105517*z^14+387677/313167245538822961*z^15:\nDigi ts := 25:\npts := []: z0 := 0:\nfor ct from 0 to 95 do\n zz := newto n(R(z)=exp(ct*Pi/100*I),z=z0):\n z0 := zz:\n pts := [op(pts),[surd (Re(zz),11),Im(zz)]]:\nend do:\nplot(pts,color=COLOR(RGB,.9,.4,0),thic kness=2,font=[HELVETICA,9]);\nDigits := 10:" }}{PARA 13 "" 1 "" {GLPLOT2D 287 308 308 {PLOTDATA 2 "6(-%'CURVESG6#7\\q7$$\"\"!F)F(7$$\" :kmvMpq))*Rfodw!#F$\":_EYQKz*e`EfTJ!#E7$$\":!)=!p<;H?VkbQ9F0$\":)oUpZ' ezrI&=$G'F07$$\":y5ySxG()y9h53#F0$\":no)orz$p2'zxC%*F07$$\":x&='p=E#Q, B]0FF0$\":8L\"*)H#F@7$$\":l9&p=O(ofC l*R*p&F0$\":#=U3#)Q3B)QLu#GF@7$$\":dL$49XB;V^5'G'F0$\":U*oT;r)*e`EfTJF @7$$\":8))*)*Hg-`W4SroF0$\":4n9r-**[*=>vbMF@7$$\":ihei`Q'pbX+cuF0$\":: #\\'>aI3V=6*pPF@7$$\":v)Q\"fU2+V0:0/)F0$\":Su)pZI!o'\\/2%3%F@7$$\":A93 g)G#\\Y'\\VD')F0$\":$)3&4.d&G]rH#)R%F@7$$\":d^;WQinI>'=6#*F0$\":(GN!31 c!R!)*)Q7ZF@7$$\":$ebB&>=X53A\")z*F0$\":fVI%GW^vX#[l-&F@7$$\":q4G2nkTC o`'Q5F@$\":x:3!=p*\\wn'[l& F@7$$\":/!*zI2E_0!G(o:\"F@$\":dN/]Dm&)=/E!pfF@7$$\":(QLVdO8.&)eG;7F@$ \":v1wn_['G2`=$G'F@7$$\":'[@miTu7-(=fF\"F@$\":g?]Z6j2FdWtf'F@7$$\":;=K rBb,'3-yN8F@$\":^v=$[\"*[:QQ]6pF@7$$\":OAuR4gbW]weR\"F@$\":6V^80+LO5jc A(F@7$$\"::*eSPO#e876iX\"F@$\":Hm9x=(H9pB#)RvF@7$$\":A[Z;[_/\\F&y;:F@$ \":&p^\\!)=unM;)R&yF@7$$\":U3V3Ni6c3)fx:F@$\":)e9m_CG@+49o\")F@7$$\":: LuW-&\\csnkQ;F@$\":xF#[-Itpl,I#[)F@7$$\":VVZJ^m8H#o#**p\"F@$\":6E.I$)3 K5Vfkz)F@7$$\":*fT!p<@z)R@VhU+crN/'p=16*F@7$$\":wj'3\"\\*o_r^ :B=F@$\":,\"[@PyLWgzxC%*F@7$$\":i[[r%e#>A-(3&)=F@$\":*G'e(eS5xBs$*Q(*F @7$$\":]f!)o!yfR[v@Z>F@$\":dHJ3HSJ&['4`+\"!#C7$$\":qVq!*fM\"4Sa`4?F@$ \":bWq\\mv*RubsO5Fgu7$$\":l\"R\\4CgR*GG?2#F@$\":S'>\"*)=\\B)*\\T\"o5Fg u7$$\":-U()eG&p)ok#oM@F@$\":&eC0k]EdCub*4\"Fgu7$$\":`*eD?gF)o1%[(>#F@$ \":2a-^&Q$>$[L(48\"Fgu7$$\":3dEnxsFA90Q>\"Fgu7$$\":\"Q:Ktf%)pi>h'Q#F@$\":C\" Q/2hc)z5@_A\"Fgu7$$\":bLG/%\\lyi#Q)\\CF@$\":%e&3=js?5-PmD\"Fgu7$$\":%R _,ug0UJ`78DF@$\":j'=TAxPHGH0)G\"Fgu7$$\":hdmR%)H-29`kd#F@$\":/AbA\"zNo F)o%>8Fgu7$$\":QE2j(>0u62!)REF@$\":x#)\\^#oNR;Z)3N\"Fgu7$$\":5K*fdH%>T 3YJq#F@$\":b/N$3row!f+BQ\"Fgu7$$\":5exs%o]^viYmFF@$\":)\\CBo1$egW;PT\" Fgu7$$\":t)f\"e)4#p3BP(HGF@$\":)>$o-#*4bhFK^W\"Fgu7$$\":bHv$=?EMgP$H*G F@$\":Q.J^ByOK2[lZ\"Fgu7$$\":\\bL@gZd]YHg&HF@$\":$)Rz$[1MOFQ'z]\"Fgu7$ $\":.dFg>h&R\\m**=IF@$\":j@9w'f'zf_z$R:Fgu7$$\":wtefoyO$=i!=3$F@$\":&= %\\9k)3K`^zq:Fgu7$$\":!HL=&)y*))zbFW9$F@$\":a4uqQB0(*o5Ag\"Fgu7$$\":)e =H&)*)y1(RGo?$F@$\"::s.O;G%p5hiL;Fgu7$$\":is\"z\")G*3mku*oKF@$\":%o(H0 c_-hQT]m\"Fgu7$$\":G*\\Z9P4i9-$3L$F@$\":RtPyU+M)ykX'p\"Fgu7$$\":T`:U?r *o\"zcBR$F@$\":-ag'>&>ANMrys\"Fgu7$$\":'[O321dn-O^`MF@$\":i9-$)e4Z\\#f GfoI7VJMOF@$\":\"H9p(\\N?\"Hs_`=Fgu7$ $\":()*)p4#[P>aQ_$p$F@$\":]m;oq$[)y&)R\\)=Fgu7$$\":*)pSOW**R5$36_PF@$ \":.$o)[H`LZq^j\">Fgu7$$\":JT[k=(=J5;,5QF@$\":H:G0iundgix%>Fgu7$$\":7: yGo91Socr'QF@$\":\")))R$)R$GFnBFgu7$$\":eU'[E#y>2wpM#RF@$\":AY%Q7E] ;h2e5?Fgu7$$\":4#*3Q(=5+sm')yRF@$\":.k9I)\\h*)>v)>/#Fgu7$$\":A4@.n%G;X SDLSF@$\":[<+$=!Rx2L#Rt?Fgu7$$\":\\i%\\N%>Z%zw_'3%F@$\":Q@Bc'>]oH[z/@F gu7$$\":;)=K]2/BT0dQTF@$\":X.vx**)R6%f%>O@Fgu7$$\":;8$p\"**e\\TE]#*=%F @$\":-\\;3.g\\b8\"fn@Fgu7$$\":T*3%zs7Ru*fTQUF@$\":(pQ$)>^L8\"*Q)*)>#Fg u7$$\":\\mKX*Q#fQQ%*eG%F@$\":l1LSQy1Y@s.B#Fgu7$$\":'GLd;PX)oN%[JVF@$\" :C#\\!3[!o_m`vhAFgu7$$\":Gnd&==t03.&\\P%F@$\":+1*on2zv.D8$H#Fgu7$$\":8 Ll5L@u/'H,;WF@$\":z<&3R\"*4EoE]CBFgu7$$\":OH[;v+kD%pLaWF@$\":#4)=,ED:_ xkeN#Fgu7$$\":Mk#yc[sH'\\8&*[%F@$\":O])y\"R#*G0gB'e%F@$ \":[7j\\Fu9())f]7DFgu7$$\":fSP6v'>\")4#QVf%F@$\":Wn<_8[CzB)yVDFgu7$$\" :>;YXp')RL6vCf%F@$\":;Z'\\ba+d%=]]d#Fgu7$$\":n(\\P\")e,mqOExXF@$\":E#> Lw\"\\)=^#*G1EFgu7$$\":x#3tJ&G!R$)=5VXF@$\":bM!*f'ytp8E]PEFgu7$$\":;f4 bo)3M#))\\&zWF@$\":k/=.p([()*=(ooEFgu7$$\":j)3Qhiu&[4cIO%F@$\":\"**=J_ c,0H'R)*p#Fgu7$$\":=&)**p`1ASJ8^6%F@$\":1W&yU3xB.j&4t#Fgu7$$!:*oX\\8zB tBeQ#f$F@$\":F'G`snN\"**HL?w#Fgu7$$!:$pV7$QxCL8h*[VF@$\":\"e>c2SSd;k1$ z#Fgu7$$!:+k0*f.2MFadeYF@$\":Vqi%f8Y8d60CGFgu7$$!:6pmA)R;em4Q\")[F@$\" :q-i4FY:,t#)\\&GFgu7$$!:RZXIlmmz#Q6l]F@$\":YB9P(*Q=%[g&e)GFgu7$$!:HTwa EuL096kA&F@$\":.![el9xKIdm;HFgu7$$!:q1e_j3\"*QqpIP&F@$\":*RDmTCf;,hSZH Fgu7$$!:t/i@n\"ot'=q$4bF@$\":L,I2\\g2[>r!yHFgu-%*THICKNESSG6#\"\"#-%&C OLORG6&%$RGBG$\"\"*!\"\"$\"\"%F]jlF(-%%FONTG6$%*HELVETICAGF\\jl-%+AXES LABELSG6$Q!6\"Fgjl-%%VIEWG6$%(DEFAULTGF\\[m" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 123 "The relevant intersection poin t of the boundary curve with the imaginary axis can be determined more accurately as follows." }}{PARA 0 "" 0 "" {TEXT -1 86 "First we look \+ for points on the boundary curve either side of the intersection point . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "Digits := 15:\nz0 := 2.7*I:\nfor ct from 86 to 89 do \n print(newton(R(z)=exp(ct*Pi/100*I),z=z0));\nend do;\nDigits := 10 :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0RZ.b$o!4\"!#=$\"0;]!H'R)*p# !#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0RA*zefId!#>$\"0rPKIc4t#!# 9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0/$QMj\"fG\"!#>$\"0d8**HL?w#! #9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0%QZ&GXD0\"!#=$\"0/ulTmIz#!# 9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "Th en we apply the bisection method to calculate the parameter value asso ciated with the intersection point." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 " real_part := proc(u)\n Re(newton(R(z)=exp(u*Pi*I ),z=2.7*I))\nend proc:\nDigits := 15:\nu0 := bisect('real_part'(u),u=0 .86..0.89);\nnewton(R(z)=exp(u0*Pi*I),z=2.7*I);\nDigits := 10:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"0@hQG#o$y)!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0T2RfN4m'!#H$\"0_C\\7lpv#!#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, 2.7570];" "6#7$\"\"!-%&FloatG6$\"&qv#!\"%" }{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 97 "The stability func tion R* for the 16 stage, order 8 scheme is given (approximately) as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "expand(subs(ee ,subs(b=`b*`,StabilityFunction(8,16,'expanded')))):\nmap(convert,evalf [28](%),rational,24):\n`R*` := unapply(%,z):\n'`R*`(z)'=`R*`(z);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#/-%#R*G6#%\"zG,@\"\"\"F)F'F)*&#F)\"\"# F)*$)F'F,F)F)F)*&#F)\"\"'F)*$)F'\"\"$F)F)F)*&#F)\"#CF)*$)F'\"\"%F)F)F) *&#F)\"$?\"F)*$)F'\"\"&F)F)F)*&#F)\"$?(F)*$)F'F1F)F)F)*&#F)\"%S]F)*$)F '\"\"(F)F)F)*&#F)\"&?.%F)*$)F'\"\")F)F)F)*&#\"*qi]c%\"0p_4dnya\"F)*$)F '\"\"*F)F)F)*&#\"*'=-.J\"0vGW%[-/SF)*$)F'\"#5F)F)F)*&#\"*E`.:#\"12906t G^9F)*$)F'\"#6F)F)F)*&#\"*zb,B\"\"1S]Jz7/8=F)*$)F'\"#7F)F)!\"\"*&#\")W iF8\"1&HUTH$\\`IF)*$)F'\"#8F)F)F)*&#\"(5L'=\"2FfT2agDZ)F)*$)F'\"#9F)F) F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "W e can find the point where the boundary of the stability region inters ects the negative real axis by solving the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`R*`(z) = -1;" "6#/-%#R*G6#%\"zG,$ \"\"\"!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "z_0 := newton(`R*`(z)=-1,z=- 4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$z_0G$!+VpNzQ!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 322 "z _0 := newton(`R*`(z)=-1,z=-4):\np_1 := plot([`R*`(z),-1],z=-4.39..0.49 ,color=[red,blue]):\np_2 := plot([[[z_0,-1]]$3],style=point,symbol=[ci rcle,cross,diamond],color=black):\np_3 := plot([[z_0,0],[z_0,-1]],line style=3,color=COLOR(RGB,0,.5,0)):\nplots[display]([p_1,p_2,p_3],view=[ -4.39..0.49,-1.47..1.47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 369 264 264 {PLOTDATA 2 "6+-%'CURVESG6$7Y7$$!3o************* Q%!#<$!3aHx27c>EZF*7$$!3Amm\">Z2MO%F*$!3/k7J)[mDQ%F*7$$!3lLL$Q%\\\"oL% F*$!3O,Tx>'R=1%F*7$$!3?++v:CA5VF*$!3509?;ohiPF*7$$!3ummm())HOG%F*$!3c3 Tnard$[$F*7$$!3\")**\\naQNPUF*$!3_$z!>:oWUIF*7$$!3'GL$o@y2\">%F*$!3O]0 Cd[h_EF*7$$!3h**\\s#eN!RTF*$!3[rNfzwnoAF*7$$!3MmmwVL*p3%F*$!3!o[[)z^nN >F*7$$!3Vmm1NhgMSF*$!3m'>*=+OfX;F*7$$!3_mmOE*=A)RF*$!3)y(zx(3]^R\"F*7$ $!3#HL$[$[Uz(QF*$!3%HZ$G***HM&**!#=7$$!3Ammhs[E\"y$F*$!3anBJlP&y=(F^o7 $$!3U***\\j!3;\"o$F*$!3AR?1DCK^]F^o7$$!3Imm\"y.Lwd$F*$!3#o&4jeOeIMF^o7 $$!3a***\\TGPWZ$F*$!3O*z5\\6t\"fAF^o7$$!3)GLLrw(GoLF*$!3O+n*\\$)3NR\"F ^o7$$!3Cmmw97zuKF*$!3MwY@s6x&R)!#>7$$!3W****RTi`pJF*$!3%\\&p\\i[YXQF]q 7$$!3')*****43\\Q1$F*$!3a=80^bSUb!#?7$$!3m****f=***>'HF*$\"3O!4t)zD,#y \"F]q7$$!3Amm\"=76&pGF*$\"3&*pn-[9R>MF]q7$$!3'GLLn*H`fFF*$\"3)4Vs)zw7u \\F]q7$$!3yKL$*)4njm#F*$\"33?_Z2%QK3'F]q7$$!3!)***\\:,$*zb#F*$\"3@zTc/ .(=H !y<0$*F]q7$$!3S***\\E5\"fcAF*$\"3\\`LB6w)y.\"F^o7$$!3'HLL3C>?:#F*$\"3; QrtIpyd6F^o7$$!3sKL)yi*)f0#F*$\"3#)RE&4bOsF\"F^o7$$!3?mm;<(3C&>F*$\"3d 1&*=Y(4#=9F^o7$$!3\\mm\"=E<[%=F*$\"3Ic'R\\***4!e\"F^o7$$!3m***\\Oee6v \"F*$\"3:U8$G%\\bNF^o7$$!3))****>HF]X :F*$\"3Y6&4stZ?8#F^o7$$!3b*****=*zEV9F*$\"3%\\g\\](*e:O#F^o7$$!3$)*** \\ni]VM\"F*$\"3s/EP*z*42EF^o7$$!3y****pd(>XB\"F*$\"3&pOPm$4v4HF^o7$$!3 MLL$p$=$e8\"F*$\"3H(oQ\\$>b6KF^o7$$!3r*****pTh/.\"F*$\"3'GH;W7C%oNF^o7 $$!3qlm;a:!)\\$*F^o$\"3azkvQr$f#RF^o7$$!3r$****f;RfI)F^o$\"3%p*HRSS!zN %F^o7$$!3$RLL)3guBtF^o$\"3TZ_x3,m2[F^o7$$!3o&***\\`62(H'F^o$\"3#))\\kl 7yuK&F^o7$$!3=HLLB)3LH&F^o$\"335!*Rv%*)**)eF^o7$$!36(***\\(oiCC%F^o$\" 3J=d'fPFEa'F^o7$$!3jcmmiHPIKF^o$\"3CyoZL&o%RsF^o7$$!3_jmmR]O&>#F^o$\"3 _h?d%[3*G!)F^o7$$!3kBL$[]F*o6F^o$\"3UO!='eg!o*))F^o7$$!3%)=****>,QdAF] q$\"3YOQT13zw(*F^o7$$\"3Ozmmcej_&)F]q$\"3%o%f4t-H*3\"F*7$$\"3-TLL,V7A= F^o$\"3'fB_^0p)*>\"F*7$$\"330+]%[**H&GF^o$\"3U,(4![4;I8F*7$$\"3C4+]r&y 'RQF^o$\"3ysT8_#)4o9F*7$$\"3!***************[F^o$\"3HPTq?iJK;F*-%'COLO URG6&%$RGBG$\"*++++\"!\")$\"\"!F`]lF_]l-F$6$7S7$F($!\"\"F`]l7$F=Fe]l7$ FGFe]l7$FQFe]l7$FenFe]l7$FjnFe]l7$F`oFe]l7$FeoFe]l7$FjoFe]l7$F_pFe]l7$ FdpFe]l7$FipFe]l7$F_qFe]l7$FdqFe]l7$FjqFe]l7$F_rFe]l7$FdrFe]l7$FirFe]l 7$F^sFe]l7$FcsFe]l7$FhsFe]l7$F]tFe]l7$FbtFe]l7$FgtFe]l7$F\\uFe]l7$FauF e]l7$FfuFe]l7$F[vFe]l7$F`vFe]l7$FevFe]l7$FjvFe]l7$F_wFe]l7$FdwFe]l7$Fi wFe]l7$F^xFe]l7$FcxFe]l7$FhxFe]l7$F]yFe]l7$FbyFe]l7$FgyFe]l7$F\\zFe]l7 $FazFe]l7$FfzFe]l7$F[[lFe]l7$F`[lFe]l7$Fe[lFe]l7$Fj[lFe]l7$F_\\lFe]l7$ Fd\\lFe]l-Fi\\l6&F[]lF_]lF_]lF\\]l-F$6&7#7$$!3))*****H%pNzQF*Fe]l-%'SY MBOLG6#%'CIRCLEG-Fi\\l6&F[]lF`]lF`]lF`]l-%&STYLEG6#%&POINTG-F$6&F[al-F `al6#%&CROSSGFcalFeal-F$6&F[al-F`al6#%(DIAMONDGFcalFeal-F$6%7$7$F]alF_ ]lF\\al-%&COLORG6&F[]lF_]l$\"\"&Ff]lF_]l-%*LINESTYLEG6#\"\"$-%%FONTG6$ %*HELVETICAG\"\"*-%+AXESLABELSG6%Q\"z6\"Q!Ficl-Facl6#%(DEFAULTG-%%VIEW G6$;$!$R%!\"#$\"#\\Fddl;$!$Z\"Fddl$\"$Z\"Fddl" 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 16 stage, order 8 scheme. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1465 "`R*` := z -> add(z^j/j!,j= 0..8)+\n 456506270/154786757095269*z^9+310302186/400402484442875*z ^10+\n 215035326/1451287311051407*z^11-123015579/1813041279315040* z^12+\n 13276244/3053493294142295*z^13+1863310/84725605407415927*z ^14:\npts := []: z0 := 0:\nfor ct from 0 to 280 do\n zz := newton(`R *`(z)=exp(ct*Pi/20*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz) ,Im(zz)]]:\nend do:\np_1 := plot(pts,color=COLOR(RGB,.3,.15,0)):\np_2 \+ := plots[polygonplot]([seq([pts[i-1],pts[i],[-1.9,0]],i=2..nops(pts))] ,\n style=patchnogrid,color=COLOR(RGB,.75,.32,0)):\npts := [] : z0 := 0.9+4.2*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:\np_3 := plot(pts,color=COLOR(RGB,.3,.15,0)):\np_4 := plots [polygonplot]([seq([pts[i-1],pts[i],[.82,4.02]],i=2..nops(pts))],\n \+ style=patchnogrid,color=COLOR(RGB,.75,.32,0)):\npts := []: z0 : = 0.9-4.2*I:\nfor ct from 0 to 60 do\n zz := newton(`R*`(z)=exp(ct*P i/30*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nen d do:\np_5 := plot(pts,color=COLOR(RGB,.3,.15,0)):\np_6 := plots[polyg onplot]([seq([pts[i-1],pts[i],[.82,-4.02]],i=2..nops(pts))],\n \+ style=patchnogrid,color=COLOR(RGB,.75,.32,0)):\np_7 := plot([[[-4.59 ,0],[1.49,0]],[[0,-4.79],[0,4.79]]],color=black,linestyle=3):\nplots[d isplay]([p_||(1..7)],view=[-4.59..1.49,-4.79..4.79],font=[HELVETICA,9] ,\n labels=[`Re(z)`,`Im(z)`],axes=boxed,scaling=constrained );" }}{PARA 13 "" 1 "" {GLPLOT2D 396 515 515 {PLOTDATA 2 "6/-%'CURVESG 6$7e\\l7$$\"\"!F)F(7$F($\"3++++Fjzq:!#=7$F($\"3)******Rl#fTJF-7$$\"3)) *****>s(*>h\"!#F$\"3#******p(*)Q7ZF-7$$\"3++++FGp#y#!#E$\"3M+++f_=$G'F -7$$\"3))******\\%3!oC!#D$\"3W*****f>\")R&yF-7$$\"3*)*****4im*49!#C$\" 3j*****p9vZU*F-7$$\"3))******ySN&y&FF$\"3-+++Igb*4\"!#<7$$\"3%******4% yXzb]^)RFR$\"3&******4p'p89FN7$$\"3B++ +S8*4$\\FR$\"3-+++$*[tq:FN7$$!3_*****\\S/Bs(FR$\"3%******HD2xs\"FN7$$! 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9495689364696426720161112012414227752328969720658987315654179873760357 725235734000399440000000/265558296661064021061274102756797754810572081 5295466925220401896406182066930815061332065006629830010472987876198274 11375675716583283801757*6^(1/2),\na[12,11]=\n1234617126598879151772713 3939660686081047902877786934801487045060626091401956028566128821249812 8400476015695960341952/\n281629106670320674754245209358840703704235147 3078388967410755112208260568290472056143249782532261762750789227161324 61,\na[13,1]=-56042772675322042139227629978042586330633622706053363946 766144416933631/\n 5880854077232319052559012261322343050735211 8534557342666015625000000000+\n 281404579734699232141455524604 487724159024972527/\n 1478009944832743180452316204077188415527 343750000*6^(1/2),\na[13,2]=0,\na[13,3]=0,\na[13,4]=0,\na[13,5]=0,\na[ 13,6]=-1027163900229750356561238237947225332675621517/1792618944311326 64078747698292867431640625000-\n 27452923916412025253731039793 36813513372321/11702216468464340311060649744558385937500000*6^(1/2),\n a[13,7]=-157229999853748227305165773364426925282378072238332930121/\n \+ 36699907367985458573273204094330716033963413238525390625+\n \+ 5757606442802795095318986067317837904184278650664590252101/\n \+ 3523191107326604023034227593055748739260487670898437500000*6^(1/ 2),\na[13,8]=-93114481685939341460159650199040136021338029433258183466 22781285907057/\n 42559708490101242171931354496687399854013133 63005576159362792968750000-\n 84421373920409769642436657381346 3172477074917581/\n 421018835994657833697686816496616302490234 3750000*6^(1/2),\na[13,9]=\n885774233856672590222951867695327816457340 130391639153070521335485617578/\n3010985413802950110154692484654652901 12505656143757799934635162353515625-\\\n281404579734699232141455524604 487724159024972527/\n284481916364737983221402322504830303192138671875* 6^(1/2),\na[13,10]=315479116729780153956412124052199685097744239386639 023787359107959254802182/\n 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748401972838404/\n 2560337247282641848992620902543472728729248 046875*6^(1/2),\na[14,10]=-2784376447126269318936520113562067049032847 5323282820219474851621693895769527094334687108984/\n 12257041 0662851642220025943006055939294341391930221663178021214129993570247045 96261133984375+\n 5747743002719985986838731141054720166992414 95055292/\n 1049352151254569101542262489932969253892183788671 875*6^(1/2),\na[14,11]=-3424113435184824562423280943767688900943193050 3529853032576417589898516/\n 56133478243586519811009850090242 81007603230062439942682713165283203125,\na[14,12]=\n-34320443758939323 7810236856805228650103385091051699920208853270521163343279392054770280 0961532438008401883737341854688972639605334600163938610268855705742764 072609/\n1143174106341682260971647690410567292143926198650927778920823 2674611113712759075998017148701658133941475190682109317668444949946165 80258435518181434575195312500000,\na[14,13]=47469308760239193350794516 12726717649218264199984/\n 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22875835719955334334798191459879782123534889390467935109772/\n \+ 881062690195483524567227513129587089250371395751217068145330081498841 7642493125-\n 176324722711694310025656085505072661620440362221 411708/\n 2856194067198291074857712070420401334654201499645556 25*6^(1/2),\na[15,9]=1776944872251389834227683749066509728692760724707 3335618566987143467294900183033216/\n 2551217008137889615056342 146084561867122485163596619283719957742418751029506356875-\n 19 748368943709762722873481576568138101489320568798111296/\n 64845 54262322259071286545935997129135111813687175650625*6^(1/2),\na[15,10]= 9765926613912407481819326480192954778165992654378638151019095418421857 0746215033823993530000000/\n 185600766544697062059634829087870 56850812308205603127326855360961727608242796551101182080033599-\n \+ 85297084611782122474911131363078900058888025224607913745000000/\n \+ 69210659450201393843166746722954036326338355649915383851733911 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/&F%6#\"\"$F//&F%6#\"\"%F//&F%6#\"\"&F//&F%6#\"\"'F//&F%6#\"\"(F//&F%6 #\"\")#!H+++SP@Y9G8)Q)GM2$\\bcX\"Ipy\\7t$eAk/Ch0\"4:6!3Q;\"/&F%6#\"\"* #\"M+++?Zh:puv]8\"33gq[O;l*>\"M*fWl^KCy0io%Roo3P0>/WR')/&F%6#\"#5#\"Y+ ++++++++++vVV!3(pc0(=9)*>z5J#*)\"Z\"R&)f<$e$p(fi01Xk&)zYfXWnN$))4()z** p/&F%6#\"#6#\"D#\\V6(Q\"\\=8n!f%\\&RF/r%\"EnMw#\\86jM>gRB7RYo4#/&F%6# \"#7#\"\\u$3WG!G[2>JRT_Ug8crny;QYri(Q>J]zstM2us4Li5JZOn'>J2t%H^jA%3$Qq $)zw@$3l]nzSZe5SY+XS@W+X3#\"\\u3'*Q#f'yw>cy)Rbv\\\"eQf?TbrBY-7KUNv()fv %GpJ5OjY\"o/w%[0enem490j#>]=Z901D+^EG`AQ--tj8\"*)\\3nO/&F%6#\"#8#\"O++ ++voz\"4pG6fz4lIU@GPI0'\"Q*)p7D(RF;vPdx+HbM1=0Nd#**Q5/&F%6#\"#9#\"N+v= nQYDn41%e.O)\\$*4#\\5,u\"*\"Oj9b*)HQ2OXP2z:LlY8\"f\")\\Us'/&F%6#\"#:# \"CdR\\'f*pr8&*o_md&\\ae#\"DJvU[gg3K;STwg^MuX)/&F%6#\"#;F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 39 "weights for th e 16 stage order 8 scheme" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "seq(`b*`[i]=subs(ee,`b*`[i]),i=1..16);" }}{PARA 12 " " 1 "" {XPPMATH 20 "62/&%#b*G6#\"\"\"#\"?`Uo1yJR4=37.iDb\"Ak9/WL_eymMx Da'pw#/&F%6#\"\"#\"\"!/&F%6#\"\"$F//&F%6#\"\"%F//&F%6#\"\"&F//&F%6#\" \"'F//&F%6#\"\"(F//&F%6#\"\")#\"E+++!p,')*)*e(fB9j)QB7A\"Ehu7nY2&oa+4( e)zOh%45/&F%6#\"\"*#\"H+++?>c7yl%)zH08z3a$=5\"IR,578C]m]w*fPu+K$Qw\\6/ &F%6#\"#5#\"R+++++++++]P%)RKX9+jx!4.s88\"SLr6>dRPqV26`mk7*)4^:K#>?=:\" /&F%6#\"#6#\"@G<:'33X]S9qK7hL[\"A.X_hm>I_9a9C@83>/&F%6#\"#7#!fsd**eSB^ D!4x&)Qi>8\"f$>FQD)pm?*f5S9ZOZ7Z'R\"e-JFp5[YK#euBmH @\"fso03L#*)eUdOzNuG&[jI7$e;x4!QyTz\")3n#HsH)=&[b()ff#fKt?=\"G(o7*)=s4 ,ia!>&)33nBFRPaN5/&F%6#\"#8#\"L+++]P%['R1eV%e&3k`2\\7.'[&*G=$/&F%6#\"#9F//&F%6#\"#:F//&F%6#\"#;#\"@%y+^*y7())\\11r +>R=\"A(R:a(3U^Vj5N_R^/Q" }}}{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 30 "#===== ========================" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 15 "Sharp 's scheme" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 " " 0 "" {TEXT -1 52 "This scheme is based on a scheme by Philip W. Shar p." }}{PARA 0 "" 0 "" {TEXT -1 79 "See: Journal of Applied Mathematics & Decision Sciences, 4(2), 183-192 (2000), " }}{PARA 0 "" 0 "" {TEXT -1 95 " \"High order explicit Runge-Kutta pairs for ephemerides o f the Solar System and the Moon\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 15 "The nodes are: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[2] = 1/50" "6#/&%\"cG6#\"\"#*&\"\"\"F )\"#]!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3] = 3837236/48429375+ 1031368/145288125*sqrt(6);" "6#/&%\"cG6#\"\"$,&*&\"(Os$Q\"\"\"\")v$H%[ !\"\"F+*(\"(o8.\"F+\"*D\")GX\"F--%%sqrtG6#\"\"'F+F+" }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "c[4] = 1918618/16143125+515684/48429375*sqrt(6);" "6 #/&%\"cG6#\"\"%,&*&\"(='=>\"\"\"\")DJ9;!\"\"F+*(\"'%o:&F+\")v$H%[F--%% sqrtG6#\"\"'F+F+" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5] = 14/45" "6#/ &%\"cG6#\"\"&*&\"#9\"\"\"\"#X!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 " c[6] = 156/625+26/625*sqrt(6);" "6#/&%\"cG6#\"\"',&*&\"$c\"\"\"\"\"$D' !\"\"F+*(\"#EF+F,F--%%sqrtG6#F'F+F+" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7] = 156/625-26/625*sqrt(6);" "6#/&%\"cG6#\"\"(,&*&\"$c\"\"\"\"\"$D '!\"\"F+*(\"#EF+F,F--%%sqrtG6#\"\"'F+F-" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[8] = 52/125" "6#/&%\"cG6#\"\" )*&\"#_\"\"\"\"$D\"!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[9] = 39/ 125" "6#/&%\"cG6#\"\"**&\"#R\"\"\"\"$D\"!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[10] = 21/200" "6#/&%\"cG6#\"#5*&\"#@\"\"\"\"$+#!\"\" " }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[11] = 280/477" "6#/&%\"cG6#\"#6* &\"$!G\"\"\"\"$x%!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[12] = 3658 227035053715/5349704719299032" "6#/&%\"cG6#\"#7*&\"1:P0NqAeO\"\"\"\"1K !*H>Zq\\`!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[13] = 247/281" "6# /&%\"cG6#\"#8*&\"$Z#\"\"\"\"$\"G!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[14] = 229/250" "6#/&%\"cG6#\"#9*&\"$H#\"\"\"\"$]#!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[15] = 1" "6#/&%\"cG6#\"#:\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[16] = 1" "6#/&%\"cG6#\"#;\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "#---------------------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "checking the combined scheme" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "coefficients of the combined sche me " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13775 "ee := \{c[2]=1/50,\nc[3]=3837236/48429375+1031368/ 145288125*6^(1/2),\nc[4]=1918618/16143125+515684/48429375*6^(1/2),\nc[ 5]=14/45,\nc[6]=156/625+26/625*6^(1/2),\nc[7]=156/625-26/625*6^(1/2), \nc[8]=52/125,\nc[9]=39/125,\nc[10]=21/200,\nc[11]=280/477,\nc[12]=365 8227035053715/5349704719299032,\nc[13]=247/281,\nc[14]=229/250,\nc[15] =1,\nc[16]=1,\n\na[2,1]=1/50,\na[3,1]=-24000387317036/281448523546875- 5917264532296/281448523546875*6^(1/2),\na[3,2]=46300580261936/28144852 3546875+7915204837696/281448523546875*6^(1/2),\na[4,1]=959309/32286250 +128921/48429375*6^(1/2),\na[4,2]=0,\na[4,3]=2877927/32286250+128921/1 6143125*6^(1/2),\na[5,1]=2826523628723851/5953434698904030-68459492317 475/595343469890403*6^(1/2),\na[5,2]=0,\na[5,3]=-704240024458145/39689 5646593602+91277530807085/198447823296801*6^(1/2),\na[5,4]=95892564222 5180/595343469890403-205373100103780/595343469890403*6^(1/2),\na[6,1]= 376341108/9406484375+207933466/65845390625*6^(1/2),\na[6,2]=0,\na[6,3] =0,\na[6,4]=4343545768844529/27892881885795625+469265141246109/2789288 1885795625*6^(1/2),\na[6,5]=1559927818449/28957835234375+4382126882523 /202704846640625*6^(1/2),\na[7,1]=11781705468/235162109375+2328587014/ 1646134765625*6^(1/2),\na[7,2]=0,\na[7,3]=0,\na[7,4]=23459106068523828 440829/354298872323611753203125+7870375504052283205581/354298872323611 753203125*6^(1/2),\na[7,5]=146263465360621089/7558718942052734375-1881 455818308499953/52911032594369140625*6^(1/2),\na[7,6]=9444124356888/82 889304453125-2459298027368/82889304453125*6^(1/2),\na[8,1]=52/1125,\na [8,2]=0,\na[8,3]=0,\na[8,4]=0,\na[8,5]=0,\na[8,6]=208/1125-13/1125*6^( 1/2),\na[8,7]=208/1125+13/1125*6^(1/2),\na[9,1]=741/16000,\na[9,2]=0, \na[9,3]=0,\na[9,4]=0,\na[9,5]=0,\na[9,6]=2301/16000-897/32000*6^(1/2) ,\na[9,7]=2301/16000+897/32000*6^(1/2),\na[9,8]=-351/16000,\na[10,1]=3 5291978967/748709478400,\na[10,2]=0,\na[10,3]=0,\na[10,4]=0,\na[10,5]= 0,\na[10,6]=23154511989/149741895680+39398793/1772093440*6^(1/2),\na[1 0,7]=23154511989/149741895680-39398793/1772093440*6^(1/2),\na[10,8]=-6 251205429/149741895680,\na[10,9]=-981041103/4679434240,\na[11,1]=16015 89807329134144752443/16639785968494158002257920,\na[11,2]=0,\na[11,3]= 0,\na[11,4]=0,\na[11,5]=0,\na[11,6]=-1736562342312744743536201/1109319 064566277200150528-\n 360257484908262597335743/511993414415204 861607936*6^(1/2),\na[11,7]=-1736562342312744743536201/110931906456627 7200150528+\n 360257484908262597335743/51199341441520486160793 6*6^(1/2),\na[11,8]=512032742176678555764127/369773021522092400050176, \na[11,9]=248233526294563631278471/103998662303088487514112,\na[11,10] =-3/20,\na[12,1]=-1319870176087866963572254233875946356127193892061286 06880670434178321331969627889057541436355642743061150672386594396559/3 1875392608799555501514792620161201024044722829578946248679811622147609 3939683123897279961564118214685494052121351290880,\na[12,2]=0,\na[12,3 ]=0,\na[12,4]=0,\na[12,5]=0,\na[12,6]=\n-58103861922516087620385683462 9458675128926705143465192450716448466169075797359178616021045291080972 121429188543592047/101191722567617636512745373397337146108078485173266 4960275549575306273314094232139356444322425772110112679530543972352+\n 7134827980789881496508823372973790665635109665954131159198195323855274 4805776480028282602780942939988708855996996031/51893191060316736673202 7555883780236451684539350084595013102346310909391843195968900740678167 06262057060488745844736*6^(1/2),\na[12,7]=\n-5810386192251608762038568 3462945867512892670514346519245071644846616907579735917861602104529108 0972121429188543592047/10119172256761763651274537339733714610807848517 3266496027554957530627331409423213935644432242577211011267953054397235 2-\n713482798078988149650882337297379066563510966595413115919819532385 52744805776480028282602780942939988708855996996031/5189319106031673667 3202755588378023645168453935008459501310234631090939184319596890074067 816706262057060488745844736*6^(1/2),\na[12,8]=-18935700826260772432168 3086336517345228379250897103291049044350530935228180690663776657891613 652665009511679250229667441/104902085728430283184879370421906174798708 0296296196008818986393067503335611020651132847280914717087483477779997 25133824,\na[12,9]=-16183509927928156539922841522541118273994265340148 47245801101845172567304269189800544372100050869595166981551925667441/1 9637518660778297585754649024920739916598981028937029385347383945787366 501641192454385997632075140011874187139618963456,\na[12,10]=6883437842 7149827544141552835305430278000101566001470691198893507717914313664393 29656536871565378282089012991331513/1827181489551794784669860898707808 3524232186538856420801802422529600115450739992009460663708366411323198 80849653760,\na[12,11]=11559027144071691256623556623388974609716247980 4636463234298604185457969653794053637008425503953091180886565/31536133 3249071836330411464879245754163964468656963767284078726004630975015921 697525870414526347596936773632,\na[13,1]=52151747835589184079975834686 35543407988332719241764605769949554629/\n 202831326132148120646 85094275151111714651171227532533713038580121600,\na[13,2]=0,\na[13,3]= 0,\na[13,4]=0,\na[13,5]=0,\na[13,6]=2843598186227456480865065344408178 581293412110128603/\n 79207505317500213926584433527282071629335 5019960320\n -1822707089022686744784094266679051232342258554425 7/\n 121857700488461867579360666965049340968208464609280*6^(1/2 ),\na[13,7]=2843598186227456480865065344408178581293412110128603/\n \+ 792075053175002139265844335272820716293355019960320+\n 182 27070890226867447840942666790512323422585544257/\n 121857700488 461867579360666965049340968208464609280*6^(1/2),\na[13,8]=932682946442 2062118248457481351539504275339476759467047326605595685633/\n 4 901901791858228863857691041029309678010355547895721285919177263022080, \na[13,9]=-74160415509054246685621323607237420625123561706830476231646 5738169791/\n 141551673163321136844445993892555326037025917405 403892742525852712960,\na[13,10]=-605850486644121965559554861876248539 9974773685307046001179355536003/\n 22522757208153961727264006 94965157641073696835574259179818290290400,\na[13,11]=-7291704718646518 3128180555150230405657138451692847535142343993/\n 44661747288 016218276854771442831738093234145203222656783563600,\na[13,12]=2736153 9205409276437741331476352964869466609155582532859837420204888872968492 41173151960647763453239551016003889152/2485672110698341015290264470463 9392039558692496183754067871690180096884577498661778268011927103452628 47046284166825,\na[14,1]=196143162589031568706314157581823240552254589 8155499982338718373117379429883/\n 4800566471670774299905935680 55406093586176318669944422673481728000000000000,\na[14,2]=0,\na[14,3]= 0,\na[14,4]=0,\na[14,5]=0,\na[14,6]=-868852560614631553002241458034639 2155721271039/\n 223867381187544331818146076024811762483200000 00-\n 10256190098435854298655077997613296122112139953/\n \+ 1148037852243817086246902953973393653760000000*6^(1/2),\na[14,7]=-8 688525606146315530022414580346392155721271039/\n 2238673811875 4433181814607602481176248320000000+\n 102561900984358542986550 77997613296122112139953/\n 11480378522438170862469029539733936 53760000000*6^(1/2),\na[14,8]=-108151392092290424953498836380059772609 736403739434481043071361807712075869481/\n 8600735495194563448 316261478331353993230137106756553600705167485829120000000,\na[14,9]=68 3210554257935462600257975958139742203919396113084127371502375524416129 719/\n 26895337200565243662247103690698994332502640106760065066 162305761280000000,\na[14,10]=-125971034051203704183074450363446847441 594334546885083244594242327104115033/\n 506604993469836348869 8655054901069679084758735799331062593807151200000000,\na[14,11]=432233 8495495152743252505005837177994220267688026960252214552638944423/\n \+ 2368676257875084221529581671796767575351429990009243576305000000 00000,\na[14,12]=-8868241439418361942544164786624338811291728923916146 3940944492930492112547171652363240146123589908870567811533658125375935 101390832/940510477623017606720238368944449968423863982309765008995433 3383377958770491999076242219980453370248022825420814384818872314453125 ,\na[14,13]=26235475641986625187247554297838197168935151270802587/\n \+ 31781620957198174033817415268740604591106877500000000,\na[15,1] =-29336887686855537371939221904429024146385699071658194269998471518947 47/\n 14239678548131378023505167950652012589301076964702261708 13903745843200,\na[15,2]=0,\na[15,3]=0,\na[15,4]=0,\na[15,5]=0,\na[15, 6]=-279050827135618188106138704976571118076242172562777/\n 269 80717750745660055932121988692169249262917386240+\n 59017804198 407615229179283246229064921710388893173/\n 1798714516716377337 0621414659128112832841944924160*6^(1/2),\na[15,7]=-2790508271356181881 06138704976571118076242172562777/\n 26980717750745660055932121 988692169249262917386240-\n 5901780419840761522917928324622906 4921710388893173/\n 179871451671637733706214146591281128328419 44924160*6^(1/2),\na[15,8]=6824047782391855906055099601316677053574344 6467404475965020846786328901/\n 6962862590082277531624785771628 3393138506964199995817446989013471723520,\na[15,9]=4853160486533574344 0838806675493568975092395234916265724406574203650554879/\n 7529 075569049450724715447951105730327391304314424323559158913160761835520, \na[15,10]=73158981980491143736917790272372062352348938687470903172269 10860963581499/\n 43240021937913168418365551740095680586702478 3959757582533074834771793600,\na[15,11]=-87035912584683752124645187592 152267644073875904388006117245587111831/\n 414685325327230536 63401983927439573730970639521941633396843682248800,\na[15,12]=79300605 4328041651061360131256412474400253089909554005378332728214806108995212 138291759017448087224471716436232175864384424753159293287828190208/600 6353055070484301705313231740079158134464998533481863014445357229325524 4407178977557750808395528025807084478052037198704114905334161770421932 5,\na[15,13]=190132386928877842671649814278676303562620818706009464227 01364458/\n 14651630863314419811073580576240090560646373319184 0985648075179899,\na[15,14]=368176545506575596342007241113258886329861 009608750000000/\n 7515329389098801941975451526298754679007062 667248055263091,\na[16,1]=-3638865833016212476220002370307465537936296 1851837455245313588466117/\n 299291302137383314536268908335335 078201218487321808786511993451315200,\na[16,2]=0,\na[16,3]=0,\na[16,4] =0,\na[16,5]=0,\na[16,6]=-50172940999755801588626680312841970430535915 31405121/\n 43287413492571995160823628773996668186239908708352 0-\n 115700422823857939498444446575144266776173664871303/\n \+ 199788062273409208434570594341523083936491886346240*6^(1/2),\na[ 16,7]=-5017294099975580158862668031284197043053591531405121/\n \+ 432874134925719951608236287739966681862399087083520+\n 115700 422823857939498444446575144266776173664871303/\n 1997880622734 09208434570594341523083936491886346240*6^(1/2),\na[16,8]=-918693847066 17020415871523809581333688744319256441669606452442156503951243/\n \+ 100539905324968707856110496731312310121511682255963296510778796467 97168640,\na[16,9]=528629993814031198075094729787439820568787345401711 87101666495333163485251/\n 239990528147496105898876739352244665 5063434938667458594324573249770618880,\na[16,10]=119649658612944343374 27534231330501089458731146841410298258149571218167/\n 19740179 23973741961134058915802491373563600428420464580517659076868800,\na[16, 11]=37167680872257703003686692191635149388479305578570534942584948859/ \n 39133905242722475581980638159200847158580465048660062202439 36800,\na[16,12]=-2944395586705434775334102612104558957897818546022036 9427665634428174791788280117223531690217195051250854008448256016995751 289856/613836440653194383209173469818173591961818242299772672817363925 5839442923939727454711783369110119257425239678616468261908910975,\na[1 6,13]=407816748385172686498153181346812432791118177175769818363629863/ \n 62616239788238609519620162976830375962825007575299712016258 0989,\na[16,14]=0,\na[16,15]=0,\n\nb[1]=30703843389361946002220520407/ 1036329015084155723633962896000,\nb[2]=0,\nb[3]=0,\nb[4]=0,\nb[5]=0,\n b[6]=0,\nb[7]=0,\nb[8]=1516681888913470906364013671875/194237682145824 39936604117641536,\nb[9]=1929922737998470573359614532470703125/9295447 834009061726737853188569292704,\nb[10]=2707239736812920996807243315200 0000000/159540891067276798629433718421290211669,\nb[11]=34166762877384 48149119878197304164096817920457/\n 22521752441211566270536786917 243920830369456000,\nb[12]=9090349007494116456314399912605249779168865 9150254835513033014882906689676415155529203822233336681699355686093564 6735988456500531298304/63019787491889793176593803558822113711881465060 6654322610722621749344398603131630645015192260062053457910450104233769 0306078523205079625,\nb[13]=916089774614920438365328235274780485842357 1/\n 54934119002888850773584011583391921191449440,\nb[14]=3769686 146953412690297035156250000/195792979665408643382362918863397227,\nb[1 5]=50782110772148063247179059/1538266148871578545201811280,\nb[16]=0, \n\n`b*`[1]=135131455470598097879473933/4525454214341291369580624000, \n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=0,\n`b*`[5]=0,\n`b*`[6]=0,\n`b*`[7]= 0,\n`b*`[8]=2518169234679274570156341552734375/38284247150941989115046 715871467456,\n`b*`[9]=13171020424136540706261627197265625/61559257178 867958455217570785227104,\n`b*`[10]=3319111100314426409898627200000000 0/196721197370254992144801132455351679,\n`b*`[11]=98603841096694858013 088556726735239713679/\n 5740512436268336928233065765361792580 32000,\n`b*`[12]=50936357685385761074157893008911450543340655548467600 39946162068969260804962050605021707042060987973190001839145435136/4058 9106364182299226269510371318864016087966429387805447847648739509460272 079710913035195393529050583646417571607929125,\n`b*`[13]=1080107210965 23379193662759959856611609133/5706891931812231515532005820513974113771 20,\n`b*`[14]=0,\n`b*`[15]=0,\n`b*`[16]=26859551018855966185191031/763 900876650511794556001520\}:" }}}{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 0 357 "subs(ee,mat rix([seq([c[i],seq(a[i,j],j=1..i-1),``$(4-i)],i=2..4)]));\nfor ii from 5 to 16 do\n print(``);\n print(c[ii]=subs(ee,c[ii])); \n for j j to ii-1 do\n print(a[ii,jj]=subs(ee,a[ii,jj]));\n end do:\nen d do:print(``);\nfor ii to 16 do\n print(b[ii]=subs(ee,b[ii]));\nend do:print(``);\nfor ii to 16 do\n print(`b*`[ii]=subs(ee,`b*`[ii])); \nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7%7&#\"\"\"\" #]F(%!GF+7&,&#\"(Os$Q\")v$H%[F)*(\"(o8.\"F)\"*D\")GX\"!\"\"\"\"'#F)\" \"#F),&#\"/OqJ(Q+S#\"0voaB&[9GF4*(\".'HKXEE!e+j% F;F)*(\".'pP[?:zF)F;F4F5F6F)F+7&,&#\"(='=>\")DJ9;F)*(\"'%o:&F)F0F4F5F6 F),&#\"'4$f*\")]iGKF)*(\"'@*G\"F)F0F4F5F6F)\"\"!,&#\"(Fz(GFMF)*(FOF)FG F4F5F6F)Q)pprint106\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"&#\"#9\"#X" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"\",&#\"1^QsGO_EG\"1IS!*)pMM&fF(*(\"/v uJ#\\f%oF(\"0./*)pMM&f!\"\"\"\"'#F(\"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"$,&#\"0X\"eW-SUq\"0-OfYc*oR!\"\"*(\"/&323`x7* \"\"\"\"0,oHByW)>F-\"\"'#F0\"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 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"6#/&%\"bG6#\"#5#\"G++++?:LC2o*4#H\"otRsq#\"Hp; @!H@%=PVH')zws1\"*3af\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\" #6#\"Od/#z\"o4kTI(>y)>\"\\\"[%QxGwmT$\"P+gXp.$3#RC:]kI;8.')RW$\\I'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#8#\"LrNUe[!yu_BG` OQ/#\\hu(*3;*\"MS%\\9>@>R$e6Set2&)))G+>T$\\&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#9#\"C++Dc^.(H!p7M&p9'opP\"EFsRj)=HO#QV'3amz Hz&>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#:#\";f!zrCj![@x5@y ]\"=!G6=?X&y:()[hEQ:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#; \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"\"#\"% 4:ZUGQ" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"*#\"DDcE(>F;E12 aOTU?5<8\"D/rA&yqv@b%ez')yrDf:'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% #b*G6#\"#5#\"D++++?F')*)4kU9.56\">L\"Ez;NbC8,[9#*\\Dqt>@n>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#6#\"JzOrR_tEnb)38!e[p'4TQg)*\"K+ ?.e#zh`wlIBGpLoiV70u&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#7 #\"arO^VX\"R=+!>tz)41Uqq@]g]?'\\!3Ep*o?;Y*R+wY[blSLa]9\"*3I*y:u5w&Q&od j$4&\"brD\"HzgrvTYOe]!HNR&>NI\"4rz?Fg%4&R([w%yW0yQHk'z3;S')=8P5&piA*H# =kj5*eS" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#8#\"KL\"4;hc)f* fFm$>zL_'4@2,3\"\"K?rP6uR^?e+Kb^JA\"=$>*oq&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#9\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ &%#b*G6#\"#:\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#;#\"; J5>&=mf&)=5bfo#\" " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "RK9_16eqs := [op( RowSumConditions(16,'expanded')),op(OrderConditions(9,16,'expanded'))] :\n`RK8_16eqs*` := subs(b=`b*`,OrderConditions(8,16,'expanded')):" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Check:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "expand(subs(ee,RK9_16eqs)): \nmap(u->lhs(u)-rhs(u),%);\nnops(%);\nexpand(subs(ee,`RK8_16eqs*`)):\n map(u->lhs(u)-rhs(u),%);\nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# 7ajl\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$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 12 "" 1 "" {XPPMATH 20 "6#7dw\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$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 257 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 "Next we set-up stage-order conditions to check for stage-orders from 2 to 6 i nclusive. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "for ct from 2 \+ to 6 do\n so||ct||_16 := StageOrderConditions(ct,16,'expanded');\nen d do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 138 "Stage 3 has stage-order 2, stages 4 and 5 have stage-order 3, sta ges 6 and 7 have stage-order 4, while stages 8 to 16 have stage-order \+ 5. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 175 "[seq([seq(expand(sub s(ee,so||i||_16[j])),i=2..6)],j=1..14)]:\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#70\"\"#\"\"$F%\" \"%F&\"\"&F'F'F'F'F'F'F'F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The simplifying conditions:" }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j+1 .. 16) = b[j] *(1-c[j]);" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F 0F,F,F,\"#;*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 3 ", " } {XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 9 " . . 15, " }}{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 0 151 "[Sum(b[i]*a[i,1], i=1+1..16)=b[1],seq(Sum(b[i]*a[i,j],i=j+1..16)=b[j]*(1-c[j]),j=2..15)] :\nmap(u->lhs(u)-rhs(u),subs(ee,eval(subs(Sum=add,%))));\nnops(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#71\"\"!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 27 "The simplifying conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(`b*`[i]*a[i,j],i = j+1 \+ .. 16) = `b*`[j]*(1-c[j]);" "6#/-%$SumG6$*&&%#b*G6#%\"iG\"\"\"&%\"aG6$ F+%\"jGF,/F+;,&F0F,F,F,\"#;*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 9 " . . 15, " }}{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 0 163 "[Sum(`b* `[i]*a[i,1],i=1+1..16)=`b*`[1],seq(Sum(`b*`[i]*a[i,j],i=j+1..16)=`b*`[ j]*(1-c[j]),j=2..15)]:\nmap(u->lhs(u)-rhs(u),subs(ee,eval(subs(Sum=add ,%))));\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#71\"\"!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 93 "We can calculat e the principal error norm, that is, the 2-norm of the principal error terms. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "errterms9_16 := PrincipalErrorTerms(9,16,'expanded'):\nsm := 0:\nfor ct to nops(errte rms9_16) do\n sm := sm+(evalf(subs(ee,errterms9_16[ct])))^2;\nend do :\nsqrt(sm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+cCchu!#;" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "We can al so calculate the principal error norm of the order 8 embedded scheme. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "`errterms8_16*` := subs (b=`b*`,PrincipalErrorTerms(8,16,'expanded')):\nsm := 0:\nfor ct to no ps(`errterms8_16*`) do\n sm := sm+(evalf(subs(ee,`errterms8_16*`[ct] )))^2;\nend do:\nsqrt(sm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+VWb@ 7!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#----------------- ----------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 34 "construction of the order 9 scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 94 "First we construct a 15 stage orde r 9 scheme starting with a consideration of stages 2 to 11. " }}{PARA 0 "" 0 "" {TEXT -1 21 "We specify the nodes:" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "c[2] = 1/50;" "6#/&%\"cG6#\"\"#*&\"\" \"F)\"#]!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5] = 28/90;" "6#/&% \"cG6#\"\"&*&\"#G\"\"\"\"#!*!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c [8] = 52/125;" "6#/&%\"cG6#\"\")*&\"#_\"\"\"\"$D\"!\"\"" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "c[10] = 21/200;" "6#/&%\"cG6#\"#5*&\"#@\"\"\"\"$+ #!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[11] = 280/477;" "6#/&%\"cG 6#\"#6*&\"$!G\"\"\"\"$x%!\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 35 "and the zero linking coefficients: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[4,2]=0" "6#/&%\"aG6$\"\"%\"\"#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[5,2]=0" "6#/&%\"aG6$\"\"&\"\"#\"\"! " }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[6,2]=0" "6#/&%\"aG6$\"\"'\"\"#\" \"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[6,3]=0" "6#/&%\"aG6$\"\"'\"\"$ \"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[7,2]=0" "6#/&%\"aG6$\"\"(\" \"#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[7,3]=0" "6#/&%\"aG6$\"\"( \"\"$\"\"!" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "a[8,2]=0" "6#/&%\"aG6$\"\")\"\"#\"\"!" }{TEXT -1 2 ", \+ " }{XPPEDIT 18 0 "a[8,3]=0" "6#/&%\"aG6$\"\")\"\"$\"\"!" }{TEXT -1 2 " , " }{XPPEDIT 18 0 "a[8,4]=0" "6#/&%\"aG6$\"\")\"\"%\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[8,5]=0" "6#/&%\"aG6$\"\")\"\"&\"\"!" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[9,2]=0 " "6#/&%\"aG6$\"\"*\"\"#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[9,3] =0" "6#/&%\"aG6$\"\"*\"\"$\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[9, 4]=0" "6#/&%\"aG6$\"\"*\"\"%\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[ 9,5]=0" "6#/&%\"aG6$\"\"*\"\"&\"\"!" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[10,2]=0" "6#/&%\"aG6$\"#5\"\"#\" \"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[10,3]=0" "6#/&%\"aG6$\"#5\"\"$ \"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[10,4]=0" "6#/&%\"aG6$\"#5\" \"%\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[10,5]=0" "6#/&%\"aG6$\"#5 \"\"&\"\"!" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "a[11,2]=0" "6#/&%\"aG6$\"#6\"\"#\"\"!" }{TEXT -1 2 ", \+ " }{XPPEDIT 18 0 "a[11,3]=0" "6#/&%\"aG6$\"#6\"\"$\"\"!" }{TEXT -1 2 " , " }{XPPEDIT 18 0 "a[11,4]=0" "6#/&%\"aG6$\"#6\"\"%\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[11,5]=0" "6#/&%\"aG6$\"#6\"\"&\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 20 " in stages 2 to 11. " }}{PARA 0 "" 0 "" {TEXT -1 17 "We also specify " }{XPPEDIT 18 0 "a[11,10] = - 3/20;" "6#/&%\"aG6$\"#6\"#5,$*&\"\"$\"\"\"\"#?!\"\"F." }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 119 ": The rational v alues for the nodes and linking coefficients given here are close to t he decimal values given by Sharp." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 69 "The stage-order equations relating to the se stages, such that stages " }{XPPEDIT 18 0 "3,4,5,6,7,8,9,10,11;" "6 +\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#5\"#6" }{TEXT -1 19 " have stag e orders " }{XPPEDIT 18 0 "2,3,3,4,4,5,5,5,5;" "6+\"\"#\"\"$F$\"\"%F% \"\"&F&F&F&" }{TEXT -1 96 " respectively taken together with the row-s um conditions can then be solved to obtain the nodes " }{XPPEDIT 18 0 "c[3],c[4],c[6],c[7],c[9],c[11];" "6(&%\"cG6#\"\"$&F$6#\"\"%&F$6#\"\"' &F$6#\"\"(&F$6#\"\"*&F$6#\"#6" }{TEXT -1 67 " and the remaining non-ze ro linking coefficients for these stages. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "The computation is made more ef ficient by " }{TEXT 260 48 "including explicitly relations between the nodes" }{TEXT -1 42 " arising from the stage-order conditions: " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "c[3]=2/3" "6#/&%\"cG6#\"\"$*&\"\"#\"\"\"F'!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 8 ", \+ " }{XPPEDIT 18 0 "c[4] = (6+sqrt(6))*(9*c[8]-24*c[5]+4*c[5]*sqrt (6))*c[8]/(60*(2*c[8]-6*c[5]+c[5]*sqrt(6)));" "6#/&%\"cG6#\"\"%**,&\" \"'\"\"\"-%%sqrtG6#F*F+F+,(*&\"\"*F+&F%6#\"\")F+F+*&\"#CF+&F%6#\"\"&F+ !\"\"*(F'F+&F%6#F9F+-F-6#F*F+F+F+&F%6#F4F+*&\"#gF+,(*&\"\"#F+&F%6#F4F+ F+*&F*F+&F%6#F9F+F:*&&F%6#F9F+-F-6#F*F+F+F+F:" }{TEXT -1 2 ", " }} {PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "c[6] = (6+sqrt(6))*c[8]/10;" "6#/&%\"cG6#\"\"'*(,&F'\" \"\"-%%sqrtG6#F'F*F*&F%6#\"\")F*\"#5!\"\"" }{TEXT -1 8 ", " } {XPPEDIT 18 0 "c[7] = (6-sqrt(6))*c[8]/10;" "6#/&%\"cG6#\"\"(*(,&\"\"' \"\"\"-%%sqrtG6#F*!\"\"F+&F%6#\"\")F+\"#5F/" }{TEXT -1 8 ", " } {XPPEDIT 18 0 "c[9]=3/4" "6#/&%\"cG6#\"\"**&\"\"$\"\"\"\"\"%!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "c[8]" "6#&%\"cG6#\"\")" }{TEXT -1 2 ". \+ " }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "The equations that lead to " }{XPPEDIT 18 0 "c[7] < c[6];" "6#2&%\"cG6# \"\"(&F%6#\"\"'" }{TEXT -1 19 ", have been chosen." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 821 "RSeqs := Ro wSumConditions(11,'expanded'):\nSOeqs := [op(StageOrderConditions(2,11 ,'expanded')),\n op(StageOrderConditions(3,4..11,'expand ed')),\n op(StageOrderConditions(4,6..11,'expanded')),\n op(StageOrderConditions(5,8..11,'expanded'))]:\nnode_eq sB := [c[3]=2/3*c[4],c[9]=3/4*c[8],c[6]=1/10*(6+6^(1/2))*c[8],c[7]=1/1 0*(6-6^(1/2))*c[8],\n c[4]=1/60*(6+6^(1/2))*(9*c[8]-24*c[ 5]+4*c[5]*6^(1/2))*c[8]/(2*c[8]-6*c[5]+c[5]*6^(1/2))]:\n\ne1 := \{c[2] =1/50,c[5]=28/90,c[8]=52/125,c[10]=21/200,c[11]=280/477,\n \+ seq(a[i,2]=0,i=4..11),seq(a[i,3]=0,i=6..11),\n seq(a[i,4]=0 ,i=8..11),seq(a[i,5]=0,i=8..11),a[11,10]=-3/20\}:\neqns := expand(rati onalize(subs(e1,[op(RSeqs),op(SOeqs),op(node_eqsB)]))):\nconvert(ListT ools[Enumerate](%),matrix);\n``;\nindets(eqns);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7L7$\"\"\"/&%\"aG6$\"\"#F(#F(\"#]7$ F-/,&&F+6$\"\"$F(F(&F+6$F5F-F(&%\"cG6#F57$F5/,&&F+6$\"\"%F(F(&F+6$F@F5 F(&F96#F@7$F@/,(&F+6$\"\"&F(F(&F+6$FJF5F(&F+6$FJF@F(#\"#9\"#X7$FJ/,(&F +6$\"\"'F(F(&F+6$FWF@F(&F+6$FWFJF(&F96#FW7$FW/,*&F+6$\"\"(F(F(&F+6$F]o F@F(&F+6$F]oFJF(&F+6$F]oFWF(&F96#F]o7$F]o/,(&F+6$\"\")F(F(&F+6$F[pFWF( &F+6$F[pF]oF(#\"#_\"$D\"7$F[p/,*&F+6$\"\"*F(F(&F+6$FhpFWF(&F+6$FhpF]oF (&F+6$FhpF[pF(&F96#Fhp7$Fhp/,,&F+6$\"#5F(F(&F+6$FfqFWF(&F+6$FfqF]oF(&F +6$FfqF[pF(&F+6$FfqFhpF(#\"#@\"$+#7$Ffq/,.&F+6$\"#6F(F(#F5\"#?!\"\"&F+ 6$FgrFWF(&F+6$FgrF]oF(&F+6$FgrF[pF(&F+6$FgrFhpF(#\"$!G\"$x%7$Fgr/,$*&F .F(F6F(F(,$*&#F(F-F(*$)F8F-F(F(F(7$\"#7/*&FAF(F8F(,$*&F\\tF(*$)FCF-F(F (F(7$\"#8/,&*&FKF(F8F(F(*&FMF(FCF(F(#\"#)*\"%D?7$FP/,&*&FXF(FCF(F(*&FO F(FZF(F(,$*&F\\tF(*$)FfnF-F(F(F(7$\"#:/,(*&F^oF(FCF(F(*&FOF(F`oF(F(*&F boF(FfnF(F(,$*&F\\tF(*$)FdoF-F(F(F(7$\"#;/,&*&F\\pF(FfnF(F(*&F^pF(FdoF (F(#\"%_8\"&Dc\"7$\"#/,,#\"#j\"%+SFjr*&F[sF(FfnF(F(*&F]sF (FdoF(F(*&F`pF(F_sF(F(*&FasF(F_qF(F(#\"&+#R\"'HvA7$Fir/*&FAF(F^tF(,$*& #F(F5F(*$)FCF5F(F(F(7$F`r/,&*&FKF(F^tF(F(*&FMF(FftF(F(#\"%WF\"'vLF7$\" #A/,&*&FXF(FftF(F(*&#\"$'>F_uF(FZF(F(,$*&FfyF(*$)FfnF5F(F(F(7$\"#B/,(* &F^oF(FftF(F(*&FgzF(F`oF(F(*&FboF(FhuF(F(,$*&FfyF(*$)FdoF5F(F(F(7$\"#C /,&*&F\\pF(FhuF(F(*&F^pF(FcvF(F(#\"'319\"(v$fe7$\"#D/,(*&FipF(FhuF(F(* &F[qF(FcvF(F(*&#\"%/FF\\wF(F]qF(F(,$*&FfyF(*$)F_qF5F(F(F(7$\"#E/,**&Fg qF(FhuF(F(*&FiqF(FcvF(F(*&Fh\\lF(F[rF(F(*&F]rF(FgwF(F(#\"%(3$\"(+++)7$ \"#F/,,#\"%B8\"'++!)Fjr*&F[sF(FhuF(F(*&F]sF(FcvF(F(*&Fh\\lF(F_sF(F(*&F asF(FgwF(F(#\")+?&>#\"***RfD$7$\"#G/,&*&FXF(FhyF(F(*&#F_z\"&D6*F(FZF(F (,$*&#F(F@F(*$)FfnF@F(F(F(7$\"#H/,(*&F^oF(FhyF(F(*&F]_lF(F`oF(F(*&FboF (F\\[lF(F(,$*&Fa_lF(*$)FdoF@F(F(F(7$\"#I/,&*&F\\pF(F\\[lF(F(*&F^pF(Fg[ lF(F(#\"(/z#=\"*D19W#7$\"#J/,(*&FipF(F\\[lF(F(*&F[qF(Fg[lF(F(*&#F_\\l \"(DJ&>F(F]qF(F(,$*&Fa_lF(*$)F_qF@F(F(F(7$\"#K/,**&FgqF(F\\[lF(F(*&Fiq F(Fg[lF(F(*&F_alF(F[rF(F(*&F]rF(F]]lF(F(#\"'\"[%>\"+++++k7$\"#L/,,#\"& $yF\"*+++g\"Fjr*&F[sF(F\\[lF(F(*&F]sF(Fg[lF(F(*&F_alF(F_sF(F(*&FasF(F] ]lF(F(#\"+++kO:\",TeWp<&7$\"#M/,&*&F\\pF(Fc_lF(F(*&F^pF(F^`lF(F(#\"*KS ?!Q\"-D1*ye_\"7$\"#N/,(*&FipF(Fc_lF(F(*&F[qF(F^`lF(F(*&#\"(;;J(Fg`lF(F ]qF(F(,$*&#F(FJF(*$)F_qFJF(F(F(7$\"#O/,**&FgqF(Fc_lF(F(*&FiqF(F^`lF(F( *&F^dlF(F[rF(F(*&F]rF(FdalF(F(#\"(,T3%\".+++++g\"7$\"#P/,,#\"'VMe\",++ ++?$Fjr*&F[sF(Fc_lF(F(*&F]sF(F^`lF(F(*&F^dlF(F_sF(F(*&FasF(FdalF(F(#\" -++O2UM\"/dhmDSpC7$\"#Q/F8,$*&#F-F5F(FCF(F(7$\"#R/F_q#FeflFbp7$\"#S/Ff n,&#\"$c\"\"$D'F(*(F_]lF(F^glFjrFWF\\tF(7$\"#T/Fdo,&F\\glF(*(F_]lF(F^g lFjrFWF\\tFjr7$\"#U/FC,&#\"(='=>\")DJ9;F(*(\"'%o:&F(\")v$H%[FjrFWF\\tF (Q)pprint116\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# " 0 "" {MPLTEXT 1 0 22 "infol evel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "e2 := \+ expand(rationalize(solve(\{op(eqns)\}))):\ne3 := `union`(e1,e2):\ninfo level[solve] := 0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 260 4 "Note" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 106 " If the equations giving the relations between the nodes are omitted w e need to select the solution with " }{XPPEDIT 18 0 "c[7] < c[6];" "6 #2&%\"cG6#\"\"(&F%6#\"\"'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 23 " Thus we require that " }{XPPEDIT 18 0 "c[6]=156/625+26/625*sqrt( 6)" "6#/&%\"cG6#\"\"',&*&\"$c\"\"\"\"\"$D'!\"\"F+*(\"#EF+F,F--%%sqrtG6 #F'F+F+" }{TEXT -1 1 " " }{TEXT 272 1 "~" }{TEXT -1 18 " 0.35149877 a nd " }{XPPEDIT 18 0 "c[7]=156/625-26/625*sqrt(6)" "6#/&%\"cG6#\"\"(,& *&\"$c\"\"\"\"\"$D'!\"\"F+*(\"#EF+F,F--%%sqrtG6#\"\"'F+F-" }{TEXT -1 1 " " }{TEXT 273 1 "~" }{TEXT -1 46 " 0.14770123, rather than the othe r way round. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "The following commands achieve this." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[so lve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 185 "eqns := subs (e1,[op(RSeqs),op(SOeqs)]):\nsol := solve(\{op(eqns)\}):\ne2 := op(sel ect(u_->evalf(subs(u_,c[6]) " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e3 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2523 "e3 := \{c[10] = 21/200, c[9] = 39 /125, c[8] = 52/125, c[5] = 14/45, a[2,1] = 1/50, a[5,2] = 0, a[4,2] = 0, a[6,3] = 0, a[6,2] = 0, c[11] = 280/477, c[2] = 1/50, a[8,3] = 0, \+ a[8,2] = 0, a[8,1] = 52/1125, a[7,3] = 0, a[7,2] = 0, a[9,5] = 0, a[9, 4] = 0, a[9,3] = 0, a[9,2] = 0, a[9,1] = 741/16000, a[8,5] = 0, a[8,4] = 0, a[10,3] = 0, a[10,2] = 0, a[10,1] = 35291978967/748709478400, a[ 9,8] = -351/16000, a[7,4] = 23459106068523828440829/354298872323611753 203125+7870375504052283205581/354298872323611753203125*6^(1/2), a[11,3 ] = 0, a[11,2] = 0, a[10,9] = -981041103/4679434240, a[10,8] = -625120 5429/149741895680, a[10,5] = 0, a[10,4] = 0, a[11,10] = -3/20, a[11,5] = 0, a[11,4] = 0, a[6,4] = 4343545768844529/27892881885795625+4692651 41246109/27892881885795625*6^(1/2), a[7,5] = 146263465360621089/755871 8942052734375-1881455818308499953/52911032594369140625*6^(1/2), c[7] = 156/625-26/625*6^(1/2), a[8,6] = 208/1125-13/1125*6^(1/2), a[8,7] = 2 08/1125+13/1125*6^(1/2), a[10,6] = 23154511989/149741895680+39398793/1 772093440*6^(1/2), a[10,7] = 23154511989/149741895680-39398793/1772093 440*6^(1/2), a[9,7] = 2301/16000+897/32000*6^(1/2), a[7,6] = 944412435 6888/82889304453125-2459298027368/82889304453125*6^(1/2), a[5,3] = -70 4240024458145/396895646593602+91277530807085/198447823296801*6^(1/2), \+ a[5,1] = 2826523628723851/5953434698904030-68459492317475/595343469890 403*6^(1/2), a[3,2] = 46300580261936/281448523546875+7915204837696/281 448523546875*6^(1/2), a[4,1] = 959309/32286250+128921/48429375*6^(1/2) , a[3,1] = -24000387317036/281448523546875-5917264532296/2814485235468 75*6^(1/2), a[11,6] = -1736562342312744743536201/110931906456627720015 0528-360257484908262597335743/511993414415204861607936*6^(1/2), a[9,6] = 2301/16000-897/32000*6^(1/2), a[11,7] = -1736562342312744743536201/ 1109319064566277200150528+360257484908262597335743/5119934144152048616 07936*6^(1/2), c[4] = 1918618/16143125+515684/48429375*6^(1/2), a[5,4] = 958925642225180/595343469890403-205373100103780/595343469890403*6^( 1/2), a[6,5] = 1559927818449/28957835234375+4382126882523/202704846640 625*6^(1/2), a[4,3] = 2877927/32286250+128921/16143125*6^(1/2), c[3] = 3837236/48429375+1031368/145288125*6^(1/2), a[6,1] = 376341108/940648 4375+207933466/65845390625*6^(1/2), a[11,8] = 512032742176678555764127 /369773021522092400050176, a[11,9] = 248233526294563631278471/10399866 2303088487514112, a[11,1] = 1601589807329134144752443/1663978596849415 8002257920, a[7,1] = 11781705468/235162109375+2328587014/1646134765625 *6^(1/2), c[6] = 156/625+26/625*6^(1/2)\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "printed coefficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 219 "subs(e3,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(4-i)],i=2..4 )]));\nfor ii from 5 to 11 do\n print(``);\n print(c[ii]=subs(e3,c [ii])); \n for jj to ii-1 do\n print(a[ii,jj]=subs(e3,a[ii,jj]) );\n end do:\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6 #7%7&#\"\"\"\"#]F(%!GF+7&,&#\"(Os$Q\")v$H%[F)*(\"(o8.\"F)\"*D\")GX\"! \"\"\"\"'#F)\"\"#F),&#\"/OqJ(Q+S#\"0voaB&[9GF4*(\".'HKXEE!e+j%F;F)*(\".'pP[?:zF)F;F4F5F6F)F+7&,&#\"(='=>\")DJ9;F)*(\"' %o:&F)F0F4F5F6F),&#\"'4$f*\")]iGKF)*(\"'@*G\"F)F0F4F5F6F)\"\"!,&#\"(Fz (GFMF)*(FOF)FGF4F5F6F)Q(pprint66\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"&#\"#9\"#X" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"\",&#\"1^QsGO_EG\"1I S!*)pMM&fF(*(\"/vuJ#\\f%oF(\"0./*)pMM&f!\"\"\"\"'#F(\"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"#\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"$,&#\"0X\"eW-SUq\"0-OfYc*oR!\"\"*( \"/&323`x7*\"\"\"\"0,oHByW)>F-\"\"'#F0\"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"%,&#\"0!=DAkD*e*\"0./*)pMM&f\"\"\"*( \"0!y.,5t`?F-F,!\"\"\"\"'#F-\"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"',&#\"$c\"\"$D'\" \"\"*(\"#EF,F+!\"\"F'#F,\"\"#F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"\"'\"\"\",&#\"*36Mw$\"+vV[1%*F(*(\"*mM$z?F(\",D1RXe'!\"\"F'#F( \"\"#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"#\"\"!" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"%,&#\"1HX%)odaVV\"2Dcz&)=)G*y #\"\"\"*(\"04hCT^Ep%F-F,!\"\"F'#F-\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"&,&#\".\\%=y#*f:\"/vVBNy&*G\"\"\"*(\" .BD)o7#Q%F-\"0D1kY[q-#!\"\"F'#F-\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"(,&#\"$c\"\" $D'\"\"\"*(\"#EF,F+!\"\"\"\"'#F,\"\"#F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"\",&#\",oaq\"y6\"-v$4@;N#F(*(\"+9qeGBF(\".DcwM hk\"!\"\"\"\"'#F(\"\"#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"\"(\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"$ \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"%,&#\"8H3WG Q_og5fM#\"9DJ?`HN\"-+%y%4([(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5 \"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"$\"\"! " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"',&#\",*)>^aJ#\"-!o&*=u\\\"\"\"\"*(\") $z)RRF-\"+SM4s^aJ#\"-!o&*=u\\\"\"\"\"*(\")$z)RRF-\"+SM4s$46!\"\"*(\"9 VdL(fi#3\\[d-O\"\"\"\"9Ozgh[?:WT$*>^F-F(#F0\"\"#F-" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"(,&#\":,i`VZu7BMilt\"\":G0:+sFmX1>$4 6!\"\"*(\"9VdL(fi#3\\[d-O\"\"\"\"9Ozgh[?:WT$*>^F-\"\"'#F0\"\"#F0" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\")#\"9FTwb&ym " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 27 "We do not need to specify " } {XPPEDIT 18 0 "c[12]" "6#&%\"cG6#\"#7" }{TEXT -1 43 ", because, accord ing to Verner, the nodes " }{XPPEDIT 18 0 "c[8]" "6#&%\"cG6#\"\")" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "c[9]" "6#&%\"cG6#\"\"*" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "c[10]" "6#&%\"cG6#\"#5" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "c[11]" "6#&%\"cG6#\"#6" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "c[12]" "6#&%\"cG6#\"#7" }{TEXT -1 30 " 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),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$*&-%\"p G6#%\"xG\"\"\"-%!G6#*&,&F-F-F,!\"\"\"\"$-%*factorialG6#F4F3F-/F,;\"\"! F-F--F&6$*&-%\"qG6#F,F-,&F-F-F,F3F-/F,;F:F-F-*&-F&6$*&-F*6#F,F--F/6#*& ,&F-F-F,F3\"\"#-F66#FNF3F-/F,;F:F-F--F&6$*&-F?6#F,F--F/6#*&,&F-F-F,F3F N-F66#FNF3F-/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[ 8])*(x-c[9])*(x-c[10])*(x-c[11])" "6#/-%\"pG6#%\"xG*,F'\"\"\",&F'F)&% \"cG6#\"\")!\"\"F),&F'F)&F,6#\"\"*F/F),&F'F)&F,6#\"#5F/F),&F'F)&F,6#\" #6F/F)" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "q(x)=(x-c[12])*p(x)" "6# /-%\"qG6#%\"xG*&,&F'\"\"\"&%\"cG6#\"#7!\"\"F*-%\"pG6#F'F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 155 "See: J.H. Verner, SIAM Journal of Numerical Analysis 1978, 772-79 0, \"Explicit Runge-Kutta methods with estimates of the Local Truncati on Error.\" (page 780)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "p := x -> x*(x-c[8])*(x-c[9])*(x-c [10])*(x-c[11]):\n'p(x)'=p(x);\nq := x -> (x-c[12])*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),&F'F)&F,6#\"#5F/F),&F'F)&F,6#\" #6F/F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"qG6#%\"xG*.,&F'\"\"\"&% \"cG6#\"#7!\"\"F*F'F*,&F'F*&F,6#\"\")F/F*,&F'F*&F,6#\"\"*F/F*,&F'F*&F, 6#\"#5F/F*,&F'F*&F,6#\"#6F/F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "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)*I nt('q(x)'*(1-x)^2/2!,x=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$Ie qG/*&-%$IntG6$,$*&#\"\"\"\"\"'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-F8F--F(6$,$*&FEF-*&F>F-FHF-F-F-F8F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "e4 := solve( subs(e3,value(Ieq)),\{c[12]\}):\nc[12]=subs(e4,c[12]);\ne5 := `union`( e3,e4):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#7#\"1:P0NqAeO\" 1K!*H>Zq\\`" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Thus " }{XPPEDIT 18 0 "c[12] = 3658227035053715/534970471929903 2" "6#/&%\"cG6#\"#7*&\"1:P0NqAeO\"\"\"\"1K!*H>Zq\\`!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 99 "Now we can use the quadrature equa tions to find the weights once the remaining nodes once the nodes" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[13] = 247/281;" "6# /&%\"cG6#\"#8*&\"$Z#\"\"\"\"$\"G!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[14]=229/250" "6#/&%\"cG6#\"#9*&\"$H#\"\"\"\"$]#!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[15]=1" "6#/&%\"cG6#\"#:\"\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 38 "are specified along with the weights " }{XPPEDIT 18 0 "b[2]=0" "6#/&%\"bG6#\"\"#\"\"!" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "b[3]=0" "6#/&%\"bG6#\"\"$\"\"!" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "b[4]=0" "6#/&%\"bG6#\"\"%\"\"!" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "b[5]=0" "6#/&%\"bG6#\"\"&\"\"!" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "b[6]=0" "6#/&%\"bG6#\"\"'\"\"!" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "b[7]=0" "6#/&%\"bG6#\"\"(\"\"!" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 292 "Qeqs := QuadratureConditions(9,15,'expanded'):\ne6 := \{seq(b[i]= 0,i=2..7),c[13]=247/281,c[14]=229/250,c[15]=1\}:\ne7 := `union`(e5,e6) :\nquadeqns := subs(e7,Qeqs):\nfor ct to nops(quadeqns) do\n print(` equation `||ct); print(``);print(quadeqns[ct]);print(``);\nend do:\n indets(quadeqns);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equat ion~~~1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,4&%\"bG6#\"\"\"F(&F&6#\"\")F(&F&6#\"\"*F(&F&6#\"#5F(& F&6#\"#6F(&F&6#\"#7F(&F&6#\"#8F(&F&6#\"#9F(&F&6#\"#:F(F(" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equatio n~~~2G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,2*&#\"#_\"$D\"\"\"\"&%\"bG6#\"\")F)F)*&#\"#RF(F)&F+6# \"\"*F)F)*&#\"#@\"$+#F)&F+6#\"#5F)F)*&#\"$!G\"$x%F)&F+6#\"#6F)F)*&#\"1 :P0NqAeO\"1K!*H>Zq\\`F)&F+6#\"#7F)F)*&#\"$Z#\"$\"GF)&F+6#\"#8F)F)*&#\" $H#\"$]#F)&F+6#\"#9F)F)&F+6#\"#:F)#F)\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~3G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,2*&#\"%/F\"&Dc\"\"\"\"&%\"bG6#\"\")F)F)*&#\"%@:F(F)&F+6#\"\"*F )F)*&#\"$T%\"&++%F)&F+6#\"#5F)F)*&#\"&+%y\"'HvAF)&F+6#\"#6F)F)*&#\"AD7 INHPbX*y**R]i#Q8\"ACq8OM:kZL!p$eS$>'GF)&F+6#\"#7F)F)*&#\"&45'\"&h*yF)& F+6#\"#8F)F)*&#\"&TC&\"&+D'F)&F+6#\"#9F)F)&F+6#\"#:F)#F)\"\"$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%-equation~~~4G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/,2*&#\"'319\"(DJ&>\"\"\"&%\"bG6#\"\")F)F)*&#\"& >$fF(F)&F+6#\"\"*F)F)*&#\"%h#*\"(+++)F)&F+6#\"#5F)F)*&#\")+?&>#\"*L8`3 \"F)&F+6#\"#6F)F)*&#\"Pv3I!eP3-*GKL1o&4>528s!oc*[\"Qo2c#=JPv'\\FvZ.f(p \\%z$Q@]5`\"F)&F+6#\"#7F)F)*&#\")B#p]\"\")T!)=AF)&F+6#\"#8F)F)*&#\")*) *3?\"\")+]i:F)&F+6#\"#9F)F)&F+6#\"#:F)#F)\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~5G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,2*&#\"(;;J(\"*D19W#\"\"\"&%\"bG6#\"\")F)F)*&#\"(TMJ#F(F)&F+6# \"\"*F)F)*&#\"'\"[%>\"+++++;F)&F+6#\"#5F)F)*&#\"+++cYh\",TeWp<&F)&F+6# \"#6F)F)*&#\"jnD1]'[\"y\\')R0&3O)HLWQVDq-[)['y6'Hl%4z\"\"jnwld.x%o\"G0 71*[Q/#eq.**3m_8jk_Wbm1>)F)&F+6#\"#7F)F)*&#\"+\"3)4AP\"+@&R[B'F)&F+6# \"#8F)F)*&#\"+\"[e+v#\"+++D1RF)&F+6#\"#9F)F)&F+6#\"#:F)#F)\"\"&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%-equation~~~6G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#/,2*&#\"*KS?!Q\",D\"yv^I\"\"\"&%\"bG6#\"\")F)F)* &#\")*>C-*F(F)&F+6#\"\"*F)F)*&#\"(,T3%\"-+++++KF)&F+6#\"#5F)F)*&#\".++ !o.@<\"/dhmDSpCF)&F+6#\"#6F)F)*&#\"iov=2cU%>$paZ#)*Rh)\\^YzdLd9Pfv>yLW 'zgYhH,*o^l\"joKWna!evS\\e!f!p*\".,a!**)>v\"F)&F+6#\"#8F)F)*&#\"-\\@Rj(H'\"-++]i l(*F)&F+6#\"#9F)F)&F+6#\"#:F)#F)\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~7G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,2*&#\" ,k'41x>\".DcE(p9Q\"\"\"&%\"bG6#\"\")F)F)*&#\"+hPu=NF(F)&F+6#\"\"*F)F)* &#\")@hw&)\"/++++++kF)&F+6#\"#5F)F)*&#\"0+++/.*=[\"2*)ovU-0z<\"F)&F+6# \"#6F)F)*&#\"ipDcwDb&QTX&e'\\cyfo'Q7Lpa)R+:**\\rU(>Ast%\\g@6gC+[secnR# \"jpC)\\F,]#z(4$=t^VeW.)f#)3nW&))[s:b->n?wi]F\\ec=q@$HdZ6WBF)&F+6#\"#7 F)F)*&#\"0HP#=[\"3F#\"0\"o\"f[C?&4ZUCHq!yP\"7\"G(>KUoAb2!o#F)&F+6#\"#6F)F)*&#\"hrD1RD \"3\"4>YYf'[:jPOtwcgKb88)\\J.U^6ZBqduH**oi'G5#G%R%yl9qf*Ruh,&G>Z*[2K\" irwP)eHUCax1&*yyi\"f(yXiE(G\\K.'=Zb@a.ve\\=Hig =q3nF)&F+6#\"#7F)F)*&#\"5hD)QeCT,aQ\"\"5T%4Ni_QAt)QF)&F+6#\"#8F)F)*&# \"4ht-?*[;#Gc(\"5++++D1*ye_\"F)&F+6#\"#9F)F)&F+6#\"#:F)#F)F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<+&% \"bG6#\"#:&F%6#\"#9&F%6#\"#7&F%6#\"#6&F%6#\"#5&F%6#\"\"*&F%6#\"\")&F%6 #\"#8&F%6#\"\"\"" }}{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 73 "e8 := solve(\{op(quadeqns)\}):\ne9 := `union`(e7,e8):\ninfolevel[solve] := \+ 0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "for ii to 15 do\n wt_val := subs(e9,b[ii]);\n if wt_val<>0 then print(b[ii]=wt_val) end if;\nend do:\n``;\nevalf[8](su bs(e9,[seq(b[i],i=1..15)]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"b G6#\"\"\"#\">2/_?A+Y>O*QVQqI\"@+g*G'RjBd:%3:!Hj.\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"bG6#\"\")#\"@v=n8SO14Z8*))=o;:\"AO:k" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"*#\"FDJqqC`9'fLdq%) *ztA*H>\"F/FHp&)=`ytE<14S$yW&H*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"bG6#\"#5#\"G++++?:LC2o*4#H\"otRsq#\"Hp;@!H@%=PVH')zws1\"*3af\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#6#\"Od/#z\"o4kTI(>y)>\"\\ \"[%QxGwmT$\"P+gXp.$3#RC:] kI;8.')RW$\\I'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#8#\"LrNUe[!yu_BG`OQ/#\\hu(*3;*\"MS%\\9>@>R$ e6Set2&)))G+>T$\\&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#9#\" C++Dc^.(H!p7M&p9'opP\"EFsRj)=HO#QV'3amzHz&>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#:#\";f!zrCj![@x5@y]\"=!G6=?X&y:()[hEQ:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #71$\")0viH!\"*$\"\"!F(F'F'F'F'F'$\")8Q3yF&$\")??w?!\")$\")S*op\"F-$\" )h0<:F-$\")&fCW\"F-$\")`hn;F-$\")HMD>F&$\")lD,LF&" }}}{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 3616 "e9 := \{c[10] = 21/200, c[ 9] = 39/125, c[8] = 52/125, c[5] = 14/45, a[2,1] = 1/50, c[15] = 1, c[ 14] = 229/250, c[13] = 247/281, a[5,2] = 0, a[4,2] = 0, a[6,3] = 0, a[ 6,2] = 0, c[11] = 280/477, c[2] = 1/50, a[8,3] = 0, a[8,2] = 0, a[8,1] = 52/1125, a[7,3] = 0, a[7,2] = 0, a[9,5] = 0, a[9,4] = 0, a[9,3] = 0 , a[9,2] = 0, a[9,1] = 741/16000, a[8,5] = 0, a[8,4] = 0, a[10,3] = 0, a[10,2] = 0, a[10,1] = 35291978967/748709478400, a[9,8] = -351/16000, a[7,4] = 23459106068523828440829/354298872323611753203125+78703755040 52283205581/354298872323611753203125*6^(1/2), a[11,3] = 0, a[11,2] = 0 , a[10,9] = -981041103/4679434240, a[10,8] = -6251205429/149741895680, a[10,5] = 0, a[10,4] = 0, a[11,10] = -3/20, a[11,5] = 0, a[11,4] = 0, b[4] = 0, b[3] = 0, b[2] = 0, b[7] = 0, b[6] = 0, b[5] = 0, b[14] = 3 769686146953412690297035156250000/195792979665408643382362918863397227 , a[6,4] = 4343545768844529/27892881885795625+469265141246109/27892881 885795625*6^(1/2), a[7,5] = 146263465360621089/7558718942052734375-188 1455818308499953/52911032594369140625*6^(1/2), b[15] = 507821107721480 63247179059/1538266148871578545201811280, c[7] = 156/625-26/625*6^(1/2 ), a[8,6] = 208/1125-13/1125*6^(1/2), a[8,7] = 208/1125+13/1125*6^(1/2 ), a[10,6] = 23154511989/149741895680+39398793/1772093440*6^(1/2), a[1 0,7] = 23154511989/149741895680-39398793/1772093440*6^(1/2), a[9,7] = \+ 2301/16000+897/32000*6^(1/2), a[7,6] = 9444124356888/82889304453125-24 59298027368/82889304453125*6^(1/2), a[5,3] = -704240024458145/39689564 6593602+91277530807085/198447823296801*6^(1/2), a[5,1] = 2826523628723 851/5953434698904030-68459492317475/595343469890403*6^(1/2), a[3,2] = \+ 46300580261936/281448523546875+7915204837696/281448523546875*6^(1/2), \+ a[4,1] = 959309/32286250+128921/48429375*6^(1/2), a[3,1] = -2400038731 7036/281448523546875-5917264532296/281448523546875*6^(1/2), b[13] = 91 60897746149204383653282352747804858423571/5493411900288885077358401158 3391921191449440, a[11,6] = -1736562342312744743536201/110931906456627 7200150528-360257484908262597335743/511993414415204861607936*6^(1/2), \+ a[9,6] = 2301/16000-897/32000*6^(1/2), a[11,7] = -17365623423127447435 36201/1109319064566277200150528+360257484908262597335743/5119934144152 04861607936*6^(1/2), c[4] = 1918618/16143125+515684/48429375*6^(1/2), \+ a[5,4] = 958925642225180/595343469890403-205373100103780/5953434698904 03*6^(1/2), a[6,5] = 1559927818449/28957835234375+4382126882523/202704 846640625*6^(1/2), a[4,3] = 2877927/32286250+128921/16143125*6^(1/2), \+ c[3] = 3837236/48429375+1031368/145288125*6^(1/2), a[6,1] = 376341108/ 9406484375+207933466/65845390625*6^(1/2), b[9] = 192992273799847057335 9614532470703125/9295447834009061726737853188569292704, b[12] = 909034 9007494116456314399912605249779168865915025483551303301488290668967641 51555292038222333366816993556860935646735988456500531298304/6301978749 1889793176593803558822113711881465060665432261072262174934439860313163 06450151922600620534579104501042337690306078523205079625, b[11] = 3416 676287738448149119878197304164096817920457/225217524412115662705367869 17243920830369456000, b[1] = 30703843389361946002220520407/10363290150 84155723633962896000, a[11,8] = 512032742176678555764127/3697730215220 92400050176, b[8] = 1516681888913470906364013671875/194237682145824399 36604117641536, a[11,9] = 248233526294563631278471/1039986623030884875 14112, a[11,1] = 1601589807329134144752443/16639785968494158002257920, b[10] = 27072397368129209968072433152000000000/1595408910672767986294 33718421290211669, c[12] = 3658227035053715/5349704719299032, a[7,1] = 11781705468/235162109375+2328587014/1646134765625*6^(1/2), c[6] = 156 /625+26/625*6^(1/2)\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 109 "It remains to determine \+ the linking coefficients in stages 12 to 15. We have the following zer o coefficients." }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "a [12,2]=0" "6#/&%\"aG6$\"#7\"\"#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[12,3]=0" "6#/&%\"aG6$\"#7\"\"$\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[12,4]=0" "6#/&%\"aG6$\"#7\"\"%\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[12,5]=0" "6#/&%\"aG6$\"#7\"\"&\"\"!" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "a[13,2]=0" "6#/&%\"aG6$\"# 8\"\"#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[13,3]=0" "6#/&%\"aG6$ \"#8\"\"$\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[13,4]=0" "6#/&%\"aG 6$\"#8\"\"%\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[13,5]=0" "6#/&%\" aG6$\"#8\"\"&\"\"!" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "a[14,2]=0" "6#/&%\"aG6$\"#9\"\"#\"\"!" }{TEXT -1 2 " , " }{XPPEDIT 18 0 "a[14,3]=0" "6#/&%\"aG6$\"#9\"\"$\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[14,4]=0" "6#/&%\"aG6$\"#9\"\"%\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[14,5]=0" "6#/&%\"aG6$\"#9\"\"&\"\"!" } {TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[1 5,2]=0" "6#/&%\"aG6$\"#:\"\"#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a [15,3]=0" "6#/&%\"aG6$\"#:\"\"$\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[15,4]=0" "6#/&%\"aG6$\"#:\"\"%\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[15,5]=0" "6#/&%\"aG6$\"#:\"\"&\"\"!" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 154 "We make use of th e stage-order conditions for stages 12 to 15 so that all these stages \+ all have stage-order 5 and incorporate the simplifying conditions: " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = \+ j+1 .. 15) = b[j]*(1-c[j]);" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6 $F+%\"jGF,/F+;,&F0F,F,F,\"#:*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "for " }{XPPEDIT 18 0 "j= 1" "6#/%\"jG\"\"\"" }{TEXT -1 10 " ( with " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\"\"!" }{TEXT -1 5 " ), " }{XPPEDIT 18 0 "j=7" "6#/ %\"jG\"\"(" }{TEXT -1 8 " . . 13." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 185 "Then it turns out that the following col lection of \"simple\" order conditions (given in abreviated form) is s ufficient to determine the remaining linking coefficients in stages 11 to 15." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 194 "SO9 := SimpleOrderConditions(9):\n[seq([i,SO9[i]],i= [102,106,125,212,223,239,245,251,253])]:\nlinalg[augment](linalg[delco ls](%,2..2),matrix([[` `]$(linalg[rowdim](%))]),linalg[delcols](%,1.. 1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7+7%\"$-\"%#~~G/*( %\"bG\"\"\"%\"cGF--%!G6#*&%\"aGF--F06#*&F3F--F06#*&)F.\"\"$F-F3F-F-F-F -#F-\"$g*7%\"$1\"F)/*&F,F--F06#*&F3F--F06#*&F3F--F06#*&)F.\"\"%F-F3F-F -F-F-#F-\"%!o\"7%\"$D\"F)/*(F,F-F.F--F06#*&)F.\"\"&F-F3F-F-#F-\"#[7%\" $7#F)/*(F,F-)F.\"\"#F-F/F-#F-\"%!3\"7%\"$B#F)/*(F,F-F.F-FBF-#F-\"%!*=7 %\"$R#F)/*(F,F-FhnF-FEF-#F-\"$q#7%\"$X#F)/*(F,F-F.F--F06#*&F3F-FSF-F-# F-\"$y$7%\"$^#F)/*(F,F-FhnF-FSF-#F-\"#a7%\"$`#F)/*(F,F-F.F--F06#*&)F. \"\"'F-F3F-F-#F-\"#jQ(pprint16\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 20 "The associated trees" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 456 "AST9 := AllSimpleTrees(9): \nwhch := [102,106,125,212,223,239,245,251,253]:\nm := 3: ## number of trees per row\nnn := nops(whch): q := iquo(nn,m,'r'):\nfor i to nn do \n p||i := DrawTree(AST9[whch[i]],height=4,width=2,show_orderconditi on=true,\n color=COLOR(RGB,.5,0,.9),font_color=black);\nend do:\npp := plot([[1,1]],style=line,axes=none):\nplots[display](convert([seq([ p||((k-1)*m+1..m*k)],k=1..q),\n `if`(r>0,[p||(m*q+1..nn),pp$(m*(q+1)- nn)],NULL)],array));" }}{PARA 13 "" 1 "" {GLPLOT2D 782 948 948 {PLOTDATA 2 "6fu-%%TEXTG6&7$$\"#g\"\"!$!+E!G3\"R!\")Q5b~c~~(a~(a~c~))~ =~~~6\"-%'COLOURG6&%$RGBGF)F)F)-%%FONTG6$%(COURIERG\"#5-F$6&7$F'$!+uu) G%QF,Q7~~~~~~~~~~~~~~~~~~~~~1F.F/F3-F$6&7$F'$!+(*)=T'QF,Q9~~~~~~~~~~~~ ~~~~~~~~~___F.F/F3-F$6&7$F'$!+,+++SF,Q9~~~~~~~~~~~~~~~~~~~~~270F.F/F3- F$6&7$F'$!+0\\RMQF,Q7~~~~2~~~~~~~~4~~~~~~~~F.F/F3-%'CURVESG6&7+7$F'$!+ ,Q^)p$F,7$$\"+++++bF,$!+^`)QF$F,7$F'FZ7$$\"+++++lF,FZ7$Fhn$!+,pD\\GF,7 $$\"+LLLLjF,$!+]%GYU#F,7$$\"+WWWWkF,F`o7$$\"+cbbblF,F`o7$$\"+nmmmmF,F` o-%'SYMBOLG6#%'CIRCLEG-%&COLORG6&F2$F)F)FbpFbp-%&STYLEG6#%&POINTG-FQ6& FS-F\\p6#%(DIAMONDGF_pFcp-FQ6&FS-F\\p6#%&CROSSGF_pFcp-FQ6%7$FTFW-F`p6& F2$\"\"&!\"\"Fbp$\"\"*Fhq-%*THICKNESSG6#\"\"#-FQ6%7$FTFfnFdqF[r-FQ6%7$ FTFgnFdqF[r-FQ6%7$FgnFjnFdqF[r-FQ6%7$FjnF]oFdqF[r-FQ6%7$FjnFboFdqF[r-F Q6%7$FjnFeoFdqF[r-FQ6%7$FjnFhoFdqF[r-FQ6%7#7$F'$!+++++?F,-F06&F2$F7Fhq FbpFbp-Fdp6#%%LINEG-FQ6%7#7$$\"+++++]F,FUFjsF]t-FQ6%7#7$$\"+++++qF,FUF jsF]t-F$6&7$$\"#IF)$!+d-G3\"*!\"*Q7b~(a~(a~(a~c~)))~=~~~~F.F/F3-F$6&7$ F_u$!+PZ()G%)FcuQ9~~~~~~~~~~~~~~~~~~~~~~~1F.F/F3-F$6&7$F_u$!+i*)=T')Fc uQ<~~~~~~~~~~~~~~~~~~~~~~~____F.F/F3-F$6&7$F_u$!+++++5F,Q<~~~~~~~~~~~~ ~~~~~~~~~~~1680F.F/F3-F$6&7$F_u$!+Z!\\RM)FcuQ9~~~~~~~~~~~~~4~~~~~~~~~~ F.F/F3-FQ6&7*7$F_u$!+1!Q^)pFcu7$F_u$!+/N&)QFFcu7$F_u$\"+)*4V2:Fcu7$F_u $\"+/br`dFcu7$$\"++++DEF,$\"+++++5F,7$$\"++++vGF,F_x7$$\"++++DJF,F_x7$ $\"++++vLF,F_xF[pF_pFcp-FQ6&F_wFipF_pFcp-FQ6&F_wF^qF_pFcp-FQ6%7$F`wFcw FdqF[r-FQ6%7$FcwFfwFdqF[r-FQ6%7$FfwFiwFdqF[r-FQ6%7$FiwF\\xFdqF[r-FQ6%7 $FiwFaxFdqF[r-FQ6%7$FiwFdxFdqF[r-FQ6%7$FiwFgxFdqF[r-FQ6%7#7$F_uF_xFjsF ]t-FQ6%7#7$$\"+++++?F,FawFjsF]t-FQ6%7#7$$\"+++++SF,FawFjsF]t-FQ6#-%'LE GENDG6#QB__never_display_this_legend_entryF.-F$6&7$FbpF*Q:b~c~~(a~(a~( a~c~)))~=~~~~F.F/F3-F$6&7$FbpF;Q<~~~~~~~~~~~~~~~~~~~~~~~~~~1F.F/F3-F$6 &7$FbpFAQ?~~~~~~~~~~~~~~~~~~~~~~~~~~____F.F/F3-F$6&7$FbpFGQ?~~~~~~~~~~ ~~~~~~~~~~~~~~~~1080F.F/F3-F$6&7$FbpFMQ<~~~~2~~~~~~~~~~~3~~~~~~~~~~F.F /F3-FQ6&7+7$FbpFU7$$!+++++]FcuFZ7$FbpFZ7$$FetFcuFZ7$Ff]lF[o7$Ff]lF`o7$ $\"+LLLLLFcuFhs7$Ff]lFhs7$$FjoFcuFhsF[pF_pFcp-FQ6&F_]lFipF_pFcp-FQ6&F_ ]lF^qF_pFcp-FQ6%7$F`]lFa]lFdqF[r-FQ6%7$F`]lFd]lFdqF[r-FQ6%7$F`]lFe]lFd qF[r-FQ6%7$Fe]lFg]lFdqF[r-FQ6%7$Fg]lFh]lFdqF[r-FQ6%7$Fh]lFi]lFdqF[r-FQ 6%7$Fh]lF\\^lFdqF[r-FQ6%7$Fh]lF]^lFdqF[r-FQ6%7#7$FbpFhsFjsF]t-FQ6%7#7$ FdvFUFjsF]t-FQ6%7#7$F_xFUFjsF]t-F$6&7$F_u$!+E!G3\"pF,Q0b~c~~(a~c~)~=~~ F.F/F3-F$6&7$F_u$!+uu)G%oF,Q2~~~~~~~~~~~~~~~~1F.F/F3-F$6&7$F_u$!+(*)=T 'oF,Q3~~~~~~~~~~~~~~~~__F.F/F3-F$6&7$F_u$!+,+++qF,Q3~~~~~~~~~~~~~~~~54 F.F/F3-F$6&7$F_u$!+0\\RMoF,Q2~~~~2~~~~~5~~~~~~F.F/F3-FQ6&7+7$F_u$!+,Q^ )p'F,7$$\"+++++DF,$!+^`)QF'F,7$F_uF^cl7$$\"+++++NF,F^cl7$$F[^lF,$!+,pD \\eF,7$$\"+nmm;MF,Ffcl7$FbclFfcl7$$\"+LLL$e$F,Ffcl7$$\"+nmmmOF,FfclF[p F_pFcp-FQ6&FgblFipF_pFcp-FQ6&FgblF^qF_pFcp-FQ6%7$FhblF[clFdqF[r-FQ6%7$ FhblF`clFdqF[r-FQ6%7$FhblFaclFdqF[r-FQ6%7$FaclFdclFdqF[r-FQ6%7$FaclFhc lFdqF[r-FQ6%7$FaclF[dlFdqF[r-FQ6%7$FaclF\\dlFdqF[r-FQ6%7$FaclF_dlFdqF[ r-FQ6%7#7$F_u$Fc]lF,FjsF]t-FQ6%7#7$F[[lFiblFjsF]t-FQ6%7#7$Fa[lFiblFjsF ]t-F$6&7$F'Fj`lQ/b~c~(a~c~)~=~~F.F/F3-F$6&7$F'F`alQ1~~~~~~~~~~~~~~~1F. 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7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Cu rve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve \+ 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" "Curve 24" "Curve 25" "Curve 26" "Curve 27" "Curve 28" "Curve 29" "Curve 30" "Curve 31" "Cur ve 32" "Curve 33" "Curve 34" "Curve 35" "Curve 36" "Curve 37" "Curve 3 8" "Curve 39" "Curve 40" "Curve 41" "Curve 42" "Curve 43" "Curve 44" " Curve 45" "Curve 46" "Curve 47" "Curve 48" "Curve 49" "Curve 50" "Curv e 51" "Curve 52" "Curve 53" "Curve 54" "Curve 55" "Curve 56" "Curve 57 " "Curve 58" "Curve 59" "Curve 60" "Curve 61" "Curve 62" "Curve 63" "C urve 64" "Curve 65" "Curve 66" "Curve 67" "Curve 68" "Curve 69" "Curve 70" "Curve 71" "Curve 72" "Curve 73" "Curve 74" "Curve 75" "Curve 76 " "Curve 77" "Curve 78" "Curve 79" "Curve 80" "Curve 81" "Curve 82" "C urve 83" "Curve 84" "Curve 85" "Curve 86" "Curve 87" "Curve 88" "Curve 89" "Curve 90" "Curve 91" "Curve 92" "Curve 93" "Curve 94" "Curve 95 " "Curve 96" "Curve 97" "Curve 98" "Curve 99" "Curve 100" "Curve 101" "Curve 102" "Curve 103" "Curve 104" "Curve 105" "Curve 106" "Curve 107 " "Curve 108" "Curve 109" "Curve 110" "Curve 111" "Curve 112" "Curve 1 13" "Curve 114" "Curve 115" "Curve 116" "Curve 117" "Curve 118" "Curve 119" "Curve 120" "Curve 121" "Curve 122" "Curve 123" "Curve 124" "Cur ve 125" "Curve 126" "Curve 127" "Curve 128" "Curve 129" "Curve 130" "C urve 131" "Curve 132" "Curve 133" "Curve 134" "Curve 135" "Curve 136" "Curve 137" "Curve 138" "Curve 139" "Curve 140" "Curve 141" "Curve 142 " "Curve 143" "Curve 144" "Curve 145" "Curve 146" "Curve 147" "Curve 1 48" "Curve 149" "Curve 150" "Curve 151" "Curve 152" "Curve 153" "Curve 154" "Curve 155" "Curve 156" "Curve 157" "Curve 158" "Curve 159" "Cur ve 160" "Curve 161" "Curve 162" "Curve 163" "Curve 164" "Curve 165" "C urve 166" "Curve 167" "Curve 168" "Curve 169" }}}}{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 49 "SO 9_15 := SimpleOrderConditions(9,15,'expanded'):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 485 "SOeqs2 := [seq(add(a[i,j],j= 1..i-1)=c[i],i=1 2..15),op(StageOrderConditions(2,12..15,'expanded')),\n op(Sta geOrderConditions(3,12..15,'expanded')),op(StageOrderConditions(4,12.. 15,'expanded')),\n op(StageOrderConditions(5,12..15,'exp anded'))]:\nord_cdns := [seq(SO9_15[i],i=[102,106,125,212,223,239,245, 251,253])]:\nsimp_eqs := [add(b[i]*a[i,1],i=1+1..15)=b[1],seq(add(b[i] *a[i,j],i=j+1..15)=b[j]*(1-c[j]),j=[$7..13])]:\ncdns := [op(SOeqs2),op (ord_cdns),op(simp_eqs)]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 127 "It is possible to manage with fewer equations, but the computa tion may be less efficient if the number of equations is reduced." }} {PARA 0 "" 0 "" {TEXT -1 50 "For example, the simplifying conditions g iven by " }{XPPEDIT 18 0 "j=8" "6#/%\"jG\"\")" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "j=10" "6#/%\"jG\"#5" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "j=12" "6#/%\"jG\"#7" }{TEXT -1 17 " may be omitted." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "e10 \+ :=\{seq(seq(a[i,j]=0,i=12..15),j=2..5)\}:\ne11 := `union`(e9,e10):\neq ns2 := subs(e11,cdns):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "There are 37 equations and 34 unknowns." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "nops(eqns2);\nindets(eqns2);\nnops( %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#P" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#&F%6$F,FA&F%6$F,F2&F%6$F,F (&F%6$F,FK&F%6$F+FD&F%6$F+F2&F%6$F+F(&F%6$F+FK&F%6$F+F9&F%6$F'FD&F%6$F 'FK&F%6$F,FD&F%6$F'F9&F%6$F+F>&F%6$F+FA&F%6$F'F+&F%6$F'F,&F%6$F'F>&F%6 $F'FA" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#M" }}}{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 74 "e12 := solve( \{op(eqns2)\}):\ninfolevel[solve] := 0:\ne13 := `union`(e11,e12):" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "e13 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10910 "e13 := \{ c[10] = 21/200, c[9] = 39/125, c[8] = 52/125, c[5] = 14/45, a[2,1] = 1 /50, c[15] = 1, c[14] = 229/250, c[13] = 247/281, a[5,2] = 0, a[4,2] = 0, a[6,3] = 0, a[6,2] = 0, c[11] = 280/477, c[2] = 1/50, a[8,3] = 0, \+ a[8,2] = 0, a[8,1] = 52/1125, a[7,3] = 0, a[7,2] = 0, a[9,5] = 0, a[9, 4] = 0, a[9,3] = 0, a[9,2] = 0, a[9,1] = 741/16000, a[8,5] = 0, a[8,4] = 0, a[10,3] = 0, a[10,2] = 0, a[10,1] = 35291978967/748709478400, a[ 9,8] = -351/16000, a[7,4] = 23459106068523828440829/354298872323611753 203125+7870375504052283205581/354298872323611753203125*6^(1/2), a[11,3 ] = 0, a[11,2] = 0, a[10,9] = -981041103/4679434240, a[10,8] = -625120 5429/149741895680, a[10,5] = 0, a[10,4] = 0, a[12,3] = 0, a[12,2] = 0, a[11,10] = -3/20, a[11,5] = 0, a[11,4] = 0, a[13,2] = 0, a[12,5] = 0, a[12,4] = 0, a[13,5] = 0, a[13,4] = 0, a[13,3] = 0, a[14,5] = 0, a[14 ,4] = 0, a[14,3] = 0, a[14,2] = 0, a[15,5] = 0, a[15,4] = 0, a[15,3] = 0, a[15,2] = 0, b[4] = 0, b[3] = 0, b[2] = 0, b[7] = 0, b[6] = 0, b[5 ] = 0, b[14] = 3769686146953412690297035156250000/19579297966540864338 2362918863397227, a[6,4] = 4343545768844529/27892881885795625+46926514 1246109/27892881885795625*6^(1/2), a[7,5] = 146263465360621089/7558718 942052734375-1881455818308499953/52911032594369140625*6^(1/2), b[15] = 50782110772148063247179059/1538266148871578545201811280, c[7] = 156/6 25-26/625*6^(1/2), a[8,6] = 208/1125-13/1125*6^(1/2), a[8,7] = 208/112 5+13/1125*6^(1/2), a[10,6] = 23154511989/149741895680+39398793/1772093 440*6^(1/2), a[10,7] = 23154511989/149741895680-39398793/1772093440*6^ (1/2), a[9,7] = 2301/16000+897/32000*6^(1/2), a[7,6] = 9444124356888/8 2889304453125-2459298027368/82889304453125*6^(1/2), a[5,3] = -70424002 4458145/396895646593602+91277530807085/198447823296801*6^(1/2), a[5,1] = 2826523628723851/5953434698904030-68459492317475/595343469890403*6^ (1/2), a[3,2] = 46300580261936/281448523546875+7915204837696/281448523 546875*6^(1/2), a[4,1] = 959309/32286250+128921/48429375*6^(1/2), a[3, 1] = -24000387317036/281448523546875-5917264532296/281448523546875*6^( 1/2), b[13] = 9160897746149204383653282352747804858423571/549341190028 88850773584011583391921191449440, a[11,6] = -1736562342312744743536201 /1109319064566277200150528-360257484908262597335743/511993414415204861 607936*6^(1/2), a[9,6] = 2301/16000-897/32000*6^(1/2), a[11,7] = -1736 562342312744743536201/1109319064566277200150528+3602574849082625973357 43/511993414415204861607936*6^(1/2), c[4] = 1918618/16143125+515684/48 429375*6^(1/2), a[5,4] = 958925642225180/595343469890403-2053731001037 80/595343469890403*6^(1/2), a[6,5] = 1559927818449/28957835234375+4382 126882523/202704846640625*6^(1/2), a[4,3] = 2877927/32286250+128921/16 143125*6^(1/2), c[3] = 3837236/48429375+1031368/145288125*6^(1/2), a[6 ,1] = 376341108/9406484375+207933466/65845390625*6^(1/2), a[13,1] = 52 15174783558918407997583468635543407988332719241764605769949554629/2028 3132613214812064685094275151111714651171227532533713038580121600, a[12 ,7] = -581038619225160876203856834629458675128926705143465192450716448 466169075797359178616021045291080972121429188543592047/101191722567617 6365127453733973371461080784851732664960275549575306273314094232139356 444322425772110112679530543972352-713482798078988149650882337297379066 5635109665954131159198195323855274480577648002828260278094293998870885 5996996031/51893191060316736673202755588378023645168453935008459501310 234631090939184319596890074067816706262057060488745844736*6^(1/2), b[9 ] = 1929922737998470573359614532470703125/9295447834009061726737853188 569292704, b[12] = 909034900749411645631439991260524977916886591502548 3551303301488290668967641515552920382223333668169935568609356467359884 56500531298304/6301978749188979317659380355882211371188146506066543226 1072262174934439860313163064501519226006205345791045010423376903060785 23205079625, a[14,7] = 10256190098435854298655077997613296122112139953 /1148037852243817086246902953973393653760000000*6^(1/2)-86885256061463 15530022414580346392155721271039/2238673811875443318181460760248117624 8320000000, a[15,7] = -27905082713561818810613870497657111807624217256 2777/26980717750745660055932121988692169249262917386240-59017804198407 615229179283246229064921710388893173/179871451671637733706214146591281 12832841944924160*6^(1/2), a[13,6] = -18227070890226867447840942666790 512323422585544257/121857700488461867579360666965049340968208464609280 *6^(1/2)+2843598186227456480865065344408178581293412110128603/79207505 3175002139265844335272820716293355019960320, a[14,6] = -10256190098435 854298655077997613296122112139953/114803785224381708624690295397339365 3760000000*6^(1/2)-8688525606146315530022414580346392155721271039/2238 6738118754433181814607602481176248320000000, b[11] = 34166762877384481 49119878197304164096817920457/2252175244121156627053678691724392083036 9456000, b[1] = 30703843389361946002220520407/103632901508415572363396 2896000, a[11,8] = 512032742176678555764127/369773021522092400050176, \+ b[8] = 1516681888913470906364013671875/1942376821458243993660411764153 6, a[11,9] = 248233526294563631278471/103998662303088487514112, a[11,1 ] = 1601589807329134144752443/16639785968494158002257920, b[10] = 2707 2397368129209968072433152000000000/15954089106727679862943371842129021 1669, c[12] = 3658227035053715/5349704719299032, a[7,1] = 11781705468/ 235162109375+2328587014/1646134765625*6^(1/2), c[6] = 156/625+26/625*6 ^(1/2), a[14,12] = -88682414394183619425441647866243388112917289239161 4639409444929304921125471716523632401461235899088705678115336581253759 35101390832/9405104776230176067202383689444499684238639823097650089954 3333833779587704919990762422199804533702480228254208143848188723144531 25, a[13,12] = 2736153920540927643774133147635296486946660915558253285 983742020488887296849241173151960647763453239551016003889152/248567211 0698341015290264470463939203955869249618375406787169018009688457749866 177826801192710345262847046284166825, a[12,6] = -581038619225160876203 8568346294586751289267051434651924507164484661690757973591786160210452 91080972121429188543592047/1011917225676176365127453733973371461080784 8517326649602755495753062733140942321393564443224257721101126795305439 72352+7134827980789881496508823372973790665635109665954131159198195323 8552744805776480028282602780942939988708855996996031/51893191060316736 6732027555883780236451684539350084595013102346310909391843195968900740 67816706262057060488745844736*6^(1/2), a[14,10] = -1259710340512037041 83074450363446847441594334546885083244594242327104115033/5066049934698 363488698655054901069679084758735799331062593807151200000000, a[13,7] \+ = 2843598186227456480865065344408178581293412110128603/792075053175002 139265844335272820716293355019960320+182270708902268674478409426667905 12323422585544257/121857700488461867579360666965049340968208464609280* 6^(1/2), a[15,10] = 73158981980491143736917790272372062352348938687470 90317226910860963581499/4324002193791316841836555174009568058670247839 59757582533074834771793600, a[15,6] = -2790508271356181881061387049765 71118076242172562777/2698071775074566005593212198869216924926291738624 0+59017804198407615229179283246229064921710388893173/17987145167163773 370621414659128112832841944924160*6^(1/2), a[13,9] = -7416041550905424 66856213236072374206251235617068304762316465738169791/1415516731633211 36844445993892555326037025917405403892742525852712960, a[13,8] = 93268 29464422062118248457481351539504275339476759467047326605595685633/4901 901791858228863857691041029309678010355547895721285919177263022080, a[ 15,8] = 68240477823918559060550996013166770535743446467404475965020846 786328901/696286259008227753162478577162833931385069641999958174469890 13471723520, a[15,14] = 3681765455065755963420072411132588863298610096 08750000000/7515329389098801941975451526298754679007062667248055263091 , a[12,11] = 115590271440716912566235566233889746097162479804636463234 298604185457969653794053637008425503953091180886565/315361333249071836 3304114648792457541639644686569637672840787260046309750159216975258704 14526347596936773632, a[14,11] = 4322338495495152743252505005837177994 220267688026960252214552638944423/236867625787508422152958167179676757 535142999000924357630500000000000, a[15,11] = -87035912584683752124645 187592152267644073875904388006117245587111831/414685325327230536634019 83927439573730970639521941633396843682248800, a[12,10] = 6883437842714 9827544141552835305430278000101566001470691198893507717914313664393296 56536871565378282089012991331513/1827181489551794784669860898707808352 4232186538856420801802422529600115450739992009460663708366411323198808 49653760, a[15,13] = 1901323869288778426716498142786763035626208187060 0946422701364458/14651630863314419811073580576240090560646373319184098 5648075179899, a[12,9] = -16183509927928156539922841522541118273994265 3401484724580110184517256730426918980054437210005086959516698155192566 7441/19637518660778297585754649024920739916598981028937029385347383945 787366501641192454385997632075140011874187139618963456, a[14,13] = 262 35475641986625187247554297838197168935151270802587/3178162095719817403 3817415268740604591106877500000000, a[13,10] = -6058504866441219655595 548618762485399974773685307046001179355536003/225227572081539617272640 0694965157641073696835574259179818290290400, a[15,12] = 79300605432804 1651061360131256412474400253089909554005378332728214806108995212138291 759017448087224471716436232175864384424753159293287828190208/600635305 5070484301705313231740079158134464998533481863014445357229325524440717 89775577508083955280258070844780520371987041149053341617704219325, a[1 5,9] = 485316048653357434408388066754935689750923952349162657244065742 03650554879/7529075569049450724715447951105730327391304314424323559158 913160761835520, a[13,11] = -72917047186465183128180555150230405657138 451692847535142343993/446617472880162182768547714428317380932341452032 22656783563600, a[14,1] = 19614316258903156870631415758182324055225458 98155499982338718373117379429883/4800566471670774299905935680554060935 86176318669944422673481728000000000000, a[14,9] = 68321055425793546260 0257975958139742203919396113084127371502375524416129719/26895337200565 243662247103690698994332502640106760065066162305761280000000, a[15,1] \+ = -2933688768685553737193922190442902414638569907165819426999847151894 747/142396785481313780235051679506520125893010769647022617081390374584 3200, a[14,8] = -10815139209229042495349883638005977260973640373943448 1043071361807712075869481/86007354951945634483162614783313539932301371 06756553600705167485829120000000, a[12,8] = -1893570082626077243216830 8633651734522837925089710329104904435053093522818069066377665789161365 2665009511679250229667441/10490208572843028318487937042190617479870802 9629619600881898639306750333561102065113284728091471708748347777999725 133824, a[12,1] = -131987017608786696357225423387594635612719389206128 606880670434178321331969627889057541436355642743061150672386594396559/ 3187539260879955550151479262016120102404472282957894624867981162214760 93939683123897279961564118214685494052121351290880\}:" }}}{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 0 283 "subs(e13,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(4-i)],i=2..4) ]));\nfor ii from 5 to 15 do\n print(``);\n print(c[ii]=subs(e13,c [ii])); \n for jj to ii-1 do\n print(a[ii,jj]=subs(e13,a[ii,jj] ));\n end do:\nend do:\n``;\nfor ii to 15 do\n print(b[ii]=subs(e1 3,b[ii]));\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7%7 &#\"\"\"\"#]F(%!GF+7&,&#\"(Os$Q\")v$H%[F)*(\"(o8.\"F)\"*D\")GX\"!\"\" \"\"'#F)\"\"#F),&#\"/OqJ(Q+S#\"0voaB&[9GF4*(\".'HKXEE!e+j%F;F)*(\".'pP[?:zF)F;F4F5F6F)F+7&,&#\"(='=>\")DJ9;F)*(\"'%o: &F)F0F4F5F6F),&#\"'4$f*\")]iGKF)*(\"'@*G\"F)F0F4F5F6F)\"\"!,&#\"(Fz(GF MF)*(FOF)FGF4F5F6F)Q(pprint76\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"&#\"#9\"#X" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"\",&#\"1^QsGO_EG\"1IS!*)pM M&fF(*(\"/vuJ#\\f%oF(\"0./*)pMM&f!\"\"\"\"'#F(\"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"$,&#\"0X\"eW-SUq\"0-OfYc*oR!\"\"*(\"/ &323`x7*\"\"\"\"0,oHByW)>F-\"\"'#F0\"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"%,&#\"0!=DAkD*e*\"0./*)pMM&f\"\"\"*( \"0!y.,5t`?F-F,!\"\"\"\"'#F-\"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"',&#\"$c\"\"$D'\" \"\"*(\"#EF,F+!\"\"F'#F,\"\"#F," }}{PARA 11 "" 1 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FDJqqC`9'fLdq%)*ztA*H>\"F/FHp&)=`ytE<14S$yW&H*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#5#\"G++++?:LC2o*4#H\"otRsq#\"Hp;@!H@%=PVH') zws1\"*3af\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#6#\"Od/#z \"o4kTI(>y)>\"\\\"[%QxGwmT$\"P+gXp.$3#RC:]kI;8.')RW$\\I'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#8#\"LrNUe[!yu_BG`OQ/#\\hu( *3;*\"MS%\\9>@>R$e6Set2&)))G+>T$\\&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"bG6#\"#9#\"C++Dc^.(H!p7M&p9'opP\"EFsRj)=HO#QV'3amzHz&>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#:#\";f!zrCj![@x5@y]\"=!G6=?X&y:( )[hEQ:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "RK9_15eqs := [op(RowSumConditions(15,'expan ded')),op(OrderConditions(9,15,'expanded'))]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Check:" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 75 "expand(subs(e13,RK9_15eqs)):\nmap(u_->`if`(l hs(u_)=rhs(u_),0,1),%);\nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7 `jl\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$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 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 34 "Appendix: related order conditions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 44 "#--------------------------------------- ----" }}{PARA 0 "" 0 "" {TEXT -1 145 "unrelated 129, 130, 134, 135, 1 36, 139, 140, 141, 143, 151, 152, 156, 157, 166, 168, 173, 175, 180, 1 85, 186, 187, 212, 221, 223\n\nrelated groups" }}{PARA 0 "" 0 "" {TEXT -1 19 "132, 137, 148, 153," }}{PARA 0 "" 0 "" {TEXT -1 19 "133, \+ 138, 150, 163," }}{PARA 0 "" 0 "" {TEXT -1 59 "131, 147, 149, 192,\n14 2, 164, 170, 222,\n162, 184, 188, 227," }}{PARA 0 "" 0 "" {TEXT -1 18 "181, 197, 202, 239" }}{PARA 0 "" 0 "" {TEXT -1 38 "144, 158, 167, 177 , 193, 207, 213, 244" }}{PARA 0 "" 0 "" {TEXT -1 38 "145, 159, 169, 17 8, 194, 209, 215, 245" }}{PARA 0 "" 0 "" {TEXT -1 38 "154, 165, 172, 1 89, 206, 208, 224, 248" }}{PARA 0 "" 0 "" {TEXT -1 38 "183, 200, 204, \+ 205, 228, 231, 235, 251" }}{PARA 0 "" 0 "" {TEXT -1 78 "146, 160, 161, 174, 182, 196, 198, 201, 203, 211, 218, 230, 232, 238, 241, 253" }} {PARA 0 "" 0 "" {TEXT -1 78 "155, 171, 176, 179, 190, 195, 199, 210, 2 14, 216, 225, 229, 237, 240, 246, 255" }}{PARA 0 "" 0 "" {TEXT -1 78 " 191, 217, 219, 220, 226, 233, 234, 236, 242, 243, 247, 249, 250, 252, \+ 254, 256" }}{PARA 0 "" 0 "" {TEXT -1 43 "#---------------------------- --------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 44 "#-------------------------------------------" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 43 "construction of the order 8 embe dded scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "e13 " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10910 "e13 := \{c[10] = 21/200, c[9] = 39/125, c[8] = 52/125, c[5] = 1 4/45, a[2,1] = 1/50, c[15] = 1, c[14] = 229/250, c[13] = 247/281, a[5, 2] = 0, a[4,2] = 0, a[6,3] = 0, a[6,2] = 0, c[11] = 280/477, c[2] = 1/ 50, a[8,3] = 0, a[8,2] = 0, a[8,1] = 52/1125, a[7,3] = 0, a[7,2] = 0, \+ a[9,5] = 0, a[9,4] = 0, a[9,3] = 0, a[9,2] = 0, a[9,1] = 741/16000, a[ 8,5] = 0, a[8,4] = 0, a[10,3] = 0, a[10,2] = 0, a[10,1] = 35291978967/ 748709478400, a[9,8] = -351/16000, a[7,4] = 23459106068523828440829/35 4298872323611753203125+7870375504052283205581/354298872323611753203125 *6^(1/2), a[11,3] = 0, a[11,2] = 0, a[10,9] = -981041103/4679434240, a [10,8] = -6251205429/149741895680, a[10,5] = 0, a[10,4] = 0, a[12,3] = 0, a[12,2] = 0, a[11,10] = -3/20, a[11,5] = 0, a[11,4] = 0, a[13,2] = 0, a[12,5] = 0, a[12,4] = 0, a[13,5] = 0, a[13,4] = 0, a[13,3] = 0, a [14,5] = 0, a[14,4] = 0, a[14,3] = 0, a[14,2] = 0, a[15,5] = 0, a[15,4 ] = 0, a[15,3] = 0, a[15,2] = 0, b[4] = 0, b[3] = 0, b[2] = 0, b[7] = \+ 0, b[6] = 0, b[5] = 0, b[14] = 3769686146953412690297035156250000/1957 92979665408643382362918863397227, a[6,4] = 4343545768844529/2789288188 5795625+469265141246109/27892881885795625*6^(1/2), a[7,5] = 1462634653 60621089/7558718942052734375-1881455818308499953/52911032594369140625* 6^(1/2), b[15] = 50782110772148063247179059/15382661488715785452018112 80, c[7] = 156/625-26/625*6^(1/2), a[8,6] = 208/1125-13/1125*6^(1/2), \+ a[8,7] = 208/1125+13/1125*6^(1/2), a[10,6] = 23154511989/149741895680+ 39398793/1772093440*6^(1/2), a[10,7] = 23154511989/149741895680-393987 93/1772093440*6^(1/2), a[9,7] = 2301/16000+897/32000*6^(1/2), a[7,6] = 9444124356888/82889304453125-2459298027368/82889304453125*6^(1/2), a[ 5,3] = -704240024458145/396895646593602+91277530807085/198447823296801 *6^(1/2), a[5,1] = 2826523628723851/5953434698904030-68459492317475/59 5343469890403*6^(1/2), a[3,2] = 46300580261936/281448523546875+7915204 837696/281448523546875*6^(1/2), a[4,1] = 959309/32286250+128921/484293 75*6^(1/2), a[3,1] = -24000387317036/281448523546875-5917264532296/281 448523546875*6^(1/2), b[13] = 9160897746149204383653282352747804858423 571/54934119002888850773584011583391921191449440, a[11,6] = -173656234 2312744743536201/1109319064566277200150528-360257484908262597335743/51 1993414415204861607936*6^(1/2), a[9,6] = 2301/16000-897/32000*6^(1/2), a[11,7] = -1736562342312744743536201/1109319064566277200150528+360257 484908262597335743/511993414415204861607936*6^(1/2), c[4] = 1918618/16 143125+515684/48429375*6^(1/2), a[5,4] = 958925642225180/5953434698904 03-205373100103780/595343469890403*6^(1/2), a[6,5] = 1559927818449/289 57835234375+4382126882523/202704846640625*6^(1/2), a[4,3] = 2877927/32 286250+128921/16143125*6^(1/2), c[3] = 3837236/48429375+1031368/145288 125*6^(1/2), a[6,1] = 376341108/9406484375+207933466/65845390625*6^(1/ 2), a[13,1] = 52151747835589184079975834686355434079883327192417646057 69949554629/2028313261321481206468509427515111171465117122753253371303 8580121600, a[12,7] = -58103861922516087620385683462945867512892670514 3465192450716448466169075797359178616021045291080972121429188543592047 /101191722567617636512745373397337146108078485173266496027554957530627 3314094232139356444322425772110112679530543972352-71348279807898814965 0882337297379066563510966595413115919819532385527448057764800282826027 80942939988708855996996031/5189319106031673667320275558837802364516845 3935008459501310234631090939184319596890074067816706262057060488745844 736*6^(1/2), b[9] = 1929922737998470573359614532470703125/929544783400 9061726737853188569292704, b[12] = 90903490074941164563143999126052497 7916886591502548355130330148829066896764151555292038222333366816993556 860935646735988456500531298304/630197874918897931765938035588221137118 8146506066543226107226217493443986031316306450151922600620534579104501 042337690306078523205079625, a[14,7] = 1025619009843585429865507799761 3296122112139953/1148037852243817086246902953973393653760000000*6^(1/2 )-8688525606146315530022414580346392155721271039/223867381187544331818 14607602481176248320000000, a[15,7] = -2790508271356181881061387049765 71118076242172562777/2698071775074566005593212198869216924926291738624 0-59017804198407615229179283246229064921710388893173/17987145167163773 370621414659128112832841944924160*6^(1/2), a[13,6] = -1822707089022686 7447840942666790512323422585544257/12185770048846186757936066696504934 0968208464609280*6^(1/2)+284359818622745648086506534440817858129341211 0128603/792075053175002139265844335272820716293355019960320, a[14,6] = -10256190098435854298655077997613296122112139953/11480378522438170862 46902953973393653760000000*6^(1/2)-86885256061463155300224145803463921 55721271039/22386738118754433181814607602481176248320000000, b[11] = 3 416676287738448149119878197304164096817920457/225217524412115662705367 86917243920830369456000, b[1] = 30703843389361946002220520407/10363290 15084155723633962896000, a[11,8] = 512032742176678555764127/3697730215 22092400050176, b[8] = 1516681888913470906364013671875/194237682145824 39936604117641536, a[11,9] = 248233526294563631278471/1039986623030884 87514112, a[11,1] = 1601589807329134144752443/166397859684941580022579 20, b[10] = 27072397368129209968072433152000000000/1595408910672767986 29433718421290211669, c[12] = 3658227035053715/5349704719299032, a[7,1 ] = 11781705468/235162109375+2328587014/1646134765625*6^(1/2), c[6] = \+ 156/625+26/625*6^(1/2), a[14,12] = -8868241439418361942544164786624338 8112917289239161463940944492930492112547171652363240146123589908870567 811533658125375935101390832/940510477623017606720238368944449968423863 9823097650089954333383377958770491999076242219980453370248022825420814 384818872314453125, a[13,12] = 273615392054092764377413314763529648694 6660915558253285983742020488887296849241173151960647763453239551016003 889152/248567211069834101529026447046393920395586924961837540678716901 8009688457749866177826801192710345262847046284166825, a[12,6] = -58103 8619225160876203856834629458675128926705143465192450716448466169075797 359178616021045291080972121429188543592047/101191722567617636512745373 3973371461080784851732664960275549575306273314094232139356444322425772 110112679530543972352+713482798078988149650882337297379066563510966595 41311591981953238552744805776480028282602780942939988708855996996031/5 1893191060316736673202755588378023645168453935008459501310234631090939 184319596890074067816706262057060488745844736*6^(1/2), a[14,10] = -125 9710340512037041830744503634468474415943345468850832445942423271041150 33/5066049934698363488698655054901069679084758735799331062593807151200 000000, a[13,7] = 2843598186227456480865065344408178581293412110128603 /792075053175002139265844335272820716293355019960320+18227070890226867 447840942666790512323422585544257/121857700488461867579360666965049340 968208464609280*6^(1/2), a[15,10] = 7315898198049114373691779027237206 235234893868747090317226910860963581499/432400219379131684183655517400 956805867024783959757582533074834771793600, a[15,6] = -279050827135618 188106138704976571118076242172562777/269807177507456600559321219886921 69249262917386240+59017804198407615229179283246229064921710388893173/1 7987145167163773370621414659128112832841944924160*6^(1/2), a[13,9] = - 741604155090542466856213236072374206251235617068304762316465738169791/ 141551673163321136844445993892555326037025917405403892742525852712960, a[13,8] = 93268294644220621182484574813515395042753394767594670473266 05595685633/4901901791858228863857691041029309678010355547895721285919 177263022080, a[15,8] = 6824047782391855906055099601316677053574344646 7404475965020846786328901/69628625900822775316247857716283393138506964 199995817446989013471723520, a[15,14] = 368176545506575596342007241113 258886329861009608750000000/751532938909880194197545152629875467900706 2667248055263091, a[12,11] = 11559027144071691256623556623388974609716 2479804636463234298604185457969653794053637008425503953091180886565/31 5361333249071836330411464879245754163964468656963767284078726004630975 015921697525870414526347596936773632, a[14,11] = 432233849549515274325 2505005837177994220267688026960252214552638944423/23686762578750842215 2958167179676757535142999000924357630500000000000, a[15,11] = -8703591 2584683752124645187592152267644073875904388006117245587111831/41468532 532723053663401983927439573730970639521941633396843682248800, a[12,10] = 6883437842714982754414155283530543027800010156600147069119889350771 791431366439329656536871565378282089012991331513/182718148955179478466 9860898707808352423218653885642080180242252960011545073999200946066370 836641132319880849653760, a[15,13] = 190132386928877842671649814278676 30356262081870600946422701364458/1465163086331441981107358057624009056 06463733191840985648075179899, a[12,9] = -1618350992792815653992284152 2541118273994265340148472458011018451725673042691898005443721000508695 95166981551925667441/1963751866077829758575464902492073991659898102893 7029385347383945787366501641192454385997632075140011874187139618963456 , a[14,13] = 26235475641986625187247554297838197168935151270802587/317 81620957198174033817415268740604591106877500000000, a[13,10] = -605850 4866441219655595548618762485399974773685307046001179355536003/22522757 20815396172726400694965157641073696835574259179818290290400, a[15,12] \+ = 79300605432804165106136013125641247440025308990955400537833272821480 6108995212138291759017448087224471716436232175864384424753159293287828 190208/600635305507048430170531323174007915813446499853348186301444535 7229325524440717897755775080839552802580708447805203719870411490533416 17704219325, a[15,9] = 48531604865335743440838806675493568975092395234 916265724406574203650554879/752907556904945072471544795110573032739130 4314424323559158913160761835520, a[13,11] = -7291704718646518312818055 5150230405657138451692847535142343993/44661747288016218276854771442831 738093234145203222656783563600, a[14,1] = 1961431625890315687063141575 818232405522545898155499982338718373117379429883/480056647167077429990 593568055406093586176318669944422673481728000000000000, a[14,9] = 6832 1055425793546260025797595813974220391939611308412737150237552441612971 9/26895337200565243662247103690698994332502640106760065066162305761280 000000, a[15,1] = -293368876868555373719392219044290241463856990716581 9426999847151894747/14239678548131378023505167950652012589301076964702 26170813903745843200, a[14,8] = -1081513920922904249534988363800597726 09736403739434481043071361807712075869481/8600735495194563448316261478 331353993230137106756553600705167485829120000000, a[12,8] = -189357008 2626077243216830863365173452283792508971032910490443505309352281806906 63776657891613652665009511679250229667441/1049020857284302831848793704 2190617479870802962961960088189863930675033356110206511328472809147170 8748347777999725133824, a[12,1] = -13198701760878669635722542338759463 5612719389206128606880670434178321331969627889057541436355642743061150 672386594396559/318753926087995555015147926201612010240447228295789462 486798116221476093939683123897279961564118214685494052121351290880\}: " }}}{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 61 "We can obtain an embedded 14 stage order 8 scheme as fo llows." }}{PARA 0 "" 0 "" {TEXT -1 94 "We remove stages 14 and 15 from the 15 stage order 9 scheme and introduce a new stage 14 with:" }} {PARA 256 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "c[14] = 1;" "6#/&% \"cG6#\"#9\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[14,2] = 0;" "6#/ &%\"aG6$\"#9\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[14,3] = 0; " "6#/&%\"aG6$\"#9\"\"$\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[14,4 ]=0" "6#/&%\"aG6$\"#9\"\"%\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[1 4,5]=0" "6#/&%\"aG6$\"#9\"\"&\"\"!" }{TEXT -1 1 "," }}{PARA 256 "" 0 " " {TEXT -1 2 " " }{XPPEDIT 18 0 "`b*`[2]=0" "6#/&%#b*G6#\"\"#\"\"!" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[3]=0" "6#/&%#b*G6#\"\"$\"\"!" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[4]=0" "6#/&%#b*G6#\"\"%\"\"!" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[5]=0" "6#/&%#b*G6#\"\"&\"\"!" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[6]=0" "6#/&%#b*G6#\"\"'\"\"!" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[7] = 0;" "6#/&%#b*G6#\"\"(\"\"! " }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 56 "where the weights of the order 8 scheme are denoted by " }{XPPEDIT 18 0 "`b*`" "6#%#b*G" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 156 "We incorporate the order 8 quadrature conditions, the ro w sum conditions for this stage and stage-order conditions so that thi s new stage has stage-order 4." }}{PARA 0 "" 0 "" {TEXT -1 48 "We also incorporate the simplifying conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(`b*`[i]*a[i,j],i = j+1 .. 14) = `b*`[j]*( 1-c[j]);" "6#/-%$SumG6$*&&%#b*G6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F ,F,F,\"#9*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 5 "for " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" } {TEXT -1 10 " ( with " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\" \"!" }{TEXT -1 5 " ), " }{XPPEDIT 18 0 "j = 9;" "6#/%\"jG\"\"*" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "j = 11;" "6#/%\"jG\"#6" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "j = 13;" "6#/%\"jG\"#8" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 451 "`Qeqs*` := subs(b=`b*`,QuadratureConditions(8,14,'expanded')):\nS O_eqs2 := [add(a[14,j],j=1..13)=c[14],add(a[14,j]*c[j],j=2..13)=1/2*c[ 14]^2,\n add(a[14,j]*c[j]^2,j=2..13)=1/3*c[14]^3,add(a[14,j]*c[ j]^3,j=2..13)=1/4*c[14]^4,\n add(a[14,j]*c[j]^4,j=2..13)=1/5*c[ 14]^5]:\n`simp_eqs*` := [add(`b*`[i]*a[i,1],i=2..14)=`b*`[1],seq(add(` b*`[i]*a[i,j],i=j+1..14)=`b*`[j]*(1-c[j]),j=[9,11,13])]:\n`cdns*` := [ op(`simp_eqs*`),op(SO_eqs2),op(`Qeqs*`)]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "e14 := `union`(re move(u_->member(op(1,lhs(u_)),[14,15]) or op(0,lhs(u_))=b,e13),\n \+ \{c[14]=1,seq(a[14,i]=0,i=2..5),seq(`b*`[i]=0,i=2..7)\}):\n" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "We have 1 7 equations for the 17 unknown coefficients." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "eqns3 := subs(e14,`cdns*`):\nnops(%);\nindets(eqns 3);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<3&%\"aG6$\"#9\"#8&%#b*G6#\"\"*&F*6#\"#5&F*6#\"\") &F*6#\"\"\"&F*6#\"#7&F*6#F'&F*6#F(&F*6#\"#6&F%6$F'F8&F%6$F'F,&F%6$F'F2 &F%6$F'\"\"(&F%6$F'\"\"'&F%6$F'F5&F%6$F'F?&F%6$F'F/" }}{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 74 "e15 := solve(\{op(eqns3)\}):\ninfol evel[solve] := 0:\ne16 := `union`(e14,e15):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e16" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8954 "e16 := \{`b*`[7] = 0, `b*` [5] = 0, `b*`[6] = 0, `b*`[3] = 0, `b*`[4] = 0, `b*`[2] = 0, c[10] = 2 1/200, c[9] = 39/125, c[8] = 52/125, c[5] = 14/45, a[2,1] = 1/50, c[13 ] = 247/281, a[5,2] = 0, a[4,2] = 0, a[6,3] = 0, a[6,2] = 0, c[11] = 2 80/477, c[2] = 1/50, a[8,3] = 0, a[8,2] = 0, a[8,1] = 52/1125, a[7,3] \+ = 0, a[7,2] = 0, a[9,5] = 0, a[9,4] = 0, a[9,3] = 0, a[9,2] = 0, a[9,1 ] = 741/16000, a[8,5] = 0, a[8,4] = 0, a[10,3] = 0, a[10,2] = 0, a[14, 7] = -5017294099975580158862668031284197043053591531405121/43287413492 5719951608236287739966681862399087083520+11570042282385793949844444657 5144266776173664871303/19978806227340920843457059434152308393649188634 6240*6^(1/2), a[14,6] = -501729409997558015886266803128419704305359153 1405121/432874134925719951608236287739966681862399087083520-1157004228 23857939498444446575144266776173664871303/1997880622734092084345705943 41523083936491886346240*6^(1/2), a[10,1] = 35291978967/748709478400, a [9,8] = -351/16000, a[7,4] = 23459106068523828440829/35429887232361175 3203125+7870375504052283205581/354298872323611753203125*6^(1/2), a[11, 3] = 0, a[11,2] = 0, a[10,9] = -981041103/4679434240, a[10,8] = -62512 05429/149741895680, a[10,5] = 0, a[10,4] = 0, a[12,3] = 0, a[12,2] = 0 , a[11,10] = -3/20, a[11,5] = 0, a[11,4] = 0, a[13,2] = 0, a[12,5] = 0 , a[12,4] = 0, a[13,5] = 0, a[13,4] = 0, a[13,3] = 0, a[14,5] = 0, a[1 4,4] = 0, a[14,3] = 0, a[14,2] = 0, a[6,4] = 4343545768844529/27892881 885795625+469265141246109/27892881885795625*6^(1/2), a[7,5] = 14626346 5360621089/7558718942052734375-1881455818308499953/5291103259436914062 5*6^(1/2), `b*`[11] = 98603841096694858013088556726735239713679/574051 243626833692823306576536179258032000, c[7] = 156/625-26/625*6^(1/2), a [8,6] = 208/1125-13/1125*6^(1/2), a[8,7] = 208/1125+13/1125*6^(1/2), a [10,6] = 23154511989/149741895680+39398793/1772093440*6^(1/2), a[10,7] = 23154511989/149741895680-39398793/1772093440*6^(1/2), a[9,7] = 2301 /16000+897/32000*6^(1/2), a[7,6] = 9444124356888/82889304453125-245929 8027368/82889304453125*6^(1/2), a[5,3] = -704240024458145/396895646593 602+91277530807085/198447823296801*6^(1/2), a[5,1] = 2826523628723851/ 5953434698904030-68459492317475/595343469890403*6^(1/2), a[3,2] = 4630 0580261936/281448523546875+7915204837696/281448523546875*6^(1/2), a[4, 1] = 959309/32286250+128921/48429375*6^(1/2), `b*`[1] = 13513145547059 8097879473933/4525454214341291369580624000, a[3,1] = -24000387317036/2 81448523546875-5917264532296/281448523546875*6^(1/2), a[11,6] = -17365 62342312744743536201/1109319064566277200150528-36025748490826259733574 3/511993414415204861607936*6^(1/2), a[9,6] = 2301/16000-897/32000*6^(1 /2), a[14,8] = -918693847066170204158715238095813336887443192564416696 06452442156503951243/1005399053249687078561104967313123101215116822559 6329651077879646797168640, a[11,7] = -1736562342312744743536201/110931 9064566277200150528+360257484908262597335743/511993414415204861607936* 6^(1/2), c[4] = 1918618/16143125+515684/48429375*6^(1/2), a[5,4] = 958 925642225180/595343469890403-205373100103780/595343469890403*6^(1/2), \+ a[6,5] = 1559927818449/28957835234375+4382126882523/202704846640625*6^ (1/2), a[4,3] = 2877927/32286250+128921/16143125*6^(1/2), c[3] = 38372 36/48429375+1031368/145288125*6^(1/2), a[6,1] = 376341108/9406484375+2 07933466/65845390625*6^(1/2), a[13,1] = 521517478355891840799758346863 5543407988332719241764605769949554629/20283132613214812064685094275151 111714651171227532533713038580121600, a[12,7] = -581038619225160876203 8568346294586751289267051434651924507164484661690757973591786160210452 91080972121429188543592047/1011917225676176365127453733973371461080784 8517326649602755495753062733140942321393564443224257721101126795305439 72352-7134827980789881496508823372973790665635109665954131159198195323 8552744805776480028282602780942939988708855996996031/51893191060316736 6732027555883780236451684539350084595013102346310909391843195968900740 67816706262057060488745844736*6^(1/2), a[13,6] = -18227070890226867447 840942666790512323422585544257/121857700488461867579360666965049340968 208464609280*6^(1/2)+2843598186227456480865065344408178581293412110128 603/792075053175002139265844335272820716293355019960320, `b*`[14] = 26 859551018855966185191031/763900876650511794556001520, a[14,12] = -2944 3955867054347753341026121045589578978185460220369427665634428174791788 280117223531690217195051250854008448256016995751289856/613836440653194 3832091734698181735919618182422997726728173639255839442923939727454711 783369110119257425239678616468261908910975, a[14,11] = 371676808722577 03003686692191635149388479305578570534942584948859/3913390524272247558 198063815920084715858046504866006220243936800, a[11,8] = 5120327421766 78555764127/369773021522092400050176, a[11,9] = 2482335262945636312784 71/103998662303088487514112, `b*`[13] = 108010721096523379193662759959 856611609133/570689193181223151553200582051397411377120, a[11,1] = 160 1589807329134144752443/16639785968494158002257920, a[14,10] = 11964965 861294434337427534231330501089458731146841410298258149571218167/197401 7923973741961134058915802491373563600428420464580517659076868800, c[12 ] = 3658227035053715/5349704719299032, a[7,1] = 11781705468/2351621093 75+2328587014/1646134765625*6^(1/2), a[14,1] = -3638865833016212476220 0023703074655379362961851837455245313588466117/29929130213738331453626 8908335335078201218487321808786511993451315200, `b*`[12] = 50936357685 3857610741578930089114505433406555484676003994616206896926080496205060 5021707042060987973190001839145435136/40589106364182299226269510371318 8640160879664293878054478476487395094602720797109130351953935290505836 46417571607929125, `b*`[10] = 33191111003144264098986272000000000/1967 21197370254992144801132455351679, c[6] = 156/625+26/625*6^(1/2), a[14, 9] = 52862999381403119807509472978743982056878734540171187101666495333 163485251/239990528147496105898876739352244665506343493866745859432457 3249770618880, c[14] = 1, a[13,12] = 273615392054092764377413314763529 6486946660915558253285983742020488887296849241173151960647763453239551 016003889152/248567211069834101529026447046393920395586924961837540678 7169018009688457749866177826801192710345262847046284166825, a[12,6] = \+ -581038619225160876203856834629458675128926705143465192450716448466169 075797359178616021045291080972121429188543592047/101191722567617636512 7453733973371461080784851732664960275549575306273314094232139356444322 425772110112679530543972352+713482798078988149650882337297379066563510 9665954131159198195323855274480577648002828260278094293998870885599699 6031/51893191060316736673202755588378023645168453935008459501310234631 090939184319596890074067816706262057060488745844736*6^(1/2), a[13,7] = 2843598186227456480865065344408178581293412110128603/7920750531750021 39265844335272820716293355019960320+1822707089022686744784094266679051 2323422585544257/121857700488461867579360666965049340968208464609280*6 ^(1/2), a[14,13] = 407816748385172686498153181346812432791118177175769 818363629863/626162397882386095196201629768303759628250075752997120162 580989, a[13,9] = -741604155090542466856213236072374206251235617068304 762316465738169791/141551673163321136844445993892555326037025917405403 892742525852712960, a[13,8] = 9326829464422062118248457481351539504275 339476759467047326605595685633/490190179185822886385769104102930967801 0355547895721285919177263022080, `b*`[9] = 131710204241365407062616271 97265625/61559257178867958455217570785227104, a[12,11] = 1155902714407 1691256623556623388974609716247980463646323429860418545796965379405363 7008425503953091180886565/31536133324907183633041146487924575416396446 8656963767284078726004630975015921697525870414526347596936773632, a[12 ,10] = 688343784271498275441415528353054302780001015660014706911988935 0771791431366439329656536871565378282089012991331513/18271814895517947 8466986089870780835242321865388564208018024225296001154507399920094606 6370836641132319880849653760, a[12,9] = -16183509927928156539922841522 5411182739942653401484724580110184517256730426918980054437210005086959 5166981551925667441/19637518660778297585754649024920739916598981028937 029385347383945787366501641192454385997632075140011874187139618963456, a[13,10] = -605850486644121965559554861876248539997477368530704600117 9355536003/22522757208153961727264006949651576410736968355742591798182 90290400, a[13,11] = -729170471864651831281805551502304056571384516928 47535142343993/4466174728801621827685477144283173809323414520322265678 3563600, a[12,8] = -18935700826260772432168308633651734522837925089710 3291049044350530935228180690663776657891613652665009511679250229667441 /104902085728430283184879370421906174798708029629619600881898639306750 333561102065113284728091471708748347777999725133824, a[12,1] = -131987 0176087866963572254233875946356127193892061286068806704341783213319696 27889057541436355642743061150672386594396559/3187539260879955550151479 2620161201024044722829578946248679811622147609393968312389727996156411 8214685494052121351290880, `b*`[8] = 251816923467927457015634155273437 5/38284247150941989115046715871467456\}:" }{TEXT -1 0 "" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "printed coefficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 289 "subs(e16,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(4-i)],i=2.. 4)]));\nfor ii from 5 to 14 do\n print(``);\n print(c[ii]=subs(e16 ,c[ii])); \n for jj to ii-1 do\n print(a[ii,jj]=subs(e16,a[ii,j j]));\n end do:\nend do:\n``;\nfor ii to 14 do\n print(`b*`[ii]=su bs(e16,`b*`[ii]));\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matr ixG6#7%7&#\"\"\"\"#]F(%!GF+7&,&#\"(Os$Q\")v$H%[F)*(\"(o8.\"F)\"*D\")GX \"!\"\"\"\"'#F)\"\"#F),&#\"/OqJ(Q+S#\"0voaB&[9GF4*(\".'HKXEE!e+j%F;F)*(\".'pP[?:zF)F;F4F5F6F)F+7&,&#\"(='=>\")DJ9;F)*( \"'%o:&F)F0F4F5F6F),&#\"'4$f*\")]iGKF)*(\"'@*G\"F)F0F4F5F6F)\"\"!,&#\" (Fz(GFMF)*(FOF)FGF4F5F6F)Q(pprint86\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"&#\"#9\"#X" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"\",&#\"1^QsGO_EG\"1I S!*)pMM&fF(*(\"/vuJ#\\f%oF(\"0./*)pMM&f!\"\"\"\"'#F(\"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"#\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"$,&#\"0X\"eW-SUq\"0-OfYc*oR!\"\"*( \"/&323`x7*\"\"\"\"0,oHByW)>F-\"\"'#F0\"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"%,&#\"0!=DAkD*e*\"0./*)pMM&f\"\"\"*( \"0!y.,5t`?F-F,!\"\"\"\"'#F-\"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"',&#\"$c\"\"$D'\" \"\"*(\"#EF,F+!\"\"F'#F,\"\"#F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"\"'\"\"\",&#\"*36Mw$\"+vV[1%*F(*(\"*mM$z?F(\",D1RXe'!\"\"F'#F( \"\"#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"#\"\"!" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"%,&#\"1HX%)odaVV\"2Dcz&)=)G*y #\"\"\"*(\"04hCT^Ep%F-F,!\"\"F'#F-\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"&,&#\".\\%=y#*f:\"/vVBNy&*G\"\"\"*(\" 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'G,67MH\"ey\"3WMl]'3[cuA'=)fVG\"T?.'*>]N$H;2#GFNV%eER@+vJ0v?zF," }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"(,&#\"U.'G,67MH\"ey\" 3WMl]'3[cuA'=)fVG\"T?.'*>]N$H;2#GFNV%eER@+vJ0v?z\"\"\"*(\"SdUa&eAMK70z mE%4%yWnoA!*32F#=F-\"T!G4YY3#o4M\\]'pmg$zv'=Y)[+x&=7!\"\"\"\"'#F-\"\"# F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\")#\"aoLco&f0mKZ qYfnZR`F/&R:N\"[d%[#=@1AWYHoK*\"ao!3AIEx\">fG@d*yab.,y'4$H5/\"pdQ')G#e =z,>!\\" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"*#!`o\"zp \"QdY;Bw/$oqhN7D1UPsgB8i&oYU04bTgT(\"`ogHr_e_UF*QS0u\"f-PgKbD*Q*fWWo8@ L;t;bT\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"#5#!^o.g`b$ z6+YqI&otZ(**R&[i(='[bfb'>7Wm[]eg\"^o+/H!H=)z\"fUdNopt5kd^'\\p+ksshR:3 svAD#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"#6#!in$*RMU^`Z Gp^%Qrl0/B]^b!=GJ=lk=Zq\"H(\"in+Oc$ycEA._9MK4Q<$GWrZ&oF=i,)GZ:t6C\\oH()))[??u$)fG`#eb\"4mYp['HNw9LTxVw#4a?R:OF\"^rDo;%GYq%GEX.r#> ,o#y^'y3=K([=7?y]LN$3*oi`9LQP@I \"H*H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#9\"\"#\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#9\"\"$\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#9\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#9\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#9\"\"',&#\"U@^SJ:f`I/(>%GJ!oE')e,ev**4%H<]\"T?N3(3*R i=om*RxGO#3;&*>d#\\8uGV!\"\"*(\"T.8([mthxmU9vlWW%)\\Rz&Q#GU+d6\"\"\"\" TSiM')=\\OR3B:M%fqXV3#4MFi!)y*>F-F(#F0\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#9\"\"(,&#\"U@^SJ:f`I/(>%GJ!oE')e,ev**4%H<] \"T?N3(3*Ri=om*RxGO#3;&*>d#\\8uGV!\"\"*(\"T.8([mthxmU9vlWW%)\\Rz&Q#GU+ d6\"\"\"\"TSiM')=\\OR3B:M%fqXV3#4MFi!)y*>F-\"\"'#F0\"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#9\"\")#!eoV7&R]c@W_kgp;Wc#>Vu)oL 8e4Q_reT?qh1ZQp=*\"eoS'orzY'zy2^'HjfD#o6:75BJJn\\5h&yqo\\K0*R05" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#9\"\"*#\"eo^_[jJL&\\m;5(=r ,aM(yo0#)RuyHZ4v!)>JS\"Q**H'G&\"do!))=1x\\KdCVfeumQ\\Vj]lYC_$Rn())*e5' \\Z\"G0**R#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#9\"#5#\"bon \"=7d\\\"e#)H59%o9J(e%*3,0LJU`FuLMWHhe'\\'>\"\"ao+)oo2fw^!ek/UG/gjNP\" \\-e\"*eS8h>utR#z,u>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#9 \"#6#\"\\of)[\\eU\\`q&ybIz%)Q\\^j\">#p'o.IqdA(3onr$\"[o+o$RC?i+m[]Y!ee r%3?f\"Q1)>evCsU_!R8R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#9 \"#7#![sc)*G^d*p,c#[%3S&3D^]><-pJNA<,G)y\"zu\"GWjlwUp.Aga=y*y&*eX57E5M `xMaq'e&RWH\"jrv4\"*3>Eokhy'R_Ud#>,6pLy6ZXF(RR#HWReDROft \"=)pM<4KQ%>`1WOQh" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#9\"# 8#\"jnj)HOO=)pd&4'Q#)yRihi" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"\"#\"%4:ZUGQ" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\" \"*#\"DDcE(>F;E12aOTU?5<8\"D/rA&yqv@b%ez')yrDf:'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#5#\"D++++?F')*)4kU9.56\">L\"Ez;NbC8,[9#*\\D qt>@n>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#6#\"JzOrR_tEnb)3 8!e[p'4TQg)*\"K+?.e#zh`wlIBGpLoiV70u&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#7#\"arO^VX\"R=+!>tz)41Uqq@]g]?'\\!3Ep*o?;Y*R+wY[blSLa]9 \"*3I*y:u5w&Q&odj$4&\"brD\"HzgrvTYOe]!HNR&>NI\"4rz?Fg%4&R([w%yW0yQHk'z 3;S')=8P5&piA*H#=kj5*eS" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\" #8#\"KL\"4;hc)f*fFm$>zL_'4@2,3\"\"K?rP6uR^?e+Kb^JA\"=$>*oq&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#9#\";J5>&=mf&)=5bfo#\" " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "`RK8_14eqs*` := subs(b=`b*`,[op(RowSumCond itions(14,'expanded')),op(OrderConditions(8,14,'expanded'))]):" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Check:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "expand(subs(e16,`RK8_14eqs*` )):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7ax\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$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#\"$8#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "We can calculat e the principal error norm, that is, the 2-norm of the principal error terms. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 185 "`errterms8_14*` := subs(b=`b*`,PrincipalErrorTerms(8,14,'expanded')):\nsm := 0:\nfor \+ ct to nops(`errterms8_14*`) do\n sm := sm+(evalf(subs(e16,`errterms8 _14*`[ct])))^2;\nend do:\nsqrt(sm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #$\"+TWb@7!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 138 "We can include the new stage for the embedded scheme as \+ an additional 16th stage added to the order 9 scheme along with the co efficients " }{XPPEDIT 18 0 "a[16,14] = 0;" "6#/&%\"aG6$\"#;\"#9\"\"! " }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[16,15] = 0;" "6#/&%\"aG6$\"# ;\"#:\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 13 "The weights " }{XPPEDIT 18 0 "`b*`[i]" "6#&%#b*G6#%\"iG" }{TEXT -1 7 " for " } {XPPEDIT 18 0 "i=1" "6#/%\"iG\"\"\"" }{TEXT -1 89 " . . 13 of the 16 s tage combined scheme are those of the 14 stage scheme and the weight \+ " }{XPPEDIT 18 0 "`b*`[14];" "6#&%#b*G6#\"#9" }{TEXT -1 34 " in the 1 4 stage scheme becomes " }{XPPEDIT 18 0 "`b*`[16];" "6#&%#b*G6#\"#;" }{TEXT -1 25 " in the 16 stage scheme." }}{PARA 0 "" 0 "" {TEXT -1 8 "We set " }{XPPEDIT 18 0 "c[16] = 1;" "6#/&%\"cG6#\"#;\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[14] = 0;" "6#/&%#b*G6#\"#9\"\"!" } {TEXT -1 7 " and " }{XPPEDIT 18 0 "`b*`[15] = 0;" "6#/&%#b*G6#\"#:\" \"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 66 "We can make the or der 9 scheme into a 16 stage scheme by setting " }{XPPEDIT 18 0 "b[16 ] = 0;" "6#/&%\"bG6#\"#;\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 201 "e17 := \{c[ 16]=1,seq(a[16,i]=subs(e16,a[14,i]),i=1..13),a[16,14]=0,a[16,15]=0,b[1 6]=0,\nseq(`b*`[i]=subs(e16,`b*`[i]),i=1..13),`b*`[14]=0,`b*`[15]=0,`b *`[16]=subs(e16,`b*`[14])\}:\ne18 := `union`(e13,e17):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e18" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13588 "e18 := \{`b*`[7] = 0 , `b*`[5] = 0, `b*`[6] = 0, `b*`[3] = 0, `b*`[4] = 0, `b*`[2] = 0, b[1 6] = 0, a[16,14] = 0, `b*`[14] = 0, `b*`[15] = 0, c[10] = 21/200, c[9] = 39/125, c[8] = 52/125, c[5] = 14/45, a[2,1] = 1/50, c[16] = 1, c[15 ] = 1, c[14] = 229/250, c[13] = 247/281, a[5,2] = 0, a[4,2] = 0, a[6,3 ] = 0, a[6,2] = 0, c[11] = 280/477, c[2] = 1/50, a[8,3] = 0, a[8,2] = \+ 0, a[8,1] = 52/1125, a[7,3] = 0, a[7,2] = 0, a[9,5] = 0, a[9,4] = 0, a [9,3] = 0, a[9,2] = 0, a[9,1] = 741/16000, a[8,5] = 0, a[8,4] = 0, a[1 0,3] = 0, a[10,2] = 0, a[10,1] = 35291978967/748709478400, a[9,8] = -3 51/16000, a[7,4] = 23459106068523828440829/354298872323611753203125+78 70375504052283205581/354298872323611753203125*6^(1/2), a[11,3] = 0, a[ 11,2] = 0, a[10,9] = -981041103/4679434240, a[10,8] = -6251205429/1497 41895680, a[10,5] = 0, a[10,4] = 0, a[12,3] = 0, a[12,2] = 0, a[11,10] = -3/20, a[11,5] = 0, a[11,4] = 0, a[13,2] = 0, a[12,5] = 0, a[12,4] \+ = 0, a[13,5] = 0, a[13,4] = 0, a[13,3] = 0, a[14,5] = 0, a[14,4] = 0, \+ a[14,3] = 0, a[14,2] = 0, a[15,5] = 0, a[15,4] = 0, a[15,3] = 0, a[15, 2] = 0, a[16,5] = 0, a[16,4] = 0, a[16,3] = 0, a[16,2] = 0, b[4] = 0, \+ b[3] = 0, b[2] = 0, a[16,15] = 0, b[7] = 0, b[6] = 0, b[5] = 0, b[14] \+ = 3769686146953412690297035156250000/195792979665408643382362918863397 227, a[6,4] = 4343545768844529/27892881885795625+469265141246109/27892 881885795625*6^(1/2), a[7,5] = 146263465360621089/7558718942052734375- 1881455818308499953/52911032594369140625*6^(1/2), `b*`[11] = 986038410 96694858013088556726735239713679/5740512436268336928233065765361792580 32000, b[15] = 50782110772148063247179059/1538266148871578545201811280 , c[7] = 156/625-26/625*6^(1/2), a[8,6] = 208/1125-13/1125*6^(1/2), a[ 8,7] = 208/1125+13/1125*6^(1/2), a[10,6] = 23154511989/149741895680+39 398793/1772093440*6^(1/2), a[10,7] = 23154511989/149741895680-39398793 /1772093440*6^(1/2), a[9,7] = 2301/16000+897/32000*6^(1/2), a[7,6] = 9 444124356888/82889304453125-2459298027368/82889304453125*6^(1/2), a[5, 3] = -704240024458145/396895646593602+91277530807085/198447823296801*6 ^(1/2), a[5,1] = 2826523628723851/5953434698904030-68459492317475/5953 43469890403*6^(1/2), a[3,2] = 46300580261936/281448523546875+791520483 7696/281448523546875*6^(1/2), a[4,1] = 959309/32286250+128921/48429375 *6^(1/2), `b*`[1] = 135131455470598097879473933/4525454214341291369580 624000, a[3,1] = -24000387317036/281448523546875-5917264532296/2814485 23546875*6^(1/2), b[13] = 9160897746149204383653282352747804858423571/ 54934119002888850773584011583391921191449440, a[11,6] = -1736562342312 744743536201/1109319064566277200150528-360257484908262597335743/511993 414415204861607936*6^(1/2), a[9,6] = 2301/16000-897/32000*6^(1/2), a[1 1,7] = -1736562342312744743536201/1109319064566277200150528+3602574849 08262597335743/511993414415204861607936*6^(1/2), c[4] = 1918618/161431 25+515684/48429375*6^(1/2), a[5,4] = 958925642225180/595343469890403-2 05373100103780/595343469890403*6^(1/2), a[6,5] = 1559927818449/2895783 5234375+4382126882523/202704846640625*6^(1/2), a[4,3] = 2877927/322862 50+128921/16143125*6^(1/2), c[3] = 3837236/48429375+1031368/145288125* 6^(1/2), a[6,1] = 376341108/9406484375+207933466/65845390625*6^(1/2), \+ a[13,1] = 521517478355891840799758346863554340798833271924176460576994 9554629/20283132613214812064685094275151111714651171227532533713038580 121600, a[12,7] = -581038619225160876203856834629458675128926705143465 192450716448466169075797359178616021045291080972121429188543592047/101 1917225676176365127453733973371461080784851732664960275549575306273314 094232139356444322425772110112679530543972352-713482798078988149650882 3372973790665635109665954131159198195323855274480577648002828260278094 2939988708855996996031/51893191060316736673202755588378023645168453935 008459501310234631090939184319596890074067816706262057060488745844736* 6^(1/2), b[9] = 1929922737998470573359614532470703125/9295447834009061 726737853188569292704, b[12] = 909034900749411645631439991260524977916 8865915025483551303301488290668967641515552920382223333668169935568609 35646735988456500531298304/6301978749188979317659380355882211371188146 5060665432261072262174934439860313163064501519226006205345791045010423 37690306078523205079625, a[14,7] = 10256190098435854298655077997613296 122112139953/1148037852243817086246902953973393653760000000*6^(1/2)-86 88525606146315530022414580346392155721271039/2238673811875443318181460 7602481176248320000000, a[15,7] = -27905082713561818810613870497657111 8076242172562777/26980717750745660055932121988692169249262917386240-59 017804198407615229179283246229064921710388893173/179871451671637733706 21414659128112832841944924160*6^(1/2), a[13,6] = -18227070890226867447 840942666790512323422585544257/121857700488461867579360666965049340968 208464609280*6^(1/2)+2843598186227456480865065344408178581293412110128 603/792075053175002139265844335272820716293355019960320, a[14,6] = -10 256190098435854298655077997613296122112139953/114803785224381708624690 2953973393653760000000*6^(1/2)-868852560614631553002241458034639215572 1271039/22386738118754433181814607602481176248320000000, a[16,1] = -36 388658330162124762200023703074655379362961851837455245313588466117/299 291302137383314536268908335335078201218487321808786511993451315200, b[ 11] = 3416676287738448149119878197304164096817920457/22521752441211566 270536786917243920830369456000, b[1] = 30703843389361946002220520407/1 036329015084155723633962896000, a[11,8] = 512032742176678555764127/369 773021522092400050176, a[16,8] = -918693847066170204158715238095813336 88744319256441669606452442156503951243/1005399053249687078561104967313 1231012151168225596329651077879646797168640, a[16,9] = 528629993814031 19807509472978743982056878734540171187101666495333163485251/2399905281 474961058988767393522446655063434938667458594324573249770618880, a[16, 10] = 1196496586129443433742753423133050108945873114684141029825814957 1218167/19740179239737419611340589158024913735636004284204645805176590 76868800, a[16,11] = 3716768087225770300368669219163514938847930557857 0534942584948859/39133905242722475581980638159200847158580465048660062 20243936800, b[8] = 1516681888913470906364013671875/194237682145824399 36604117641536, a[11,9] = 248233526294563631278471/1039986623030884875 14112, `b*`[13] = 108010721096523379193662759959856611609133/570689193 181223151553200582051397411377120, a[16,12] = -29443955867054347753341 0261210455895789781854602203694276656344281747917882801172235316902171 95051250854008448256016995751289856/6138364406531943832091734698181735 9196181824229977267281736392558394429239397274547117833691101192574252 39678616468261908910975, a[16,13] = 4078167483851726864981531813468124 32791118177175769818363629863/6261623978823860951962016297683037596282 50075752997120162580989, a[11,1] = 1601589807329134144752443/166397859 68494158002257920, b[10] = 27072397368129209968072433152000000000/1595 40891067276798629433718421290211669, c[12] = 3658227035053715/53497047 19299032, a[7,1] = 11781705468/235162109375+2328587014/1646134765625*6 ^(1/2), a[16,7] = -501729409997558015886266803128419704305359153140512 1/432874134925719951608236287739966681862399087083520+1157004228238579 39498444446575144266776173664871303/1997880622734092084345705943415230 83936491886346240*6^(1/2), a[16,6] = -50172940999755801588626680312841 97043053591531405121/4328741349257199516082362877399666818623990870835 20-115700422823857939498444446575144266776173664871303/199788062273409 208434570594341523083936491886346240*6^(1/2), `b*`[12] = 5093635768538 5761074157893008911450543340655548467600399461620689692608049620506050 21707042060987973190001839145435136/4058910636418229922626951037131886 4016087966429387805447847648739509460272079710913035195393529050583646 417571607929125, `b*`[10] = 33191111003144264098986272000000000/196721 197370254992144801132455351679, c[6] = 156/625+26/625*6^(1/2), a[14,12 ] = -88682414394183619425441647866243388112917289239161463940944492930 492112547171652363240146123589908870567811533658125375935101390832/940 5104776230176067202383689444499684238639823097650089954333383377958770 491999076242219980453370248022825420814384818872314453125, a[13,12] = \+ 2736153920540927643774133147635296486946660915558253285983742020488887 296849241173151960647763453239551016003889152/248567211069834101529026 4470463939203955869249618375406787169018009688457749866177826801192710 345262847046284166825, a[12,6] = -581038619225160876203856834629458675 1289267051434651924507164484661690757973591786160210452910809721214291 88543592047/1011917225676176365127453733973371461080784851732664960275 549575306273314094232139356444322425772110112679530543972352+713482798 0789881496508823372973790665635109665954131159198195323855274480577648 0028282602780942939988708855996996031/51893191060316736673202755588378 0236451684539350084595013102346310909391843195968900740678167062620570 60488745844736*6^(1/2), a[14,10] = -1259710340512037041830744503634468 47441594334546885083244594242327104115033/5066049934698363488698655054 901069679084758735799331062593807151200000000, a[13,7] = 2843598186227 456480865065344408178581293412110128603/792075053175002139265844335272 820716293355019960320+182270708902268674478409426667905123234225855442 57/121857700488461867579360666965049340968208464609280*6^(1/2), a[15,1 0] = 73158981980491143736917790272372062352348938687470903172269108609 63581499/4324002193791316841836555174009568058670247839597575825330748 34771793600, a[15,6] = -2790508271356181881061387049765711180762421725 62777/26980717750745660055932121988692169249262917386240+5901780419840 7615229179283246229064921710388893173/17987145167163773370621414659128 112832841944924160*6^(1/2), a[13,9] = -7416041550905424668562132360723 74206251235617068304762316465738169791/1415516731633211368444459938925 55326037025917405403892742525852712960, a[13,8] = 93268294644220621182 48457481351539504275339476759467047326605595685633/4901901791858228863 857691041029309678010355547895721285919177263022080, a[15,8] = 6824047 7823918559060550996013166770535743446467404475965020846786328901/69628 625900822775316247857716283393138506964199995817446989013471723520, `b *`[9] = 13171020424136540706261627197265625/61559257178867958455217570 785227104, a[15,14] = 368176545506575596342007241113258886329861009608 750000000/7515329389098801941975451526298754679007062667248055263091, \+ a[12,11] = 11559027144071691256623556623388974609716247980463646323429 8604185457969653794053637008425503953091180886565/31536133324907183633 0411464879245754163964468656963767284078726004630975015921697525870414 526347596936773632, a[14,11] = 432233849549515274325250500583717799422 0267688026960252214552638944423/23686762578750842215295816717967675753 5142999000924357630500000000000, a[15,11] = -8703591258468375212464518 7592152267644073875904388006117245587111831/41468532532723053663401983 927439573730970639521941633396843682248800, a[12,10] = 688343784271498 2754414155283530543027800010156600147069119889350771791431366439329656 536871565378282089012991331513/182718148955179478466986089870780835242 3218653885642080180242252960011545073999200946066370836641132319880849 653760, a[15,13] = 190132386928877842671649814278676303562620818706009 46422701364458/1465163086331441981107358057624009056064637331918409856 48075179899, a[12,9] = -1618350992792815653992284152254111827399426534 0148472458011018451725673042691898005443721000508695951669815519256674 41/1963751866077829758575464902492073991659898102893702938534738394578 7366501641192454385997632075140011874187139618963456, a[14,13] = 26235 475641986625187247554297838197168935151270802587/317816209571981740338 17415268740604591106877500000000, a[13,10] = -605850486644121965559554 8618762485399974773685307046001179355536003/22522757208153961727264006 94965157641073696835574259179818290290400, a[15,12] = 7930060543280416 5106136013125641247440025308990955400537833272821480610899521213829175 9017448087224471716436232175864384424753159293287828190208/60063530550 7048430170531323174007915813446499853348186301444535722932552444071789 775577508083955280258070844780520371987041149053341617704219325, a[15, 9] = 48531604865335743440838806675493568975092395234916265724406574203 650554879/752907556904945072471544795110573032739130431442432355915891 3160761835520, a[13,11] = -7291704718646518312818055515023040565713845 1692847535142343993/44661747288016218276854771442831738093234145203222 656783563600, a[14,1] = 1961431625890315687063141575818232405522545898 155499982338718373117379429883/480056647167077429990593568055406093586 176318669944422673481728000000000000, a[14,9] = 6832105542579354626002 57975958139742203919396113084127371502375524416129719/2689533720056524 3662247103690698994332502640106760065066162305761280000000, a[15,1] = \+ -293368876868555373719392219044290241463856990716581942699984715189474 7/14239678548131378023505167950652012589301076964702261708139037458432 00, a[14,8] = -1081513920922904249534988363800597726097364037394344810 43071361807712075869481/8600735495194563448316261478331353993230137106 756553600705167485829120000000, a[12,8] = -189357008262607724321683086 3365173452283792508971032910490443505309352281806906637766578916136526 65009511679250229667441/1049020857284302831848793704219061747987080296 2961960088189863930675033356110206511328472809147170874834777799972513 3824, a[12,1] = -13198701760878669635722542338759463561271938920612860 6880670434178321331969627889057541436355642743061150672386594396559/31 8753926087995555015147926201612010240447228295789462486798116221476093 939683123897279961564118214685494052121351290880, `b*`[8] = 2518169234 679274570156341552734375/38284247150941989115046715871467456, `b*`[16] = 26859551018855966185191031/763900876650511794556001520\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "p rinted coefficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 362 "subs(e18,matrix([seq([c[i],seq(a[i,j],j=1. .i-1),``$(4-i)],i=2..4)]));\nfor ii from 5 to 16 do\n print(``);\n \+ print(c[ii]=subs(e18,c[ii])); \n for jj to ii-1 do\n print(a[i i,jj]=subs(e18,a[ii,jj]));\n end do:\nend do:print(``);\nfor ii to 1 6 do\n print(b[ii]=subs(e18,b[ii]));\nend do:print(``);\nfor ii to 1 6 do\n print(`b*`[ii]=subs(e18,`b*`[ii]));\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7%7&#\"\"\"\"#]F(%!GF+7&,&#\"(Os$Q\")v $H%[F)*(\"(o8.\"F)\"*D\")GX\"!\"\"\"\"'#F)\"\"#F),&#\"/OqJ(Q+S#\"0voaB &[9GF4*(\".'HKXEE!e+j%F;F)*(\".'pP[?:zF)F;F4F5F6 F)F+7&,&#\"(='=>\")DJ9;F)*(\"'%o:&F)F0F4F5F6F),&#\"'4$f*\")]iGKF)*(\"' @*G\"F)F0F4F5F6F)\"\"!,&#\"(Fz(GFMF)*(FOF)FGF4F5F6F)Q(pprint96\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"cG6#\"\"&#\"#9\"#X" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"\"&\"\"\",&#\"1^QsGO_EG\"1IS!*)pMM&fF(*(\"/vuJ#\\f%oF(\"0./*)pMM&f! \"\"\"\"'#F(\"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"& \"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"$,&#\" 0X\"eW-SUq\"0-OfYc*oR!\"\"*(\"/&323`x7*\"\"\"\"0,oHByW)>F-\"\"'#F0\"\" #F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"%,&#\"0!=DAkD *e*\"0./*)pMM&f\"\"\"*(\"0!y.,5t`?F-F,!\"\"\"\"'#F-\"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" cG6#\"\"',&#\"$c\"\"$D'\"\"\"*(\"#EF,F+!\"\"F'#F,\"\"#F," }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"\",&#\"*36Mw$\"+vV[1%*F(*(\"*m M$z?F(\",D1RXe'!\"\"F'#F(\"\"#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/& %\"aG6$\"\"'\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\" \"'\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"%,& #\"1HX%)odaVV\"2Dcz&)=)G*y#\"\"\"*(\"04hCT^Ep%F-F,!\"\"F'#F-\"\"#F-" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"&,&#\".\\%=y#*f:\"/ vVBNy&*G\"\"\"*(\".BD)o7#Q%F-\"0D1kY[q-#!\"\"F'#F-\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6 #\"\"(,&#\"$c\"\"$D'\"\"\"*(\"#EF,F+!\"\"\"\"'#F,\"\"#F/" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"\",&#\",oaq\"y6\"-v$4@;N#F(*( \"+9qeGBF(\".DcwMhk\"!\"\"\"\"'#F(\"\"#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 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132613214812064685094275151111714651171227532533713038580121600,\na[13 ,2]=0,\na[13,3]=0,\na[13,4]=0,\na[13,5]=0,\na[13,6]=284359818622745648 0865065344408178581293412110128603/\n 7920750531750021392658443 35272820716293355019960320\n -182270708902268674478409426667905 12323422585544257/\n 121857700488461867579360666965049340968208 464609280*6^(1/2),\na[13,7]=284359818622745648086506534440817858129341 2110128603/\n 7920750531750021392658443352728207162933550199603 20+\n 18227070890226867447840942666790512323422585544257/\n \+ 121857700488461867579360666965049340968208464609280*6^(1/2),\na[13 ,8]=932682946442206211824845748135153950427533947675946704732660559568 5633/\n 4901901791858228863857691041029309678010355547895721285 919177263022080,\na[13,9]=-7416041550905424668562132360723742062512356 17068304762316465738169791/\n 14155167316332113684444599389255 5326037025917405403892742525852712960,\na[13,10]=-60585048664412196555 95548618762485399974773685307046001179355536003/\n 2252275720 815396172726400694965157641073696835574259179818290290400,\na[13,11]=- 72917047186465183128180555150230405657138451692847535142343993/\n \+ 44661747288016218276854771442831738093234145203222656783563600,\n a[13,12]=2736153920540927643774133147635296486946660915558253285983742 020488887296849241173151960647763453239551016003889152/248567211069834 1015290264470463939203955869249618375406787169018009688457749866177826 801192710345262847046284166825,\na[14,1]=19614316258903156870631415758 18232405522545898155499982338718373117379429883/\n 480056647167 077429990593568055406093586176318669944422673481728000000000000,\na[14 ,2]=0,\na[14,3]=0,\na[14,4]=0,\na[14,5]=0,\na[14,6]=-86885256061463155 30022414580346392155721271039/\n 22386738118754433181814607602 481176248320000000-\n 1025619009843585429865507799761329612211 2139953/\n 1148037852243817086246902953973393653760000000*6^(1 /2),\na[14,7]=-8688525606146315530022414580346392155721271039/\n \+ 22386738118754433181814607602481176248320000000+\n 10256190 098435854298655077997613296122112139953/\n 1148037852243817086 246902953973393653760000000*6^(1/2),\na[14,8]=-10815139209229042495349 8836380059772609736403739434481043071361807712075869481/\n 860 0735495194563448316261478331353993230137106756553600705167485829120000 000,\na[14,9]=68321055425793546260025797595813974220391939611308412737 1502375524416129719/\n 2689533720056524366224710369069899433250 2640106760065066162305761280000000,\na[14,10]=-12597103405120370418307 4450363446847441594334546885083244594242327104115033/\n 50660 49934698363488698655054901069679084758735799331062593807151200000000, \na[14,11]=43223384954951527432525050058371779942202676880269602522145 52638944423/\n 23686762578750842215295816717967675753514299900 0924357630500000000000,\na[14,12]=-88682414394183619425441647866243388 1129172892391614639409444929304921125471716523632401461235899088705678 11533658125375935101390832/9405104776230176067202383689444499684238639 8230976500899543333833779587704919990762422199804533702480228254208143 84818872314453125,\na[14,13]=26235475641986625187247554297838197168935 151270802587/\n 3178162095719817403381741526874060459110687750 0000000,\na[15,1]=-293368876868555373719392219044290241463856990716581 9426999847151894747/\n 142396785481313780235051679506520125893 0107696470226170813903745843200,\na[15,2]=0,\na[15,3]=0,\na[15,4]=0,\n a[15,5]=0,\na[15,6]=-2790508271356181881061387049765711180762421725627 77/\n 26980717750745660055932121988692169249262917386240+\n \+ 59017804198407615229179283246229064921710388893173/\n 17 987145167163773370621414659128112832841944924160*6^(1/2),\na[15,7]=-27 9050827135618188106138704976571118076242172562777/\n 269807177 50745660055932121988692169249262917386240-\n 59017804198407615 229179283246229064921710388893173/\n 1798714516716377337062141 4659128112832841944924160*6^(1/2),\na[15,8]=68240477823918559060550996 013166770535743446467404475965020846786328901/\n 69628625900822 775316247857716283393138506964199995817446989013471723520,\na[15,9]=48 5316048653357434408388066754935689750923952349162657244065742036505548 79/\n 752907556904945072471544795110573032739130431442432355915 8913160761835520,\na[15,10]=731589819804911437369177902723720623523489 3868747090317226910860963581499/\n 432400219379131684183655517 400956805867024783959757582533074834771793600,\na[15,11]=-870359125846 83752124645187592152267644073875904388006117245587111831/\n 4 1468532532723053663401983927439573730970639521941633396843682248800,\n a[15,12]=7930060543280416510613601312564124744002530899095540053783327 2821480610899521213829175901744808722447171643623217586438442475315929 3287828190208/60063530550704843017053132317400791581344649985334818630 1444535722932552444071789775577508083955280258070844780520371987041149 053341617704219325,\na[15,13]=1901323869288778426716498142786763035626 2081870600946422701364458/\n 146516308633144198110735805762400 905606463733191840985648075179899,\na[15,14]=3681765455065755963420072 41113258886329861009608750000000/\n 75153293890988019419754515 26298754679007062667248055263091,\na[16,1]=-36388658330162124762200023 703074655379362961851837455245313588466117/\n 2992913021373833 14536268908335335078201218487321808786511993451315200,\na[16,2]=0,\na[ 16,3]=0,\na[16,4]=0,\na[16,5]=0,\na[16,6]=-501729409997558015886266803 1284197043053591531405121/\n 432874134925719951608236287739966 681862399087083520-\n 1157004228238579394984444465751442667761 73664871303/\n 19978806227340920843457059434152308393649188634 6240*6^(1/2),\na[16,7]=-5017294099975580158862668031284197043053591531 405121/\n 432874134925719951608236287739966681862399087083520+ \n 115700422823857939498444446575144266776173664871303/\n \+ 199788062273409208434570594341523083936491886346240*6^(1/2),\na[16 ,8]=-91869384706617020415871523809581333688744319256441669606452442156 503951243/\n 1005399053249687078561104967313123101215116822559 6329651077879646797168640,\na[16,9]=5286299938140311980750947297874398 2056878734540171187101666495333163485251/\n 2399905281474961058 988767393522446655063434938667458594324573249770618880,\na[16,10]=1196 4965861294434337427534231330501089458731146841410298258149571218167/\n 1974017923973741961134058915802491373563600428420464580517659 076868800,\na[16,11]=3716768087225770300368669219163514938847930557857 0534942584948859/\n 391339052427224755819806381592008471585804 6504866006220243936800,\na[16,12]=-29443955867054347753341026121045589 5789781854602203694276656344281747917882801172235316902171950512508540 08448256016995751289856/6138364406531943832091734698181735919618182422 9977267281736392558394429239397274547117833691101192574252396786164682 61908910975,\na[16,13]=40781674838517268649815318134681243279111817717 5769818363629863/\n 626162397882386095196201629768303759628250 075752997120162580989,\na[16,14]=0,\na[16,15]=0,\n\nb[1]=3070384338936 1946002220520407/1036329015084155723633962896000,\nb[2]=0,\nb[3]=0,\nb [4]=0,\nb[5]=0,\nb[6]=0,\nb[7]=0,\nb[8]=151668188891347090636401367187 5/19423768214582439936604117641536,\nb[9]=1929922737998470573359614532 470703125/9295447834009061726737853188569292704,\nb[10]=27072397368129 209968072433152000000000/159540891067276798629433718421290211669,\nb[1 1]=3416676287738448149119878197304164096817920457/\n 225217524412 11566270536786917243920830369456000,\nb[12]=90903490074941164563143999 1260524977916886591502548355130330148829066896764151555292038222333366 816993556860935646735988456500531298304/630197874918897931765938035588 2211371188146506066543226107226217493443986031316306450151922600620534 579104501042337690306078523205079625,\nb[13]=9160897746149204383653282 352747804858423571/\n 5493411900288885077358401158339192119144944 0,\nb[14]=3769686146953412690297035156250000/1957929796654086433823629 18863397227,\nb[15]=50782110772148063247179059/15382661488715785452018 11280,\nb[16]=0,\n\n`b*`[1]=135131455470598097879473933/45254542143412 91369580624000,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=0,\n`b*`[5]=0,\n`b*`[ 6]=0,\n`b*`[7]=0,\n`b*`[8]=2518169234679274570156341552734375/38284247 150941989115046715871467456,\n`b*`[9]=13171020424136540706261627197265 625/61559257178867958455217570785227104,\n`b*`[10]=3319111100314426409 8986272000000000/196721197370254992144801132455351679,\n`b*`[11]=98603 841096694858013088556726735239713679/\n 5740512436268336928233 06576536179258032000,\n`b*`[12]=50936357685385761074157893008911450543 3406555484676003994616206896926080496205060502170704206098797319000183 9145435136/40589106364182299226269510371318864016087966429387805447847 648739509460272079710913035195393529050583646417571607929125,\n`b*`[13 ]=108010721096523379193662759959856611609133/5706891931812231515532005 82051397411377120,\n`b*`[14]=0,\n`b*`[15]=0,\n`b*`[16]=268595510188559 66185191031/763900876650511794556001520\}:" }}}{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[9,16];" "6#&%\"TG6$\"\"*\"#;" }{TEXT -1 129 " denote the vector \+ whose components are the principal error terms of the 16 stage, order \+ 9 scheme (the error terms of order 10)." }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "`T*`[8,16];" "6#&%#T*G6$\"\")\"#;" }{TEXT -1 146 " denote the vector whose components are the principal error term s of the embedded 16 stage, order 8 scheme (the error terms of order 9 ) and let " }{XPPEDIT 18 0 "`T*`[9, 16];" "6#&%#T*G6$\"\"*\"#;" } {TEXT -1 100 " denote the vector whose components are the error terms of order 10 of the embedded order 8 scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Denote the 2-norm of these vect ors by " }{XPPEDIT 18 0 "abs(abs(T[9,16]));" "6#-%$absG6#-F$6#&%\"TG6 $\"\"*\"#;" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "abs(abs(`T*`[8,16]));" " 6#-%$absG6#-F$6#&%#T*G6$\"\")\"#;" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "abs(abs(`T*`[9,16]));" "6#-%$absG6#-F$6#&%#T*G6$\"\"*\"#;" }{TEXT -1 15 " respectively." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Define: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "A[10] = abs(abs(T[9, 16]));" "6#/&%\"AG6#\"#5-%$absG6#- F)6#&%\"TG6$\"\"*\"#;" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "B[10] = ab s(abs(`T*`[9,16]))/abs(abs(`T*`[8,16]));" "6#/&%\"BG6#\"#5*&-%$absG6#- F*6#&%#T*G6$\"\"*\"#;\"\"\"-F*6#-F*6#&F/6$\"\")F2!\"\"" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "C[10] = abs(abs(`T*`[9,16]-T[9,16]))/abs(abs(`T *`[8,16]));" "6#/&%\"CG6#\"#5*&-%$absG6#-F*6#,&&%#T*G6$\"\"*\"#;\"\"\" &%\"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 Pri nce have suggested that as well as attempting to ensure that " } {XPPEDIT 18 0 "A[10];" "6#&%\"AG6#\"#5" }{TEXT -1 73 " is a minimum, \+ if the embedded scheme is to be used for error control, " }{XPPEDIT 18 0 "B[10];" "6#&%\"BG6#\"#5" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "C [10];" "6#&%\"CG6#\"#5" }{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 0 168 "errterms9_16 := PrincipalEr rorTerms(9,16,'expanded'):\n`errterms9_16*` :=subs(b=`b*`,errterms9_16 ):\n`errterms8_16*` := subs(b=`b*`,PrincipalErrorTerms(8,16,'expanded' )):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 468 "nmB := 0: \nfor ct to nops(`errterms9_16*`) do\n \+ nmB := nmB+evalf(subs(ee,`errterms9_16*`[ct]))^2;\nend do:\nsnmB := sq rt(nmB):\ndnB := 0:\nfor ct to nops(`errterms8_16*`) do\n dnB := dnB +evalf(subs(ee,`errterms8_16*`[ct]))^2;\nend do:\nsdnB := sqrt(dnB):\n nmC := 0:\nfor ct to nops(errterms9_16) do\n nmC := nmC+(evalf(subs( ee,`errterms9_16*`[ct]))-evalf(subs(ee,errterms9_16[ct])))^2;\nend do: \nsnmC := sqrt(nmC):\n'B[10]'= evalf[8](snmB/sdnB);\n'C[10]'= evalf[8] (snmC/sdnB);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"BG6#\"#5$\")EFN " 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 36 "co efficients of the combined scheme " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13775 "ee := \{c[2]=1/50,\nc[3]= 3837236/48429375+1031368/145288125*6^(1/2),\nc[4]=1918618/16143125+515 684/48429375*6^(1/2),\nc[5]=14/45,\nc[6]=156/625+26/625*6^(1/2),\nc[7] =156/625-26/625*6^(1/2),\nc[8]=52/125,\nc[9]=39/125,\nc[10]=21/200,\nc [11]=280/477,\nc[12]=3658227035053715/5349704719299032,\nc[13]=247/281 ,\nc[14]=229/250,\nc[15]=1,\nc[16]=1,\n\na[2,1]=1/50,\na[3,1]=-2400038 7317036/281448523546875-5917264532296/281448523546875*6^(1/2),\na[3,2] =46300580261936/281448523546875+7915204837696/281448523546875*6^(1/2), \na[4,1]=959309/32286250+128921/48429375*6^(1/2),\na[4,2]=0,\na[4,3]=2 877927/32286250+128921/16143125*6^(1/2),\na[5,1]=2826523628723851/5953 434698904030-68459492317475/595343469890403*6^(1/2),\na[5,2]=0,\na[5,3 ]=-704240024458145/396895646593602+91277530807085/198447823296801*6^(1 /2),\na[5,4]=958925642225180/595343469890403-205373100103780/595343469 890403*6^(1/2),\na[6,1]=376341108/9406484375+207933466/65845390625*6^( 1/2),\na[6,2]=0,\na[6,3]=0,\na[6,4]=4343545768844529/27892881885795625 +469265141246109/27892881885795625*6^(1/2),\na[6,5]=1559927818449/2895 7835234375+4382126882523/202704846640625*6^(1/2),\na[7,1]=11781705468/ 235162109375+2328587014/1646134765625*6^(1/2),\na[7,2]=0,\na[7,3]=0,\n a[7,4]=23459106068523828440829/354298872323611753203125+78703755040522 83205581/354298872323611753203125*6^(1/2),\na[7,5]=146263465360621089/ 7558718942052734375-1881455818308499953/52911032594369140625*6^(1/2), \na[7,6]=9444124356888/82889304453125-2459298027368/82889304453125*6^( 1/2),\na[8,1]=52/1125,\na[8,2]=0,\na[8,3]=0,\na[8,4]=0,\na[8,5]=0,\na[ 8,6]=208/1125-13/1125*6^(1/2),\na[8,7]=208/1125+13/1125*6^(1/2),\na[9, 1]=741/16000,\na[9,2]=0,\na[9,3]=0,\na[9,4]=0,\na[9,5]=0,\na[9,6]=2301 /16000-897/32000*6^(1/2),\na[9,7]=2301/16000+897/32000*6^(1/2),\na[9,8 ]=-351/16000,\na[10,1]=35291978967/748709478400,\na[10,2]=0,\na[10,3]= 0,\na[10,4]=0,\na[10,5]=0,\na[10,6]=23154511989/149741895680+39398793/ 1772093440*6^(1/2),\na[10,7]=23154511989/149741895680-39398793/1772093 440*6^(1/2),\na[10,8]=-6251205429/149741895680,\na[10,9]=-981041103/46 79434240,\na[11,1]=1601589807329134144752443/1663978596849415800225792 0,\na[11,2]=0,\na[11,3]=0,\na[11,4]=0,\na[11,5]=0,\na[11,6]=-173656234 2312744743536201/1109319064566277200150528-\n 3602574849082625 97335743/511993414415204861607936*6^(1/2),\na[11,7]=-17365623423127447 43536201/1109319064566277200150528+\n 360257484908262597335743 /511993414415204861607936*6^(1/2),\na[11,8]=512032742176678555764127/3 69773021522092400050176,\na[11,9]=248233526294563631278471/10399866230 3088487514112,\na[11,10]=-3/20,\na[12,1]=-1319870176087866963572254233 8759463561271938920612860688067043417832133196962788905754143635564274 3061150672386594396559/31875392608799555501514792620161201024044722829 5789462486798116221476093939683123897279961564118214685494052121351290 880,\na[12,2]=0,\na[12,3]=0,\na[12,4]=0,\na[12,5]=0,\na[12,6]=\n-58103 8619225160876203856834629458675128926705143465192450716448466169075797 359178616021045291080972121429188543592047/101191722567617636512745373 3973371461080784851732664960275549575306273314094232139356444322425772 110112679530543972352+\n7134827980789881496508823372973790665635109665 9541311591981953238552744805776480028282602780942939988708855996996031 /518931910603167366732027555883780236451684539350084595013102346310909 39184319596890074067816706262057060488745844736*6^(1/2),\na[12,7]=\n-5 8103861922516087620385683462945867512892670514346519245071644846616907 5797359178616021045291080972121429188543592047/10119172256761763651274 5373397337146108078485173266496027554957530627331409423213935644432242 5772110112679530543972352-\n713482798078988149650882337297379066563510 9665954131159198195323855274480577648002828260278094293998870885599699 6031/51893191060316736673202755588378023645168453935008459501310234631 090939184319596890074067816706262057060488745844736*6^(1/2),\na[12,8]= -189357008262607724321683086336517345228379250897103291049044350530935 228180690663776657891613652665009511679250229667441/104902085728430283 1848793704219061747987080296296196008818986393067503335611020651132847 28091471708748347777999725133824,\na[12,9]=-16183509927928156539922841 5225411182739942653401484724580110184517256730426918980054437210005086 9595166981551925667441/19637518660778297585754649024920739916598981028 9370293853473839457873665016411924543859976320751400118741871396189634 56,\na[12,10]=68834378427149827544141552835305430278000101566001470691 19889350771791431366439329656536871565378282089012991331513/1827181489 5517947846698608987078083524232186538856420801802422529600115450739992 00946066370836641132319880849653760,\na[12,11]=11559027144071691256623 5566233889746097162479804636463234298604185457969653794053637008425503 953091180886565/315361333249071836330411464879245754163964468656963767 284078726004630975015921697525870414526347596936773632,\na[13,1]=52151 74783558918407997583468635543407988332719241764605769949554629/\n \+ 2028313261321481206468509427515111171465117122753253371303858012160 0,\na[13,2]=0,\na[13,3]=0,\na[13,4]=0,\na[13,5]=0,\na[13,6]=2843598186 227456480865065344408178581293412110128603/\n 79207505317500213 9265844335272820716293355019960320\n -1822707089022686744784094 2666790512323422585544257/\n 1218577004884618675793606669650493 40968208464609280*6^(1/2),\na[13,7]=2843598186227456480865065344408178 581293412110128603/\n 79207505317500213926584433527282071629335 5019960320+\n 1822707089022686744784094266679051232342258554425 7/\n 121857700488461867579360666965049340968208464609280*6^(1/2 ),\na[13,8]=9326829464422062118248457481351539504275339476759467047326 605595685633/\n 49019017918582288638576910410293096780103555478 95721285919177263022080,\na[13,9]=-74160415509054246685621323607237420 6251235617068304762316465738169791/\n 141551673163321136844445 993892555326037025917405403892742525852712960,\na[13,10]=-605850486644 1219655595548618762485399974773685307046001179355536003/\n 22 52275720815396172726400694965157641073696835574259179818290290400,\na[ 13,11]=-72917047186465183128180555150230405657138451692847535142343993 /\n 446617472880162182768547714428317380932341452032226567835 63600,\na[13,12]=27361539205409276437741331476352964869466609155582532 85983742020488887296849241173151960647763453239551016003889152/2485672 1106983410152902644704639392039558692496183754067871690180096884577498 66177826801192710345262847046284166825,\na[14,1]=196143162589031568706 3141575818232405522545898155499982338718373117379429883/\n 4800 5664716707742999059356805540609358617631866994442267348172800000000000 0,\na[14,2]=0,\na[14,3]=0,\na[14,4]=0,\na[14,5]=0,\na[14,6]=-868852560 6146315530022414580346392155721271039/\n 223867381187544331818 14607602481176248320000000-\n 10256190098435854298655077997613 296122112139953/\n 1148037852243817086246902953973393653760000 000*6^(1/2),\na[14,7]=-8688525606146315530022414580346392155721271039/ \n 22386738118754433181814607602481176248320000000+\n \+ 10256190098435854298655077997613296122112139953/\n 11480378522 43817086246902953973393653760000000*6^(1/2),\na[14,8]=-108151392092290 424953498836380059772609736403739434481043071361807712075869481/\n \+ 86007354951945634483162614783313539932301371067565536007051674858 29120000000,\na[14,9]=683210554257935462600257975958139742203919396113 084127371502375524416129719/\n 26895337200565243662247103690698 994332502640106760065066162305761280000000,\na[14,10]=-125971034051203 704183074450363446847441594334546885083244594242327104115033/\n \+ 5066049934698363488698655054901069679084758735799331062593807151200 000000,\na[14,11]=4322338495495152743252505005837177994220267688026960 252214552638944423/\n 2368676257875084221529581671796767575351 42999000924357630500000000000,\na[14,12]=-8868241439418361942544164786 6243388112917289239161463940944492930492112547171652363240146123589908 870567811533658125375935101390832/940510477623017606720238368944449968 4238639823097650089954333383377958770491999076242219980453370248022825 420814384818872314453125,\na[14,13]=2623547564198662518724755429783819 7168935151270802587/\n 317816209571981740338174152687406045911 06877500000000,\na[15,1]=-29336887686855537371939221904429024146385699 07165819426999847151894747/\n 14239678548131378023505167950652 01258930107696470226170813903745843200,\na[15,2]=0,\na[15,3]=0,\na[15, 4]=0,\na[15,5]=0,\na[15,6]=-279050827135618188106138704976571118076242 172562777/\n 2698071775074566005593212198869216924926291738624 0+\n 59017804198407615229179283246229064921710388893173/\n \+ 17987145167163773370621414659128112832841944924160*6^(1/2),\na[15 ,7]=-279050827135618188106138704976571118076242172562777/\n 26 980717750745660055932121988692169249262917386240-\n 5901780419 8407615229179283246229064921710388893173/\n 179871451671637733 70621414659128112832841944924160*6^(1/2),\na[15,8]=6824047782391855906 0550996013166770535743446467404475965020846786328901/\n 6962862 5900822775316247857716283393138506964199995817446989013471723520,\na[1 5,9]=48531604865335743440838806675493568975092395234916265724406574203 650554879/\n 75290755690494507247154479511057303273913043144243 23559158913160761835520,\na[15,10]=73158981980491143736917790272372062 35234893868747090317226910860963581499/\n 43240021937913168418 3655517400956805867024783959757582533074834771793600,\na[15,11]=-87035 912584683752124645187592152267644073875904388006117245587111831/\n \+ 4146853253272305366340198392743957373097063952194163339684368224 8800,\na[15,12]=793006054328041651061360131256412474400253089909554005 3783327282148061089952121382917590174480872244717164362321758643844247 53159293287828190208/6006353055070484301705313231740079158134464998533 4818630144453572293255244407178977557750808395528025807084478052037198 7041149053341617704219325,\na[15,13]=190132386928877842671649814278676 30356262081870600946422701364458/\n 14651630863314419811073580 5762400905606463733191840985648075179899,\na[15,14]=368176545506575596 342007241113258886329861009608750000000/\n 7515329389098801941 975451526298754679007062667248055263091,\na[16,1]=-3638865833016212476 2200023703074655379362961851837455245313588466117/\n 299291302 137383314536268908335335078201218487321808786511993451315200,\na[16,2] =0,\na[16,3]=0,\na[16,4]=0,\na[16,5]=0,\na[16,6]=-50172940999755801588 62668031284197043053591531405121/\n 43287413492571995160823628 7739966681862399087083520-\n 115700422823857939498444446575144 266776173664871303/\n 1997880622734092084345705943415230839364 91886346240*6^(1/2),\na[16,7]=-501729409997558015886266803128419704305 3591531405121/\n 432874134925719951608236287739966681862399087 083520+\n 115700422823857939498444446575144266776173664871303/ \n 199788062273409208434570594341523083936491886346240*6^(1/2) ,\na[16,8]=-9186938470661702041587152380958133368874431925644166960645 2442156503951243/\n 100539905324968707856110496731312310121511 68225596329651077879646797168640,\na[16,9]=528629993814031198075094729 78743982056878734540171187101666495333163485251/\n 239990528147 4961058988767393522446655063434938667458594324573249770618880,\na[16,1 0]=1196496586129443433742753423133050108945873114684141029825814957121 8167/\n 197401792397374196113405891580249137356360042842046458 0517659076868800,\na[16,11]=371676808722577030036866921916351493884793 05578570534942584948859/\n 39133905242722475581980638159200847 15858046504866006220243936800,\na[16,12]=-2944395586705434775334102612 1045589578978185460220369427665634428174791788280117223531690217195051 250854008448256016995751289856/613836440653194383209173469818173591961 8182422997726728173639255839442923939727454711783369110119257425239678 616468261908910975,\na[16,13]=4078167483851726864981531813468124327911 18177175769818363629863/\n 62616239788238609519620162976830375 9628250075752997120162580989,\na[16,14]=0,\na[16,15]=0,\n\nb[1]=307038 43389361946002220520407/1036329015084155723633962896000,\nb[2]=0,\nb[3 ]=0,\nb[4]=0,\nb[5]=0,\nb[6]=0,\nb[7]=0,\nb[8]=15166818889134709063640 13671875/19423768214582439936604117641536,\nb[9]=192992273799847057335 9614532470703125/9295447834009061726737853188569292704,\nb[10]=2707239 7368129209968072433152000000000/15954089106727679862943371842129021166 9,\nb[11]=3416676287738448149119878197304164096817920457/\n 22521 752441211566270536786917243920830369456000,\nb[12]=9090349007494116456 3143999126052497791688659150254835513033014882906689676415155529203822 2333366816993556860935646735988456500531298304/63019787491889793176593 8035588221137118814650606654322610722621749344398603131630645015192260 0620534579104501042337690306078523205079625,\nb[13]=916089774614920438 3653282352747804858423571/\n 549341190028888507735840115833919211 91449440,\nb[14]=3769686146953412690297035156250000/195792979665408643 382362918863397227,\nb[15]=50782110772148063247179059/1538266148871578 545201811280,\nb[16]=0,\n\n`b*`[1]=135131455470598097879473933/4525454 214341291369580624000,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=0,\n`b*`[5]=0, \n`b*`[6]=0,\n`b*`[7]=0,\n`b*`[8]=2518169234679274570156341552734375/3 8284247150941989115046715871467456,\n`b*`[9]=1317102042413654070626162 7197265625/61559257178867958455217570785227104,\n`b*`[10]=331911110031 44264098986272000000000/196721197370254992144801132455351679,\n`b*`[11 ]=98603841096694858013088556726735239713679/\n 574051243626833 692823306576536179258032000,\n`b*`[12]=5093635768538576107415789300891 1450543340655548467600399461620689692608049620506050217070420609879731 90001839145435136/4058910636418229922626951037131886401608796642938780 5447847648739509460272079710913035195393529050583646417571607929125,\n `b*`[13]=108010721096523379193662759959856611609133/570689193181223151 553200582051397411377120,\n`b*`[14]=0,\n`b*`[15]=0,\n`b*`[16]=26859551 018855966185191031/763900876650511794556001520\}:" }}}{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 stability f unction R for the 16 stage, order 9 scheme is given (approximately) as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "expand(subs(ee,StabilityFunction(9,16,'expanded'))): \nmap(convert,evalf[28](%),rational,24):\nR := unapply(%,z):\n'R(z)'=R (z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"RG6#%\"zG,B\"\"\"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)*&#F)\"'!)GOF)*$)F' \"\"*F)F)F)*&#\"*]\"zu:\"0$[#49A+H'F)*$)F'\"#5F)F)F)*&#\")i`bF\"1hFG6l N_WF)*$)F'\"#6F)F)F)*&#\")_%y)=\"1`ae([&)R6(F)*$)F'\"#7F)F)F)*&#\"(6Z[ (\"2z,\")e4Kkx#F)*$)F'\"#8F)F)F)*&#\"''z'[\"3Tg=+H'*)4D\"F)*$)F'\"#9F) F)F)*&#\"'J46\"3*e<*3W:dw\")F)*$)F'\"#:F)F)!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can find the point wh ere the boundary of the stability region intersects the negative real \+ axis by solving the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "R(z) = 1;" "6#/-%\"RG6#%\"zG\"\"\"" }{TEXT -1 2 ". " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "z0 := newton(R(z)=1,z=-5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$!+'***o\">&!\"*" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 302 "z0 : = newton(R(z)=1,z=-5):\np1 := plot([R(z),1],z=-5.89..0.49,color=[red,b lue]):\np2 := plot([[[z0,1]]$3],style=point,symbol=[circle,cross,diamo nd],color=black):\np3 := plot([[z0,0],[z0,1]],linestyle=3,color=COLOR( RGB,0,.5,0)):\nplots[display]([p1,p2,p3],view=[-5.89..0.49,-0.07..1.47 ],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 253 253 {PLOTDATA 2 "6+-%'CURVESG6$7Z7$$!3o*************)e!#<$\"3!=TXMa?\\(RF* 7$$!3)e;/\"RNBbeF*$\"3=64\"z$o(os$F*7$$!3)HL3#yqY?eF*$\"3iU_l'49F*7$$! 3q\\P9p?r)G&F*$\"3`j6$R@Y]A\"F*7$$!3zKe/bxa?_F*$\"39$[\"G+xki5F*7$$!3O m\"Ham`T4&F*$\"32$yYDG[N7)!#=7$$!3u*\\im!*zK'\\F*$\"3&o#oj:*[46'Fho7$$ !3`m\"H\"y+$z#[F*$\"3Vc'\\1^d@_%Fho7$$!3q*\\7#=V,$p%F*$\"3-\"R4ak;+L$F ho7$$!3OLL)e(oBaXF*$\"3;NWG-b@?CFho7$$!3Sm;9H;+KWF*$\"3pPO9z^;D=Fho7$$ 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v7$$!+m'z#\\FFd`p$!+k()zq:Fb_pF`]p7%Fb^v7$F\\`p$!+!R'FKF(7$$ \"3%**************=\"FKF(-%'COLOURG6&F_\\pF)F)F)-%*LINESTYLEG6#\"\"$-F $6%7$7$F($!3A++++++!*fFK7$F($\"3A++++++!*fFKFebvFhbv-%%FONTG6$%*HELVET ICAG\"\"*-%*AXESSTYLEG6#%$BOXG-%(SCALINGG6#%,CONSTRAINEDG-%+AXESLABELS G6%%&Re(z)G%&Im(z)G-Ffcv6#%(DEFAULTG-%%VIEWG6$;$!$>'Fg\\p$\"$>\"Fg\\p; $!$*fFg\\p$\"$*fFg\\p" 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 "[-5.1917, 0];" "6#7$,$-%&FloatG6$ \"&<>&!\"%!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 115 "We can distort the boundary curve ho rizontally by taking the 11th root of the real part of points along th e curve. " }}{PARA 0 "" 0 "" {TEXT -1 33 "In this way we see that ther e is " }{TEXT 260 19 "no largest interval" }{TEXT -1 99 " on the nonne gative imaginary axis that contains the origin and lies inside the sta bility region. " }}{PARA 0 "" 0 "" {TEXT -1 120 "However, the stabili ty region intersects the nonnegative imaginary axis in an interval tha t does not contain the origin." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 483 "R := z ->add(z^j/j!,j=0..9) +\n 157479150/629002214092483*z^10+27555362/4452356511282761*z^11+ \n 18878452/7113985487585453*z^12+7484711/27764320958810179*z^13+ \n 486796/125098962900186041*z^14-110931/817657154408917589*z^15: \nDigits := 25:\npts := []: z0 := 0:\nfor ct from 0 to 255 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,.2,.7, 0),thickness=2,font=[HELVETICA,9]);\nDigits := 10:" }}{PARA 13 "" 1 " " {GLPLOT2D 287 308 308 {PLOTDATA 2 "6(-%'CURVESG6#7\\[l7$$\"\"!F)F(7$ $!:(>Hu(3#)*e!4X.x)!#F$\":iEYQKz*e`EfTJ!#E7$$!:uOO6**pLUNpqk\"F0$\":iO 'pZ'ezrI&=$G'F07$$!:xK%G&[0@3&=[\"Q#F0$\":5>s=(z$p2'zxC%*F07$$!:Yd\"y \"*pri+?'Q4$F0$\":nsK.t\"fVhqjc7!#D7$$!:jX.b4s%[;\"z/z$F0$\":]LT7n*[zE jzq:F@7$$!:^2`o%p\\&>Pn\\Z%F0$\":va(>kwQ:#fb\\)=F@7$$!:4[R,)>$o381(\\^ F0$\":7R9=\"fG^d[6*>#F@7$$!:t1eEf*=VZuP;eF0$\":1#R:`^=(G7uK^#F@7$$!:!* )3\"pN'4qvYBwkF0$\":zwtqm(3B)QLu#GF@7$$!:6\"\\qU]N'3xZ-8(F0$\":yp5(H'* **e`EfTJF@7$$!:UngmFQ^Mm)=zxF0$\":`_Ww:O\\*=>vbMF@7$$!:F5:Qbf.Il%oB%)F 0$\":\">C2B8$4V=6*pPF@7$$!:E,\\Fgs$))zMDk!*F0$\":ag:\"yo0n\\/2%3%F@7$$ !:ODg5K*GF)HF8q*F0$\":o(Q)ebbM]rH#)R%F@7$$!:\\d#RW/$=;'p_L5F@$\":$[2>c yRS!)*)Q7ZF@7$$!:CPsCT*f-h(Qm4\"F@$\":I9%zcaPyX#[l-&F@7$$!:U]&)*p&HO-8 %\\f6F@$\":\"e****e,D=6vqS`F@7$$!:v=XF%)y9\\0:@A\"F@$\":#Gv1()\\[hwn'[ l&F@7$$!:c60V()zbr/?XG\"F@$\":)Q7n#>%\\5Ug-pfF@7$$!:tk7#)pd,[dCnM\"F@$ \":$fa$H-i\"p2`=$G'F@7$$!:zVz^A`\"F@$\"::9T*f.A# e5jcA(F@7$$!:I`U4`e,[\\ePf\"F@$\":k^G2YpDEE\"Gr5b;F @$\":rD2A*pwsS;)R&yF@7$$!:zuh()o%>N_(*H;Om0h7=+B[)F@7$$!:N@?\\BE'=Q)>#Q=F@$\":U.Y4?B2aX fkz)F@7$$!:!=IN8C%H]9X*)*=F@$\":XAn')=5qOt=16*F@7$$!:\\(3TP')[V%H5&f>F @$\":!yw*ph#Hu?F@$\":!p\"oRXHD*4t$*Q(*F@7 $$!:hdCYx_8zjT,3#F@$\":sWB8*))4Lh'4`+\"!#C7$$!:^/:***zw;_c>S@F@$\":Ka_ :^y,KfDn.\"Fgu7$$!:=%3J(eE!>F_1+AF@$\":Fku6KJ\\r_T\"o5Fgu7$$!:Xi(Qz%H$ y%\\T(fAF@$\":8ZgBMz')QYd&*4\"Fgu7$$!:)y)*4:2do:X@>BF@$\":cCNtjKTVSt48 \"Fgu7$$!:eY[qhJX\"*Gt%yBF@$\":\\#G!zU24(\\$*Qi6Fgu7$$!:zAm`27pUv0vV#F @$\":/O%>k:C_,`!Q>\"Fgu7$$!:Jip#QoD)Hy)H'\\#F@$\":T+#zDv6th7AD7Fgu7$$! :nZv(RPAIk\"Q[b#F@$\":;Y^enI+GBPmD\"Fgu7$$!:dw!H_:N\\z&3Jh#F@$\":i>*y5 ,n#y@`!)G\"Fgu7$$!:pa@8u,Svc$4rEF@$\"::KkH36y1Ap%>8Fgu7$$!:F&\\nrf#*=! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 200 "Dig its := 15:\nz0 := 2.6*I:\nfor ct from 82 to 85 do\n newton(R(z)=exp( ct*Pi/100*I),z=z0);\nend do;``;\nz0 := 5.1*I:\nfor ct from 212 to 215 \+ do\n newton(R(z)=exp(ct*Pi/100*I),z=z0);\nend do;\nDigits := 10:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0LS)4=VRI!#>$\"0C3Z#)fod#!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0N@a:Rj5\"!#>$\"0j!>$3$Q3E!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0XBiq!)eW\"!#>$\"0-26W>*RE!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0>!ovPzRZ!#>$\"0*)['*=q9n#!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #^$$\"03EFw]SI\"!#;$\"0Ibt$\"0]POvs**4&!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #^$$!0rct0z\"=n!#<$\"0u!pZ\"pU5&!#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 0 328 "Digits := 15:\nreal_par t := proc(u)\n Re(newton(R(z)=exp(u*Pi*I),z=2.6*I))\nend proc:\nu0 : = bisect('real_part'(u),u=0.82..0.85);\nnewton(R(z)=exp(u0*Pi*I),z=2.6 *I);``;\nreal_part := proc(u)\n Re(newton(R(z)=exp(u*Pi*I),z=5.1*I)) \nend proc:\nu0 := bisect('real_part'(u),u=2.12..2.15);\nnewton(R(z)=e xp(u0*Pi*I),z=5.1*I);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#u0G$\"0VaMxUmM)!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"0U$e33 4BE!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"07&4%Qt)R@!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"0c!Q(\\<**4&!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 4 "The " }{TEXT 260 73 "stability region intersects th e nonegative imaginary axis in the interval" }{TEXT -1 3 " " } {XPPEDIT 18 0 "[2.6231, 5.0999];" "6#7$-%&FloatG6$\"&Ji#!\"%-F%6$\"&** 4&F(" }{TEXT -1 18 " (approximately)." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 37 "#------------------------------------ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 97 "The s tability function R* for the 16 stage, order 8 scheme is given (appr oximately) as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "e xpand(subs(ee,subs(b=`b*`,StabilityFunction(8,16,'expanded')))):\nmap( convert,evalf[28](%),rational,24):\n`R*` := unapply(%,z):\n'`R*`(z)'=` R*`(z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%#R*G6#%\"zG,@\"\"\"F)F'F )*&#F)\"\"#F)*$)F'F,F)F)F)*&#F)\"\"'F)*$)F'\"\"$F)F)F)*&#F)\"#CF)*$)F' \"\"%F)F)F)*&#F)\"$?\"F)*$)F'\"\"&F)F)F)*&#F)\"$?(F)*$)F'F1F)F)F)*&#F) \"%S]F)*$)F'\"\"(F)F)F)*&#F)\"&?.%F)*$)F'\"\")F)F)F)*&#\"*)=rDX\"0T9\\ 5[W\">F)*$)F'\"\"*F)F)F)*&#\"*%*\\O^#\"1.K&f()o?K\"F)*$)F'\"#5F)F)!\" \"*&#\")Mu`^\"1\"*4.)ep'*e\"F)*$)F'\"#6F)F)F)*&#\"(>eu)\"1eF+v!zk'>F)* $)F'\"#7F)F)F)*&#\"(HLx#\"2ES3gC;Bu$F)*$)F'\"#8F)F)F)*&#\"'m " 0 "" {MPLTEXT 1 0 33 "z_0 := newto n(`R*`(z)=-1,z=-4.4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$z_0G$!+\"3 XUT%!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 324 "z_0 := newton(`R*`(z)=-1,z=-4.4):\np_1 := plot([`R*` (z),-1],z=-4.99..0.49,color=[red,blue]):\np_2 := plot([[[z_0,-1]]$3],s tyle=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 369 264 264 {PLOTDATA 2 "6+-%'CURVESG6$7V7$ $!3A++++++!*\\!#<$!3w/'Q0^r2X$F*7$$!3cLLe(zv-$\\F*$!3W*=K7MX)eIF*7$$!3 #pmm^f^0([F*$!3VXsM2;%pq#F*7$$!3#3]()f)ee=[F*$!3*H6&QK;ZICF*7$$!3%QL3o Iz\"3ZF*$!3(QQy?)3bC>F*7$$!3imm,F%Q( \\YF*$!3%Rl')G4ukp\"F*7$$!33nm64>3KXF*$!3N#eV$[lJ38F*7$$!3\\L$3@f%)\\T %F*$!3ev)oYp;<+\"F*7$$!3jm;u*Q?kI%F*$!3wK+)*owdax!#=7$$!3E+]ZY%3S>%F*$ !3%R-W*QMb!*eFZ7$$!3%omTR&=vxSF*$!3Q\"=0;7AwP%FZ7$$!3/+]x(4o='RF*$!3Xx g&oSra?$FZ7$$!3OF*$!3gVA(*3o*o3\"FZ7$$!3?++]Qxz+NF*$!3&y(4(ph=Qi '!#>7$$!3/++5QhU'Q$F*$!3g9t8'3,9[$Fhp7$$!3mm;%zwlDG$F*$!3x$4*Q4:?N7Fhp 7$$!3[LLBUd1fJF*$\"3b8b^C&3B#))!#?7$$!3eLL$4-XW0$F*$\"3M6O.Y%[jL#Fhp7$ $!3;+]n]iuKHF*$\"3'>D>;7'phPFhp7$$!3ILL$z@A]#GF*$\"3YnyV$**z$p[Fhp7$$! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 791 "`R*` := z -> add(z^j/j!,j=0..8)+\n 452571188/191 444810491441*z^9-251364994/1322068875953203*z^10+\n 51537434/15896 69588030991*z^11+8745819/1966479075002758*z^12+\n 2773329/37423162 460084026*z^13-241766/103890057529529573*z^14:\npts := []: z0 := 0:\nf or ct from 0 to 280 do\n zz := newton(`R*`(z)=exp(ct*Pi/20*I),z=z0): \n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np_1 := \+ plot(pts,color=COLOR(RGB,0,.25,0)):\np_2 := plots[polygonplot]([seq([p ts[i-1],pts[i],[-2.4,0]],i=2..nops(pts))],\n style=patchnogri d,color=COLOR(RGB,.13,.65,0)):\np_3 := plot([[[-5.19,0],[1.19,0]],[[0, -4.99],[0,4.99]]],color=black,linestyle=3):\nplots[display]([p_||(1..3 )],view=[-5.19..1.19,-4.99..4.99],font=[HELVETICA,9],\n lab els=[`Re(z)`,`Im(z)`],axes=boxed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 396 515 515 {PLOTDATA 2 "6+-%'CURVESG6$7e\\l7$$\"\"!F)F 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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 36 "coefficients of the combined scheme " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13775 "ee := \{c[2]=1/50,\nc[3]=3837236/48429375+1031368/145288125*6 ^(1/2),\nc[4]=1918618/16143125+515684/48429375*6^(1/2),\nc[5]=14/45,\n c[6]=156/625+26/625*6^(1/2),\nc[7]=156/625-26/625*6^(1/2),\nc[8]=52/12 5,\nc[9]=39/125,\nc[10]=21/200,\nc[11]=280/477,\nc[12]=365822703505371 5/5349704719299032,\nc[13]=247/281,\nc[14]=229/250,\nc[15]=1,\nc[16]=1 ,\n\na[2,1]=1/50,\na[3,1]=-24000387317036/281448523546875-591726453229 6/281448523546875*6^(1/2),\na[3,2]=46300580261936/281448523546875+7915 204837696/281448523546875*6^(1/2),\na[4,1]=959309/32286250+128921/4842 9375*6^(1/2),\na[4,2]=0,\na[4,3]=2877927/32286250+128921/16143125*6^(1 /2),\na[5,1]=2826523628723851/5953434698904030-68459492317475/59534346 9890403*6^(1/2),\na[5,2]=0,\na[5,3]=-704240024458145/396895646593602+9 1277530807085/198447823296801*6^(1/2),\na[5,4]=958925642225180/5953434 69890403-205373100103780/595343469890403*6^(1/2),\na[6,1]=376341108/94 06484375+207933466/65845390625*6^(1/2),\na[6,2]=0,\na[6,3]=0,\na[6,4]= 4343545768844529/27892881885795625+469265141246109/27892881885795625*6 ^(1/2),\na[6,5]=1559927818449/28957835234375+4382126882523/20270484664 0625*6^(1/2),\na[7,1]=11781705468/235162109375+2328587014/164613476562 5*6^(1/2),\na[7,2]=0,\na[7,3]=0,\na[7,4]=23459106068523828440829/35429 8872323611753203125+7870375504052283205581/354298872323611753203125*6^ (1/2),\na[7,5]=146263465360621089/7558718942052734375-1881455818308499 953/52911032594369140625*6^(1/2),\na[7,6]=9444124356888/82889304453125 -2459298027368/82889304453125*6^(1/2),\na[8,1]=52/1125,\na[8,2]=0,\na[ 8,3]=0,\na[8,4]=0,\na[8,5]=0,\na[8,6]=208/1125-13/1125*6^(1/2),\na[8,7 ]=208/1125+13/1125*6^(1/2),\na[9,1]=741/16000,\na[9,2]=0,\na[9,3]=0,\n a[9,4]=0,\na[9,5]=0,\na[9,6]=2301/16000-897/32000*6^(1/2),\na[9,7]=230 1/16000+897/32000*6^(1/2),\na[9,8]=-351/16000,\na[10,1]=35291978967/74 8709478400,\na[10,2]=0,\na[10,3]=0,\na[10,4]=0,\na[10,5]=0,\na[10,6]=2 3154511989/149741895680+39398793/1772093440*6^(1/2),\na[10,7]=23154511 989/149741895680-39398793/1772093440*6^(1/2),\na[10,8]=-6251205429/149 741895680,\na[10,9]=-981041103/4679434240,\na[11,1]=160158980732913414 4752443/16639785968494158002257920,\na[11,2]=0,\na[11,3]=0,\na[11,4]=0 ,\na[11,5]=0,\na[11,6]=-1736562342312744743536201/11093190645662772001 50528-\n 360257484908262597335743/511993414415204861607936*6^( 1/2),\na[11,7]=-1736562342312744743536201/1109319064566277200150528+\n 360257484908262597335743/511993414415204861607936*6^(1/2),\na [11,8]=512032742176678555764127/369773021522092400050176,\na[11,9]=248 233526294563631278471/103998662303088487514112,\na[11,10]=-3/20,\na[12 ,1]=-13198701760878669635722542338759463561271938920612860688067043417 8321331969627889057541436355642743061150672386594396559/31875392608799 5555015147926201612010240447228295789462486798116221476093939683123897 279961564118214685494052121351290880,\na[12,2]=0,\na[12,3]=0,\na[12,4] =0,\na[12,5]=0,\na[12,6]=\n-581038619225160876203856834629458675128926 7051434651924507164484661690757973591786160210452910809721214291885435 92047/1011917225676176365127453733973371461080784851732664960275549575 306273314094232139356444322425772110112679530543972352+\n7134827980789 8814965088233729737906656351096659541311591981953238552744805776480028 282602780942939988708855996996031/518931910603167366732027555883780236 4516845393500845950131023463109093918431959689007406781670626205706048 8745844736*6^(1/2),\na[12,7]=\n-58103861922516087620385683462945867512 8926705143465192450716448466169075797359178616021045291080972121429188 543592047/101191722567617636512745373397337146108078485173266496027554 9575306273314094232139356444322425772110112679530543972352-\n713482798 0789881496508823372973790665635109665954131159198195323855274480577648 0028282602780942939988708855996996031/51893191060316736673202755588378 0236451684539350084595013102346310909391843195968900740678167062620570 60488745844736*6^(1/2),\na[12,8]=-189357008262607724321683086336517345 2283792508971032910490443505309352281806906637766578916136526650095116 79250229667441/1049020857284302831848793704219061747987080296296196008 81898639306750333561102065113284728091471708748347777999725133824,\na[ 12,9]=-161835099279281565399228415225411182739942653401484724580110184 5172567304269189800544372100050869595166981551925667441/19637518660778 2975857546490249207399165989810289370293853473839457873665016411924543 85997632075140011874187139618963456,\na[12,10]=68834378427149827544141 5528353054302780001015660014706911988935077179143136643932965653687156 5378282089012991331513/18271814895517947846698608987078083524232186538 85642080180242252960011545073999200946066370836641132319880849653760, \na[12,11]=11559027144071691256623556623388974609716247980463646323429 8604185457969653794053637008425503953091180886565/31536133324907183633 0411464879245754163964468656963767284078726004630975015921697525870414 526347596936773632,\na[13,1]=52151747835589184079975834686355434079883 32719241764605769949554629/\n 202831326132148120646850942751511 11714651171227532533713038580121600,\na[13,2]=0,\na[13,3]=0,\na[13,4]= 0,\na[13,5]=0,\na[13,6]=2843598186227456480865065344408178581293412110 128603/\n 792075053175002139265844335272820716293355019960320\n -18227070890226867447840942666790512323422585544257/\n \+ 121857700488461867579360666965049340968208464609280*6^(1/2),\na[13,7]= 2843598186227456480865065344408178581293412110128603/\n 7920750 53175002139265844335272820716293355019960320+\n 182270708902268 67447840942666790512323422585544257/\n 121857700488461867579360 666965049340968208464609280*6^(1/2),\na[13,8]=932682946442206211824845 7481351539504275339476759467047326605595685633/\n 4901901791858 228863857691041029309678010355547895721285919177263022080,\na[13,9]=-7 41604155090542466856213236072374206251235617068304762316465738169791/ \n 14155167316332113684444599389255532603702591740540389274252 5852712960,\na[13,10]=-60585048664412196555955486187624853999747736853 07046001179355536003/\n 2252275720815396172726400694965157641 073696835574259179818290290400,\na[13,11]=-729170471864651831281805551 50230405657138451692847535142343993/\n 4466174728801621827685 4771442831738093234145203222656783563600,\na[13,12]=273615392054092764 3774133147635296486946660915558253285983742020488887296849241173151960 647763453239551016003889152/248567211069834101529026447046393920395586 9249618375406787169018009688457749866177826801192710345262847046284166 825,\na[14,1]=19614316258903156870631415758182324055225458981554999823 38718373117379429883/\n 480056647167077429990593568055406093586 176318669944422673481728000000000000,\na[14,2]=0,\na[14,3]=0,\na[14,4] =0,\na[14,5]=0,\na[14,6]=-86885256061463155300224145803463921557212710 39/\n 22386738118754433181814607602481176248320000000-\n \+ 10256190098435854298655077997613296122112139953/\n 11480378 52243817086246902953973393653760000000*6^(1/2),\na[14,7]=-868852560614 6315530022414580346392155721271039/\n 223867381187544331818146 07602481176248320000000+\n 10256190098435854298655077997613296 122112139953/\n 1148037852243817086246902953973393653760000000 *6^(1/2),\na[14,8]=-10815139209229042495349883638005977260973640373943 4481043071361807712075869481/\n 860073549519456344831626147833 1353993230137106756553600705167485829120000000,\na[14,9]=6832105542579 35462600257975958139742203919396113084127371502375524416129719/\n \+ 2689533720056524366224710369069899433250264010676006506616230576128 0000000,\na[14,10]=-12597103405120370418307445036344684744159433454688 5083244594242327104115033/\n 50660499346983634886986550549010 69679084758735799331062593807151200000000,\na[14,11]=43223384954951527 43252505005837177994220267688026960252214552638944423/\n 23686 7625787508422152958167179676757535142999000924357630500000000000,\na[1 4,12]=-886824143941836194254416478662433881129172892391614639409444929 30492112547171652363240146123589908870567811533658125375935101390832/9 4051047762301760672023836894444996842386398230976500899543333833779587 70491999076242219980453370248022825420814384818872314453125,\na[14,13] =26235475641986625187247554297838197168935151270802587/\n 3178 1620957198174033817415268740604591106877500000000,\na[15,1]=-293368876 8685553737193922190442902414638569907165819426999847151894747/\n \+ 1423967854813137802350516795065201258930107696470226170813903745843 200,\na[15,2]=0,\na[15,3]=0,\na[15,4]=0,\na[15,5]=0,\na[15,6]=-2790508 27135618188106138704976571118076242172562777/\n 26980717750745 660055932121988692169249262917386240+\n 5901780419840761522917 9283246229064921710388893173/\n 179871451671637733706214146591 28112832841944924160*6^(1/2),\na[15,7]=-279050827135618188106138704976 571118076242172562777/\n 2698071775074566005593212198869216924 9262917386240-\n 590178041984076152291792832462290649217103888 93173/\n 17987145167163773370621414659128112832841944924160*6^ (1/2),\na[15,8]=682404778239185590605509960131667705357434464674044759 65020846786328901/\n 696286259008227753162478577162833931385069 64199995817446989013471723520,\na[15,9]=485316048653357434408388066754 93568975092395234916265724406574203650554879/\n 752907556904945 0724715447951105730327391304314424323559158913160761835520,\na[15,10]= 7315898198049114373691779027237206235234893868747090317226910860963581 499/\n 4324002193791316841836555174009568058670247839597575825 33074834771793600,\na[15,11]=-8703591258468375212464518759215226764407 3875904388006117245587111831/\n 41468532532723053663401983927 439573730970639521941633396843682248800,\na[15,12]=7930060543280416510 6136013125641247440025308990955400537833272821480610899521213829175901 7448087224471716436232175864384424753159293287828190208/60063530550704 8430170531323174007915813446499853348186301444535722932552444071789775 577508083955280258070844780520371987041149053341617704219325,\na[15,13 ]=19013238692887784267164981427867630356262081870600946422701364458/\n 1465163086331441981107358057624009056064637331918409856480751 79899,\na[15,14]=36817654550657559634200724111325888632986100960875000 0000/\n 751532938909880194197545152629875467900706266724805526 3091,\na[16,1]=-363886583301621247622000237030746553793629618518374552 45313588466117/\n 29929130213738331453626890833533507820121848 7321808786511993451315200,\na[16,2]=0,\na[16,3]=0,\na[16,4]=0,\na[16,5 ]=0,\na[16,6]=-5017294099975580158862668031284197043053591531405121/\n 432874134925719951608236287739966681862399087083520-\n \+ 115700422823857939498444446575144266776173664871303/\n 19978 8062273409208434570594341523083936491886346240*6^(1/2),\na[16,7]=-5017 294099975580158862668031284197043053591531405121/\n 4328741349 25719951608236287739966681862399087083520+\n 11570042282385793 9498444446575144266776173664871303/\n 199788062273409208434570 594341523083936491886346240*6^(1/2),\na[16,8]=-91869384706617020415871 523809581333688744319256441669606452442156503951243/\n 1005399 0532496870785611049673131231012151168225596329651077879646797168640,\n a[16,9]=52862999381403119807509472978743982056878734540171187101666495 333163485251/\n 23999052814749610589887673935224466550634349386 67458594324573249770618880,\na[16,10]=11964965861294434337427534231330 501089458731146841410298258149571218167/\n 1974017923973741961 134058915802491373563600428420464580517659076868800,\na[16,11]=3716768 0872257703003686692191635149388479305578570534942584948859/\n \+ 3913390524272247558198063815920084715858046504866006220243936800,\na[1 6,12]=-294439558670543477533410261210455895789781854602203694276656344 28174791788280117223531690217195051250854008448256016995751289856/6138 3644065319438320917346981817359196181824229977267281736392558394429239 39727454711783369110119257425239678616468261908910975,\na[16,13]=40781 6748385172686498153181346812432791118177175769818363629863/\n \+ 626162397882386095196201629768303759628250075752997120162580989,\na[16 ,14]=0,\na[16,15]=0,\n\nb[1]=30703843389361946002220520407/10363290150 84155723633962896000,\nb[2]=0,\nb[3]=0,\nb[4]=0,\nb[5]=0,\nb[6]=0,\nb[ 7]=0,\nb[8]=1516681888913470906364013671875/19423768214582439936604117 641536,\nb[9]=1929922737998470573359614532470703125/929544783400906172 6737853188569292704,\nb[10]=27072397368129209968072433152000000000/159 540891067276798629433718421290211669,\nb[11]=3416676287738448149119878 197304164096817920457/\n 2252175244121156627053678691724392083036 9456000,\nb[12]=909034900749411645631439991260524977916886591502548355 1303301488290668967641515552920382223333668169935568609356467359884565 00531298304/6301978749188979317659380355882211371188146506066543226107 2262174934439860313163064501519226006205345791045010423376903060785232 05079625,\nb[13]=9160897746149204383653282352747804858423571/\n 5 4934119002888850773584011583391921191449440,\nb[14]=376968614695341269 0297035156250000/195792979665408643382362918863397227,\nb[15]=50782110 772148063247179059/1538266148871578545201811280,\nb[16]=0,\n\n`b*`[1]= 135131455470598097879473933/4525454214341291369580624000,\n`b*`[2]=0, \n`b*`[3]=0,\n`b*`[4]=0,\n`b*`[5]=0,\n`b*`[6]=0,\n`b*`[7]=0,\n`b*`[8]= 2518169234679274570156341552734375/38284247150941989115046715871467456 ,\n`b*`[9]=13171020424136540706261627197265625/61559257178867958455217 570785227104,\n`b*`[10]=33191111003144264098986272000000000/1967211973 70254992144801132455351679,\n`b*`[11]=98603841096694858013088556726735 239713679/\n 574051243626833692823306576536179258032000,\n`b*` [12]=50936357685385761074157893008911450543340655548467600399461620689 69260804962050605021707042060987973190001839145435136/4058910636418229 9226269510371318864016087966429387805447847648739509460272079710913035 195393529050583646417571607929125,\n`b*`[13]=1080107210965233791936627 59959856611609133/570689193181223151553200582051397411377120,\n`b*`[14 ]=0,\n`b*`[15]=0,\n`b*`[16]=26859551018855966185191031/763900876650511 794556001520\}:" }}}{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..16);" }}{PARA 12 "" 1 " " {XPPMATH 20 "61/&%\"cG6#\"\"##\"\"\"\"#]/&F%6#\"\"$,&#\"(Os$Q\")v$H% [F)*(\"(o8.\"F)\"*D\")GX\"!\"\"\"\"'#F)F'F)/&F%6#\"\"%,&#\"(='=>\")DJ9 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@W_kgp;Wc#>Vu)oL8e4Q_reT?qh1ZQp=*\"eoS'orzY'zy2^'HjfD#o6:75BJJn\\5h&yq o\\K0*R05/&F%6$FcamFdw#\"eo^_[jJL&\\m;5(=r,aM(yo0#)RuyHZ4v!)>JS\"Q**H' G&\"do!))=1x\\KdCVfeumQ\\Vj]lYC_$Rn())*e5'\\Z\"G0**R#/&F%6$FcamFjy#\"b on\"=7d\\\"e#)H59%o9J(e%*3,0LJU`FuLMWHhe'\\'>\"\"ao+)oo2fw^!ek/UG/gjNP \"\\-e\"*eS8h>utR#z,u>/&F%6$FcamFg\\l#\"\\of)[\\eU\\`q&ybIz%)Q\\^j\"># p'o.IqdA(3onr$\"[o+o$RC?i+m[]Y!eer%3?f\"Q1)>evCsU_!R8R/&F%6$FcamF^`l#! [sc)*G^d*p,c#[%3S&3D^]><-pJNA<,G)y\"zu\"GWjlwUp.Aga=y*y&*eX57E5M`xMaq' e&RWH\"jrv4\"*3>Eokhy'R_Ud#>,6pLy6ZXF(RR#HWReDROft\"=)pM< 4KQ%>`1WOQh/&F%6$FcamF[w#\"jnj)HOO=)pd&4'Q#)yRihi/&F%6$FcamF[hlFN/&F%6$FcamFd \\mFN" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 39 "weights for the 16 stage order 9 scheme" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "seq(b[i]=subs(ee,b[i]),i=1..16);" } }{PARA 12 "" 1 "" {XPPMATH 20 "62/&%\"bG6#\"\"\"#\">2/_?A+Y>O*QVQqI\"@ +g*G'RjBd:%3:!Hj.\"/&F%6#\"\"#\"\"!/&F%6#\"\"$F//&F%6#\"\"%F//&F%6#\" \"&F//&F%6#\"\"'F//&F%6#\"\"(F//&F%6#\"\")#\"@v=n8SO14Z8*))=o;:\"AO:k< TgO*RCe9#oPU>/&F%6#\"\"*#\"FDJqqC`9'fLdq%)*ztA*H>\"F/FHp&)=`ytE<14S$yW &H*/&F%6#\"#5#\"G++++?:LC2o*4#H\"otRsq#\"Hp;@!H@%=PVH')zws1\"*3af\"/&F %6#\"#6#\"Od/#z\"o4kTI(>y)>\"\\\"[%QxGwmT$\"P+gXp.$3#RC:]kI;8.')RW $\\I'/&F%6#\"#8#\"LrNUe[!yu_BG`OQ/# \\hu(*3;*\"MS%\\9>@>R$e6Set2&)))G+>T$\\&/&F%6#\"#9#\"C++Dc^.(H!p7M&p9' opP\"EFsRj)=HO#QV'3amzHz&>/&F%6#\"#:#\";f!zrCj![@x5@y]\"=!G6=?X&y:()[h EQ:/&F%6#\"#;F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 39 "weights for the 16 stage order 8 scheme" }{TEXT -1 1 ": " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "seq(`b*`[i]=subs(ee,`b*` [i]),i=1..16);" }}{PARA 12 "" 1 "" {XPPMATH 20 "62/&%#b*G6#\"\"\"#\"%4:ZUGQ/&F%6#\"\"*#\"DDcE(>F;E12aOTU?5<8\"D/rA&yq v@b%ez')yrDf:'/&F%6#\"#5#\"D++++?F')*)4kU9.56\">L\"Ez;NbC8,[9#*\\Dqt>@ n>/&F%6#\"#6#\"JzOrR_tEnb)38!e[p'4TQg)*\"K+?.e#zh`wlIBGpLoiV70u&/&F%6# \"#7#\"arO^VX\"R=+!>tz)41Uqq@]g]?'\\!3Ep*o?;Y*R+wY[blSLa]9\"*3I*y:u5w& Q&odj$4&\"brD\"HzgrvTYOe]!HNR&>NI\"4rz?Fg%4&R([w%yW0yQHk'z3;S')=8P5&pi A*H#=kj5*eS/&F%6#\"#8#\"KL\"4;hc)f*fFm$>zL_'4@2,3\"\"K?rP6uR^?e+Kb^JA \"=$>*oq&/&F%6#\"#9F//&F%6#\"#:F//&F%6#\"#;#\";J5>&=mf&)=5bfo#\" " 0 " " {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#--------------- ------------------------------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 30 "#================= ============" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "Tsitouras' scheme A" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 133 "See: Optimized explicit \+ Runge-Kutta pairs of order 9(8), by Ch. Tsitouras,\n Applied Nu merical Mathematics, 38 (2001) 123-134." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 180 ": The scheme \+ constructed here has the same nodes as the scheme given in the precedi ng paper, but the weights of both the order 9 scheme and the embedded \+ order 8 scheme are changed." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "The nodes are: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[2] = 1/49" "6#/&%\"cG6#\"\"#*&\"\"\"F)\"#\\!\" \"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3] = 64/705-16/14805*sqrt(6)" "6#/&%\"cG6#\"\"$,&*&\"#k\"\"\"\"$0(!\"\"F+*(\"#;F+\"&0[\"F--%%sqrtG6# \"\"'F+F-" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 32/235-8/4935*sqrt (6)" "6#/&%\"cG6#\"\"%,&*&\"#K\"\"\"\"$N#!\"\"F+*(\"\")F+\"%N\\F--%%sq rtG6#\"\"'F+F-" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5] = 3/7" "6#/&%\" cG6#\"\"&*&\"\"$\"\"\"\"\"(!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[ 6] = 8/21+4/63*sqrt(6)" "6#/&%\"cG6#\"\"',&*&\"\")\"\"\"\"#@!\"\"F+*( \"\"%F+\"#jF--%%sqrtG6#F'F+F+" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7] \+ = 8/21-4/63*sqrt(6)" "6#/&%\"cG6#\"\"(,&*&\"\")\"\"\"\"#@!\"\"F+*(\"\" %F+\"#jF--%%sqrtG6#\"\"'F+F-" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[8] = 40/63" "6#/&%\"cG6#\"\")*&\"#S\" \"\"\"#j!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[9] = 10/21" "6#/&% \"cG6#\"\"**&\"#5\"\"\"\"#@!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[ 10] = 19/18" "6#/&%\"cG6#\"#5*&\"#>\"\"\"\"#=!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[11] = 7/9" "6#/&%\"cG6#\"#6*&\"\"(\"\"\"\"\"*!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[12] = 319999786/2170712113" "6#/&% \"cG6#\"#7*&\"*'y***>$\"\"\"\"+8@rq@!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[13] = 15/16" "6#/&%\"cG6#\"#8*&\"#:\"\"\"\"#;!\"\"" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "c[14] = 39/40" "6#/&%\"cG6#\"#9*&\"#R \"\"\"\"#S!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[15] = 1" "6#/&%\" cG6#\"#:\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[16] = 1" "6#/&%\"c G6#\"#;\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 52 "#---------------------------------------- -----------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "checking the combi ned scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "coefficie nts of the combined scheme " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10011 "ee := \{c[2]=1/49,\nc[3]=64/705- 16/14805*6^(1/2),\nc[4]=32/235-8/4935*6^(1/2),\nc[5]=3/7,\nc[6]=8/21+4 /63*6^(1/2),\nc[7]=8/21-4/63*6^(1/2),\nc[8]=40/63,\nc[9]=10/21,\nc[10] =19/18,\nc[11]=7/9,\nc[12]=319999786/2170712113,\nc[13]=15/16,\nc[14]= 39/40,\nc[15]=1,\nc[16]=1,\n\na[2,1]=1/49,\na[3,1]=-165952/1491075+388 96/10437525*6^(1/2),\na[3,2]=301312/1491075-7168/1491075*6^(1/2),\na[4 ,1]=8/235-2/4935*6^(1/2),\na[4,2]=0,\na[4,3]=24/235-2/1645*6^(1/2),\na [5,1]=38937/44800+171/5600*6^(1/2),\na[5,2]=0,\na[5,3]=-149931/44800-8 1/700*6^(1/2),\na[5,4]=65097/22400+477/5600*6^(1/2),\na[6,1]=176/5103- 29/5103*6^(1/2),\na[6,2]=0,\na[6,3]=0,\na[6,4]=364520/1674351+87715/50 23053*6^(1/2),\na[6,5]=1940224/15069159+779264/15069159*6^(1/2),\na[7, 1]=4336/127575+479/127575*6^(1/2),\na[7,2]=0,\na[7,3]=0,\na[7,4]=90731 944/400648275-170142739/8413613775*6^(1/2),\na[7,5]=8245504/62429373-2 2187008/437005611*6^(1/2),\na[7,6]=-3936/340025+11464/3060225*6^(1/2), \na[8,1]=40/567,\na[8,2]=0,\na[8,3]=0,\na[8,4]=0,\na[8,5]=0,\na[8,6]=1 60/567-10/567*6^(1/2),\na[8,7]=10/567*6^(1/2)+160/567,\na[9,1]=95/1344 ,\na[9,2]=0,\na[9,3]=0,\na[9,4]=0,\na[9,5]=0,\na[9,6]=295/1344-115/268 8*6^(1/2),\na[9,7]=295/1344+115/2688*6^(1/2),\na[9,8]=-15/448,\na[10,1 ]=52918819/138240000,\na[10,2]=0,\na[10,3]=0,\na[10,4]=0,\na[10,5]=0, \na[10,6]=-1453047743/103680000-4153586941/829440000*6^(1/2),\na[10,7] =-1453047743/103680000+4153586941/829440000*6^(1/2),\na[10,8]=44599023 /5120000,\na[10,9]=518179039/25920000,\na[11,1]=258780283/8618400000+5 85428803/51710400000*6^(1/2),\na[11,2]=0,\na[11,3]=0,\na[11,4]=0,\na[1 1,5]=0,\na[11,6]=19/25,\na[11,7]=1180508473123/443296800000-1364049110 99/147765600000*6^(1/2),\na[11,8]=-106856621/190800000+585428803/22896 00000*6^(1/2),\na[11,9]=-1260561943/591300000+585428803/886950000*6^(1 /2),\na[11,10]=13167297224/792049782825-9366860848/2376149348475*6^(1/ 2),\na[12,1]=307213395328582867964430765847473084972824867512957518186 088963/\n 51263642128606211329399441117103047984786333585721409 81841000000+\n 119107533326819222510639750832411974467191643469 020133053/\n 29137664905764716334007503363213354847664112851105 748578125*6^(1/2),\na[12,2]=0,\na[12,3]=0,\na[12,4]=0,\na[12,5]=0,\na[ 12,6]=10354821182100230493026667000379184955505622050895245676387169/ \n 146467548938874889512569831763151565670818095959204028052600 000-\n 32684637880879071688859023194047542236847633606098546061 86699/\n 418478711253928255750199519323290187630908845597725794 43600000*6^(1/2),\na[12,7]=5423711572609568912980111975307776991743433 35570401188235055068891/\n 715703215922030799396778727922657043 395761853183510539934169000000-\n 25571039199675674205714455677 98380600417779349808054092568810687111/\n 100198450229084311915 54902190917198607540665944569147559078366000000*6^(1/2),\na[12,8]=-892 225578009519154676238995901578841509244882985862325637048827/\n \+ 38813900468801845720831005417235164902766795429189067433939000000+\n 20367388198886087049319397392342447633889771033202442752063/ \n 22061374857221856652891395403575825813231399730122923923437 5*6^(1/2),\na[12,9]=-5506886052357703380348636429175951958250506337980 73448796494914/\n 83532274004202085425137482177422377296638445 3517335472487484375+\n 724173802627060872864689685061064804760 52519229164240896224/\n 30386421973154632748322110650208212912 5640034018674235171875*6^(1/2),\na[12,10]=1233968951154957384345492297 15587040998178289648593721222723693824/\n 16960466422214315122 555088502288155618001490986047888958907461648625-\n 8781560217 119727637264447549372070053517105489683916369731584/\n 6169685 857480653009296139869875647732994358307038155314262445125*6^(1/2),\na[ 12,11]=-241890129426298647138485610377551406165672225318019246672/\n \+ 3661688723471872237814245794078789141770452398980100701315,\na [13,1]=45077846760256141387004276823/110315894143992133591739924480+\n 1493491403898138129099/13100021190238236835840*6^(1/2),\na[13, 2]=0,\na[13,3]=0,\na[13,4]=0,\na[13,5]=0,\na[13,6]=2853073212310390018 5/9849639992660328448-\n 15062887306567756845/56283657100916162 56*6^(1/2),\na[13,7]=530875502237315716994493/24064781139210466754560- \n 8920823473649531766699837/1347627743795786138255360*6^(1/2), \na[13,8]=-155850251753928802974915857362119/1740150866053400164770402 16637440+\n 13441422635083243161891/5220309196109974077440*6^( 1/2),\na[13,9]=-493074073683718697930133408597/27602712116408194083051 274240+\n 1493491403898138129099/224694912332563742720*6^(1/2) ,\na[13,10]=200609996314078300148532240828075/101993369197964626516710 6381709312-\n 336035565877081079047275/84650664247949736775516 16*6^(1/2),\na[13,11]=-1259978731825102407292471875/947642075600343143 202947072,\na[13,12]=-193916214235317468987992391599053188049367133486 207120889311375/\n 427480923494550881113440074554172336410202 80098816132254793728,\na[14,1]=367166212120980360939350186871054255059 61/72248275402215258274603114496000000000000+\n 122146123726388 4679751555607/9994523002806272000000000000*6^(1/2),\na[14,2]=0,\na[14, 3]=0,\na[14,4]=0,\na[14,5]=0,\na[14,6]=8283471074731862302286097/75146 78949478400000000000-\n 15668946773152185221466849/429410225684 4800000000000*6^(1/2),\na[14,7]=397408075485926915758262202639/1835997 0961922048000000000000-\n 6493922587539771225254133441201/10281 58373867634688000000000000*6^(1/2),\na[14,8]=2435278493903047909370803 905780425755549361787/6951941874960961660657094977773568000000000000+ \n 10993151135374962117764000463/3982779843223552000000000000*6 ^(1/2),\na[14,9]=-558123239069103416347126929975086912148938889/341847 71406932232290260692794368000000000000+\n 12214612372638846797 51555607/171428613534976000000000000*6^(1/2),\na[14,10]=30673458616172 7173704146823378382330889519/14062668006262142258469146617375840000000 00-\n 10993151135374962117764000463/25833332595199504631200000 0000*6^(1/2),\na[14,11]=-163845778835264660255510638493965671483/11419 6173990354810149157741209600000000,\na[14,12]=-37972709869158045130412 9337662817719784451899678250630685021894233169654109523405323/\n \+ 749809437263379760530627813247160529470471033302857618530795356910 20475760640000000,\na[14,13]=-235412270220829707518634576/100049213779 82463725322265625,\na[15,1]=1564746779443331677794753119798867/2864199 537049451295689544004000000+\n 5933645037523445166666379/523267 57304999307928000000*6^(1/2),\na[15,2]=0,\na[15,3]=0,\na[15,4]=0,\na[1 5,5]=0,\na[15,6]=-57628625604267458078211/39343426545112261600000-\n \+ 6047516944575480929793/1405122376611152200000*6^(1/2),\na[15,7] =1691425739887134646566682083/96124421783968922002000000-\n 665 0904765188943408799448943/1345741904975564908028000000*6^(1/2),\na[15, 8]=4336304776662958741869045159571925984937/22710641428913956471088421 29832843500000+\n 53402805337711006499997411/208520160689094986 48000000*6^(1/2),\na[15,9]=-253787748130868165717697094592697839427/21 839824320042045467560266343639250000+\n 5933645037523445166666 379/897521918060373467750000*6^(1/2),\na[15,10]=2479701613016438411235 35377085814726576/1172015708277609400708896218214309893875-\n \+ 854444885403376103999958576/21640244757394101989947629875*6^(1/2),\na[ 15,11]=-52966635737697605455117615341/39091452325689672618552125990,\n a[15,12]=-443860399531273813022911271986650433425394696961417882001810 19629996219153064322715843/\n 9260962836182630467435788682851 407364301742047595974505212083507724183850275361816608,\na[15,13]=2546 9705993361596208461643776/2398429987672964863327432743875,\na[15,14]=- 10513898964163619809241937920000000/2553870660483628218384915527095907 53,\na[16,1]=141410754314242261138804477571/33519128225766632364045200 0000+\n 112443792511441850402669/1512551672706650408000000*6^(1 /2),\na[16,2]=0,\na[16,3]=0,\na[16,4]=0,\na[16,5]=0,\na[16,6]=-2794429 334116030797321/1137256896771917600000-\n 13769398374510385872 3/40616317741854200000*6^(1/2),\na[16,7]=27893647118108877562026663/27 78562296720245822000000-\n 103919596904457790641030873/38899872 154083441508000000*6^(1/2),\na[16,8]=101918684220971405061056556223280 61/4787568421535046196059430031281250+\n 1011994132602976653624 021/602746155289116328000000*6^(1/2),\na[16,9]=-8042480997962909835623 073494340513/1506681707334195127293078358250000+\n 11244379251 1441850402669/25943672957609370250000*6^(1/2),\na[16,10]=1712777545222 30541963559693237323184/1168212455792999305788440538350463625-\n \+ 16191906121647626457984336/625530609794743596276941125*6^(1/2),\na[ 16,11]=-11555892521024455643655681/15915117701775915252975590,\na[16,1 2]=-620367951196138613523419881305338943491581365345833611872299990616 582621/\n 196071800395784656980767050971315035686876839509793 271487889571798793056,\na[16,13]=-3540712440917044503420862464/6932876 1738078084467429795125,\na[16,14]=0,\na[16,15]=0,\n\nb[1]=173734691637 390647/4182794002754640000,\nb[2]=0,\nb[3]=0,\nb[4]=0,\nb[5]=0,\nb[6]= 0,\nb[7]=0,\nb[8]=-72263163141715044860361/169939769455665013040000,\n b[9]=14586697891849999254003/29700462390576849520000,\nb[10]=102209317 997264953344/225042304099487188475,\nb[11]=1883570537693211021/1872275 755054959100,\nb[12]=1710999041788984993956022337692530667432380407898 3341334325755071278367152457480027/\n 713814271258088281460765347 03173195056090466069876919551824450642795947233961810560,\nb[13]=-1067 8264099993989152768/2396652442219114419375,\nb[14]=1212545712242913280 000000/130535954653501897388343,\nb[15]=-369769046476619/6555223062234 0,\nb[16]=0,\n\n`b*`[1]=89673698740537/2188798536240000,\n`b*`[2]=0,\n `b*`[3]=0,\n`b*`[4]=0,\n`b*`[5]=0,\n`b*`[6]=0,\n`b*`[7]=0,\n`b*`[8]=-5 0462783649374452713/198296113717228720000,\n`b*`[9]=322482352625960258 97/70884158450064080000,\n`b*`[10]=-3511133566564032/35760734800490575 ,\n`b*`[11]=293286057408279/538164919532900,\n`b*`[12]=332632201005182 0410035389555466180704547593970934049395166930777066599/\n 137 33237974615224189190609759523239412338434762518802069756703639509120, \n`b*`[13]=-35480458347621122048/114126306772338781875,\n`b*`[14]=0,\n `b*`[15]=0,\n`b*`[16]=10688504668909/28093813123860\}:" }}}{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 "" 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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 33 "coefficients correct to 85 digits" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10631 "e85 := \{ c[2]=.2040816326530612244897959183673469387755102040816326530612244897 959183673469387755102e-1,\nc[3]=.8813293914998103008637915939241511230 858121177369086646032265578682773616755710635430e-1,\nc[4]=.1321994087 2497154512956873908862266846287181766053629969048398368024160425133565 95314,\nc[5]=.42857142857142857142857142857142857142857142857142857142 85714285714285714285714285714,\nc[6]=.53647553922432876813951009998132 64375851395225813758811703296868095847858703057159708,\nc[7]=.22542922 2680433136622394661923435467176765239323386023591575075095177118891599 0459340,\nc[8]=.634920634920634920634920634920634920634920634920634920 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6547798564,\na[15,10]=.11485998175713764042824328418581646630551072452 31727264746085904044764078740783444291,\na[15,11]=-1.35494161988168443 7267797384185067693264871735399669730726788055228483771700566556899,\n a[15,12]=-4.7928105034296099274097788701620029888166930458472381131271 58949832428594881423141325,\na[15,13]=.1061932435979635875109729020743 027072308689925543732473766614685824325160418410359628e-1,\na[15,14]=- .411684864345189495009466988081305489668081287782790342430052398568879 1171969791784246e-1,\na[16,1]=.603976967176969310356237087979864150687 3865304506299015961074216342422886891071025653,\na[16,2]=0.,\na[16,3]= 0.,\na[16,4]=0.,\na[16,5]=0.,\na[16,6]=-10.761217972824367769199315712 82414124971375832903898380547610041524956095218267732399,\na[16,7]=3.4 9515475072377998461195601473824486723405640323550882503106890197581387 1440481254240,\na[16,8]=6.24144494170242373677492564071626480276292083 0996875111584005450572491127858685073045,\na[16,9]=5.27858150077849200 4788836720457162225332452041400493051566166856270175635106977479710,\n a[16,10]=.832100209873447305970599159908380371785129795324797493273229 3528453914222696985221985e-1,\na[16,11]=-.7260953225457435281947342597 960351542527260080470531037037034726603392180112398726619,\na[16,12]=- 3.16398355063748315281225660540307052686763215512383605906258779920223 0577420298412386,\na[16,13]=-.5107133536141531692270880185912715236121 229340611367086227987862513131770800515274642e-1,\na[16,14]=0.,\na[16, 15]=0.,\n\nb[1]=.41535560088059591688958989218672019214591325556725516 71643920848892579905394945027033e-1,\nb[2]=0.,\nb[3]=0.,\nb[4]=0.,\nb[ 5]=0.,\nb[6]=0.,\nb[7]=0.,\nb[8]=-.42522808741698055893278455437035599 09471287347438636516704849004566228405918914211737,\nb[9]=.49112696294 1760884187062228629563014800083936238772432304241762918764634259835737 9221,\nb[10]=.45417824175884742543736768126380403653454508545123110379 87257309434313503153553995250,\nb[11]=1.006032649094428065183781872196 896559026233580812897577690817173850511464781870422371,\nb[12]=.239698 0714287725918428584296767300637015778481901015943085981245718488414129 283930348,\nb[13]=-4.4554912977314074662291886035328819774108595975549 53958343743900169006073451667494575,\nb[14]=9.288978775706101824452250 592593505318906312756833224753343846524732597445318251252011,\nb[15]=- 5.64083087586958235763030663567593304382535620078413536814843972488045 0621098631739385,\nb[16]=0.,\n\n`b*`[1]=.40969370755602675996424075532 57712059808390289258666760446846341226151513702275080794e-1,\n`b*`[2]= 0.,\n`b*`[3]=0.,\n`b*`[4]=0.,\n`b*`[5]=0.,\n`b*`[6]=0.,\n`b*`[7]=0.,\n `b*`[8]=-.254481959849877049212665564633749877334500786100729010991659 4299535572660392009934759,\n`b*`[9]=.454942768140698341949738475737633 0360944664988831092850020506994820232445205419794399,\n`b*`[10]=-.9818 4044208058760652553246011401132935835448470986186918797483751712812243 68195832300e-1,\n`b*`[11]=.5449743131953611946567479984094225057572337 722035168977499736380792205600223885339581,\n`b*`[12]=.242209595158858 8708919138669961560826833203358423838241214098569706190877130251687134 ,\n`b*`[13]=-.31088764151807857250663593299590313696686241495910855284 98690070283893299619981685371,\n`b*`[14]=0.,\n`b*`[15]=0.,\n`b*`[16]=. 3804575983254932988770303269652654021040941397092270762824232627895350 008519026874167\}:" }}}{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 "RK9_16eqs := [op(RowSumCond itions(16,'expanded')),op(OrderConditions(9,16,'expanded'))]:\n`RK8_16 eqs*` := subs(b=`b*`,OrderConditions(8,16,'expanded')):" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Check:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "expand(subs(ee,RK9_16eqs)):\nmap(u ->lhs(u)-rhs(u),%);\nnops(%);\nexpand(subs(ee,`RK8_16eqs*`)):\nmap(u-> lhs(u)-rhs(u),%);\nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7ajl\" \"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$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 12 "" 1 "" {XPPMATH 20 "6#7dw\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$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 257 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 "Next we set-u p stage-order conditions to check for stage-orders from 2 to 6 inclusi ve. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "for ct from 2 to 6 d o\n so||ct||_16 := StageOrderConditions(ct,16,'expanded');\nend do: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 138 "Sta ge 3 has stage-order 2, stages 4 and 5 have stage-order 3, stages 6 an d 7 have stage-order 4, while stages 8 to 16 have stage-order 5. " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 175 "[seq([seq(expand(subs(ee,so ||i||_16[j])),i=2..6)],j=1..14)]:\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,%):\nsim plify(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#70\"\"#\"\"$F%\"\"%F&\" \"&F'F'F'F'F'F'F'F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 27 "The simplifying conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j+1 .. 16) = b[j]*(1-c[j] );" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F, \"#;*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 9 " . . 15, " }}{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 0 151 "[Sum(b[i]*a[i,1],i=1+1..16) =b[1],seq(Sum(b[i]*a[i,j],i=j+1..16)=b[j]*(1-c[j]),j=2..15)]:\nmap(u-> lhs(u)-rhs(u),subs(ee,eval(subs(Sum=add,%))));\nnops(%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#71\"\"!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 27 "The simplifying conditions:" }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(`b*`[i]*a[i,j],i = j+1 .. 16) = ` b*`[j]*(1-c[j]);" "6#/-%$SumG6$*&&%#b*G6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/ F+;,&F0F,F,F,\"#;*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 9 " . . 15, " }} {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 0 163 "[Sum(`b*`[i ]*a[i,1],i=1+1..16)=`b*`[1],seq(Sum(`b*`[i]*a[i,j],i=j+1..16)=`b*`[j]* (1-c[j]),j=2..15)]:\nmap(u->lhs(u)-rhs(u),subs(ee,eval(subs(Sum=add,%) )));\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#71\"\"!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 93 "We can calculate t he principal error norm, that is, the 2-norm of the principal error te rms. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "errterms9_16 := Pr incipalErrorTerms(9,16,'expanded'):\nsm := 0:\nfor ct to nops(errterms 9_16) do\n sm := sm+(evalf(subs(ee,errterms9_16[ct])))^2;\nend do:\n sqrt(sm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+P%4Kk$!#;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "We can also cal culate the principal error norm of the order 8 embedded scheme." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "`errterms8_16*` := subs(b=` b*`,PrincipalErrorTerms(8,16,'expanded')):\nsm := 0:\nfor ct to nops(` errterms8_16*`) do\n sm := sm+(evalf(subs(ee,`errterms8_16*`[ct])))^ 2;\nend do:\nsqrt(sm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+>OY/;!#9 " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#----------------- ----------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 34 "construction of the order 9 scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 94 "First we construct a 15 stage orde r 9 scheme starting with a consideration of stages 2 to 11. " }}{PARA 0 "" 0 "" {TEXT -1 21 "We specify the nodes:" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "c[2] = 1/49;" "6#/&%\"cG6#\"\"#*&\"\" \"F)\"#\\!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5] = 3/7;" "6#/&% \"cG6#\"\"&*&\"\"$\"\"\"\"\"(!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 " c[8] = 40/63;" "6#/&%\"cG6#\"\")*&\"#S\"\"\"\"#j!\"\"" }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "c[10] = 19/18;" "6#/&%\"cG6#\"#5*&\"#>\"\"\"\"#=!\" \"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[11] = 7/9;" "6#/&%\"cG6#\"#6*& \"\"(\"\"\"\"\"*!\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 35 "and the zero linking coefficients: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[4,2]=0" "6#/&%\"aG6$\"\"%\"\"#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[5,2]=0" "6#/&%\"aG6$\"\"&\"\"#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[6,2]=0" "6#/&%\"aG6$\"\"'\"\"#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[6,3]=0" "6#/&%\"aG6$\"\"'\"\"$\"\"!" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "a[7,2]=0" "6#/&%\"aG6$\"\"(\"\"#\"\"! " }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[7,3]=0" "6#/&%\"aG6$\"\"(\"\"$\" \"!" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[8,2]=0" "6#/&%\"aG6$\"\")\"\"#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[8,3]=0" "6#/&%\"aG6$\"\")\"\"$\"\"!" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "a[8,4]=0" "6#/&%\"aG6$\"\")\"\"%\"\"!" }{TEXT -1 2 ", \+ " }{XPPEDIT 18 0 "a[8,5]=0" "6#/&%\"aG6$\"\")\"\"&\"\"!" }{TEXT -1 2 " , " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[9,2]=0" "6#/& %\"aG6$\"\"*\"\"#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[9,3]=0" "6# /&%\"aG6$\"\"*\"\"$\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[9,4]=0" " 6#/&%\"aG6$\"\"*\"\"%\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[9,5]=0 " "6#/&%\"aG6$\"\"*\"\"&\"\"!" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[10,2]=0" "6#/&%\"aG6$\"#5\"\"#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[10,3]=0" "6#/&%\"aG6$\"#5\"\"$\"\"! " }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[10,4]=0" "6#/&%\"aG6$\"#5\"\"%\" \"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[10,5]=0" "6#/&%\"aG6$\"#5\"\"& \"\"!" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[11,2]=0" "6#/&%\"aG6$\"#6\"\"#\"\"!" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "a[11,3]=0" "6#/&%\"aG6$\"#6\"\"$\"\"!" }{TEXT -1 2 ", \+ " }{XPPEDIT 18 0 "a[11,4]=0" "6#/&%\"aG6$\"#6\"\"%\"\"!" }{TEXT -1 2 " , " }{XPPEDIT 18 0 "a[11,5]=0" "6#/&%\"aG6$\"#6\"\"&\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 20 " in stages 2 to 11. " }}{PARA 0 " " 0 "" {TEXT -1 20 "Note that the node " }{XPPEDIT 18 0 "c[10]=19/18 " "6#/&%\"cG6#\"#5*&\"#>\"\"\"\"#=!\"\"" }{TEXT -1 5 " is " }{TEXT 260 23 "outside of the interval" }{TEXT -1 13 " from 0 to 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "We also specify " }{XPPEDIT 18 0 "a[11,6] = 76/100;" "6#/&%\"aG6$\"#6\"\"'*&\"#w\"\" \"\"$+\"!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 69 "The stage-order equations relating to the se stages, such that stages " }{XPPEDIT 18 0 "3,4,5,6,7,8,9,10,11;" "6 +\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#5\"#6" }{TEXT -1 19 " have stag e orders " }{XPPEDIT 18 0 "2,3,3,4,4,5,5,5,5;" "6+\"\"#\"\"$F$\"\"%F% \"\"&F&F&F&" }{TEXT -1 96 " respectively taken together with the row-s um conditions can then be solved to obtain the nodes " }{XPPEDIT 18 0 "c[3],c[4],c[6],c[7],c[9],c[11];" "6(&%\"cG6#\"\"$&F$6#\"\"%&F$6#\"\"' &F$6#\"\"(&F$6#\"\"*&F$6#\"#6" }{TEXT -1 67 " and the remaining non-ze ro linking coefficients for these stages. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "The computation is made more ef ficient by " }{TEXT 260 48 "including explicitly relations between the nodes" }{TEXT -1 42 " arising from the stage-order conditions: " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "c[3]=2/3" "6#/&%\"cG6#\"\"$*&\"\"#\"\"\"F'!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 8 ", \+ " }{XPPEDIT 18 0 "c[4] = (6+sqrt(6))*(9*c[8]-24*c[5]+4*c[5]*sqrt (6))*c[8]/(60*(2*c[8]-6*c[5]+c[5]*sqrt(6)));" "6#/&%\"cG6#\"\"%**,&\" \"'\"\"\"-%%sqrtG6#F*F+F+,(*&\"\"*F+&F%6#\"\")F+F+*&\"#CF+&F%6#\"\"&F+ !\"\"*(F'F+&F%6#F9F+-F-6#F*F+F+F+&F%6#F4F+*&\"#gF+,(*&\"\"#F+&F%6#F4F+ F+*&F*F+&F%6#F9F+F:*&&F%6#F9F+-F-6#F*F+F+F+F:" }{TEXT -1 2 ", " }} {PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "c[6] = (6+sqrt(6))*c[8]/10;" "6#/&%\"cG6#\"\"'*(,&F'\" \"\"-%%sqrtG6#F'F*F*&F%6#\"\")F*\"#5!\"\"" }{TEXT -1 8 ", " } {XPPEDIT 18 0 "c[7] = (6-sqrt(6))*c[8]/10;" "6#/&%\"cG6#\"\"(*(,&\"\"' \"\"\"-%%sqrtG6#F*!\"\"F+&F%6#\"\")F+\"#5F/" }{TEXT -1 8 ", " } {XPPEDIT 18 0 "c[9]=3/4" "6#/&%\"cG6#\"\"**&\"\"$\"\"\"\"\"%!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "c[8]" "6#&%\"cG6#\"\")" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "The e quations that lead to " }{XPPEDIT 18 0 "c[6] < c[7]" "6#2&%\"cG6#\"\" '&F%6#\"\"(" }{TEXT -1 19 ", have been chosen." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 813 "RSeqs := Ro wSumConditions(11,'expanded'):\nSOeqs := [op(StageOrderConditions(2,11 ,'expanded')),\n op(StageOrderConditions(3,4..11,'expand ed')),\n op(StageOrderConditions(4,6..11,'expanded')),\n op(StageOrderConditions(5,8..11,'expanded'))]:\nnode_eq sB := [c[3]=2/3*c[4],c[9]=3/4*c[8],c[6]=1/10*(6+6^(1/2))*c[8],c[7]=1/1 0*(6-6^(1/2))*c[8],\n c[4]=1/60*(6+6^(1/2))*(9*c[8]-24*c[ 5]+4*c[5]*6^(1/2))*c[8]/(2*c[8]-6*c[5]+c[5]*6^(1/2))]:\n\ne1 := \{c[2] =1/49,c[5]=3/7,c[8]=40/63,c[10]=19/18,c[11]=7/9,\n seq(a[i, 2]=0,i=4..11),seq(a[i,3]=0,i=6..11),\n seq(a[i,4]=0,i=8..11 ),seq(a[i,5]=0,i=8..11),a[11,6]=76/100\}:\neqns := expand(rationalize( subs(e1,[op(RSeqs),op(SOeqs),op(node_eqsB)]))):\nconvert(ListTools[Enu merate](%),matrix);\n``;\nindets(eqns);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7L7$\"\"\"/&%\"aG6$\"\"#F(#F(\"#\\7$F-/,&&F +6$\"\"$F(F(&F+6$F5F-F(&%\"cG6#F57$F5/,&&F+6$\"\"%F(F(&F+6$F@F5F(&F96# F@7$F@/,(&F+6$\"\"&F(F(&F+6$FJF5F(&F+6$FJF@F(#F5\"\"(7$FJ/,(&F+6$\"\"' F(F(&F+6$FVF@F(&F+6$FVFJF(&F96#FV7$FV/,*&F+6$FPF(F(&F+6$FPF@F(&F+6$FPF JF(&F+6$FPFVF(&F96#FP7$FP/,(&F+6$\"\")F(F(&F+6$FioFVF(&F+6$FioFPF(#\"# S\"#j7$Fio/,*&F+6$\"\"*F(F(&F+6$FfpFVF(&F+6$FfpFPF(&F+6$FfpFioF(&F96#F fp7$Ffp/,,&F+6$\"#5F(F(&F+6$FdqFVF(&F+6$FdqFPF(&F+6$FdqFioF(&F+6$FdqFf pF(#\"#>\"#=7$Fdq/,.&F+6$\"#6F(F(#F^r\"#DF(&F+6$FerFPF(&F+6$FerFioF(&F +6$FerFfpF(&F+6$FerFdqF(#FPFfp7$Fer/,$*&F.F(F6F(F(,$*&#F(F-F(*$)F8F-F( F(F(7$\"#7/*&FAF(F8F(,$*&FgsF(*$)FCF-F(F(F(7$\"#8/,&*&FKF(F8F(F(*&FMF( FCF(F(#Ffp\"#)*7$\"#9/,&*&FWF(FCF(F(*&FOF(FYF(F(,$*&FgsF(*$)FenF-F(F(F (7$\"#:/,(*&F\\oF(FCF(F(*&FOF(F^oF(F(*&F`oF(FenF(F(,$*&FgsF(*$)FboF-F( F(F(7$\"#;/,&*&FjoF(FenF(F(*&F\\pF(FboF(F(#\"$+)\"%pR7$\"#%7$\"#L/, ,*&FfrF(FazF(F(*&FhrF(F][lF(F(*&Fb`lF(FjrF(F(*&F\\sF(Fb\\lF(F(*&#F\\]l \"%KeF(F^sF(F(#\"%,C\"&Wi#7$\"#M/,&*&FjoF(Ff^lF(F(*&F\\pF(Fa_lF(F(#\") ++[?\"*VlV#**7$\"#N/,(*&FgpF(Ff^lF(F(*&FipF(Fa_lF(F(*&#\"(++c#Fj_lF(F[ qF(F(,$*&#F(FJF(*$)F]qFJF(F(F(7$\"#O/,**&FeqF(Ff^lF(F(*&FgqF(Fa_lF(F(* &FaclF(FiqF(F(*&F[rF(Fg`lF(F(#\"(*4wC\"(SyW*7$\"#P/,,*&FfrF(Fe^lF(F(*& FhrF(Fa_lF(F(*&FaclF(FjrF(F(*&F\\sF(Fg`lF(F(*&#Faal\"'w\\5F(F^sF(F(#\" &2o\"\"'X_H7$\"#Q/F8,$*&#F-F5F(FCF(F(7$\"#R/F]q#FdqFay7$F_p/Fen,&#FioF ayF(*(F@F(F`p!\"\"FVFgsF(7$\"#T/Fbo,&F^flF(*(F@F(F`pF`flFVFgsF`fl7$\"# U/FC,&#Fi`l\"$N#F(*(FioF(\"%N\\F`flFVFgsF`flQ)pprint106\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "e2 := expand(rationalize(sol ve(\{op(eqns)\}))):\ne3 := `union`(e1,e2):\ninfolevel[solve] := 0:" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" } {TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 106 " If the equations givi ng the relations between the nodes are omitted we need to select the s olution with " }{XPPEDIT 18 0 "c[7] < c[6];" "6#2&%\"cG6#\"\"(&F%6#\" \"'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 23 " Thus we require t hat " }{XPPEDIT 18 0 "c[6] = 8/21+4/63*sqrt(6);" "6#/&%\"cG6#\"\"',&* &\"\")\"\"\"\"#@!\"\"F+*(\"\"%F+\"#jF--%%sqrtG6#F'F+F+" }{TEXT -1 1 " \+ " }{TEXT 274 1 "~" }{TEXT -1 20 " 0.5364755393 and " }{XPPEDIT 18 0 "c[7] = 8/21-4/63*sqrt(6);" "6#/&%\"cG6#\"\"(,&*&\"\")\"\"\"\"#@!\"\"F +*(\"\"%F+\"#jF--%%sqrtG6#\"\"'F+F-" }{TEXT -1 1 " " }{TEXT 275 1 "~" }{TEXT -1 48 " 0.2254292227, rather than the other way round. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "The follo wing commands achieve this." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 185 "eqns := subs(e1,[op(RSeqs),op(SOeq s)]):\nsol := solve(\{op(eqns)\}):\ne2 := op(select(u_->evalf(subs(u_, c[7]) " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e3 \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1841 "e3 := \{a[11,5] = 0, c[2] = 1/49, c[5] = 3/7, c[8] = 40/63, \+ c[10] = 19/18, c[11] = 7/9, a[11,6] = 19/25, a[8,5] = 0, a[8,4] = 0, a [10,1] = 52918819/138240000, a[4,2] = 0, a[2,1] = 1/49, a[8,1] = 40/56 7, c[6] = 8/21+4/63*6^(1/2), c[7] = 8/21-4/63*6^(1/2), c[9] = 10/21, c [4] = 32/235-8/4935*6^(1/2), c[3] = 64/705-16/14805*6^(1/2), a[10,3] = 0, a[11,3] = 0, a[6,3] = 0, a[7,3] = 0, a[8,3] = 0, a[10,5] = 0, a[9, 5] = 0, a[10,4] = 0, a[10,8] = 44599023/5120000, a[11,4] = 0, a[9,4] = 0, a[4,3] = 24/235-2/1645*6^(1/2), a[7,1] = 4336/127575+479/127575*6^ (1/2), a[5,4] = 65097/22400+477/5600*6^(1/2), a[9,7] = 295/1344+115/26 88*6^(1/2), a[6,5] = 1940224/15069159+779264/15069159*6^(1/2), a[7,4] \+ = 90731944/400648275-170142739/8413613775*6^(1/2), a[10,7] = -14530477 43/103680000+4153586941/829440000*6^(1/2), a[11,7] = 1180508473123/443 296800000-136404911099/147765600000*6^(1/2), a[9,6] = 295/1344-115/268 8*6^(1/2), a[3,1] = -165952/1491075+38896/10437525*6^(1/2), a[11,1] = \+ 258780283/8618400000+585428803/51710400000*6^(1/2), a[11,10] = 1316729 7224/792049782825-9366860848/2376149348475*6^(1/2), a[7,5] = 8245504/6 2429373-22187008/437005611*6^(1/2), a[3,2] = 301312/1491075-7168/14910 75*6^(1/2), a[8,7] = 10/567*6^(1/2)+160/567, a[4,1] = 8/235-2/4935*6^( 1/2), a[5,3] = -149931/44800-81/700*6^(1/2), a[7,6] = -3936/340025+114 64/3060225*6^(1/2), a[6,4] = 364520/1674351+87715/5023053*6^(1/2), a[5 ,1] = 38937/44800+171/5600*6^(1/2), a[6,1] = 176/5103-29/5103*6^(1/2), a[8,6] = 160/567-10/567*6^(1/2), a[11,9] = -1260561943/591300000+5854 28803/886950000*6^(1/2), a[11,8] = -106856621/190800000+585428803/2289 600000*6^(1/2), a[10,6] = -1453047743/103680000-4153586941/829440000*6 ^(1/2), a[10,9] = 518179039/25920000, a[8,2] = 0, a[10,2] = 0, a[11,2] = 0, a[7,2] = 0, a[9,2] = 0, a[9,3] = 0, a[5,2] = 0, a[6,2] = 0, a[9, 1] = 95/1344, a[9,8] = -15/448\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "printed coefficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 219 "s ubs(e3,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(6-i)],i=2..6)]));\nf or ii from 7 to 11 do\n print(``);\n print(c[ii]=subs(e3,c[ii])); \+ \n for jj to ii-1 do\n print(a[ii,jj]=subs(e3,a[ii,jj]));\n e nd do:\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7(#\" \"\"\"#\\F(%!GF+F+F+7(,&#\"#k\"$0(F)*(\"#;F)\"&0[\"!\"\"\"\"'#F)\"\"#F 4,&#\"'_f;\"(v5\\\"F4*(\"&'*)QF)\")DvV5F4F5F6F),&#\"'78IF;F)*(\"%orF)F ;F4F5F6F4F+F+F+7(,&#\"#K\"$N#F)*(\"\")F)\"%N\\F4F5F6F4,&#FJFHF)*(F7F)F KF4F5F6F4\"\"!,&#\"#CFHF)*(F7F)\"%X;F4F5F6F4F+F+7(#\"\"$\"\"(,&#\"&P*Q \"&+[%F)*(\"$r\"F)\"%+cF4F5F6F)FO,&#\"'J*\\\"FfnF4*(\"#\")F)\"$+(F4F5F 6F4,&#\"&(4l\"&+C#F)*(\"$x%F)FinF4F5F6F)F+7(,&#FJ\"#@F)*(\"\"%F)\"#jF4 F5F6F),&#\"$w\"\"%.^F)*(\"#HF)F`pF4F5F6F4FOFO,&#\"'?XO\"(^Vn\"F)*(\"&: x)F)\"(`I-&F4F5F6F),&#\"(C-%>\")f\"p]\"F)*(\"'k#z(F)F]qF4F5F6F)Q(pprin t56\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"(,&#\"\")\"#@\"\"\"*(\"\"%F,\"#j!\"\"\"\"' #F,\"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"\",&# \"%OV\"'vv7F(*(\"$z%F(F,!\"\"\"\"'#F(\"\"#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"\"(\"\"%,&#\")W>t!*\"*v#[1S\"\"\"*(\"*RF9q\"F-\"+vPh8%)!\"\"\" \"'#F-\"\"#F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"&,& #\"(/bC)\")t$HC'\"\"\"*(\")3q=AF-\"*6c+P%!\"\"\"\"'#F-\"\"#F1" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"',&#\"%OR\"'D+M!\"\" *(\"&k9\"\"\"\"\"(D-1$F-F(#F0\"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\")#\"#S\"#j" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"\"#\"#S\"$n&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"#\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"\")\"\"',&#\"$g\"\"$n&\"\"\"*(\"#5F-F,!\"\"F(#F-\"\"#F0" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"(,&*(\"#5\"\"\"\"$n& !\"\"\"\"'#F,\"\"#F,#\"$g\"F-F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"*#\"#5\"#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"\"#\"#&*\"%W8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"\"*\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\" *\"\"',&#\"$&H\"%W8\"\"\"*(\"$:\"F-\"%)o#!\"\"F(#F-\"\"#F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"(,&#\"$&H\"%W8\"\"\"*(\"$: \"F-\"%)o#!\"\"\"\"'#F-\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"\"*\"\")#!#:\"$[%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#5#\"#>\"#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"\"#\")>)=H&\"*++CQ\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"#5\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\" \"',&#\"+Vx/`9\"*++o.\"!\"\"*(\"+Tpe`T\"\"\"\"*++WH)F-F(#F0\"\"#F-" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"(,&#\"+Vx/`9\"*++o.\" !\"\"*(\"+Tpe`T\"\"\"\"*++WH)F-\"\"'#F0\"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\")#\")B!*fW\"(++7&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"*#\"*R!z\"=&\")++#f#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#6 #\"\"(\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"\",&# \"*$G!ye#\"+++S=')F(*(\"*.)GaeF(\",++S5<&!\"\"\"\"'#F(\"\"#F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"#\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"#6\"\"'#\"#>\"#D" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"# 6\"\"(,&#\".BJZ30=\"\"-++!oHV%\"\"\"*(\"-*46\\SO\"F-\"-++glx9!\"\"\"\" '#F-\"\"#F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"),&#\" *@m&o5\"*++!3>!\"\"*(\"*.)Gae\"\"\"\"+++g*G#F-\"\"'#F0\"\"#F0" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"*,&#\"+V>cg7\"*++I\"f !\"\"*(\"*.)Gae\"\"\"\"*++&p))F-\"\"'#F0\"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"#5,&#\",CsHnJ\"\"-DGy\\?z\"\"\"*(\"+[3'o O*F-\".v%[$\\hP#!\"\"\"\"'#F-\"\"#F1" }}}{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 27 "We do not need to specify " }{XPPEDIT 18 0 "c[12]" "6#&%\"cG6#\"#7" }{TEXT -1 43 ", because, a ccording to Verner, the nodes " }{XPPEDIT 18 0 "c[8]" "6#&%\"cG6#\"\" )" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "c[9]" "6#&%\"cG6#\"\"*" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "c[10]" "6#&%\"cG6#\"#5" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "c[11]" "6#&%\"cG6#\"#6" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "c[12]" "6#&%\"cG6#\"#7" }{TEXT -1 30 " 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),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-F-F,F3F-/F,;F:F-F-*&-F&6$*&-F*6#F,F--F /6#*&,&F-F-F,F3\"\"#-F66#FNF3F-/F,;F:F-F--F&6$*&-F?6#F,F--F/6#*&,&F-F- F,F3FN-F66#FNF3F-/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[8])*(x-c[9])*(x-c[10])*(x-c[11])" "6#/-%\"pG6#%\"xG*,F'\"\"\",&F' F)&%\"cG6#\"\")!\"\"F),&F'F)&F,6#\"\"*F/F),&F'F)&F,6#\"#5F/F),&F'F)&F, 6#\"#6F/F)" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "q(x)=(x-c[12])*p(x) " "6#/-%\"qG6#%\"xG*&,&F'\"\"\"&%\"cG6#\"#7!\"\"F*-%\"pG6#F'F*" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 155 "See: J.H. Verner, SIAM Journal of Numerical Analysis 197 8, 772-790, \"Explicit Runge-Kutta methods with estimates of the Local Truncation Error.\" (page 780)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "p := x -> x*(x-c[8])*(x-c[9 ])*(x-c[10])*(x-c[11]):\n'p(x)'=p(x);\nq := x -> (x-c[12])*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),&F'F)&F,6#\"#5F/F),&F'F)& F,6#\"#6F/F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"qG6#%\"xG*.,&F'\" \"\"&%\"cG6#\"#7!\"\"F*F'F*,&F'F*&F,6#\"\")F/F*,&F'F*&F,6#\"\"*F/F*,&F 'F*&F,6#\"#5F/F*,&F'F*&F,6#\"#6F/F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "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);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ IeqG/*&-%$IntG6$,$*&#\"\"\"\"\"'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-F8F--F(6$,$*&FEF-*&F>F-FHF-F-F-F8F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "e4 := solve( subs(e3,value(Ieq)),\{c[12]\}):\nc[12]=subs(e4,c[12]);\ne5 := `union`( e3,e4):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#7#\"*'y***>$\"+ 8@rq@" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 " Thus " }{XPPEDIT 18 0 "c[12] = 319999786/2170712113" "6#/&%\"cG6#\"#7 *&\"*'y***>$\"\"\"\"+8@rq@!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 99 "Now we can use the quadrature equations to find the weigh ts once the remaining nodes once the nodes" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[13] = 15/16;" "6#/&%\"cG6#\"#8*&\"#:\"\"\" \"#;!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[14] = 39/40;" "6#/&%\"c G6#\"#9*&\"#R\"\"\"\"#S!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[15]= 1" "6#/&%\"cG6#\"#:\"\"\"" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[13] = 59/69;" "6#/&%\"cG6#\"#8*&\"#f\"\"\" \"#p!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[14] = 44/49;" "6#/&%\"c G6#\"#9*&\"#W\"\"\"\"#\\!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[15] =1" "6#/&%\"cG6#\"#:\"\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 38 "are specified along with the weights " }{XPPEDIT 18 0 "b[2]=0 " "6#/&%\"bG6#\"\"#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "b[3]=0" "6# /&%\"bG6#\"\"$\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "b[4]=0" "6#/&%\" bG6#\"\"%\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "b[5]=0" "6#/&%\"bG6# \"\"&\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "b[6]=0" "6#/&%\"bG6#\"\"' \"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "b[7]=0" "6#/&%\"bG6#\"\"(\"\"! " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 288 "Qeqs := QuadratureConditions(9,15,'expanded' ):\ne6 := \{seq(b[i]=0,i=2..7),c[13]=15/16,c[14]=39/40,c[15]=1\}:\ne7 \+ := `union`(e5,e6):\nquadeqns := subs(e7,Qeqs):\nfor ct to nops(quadeqn s) do\n print(`equation `||ct); print(``);print(quadeqns[ct]);prin t(``);\nend do:\nindets(quadeqns);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,4&%\"bG6#\"\"\"F(&F&6#\"\")F(&F& 6#\"\"*F(&F&6#\"#5F(&F&6#\"#6F(&F&6#\"#7F(&F&6#\"#8F(&F&6#\"#9F(&F&6# \"#:F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~2G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,2*&#\"#S\"#j\"\"\"&%\"bG6#\"\")F )F)*&#\"#5\"#@F)&F+6#\"\"*F)F)*&#\"#>\"#=F)&F+6#F0F)F)*&#\"\"(F4F)&F+6 #\"#6F)F)*&#\"*'y***>$\"+8@rq@F)&F+6#\"#7F)F)*&#\"#:\"#;F)&F+6#\"#8F)F )*&#\"#RF'F)&F+6#\"#9F)F)&F+6#FJF)#F)\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~3G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,2*&#\"%+;\"%pR\"\"\"&%\"bG6#\"\")F)F)*&#\"$+\"\"$T%F)&F+6#\"\" *F)F)*&#\"$h$\"$C$F)&F+6#\"#5F)F)*&#\"#\\\"#\")F)&F+6#\"#6F)F)*&#\"3'z X+/j)*R-\"\"4pZ#\\_x5*>r%F)&F+6#\"#7F)F)*&#\"$D#\"$c#F)&F+6#\"#8F)F)*& #\"%@:F'F)&F+6#\"#9F)F)&F+6#\"#:F)#F)\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~4G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,2*&#\"&+S'\"'Z+D\"\"\"&%\"bG6#\"\")F)F)*&#\"%+5\"%h#*F)&F+6#\" \"*F)F)*&#\"%fo\"%KeF)&F+6#\"#5F)F)*&#\"$V$\"$H(F)&F+6#\"#6F)F)*&#\";c '*>]T'RCfU$zwK\">(*o-#[biFJ$3hPG-\"F)&F+6#\"#7F)F)*&#\"%vL\"%'4%F)&F+6 #\"#8F)F)*&#\"&>$fF'F)&F+6#\"#9F)F)&F+6#\"#:F)#F)\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~ ~~5G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,2*&#\"(++c#\")hHv:\"\"\"&%\"bG6#\"\")F)F)*&#\"&++\"\" '\"[%>F)&F+6#\"\"*F)F)*&#\"'@.8\"'w\\5F)&F+6#\"#5F)F)*&#\"%,C\"%hlF)&F +6#\"#6F)F)*&#\"D;OFxt&[5\"GhLq4\"R%=T_%e,Xn9*fG?AF)&F+6# \"#7F)F)*&#\"&D1&\"&Ob'F)&F+6#\"#8F)F)*&#\"(TMJ#F'F)&F+6#\"#9F)F)&F+6# \"#:F)#F)\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~6G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,2*&#\"*++S-\"\"*VlV#**\"\"\" &%\"bG6#\"\")F)F)*&#\"'++5\"(,T3%F)&F+6#\"\"*F)F)*&#\"(*4wC\"(o&*)=F)& F+6#\"#5F)F)*&#\"&2o\"\"&\\!fF)&F+6#\"#6F)F)*&#\"LwhW$yOtj*4^WCBk!=D!) >VbL\"P$z6&f:-'3AkB(>k[F)&F+6#\"#7F)F)*&#\"'v$f(\"(w&[5F)&F +6#\"#8F)F)*&#\")*>C-*F'F)&F+6#\"#9F)F)&F+6#\"#:F)#F)\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equ ation~~~7G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,2*&#\"++++'4%\",4A]BD'\"\"\"&%\"bG6#\"\")F)F)*&#\"(++ +\"\")@hw&)F)&F+6#\"\"*F)F)*&#\")\")e/Z\")CA,MF)&F+6#\"#5F)F)*&#\"'\\w 6\"'T9`F)&F+6#\"#6F)F)*&#\"UO$=Dm/FJ[CTrvx\\;b\"GM\"=c^PP2\"\"Z4'[9V*R 6ss2paA3$)z)\\v$R1#[8yn>Y5F)&F+6#\"#7F)F)*&#\")D1R6\");sx;F)&F+6#\"#8F )F)*&#\"+hPu=NF'F)&F+6#\"#9F)F)&F+6#\"#:F)#F)\"\"(" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~8 G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,2*&#\"-+++SQ;\".n\"R1)*QR\"\"\"&%\"bG6#\"\")F)F)*&#\")+++5\"+T &)3,=F)&F+6#\"\"*F)F)*&#\"*R<(Q*)\"*K+A7'F)&F+6#\"#5F)F)*&#\"'VN#)\"(p Hy%F)&F+6#\"#6F)F)*&#\"gn'4w5upq3`:!y<$e*zpnn\"z:yiuz@vdfV$\"]o<3I$p3o _dW3^T6EL*3Y!p***zu*3(G)y)=?*4F#F)&F+6#\"#7F)F)*&#\"*v$f3<\"*caVo#F)&F +6#\"#8F)F)*&#\"-zm+Js8F'F)&F+6#\"#9F)F)&F+6#\"#:F)#F)F-" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equatio n~~~9G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,2*&#\".++++Ob'\"0@vE!yb\"[#\"\"\"&%\"bG6#\"\")F)F)*&# \"*++++\"\",h$fG#y$F)&F+6#\"\"*F)F)*&#\",TIc$)p\"\",w0'*>5\"F)&F+6#\"# 5F)F)*&#\"(,[w&\")@n/VF)&F+6#\"#6F)F)*&#\"`ocarHhU6l&*H$Qpp&yzy.L];]P#y:)f1R))p'H\\F)&F+6# \"#7F)F)*&#\"+D1*Gc#\"+'Hn\\H%F)&F+6#\"#8F)F)*&#\".\"[g#4?N&F'F)&F+6# \"#9F)F)&F+6#\"#:F)#F)F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#<+&%\"bG6#\"\"\"&F%6#\"\")&F%6#\"\"*&F %6#\"#5&F%6#\"#6&F%6#\"#7&F%6#\"#8&F%6#\"#9&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 73 "e8 := solve(\{op(quadeqns)\}):\ne9 \+ := `union`(e7,e8):\ninfolevel[solve] := 0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "for ii to 15 do\n wt_val := subs(e9,b[ii]);\n if wt_val<>0 then print(b[ii]=wt_val) end if;\nend do:\n``;\nevalf[8](subs(e9,[seq(b[i],i=1..15)]));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"\"#\"3Z1RP;pMP<\"4++kaF+% z#=%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\")#!8h.'[/:<9jJEs \"9++/8]mb%p(R*p\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"*# \"8.SD***\\=*yp'e9\"8++_\\od!Ri/qH" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%\"bG6#\"#5#\"6WL&\\E(*zJ4A5\"6v%)=([*4/B/D#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#6#\"4@5@$pP0d$)=\"4+\"f\\0bdFs=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#7#\"^pF+[dC:n$y72bdKM8M$)*yS!QKu mIDpPB-cR*\\)*)yT!**4r\"\"^pg0\"='RBZfzU1XC=b>p()pgY!4c]>tJqMl2Y\"G)3e 7F9Qr" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#8#!8oF:*)R***4k#y 1\"\"7v$>W6>AW_mR#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#9#\" :+++!G8HC7da77\"9V$)Q(*=]`Y&f`I\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ &%\"bG6#\"#:#!0>mZY!p(p$\"/SBiIAbl" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#71$\")gb`T!\"*$\"\"!F(F'F'F'F'F '$!)4G_U!\")$\")'p7\"\\F+$\")CyTXF+$\")E.15!\"($\")2)pR#F+$!)8\\bWF2$ \"))y*)G*F2$!)4$3k&F2" }}}{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 2570 "e9 := \{a[11,5] = 0, b[8] = -7226316314171504486036 1/169939769455665013040000, b[15] = -369769046476619/65552230622340, c [2] = 1/49, c[5] = 3/7, c[8] = 40/63, c[10] = 19/18, c[11] = 7/9, a[11 ,6] = 19/25, a[8,5] = 0, a[8,4] = 0, b[14] = 1212545712242913280000000 /130535954653501897388343, a[10,1] = 52918819/138240000, a[4,2] = 0, a [2,1] = 1/49, a[8,1] = 40/567, b[13] = -10678264099993989152768/239665 2442219114419375, b[12] = 17109990417889849939560223376925306674323804 078983341334325755071278367152457480027/713814271258088281460765347031 73195056090466069876919551824450642795947233961810560, c[6] = 8/21+4/6 3*6^(1/2), c[7] = 8/21-4/63*6^(1/2), c[9] = 10/21, c[4] = 32/235-8/493 5*6^(1/2), b[11] = 1883570537693211021/1872275755054959100, c[3] = 64/ 705-16/14805*6^(1/2), a[10,3] = 0, a[11,3] = 0, a[6,3] = 0, a[7,3] = 0 , a[8,3] = 0, a[10,5] = 0, a[9,5] = 0, a[10,4] = 0, a[10,8] = 44599023 /5120000, a[11,4] = 0, a[9,4] = 0, a[4,3] = 24/235-2/1645*6^(1/2), a[7 ,1] = 4336/127575+479/127575*6^(1/2), a[5,4] = 65097/22400+477/5600*6^ (1/2), a[9,7] = 295/1344+115/2688*6^(1/2), a[6,5] = 1940224/15069159+7 79264/15069159*6^(1/2), a[7,4] = 90731944/400648275-170142739/84136137 75*6^(1/2), a[10,7] = -1453047743/103680000+4153586941/829440000*6^(1/ 2), a[11,7] = 1180508473123/443296800000-136404911099/147765600000*6^( 1/2), a[9,6] = 295/1344-115/2688*6^(1/2), a[3,1] = -165952/1491075+388 96/10437525*6^(1/2), a[11,1] = 258780283/8618400000+585428803/51710400 000*6^(1/2), a[11,10] = 13167297224/792049782825-9366860848/2376149348 475*6^(1/2), a[7,5] = 8245504/62429373-22187008/437005611*6^(1/2), a[3 ,2] = 301312/1491075-7168/1491075*6^(1/2), a[8,7] = 10/567*6^(1/2)+160 /567, a[4,1] = 8/235-2/4935*6^(1/2), a[5,3] = -149931/44800-81/700*6^( 1/2), a[7,6] = -3936/340025+11464/3060225*6^(1/2), a[6,4] = 364520/167 4351+87715/5023053*6^(1/2), a[5,1] = 38937/44800+171/5600*6^(1/2), a[6 ,1] = 176/5103-29/5103*6^(1/2), a[8,6] = 160/567-10/567*6^(1/2), a[11, 9] = -1260561943/591300000+585428803/886950000*6^(1/2), a[11,8] = -106 856621/190800000+585428803/2289600000*6^(1/2), a[10,6] = -1453047743/1 03680000-4153586941/829440000*6^(1/2), a[10,9] = 518179039/25920000, b [1] = 173734691637390647/4182794002754640000, b[4] = 0, b[5] = 0, b[2] = 0, b[3] = 0, b[6] = 0, b[7] = 0, c[12] = 319999786/2170712113, a[8, 2] = 0, a[10,2] = 0, a[11,2] = 0, c[15] = 1, c[14] = 39/40, c[13] = 15 /16, a[7,2] = 0, a[9,2] = 0, a[9,3] = 0, a[5,2] = 0, a[6,2] = 0, b[9] \+ = 14586697891849999254003/29700462390576849520000, a[9,1] = 95/1344, a [9,8] = -15/448, b[10] = 102209317997264953344/225042304099487188475\} :" }{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 -1 109 "It remains to determine the linking coefficien ts in stages 12 to 15. We have the following zero coefficients." }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "a[12,2]=0" "6#/&%\"a G6$\"#7\"\"#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[12,3]=0" "6#/&% \"aG6$\"#7\"\"$\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[12,4]=0" "6#/ &%\"aG6$\"#7\"\"%\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[12,5]=0" " 6#/&%\"aG6$\"#7\"\"&\"\"!" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "a[13,2]=0" "6#/&%\"aG6$\"#8\"\"#\"\"!" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "a[13,3]=0" "6#/&%\"aG6$\"#8\"\"$\"\"! " }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[13,4]=0" "6#/&%\"aG6$\"#8\"\"%\" \"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[13,5]=0" "6#/&%\"aG6$\"#8\"\"& \"\"!" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[14,2]=0" "6#/&%\"aG6$\"#9\"\"#\"\"!" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "a[14,3]=0" "6#/&%\"aG6$\"#9\"\"$\"\"!" }{TEXT -1 2 ", \+ " }{XPPEDIT 18 0 "a[14,4]=0" "6#/&%\"aG6$\"#9\"\"%\"\"!" }{TEXT -1 2 " , " }{XPPEDIT 18 0 "a[14,5]=0" "6#/&%\"aG6$\"#9\"\"&\"\"!" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[15,2]=0" " 6#/&%\"aG6$\"#:\"\"#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[15,3]=0 " "6#/&%\"aG6$\"#:\"\"$\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[15,4] =0" "6#/&%\"aG6$\"#:\"\"%\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[15, 5]=0" "6#/&%\"aG6$\"#:\"\"&\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 154 "We make use of the stage -order conditions for stages 12 to 15 so that all these stages all hav e stage-order 5 and incorporate the simplifying conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j+1 .. \+ 15) = b[j]*(1-c[j]);" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+%\"j GF,/F+;,&F0F,F,F,\"#:*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "for " }{XPPEDIT 18 0 "j=1" "6#/%\"jG \"\"\"" }{TEXT -1 10 " ( with " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6# \"\"\"\"\"!" }{TEXT -1 5 " ), " }{XPPEDIT 18 0 "j=7" "6#/%\"jG\"\"(" }{TEXT -1 8 " . . 13." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 185 "Then it turns out that the following collection of \" simple\" order conditions (given in abreviated form) is sufficient to \+ determine the remaining linking coefficients in stages 11 to 15." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 194 "SO9 := SimpleOrderConditions(9):\n[seq([i,SO9[i]],i=[102,106,125, 212,223,239,245,251,253])]:\nlinalg[augment](linalg[delcols](%,2..2),m atrix([[` `]$(linalg[rowdim](%))]),linalg[delcols](%,1..1));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7+7%\"$-\"%#~~G/*(%\"bG\" \"\"%\"cGF--%!G6#*&%\"aGF--F06#*&F3F--F06#*&)F.\"\"$F-F3F-F-F-F-#F-\"$ g*7%\"$1\"F)/*&F,F--F06#*&F3F--F06#*&F3F--F06#*&)F.\"\"%F-F3F-F-F-F-#F -\"%!o\"7%\"$D\"F)/*(F,F-F.F--F06#*&)F.\"\"&F-F3F-F-#F-\"#[7%\"$7#F)/* (F,F-)F.\"\"#F-F/F-#F-\"%!3\"7%\"$B#F)/*(F,F-F.F-FBF-#F-\"%!*=7%\"$R#F )/*(F,F-FhnF-FEF-#F-\"$q#7%\"$X#F)/*(F,F-F.F--F06#*&F3F-FSF-F-#F-\"$y$ 7%\"$^#F)/*(F,F-FhnF-FSF-#F-\"#a7%\"$`#F)/*(F,F-F.F--F06#*&)F.\"\"'F-F 3F-F-#F-\"#jQ(pprint16\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "The associated trees" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 456 "AST9 := AllSimpleTrees(9):\nwhch : = [102,106,125,212,223,239,245,251,253]:\nm := 3: ## number of trees p er row\nnn := nops(whch): q := iquo(nn,m,'r'):\nfor i to nn do \n p|| i := DrawTree(AST9[whch[i]],height=4,width=2,show_ordercondition=true, \n color=COLOR(RGB,.5,0,.9),font_color=black);\nend do:\npp := plot ([[1,1]],style=line,axes=none):\nplots[display](convert([seq([p||((k-1 )*m+1..m*k)],k=1..q),\n `if`(r>0,[p||(m*q+1..nn),pp$(m*(q+1)-nn)],NUL L)],array));" }}{PARA 13 "" 1 "" {GLPLOT2D 782 948 948 {PLOTDATA 2 "6f u-%%TEXTG6&7$$\"#g\"\"!$!+E!G3\"R!\")Q5b~c~~(a~(a~c~))~=~~~6\"-%'COLOU RG6&%$RGBGF)F)F)-%%FONTG6$%(COURIERG\"#5-F$6&7$F'$!+uu)G%QF,Q7~~~~~~~~ ~~~~~~~~~~~~~1F.F/F3-F$6&7$F'$!+(*)=T'QF,Q9~~~~~~~~~~~~~~~~~~~~~___F.F /F3-F$6&7$F'$!+,+++SF,Q9~~~~~~~~~~~~~~~~~~~~~270F.F/F3-F$6&7$F'$!+0\\R MQF,Q7~~~~2~~~~~~~~4~~~~~~~~F.F/F3-%'CURVESG6&7+7$F'$!+,Q^)p$F,7$$\"++ +++bF,$!+^`)QF$F,7$F'FZ7$$\"+++++lF,FZ7$Fhn$!+,pD\\GF,7$$\"+LLLLjF,$!+ ]%GYU#F,7$$\"+WWWWkF,F`o7$$\"+cbbblF,F`o7$$\"+nmmmmF,F`o-%'SYMBOLG6#%' CIRCLEG-%&COLORG6&F2$F)F)FbpFbp-%&STYLEG6#%&POINTG-FQ6&FS-F\\p6#%(DIAM ONDGF_pFcp-FQ6&FS-F\\p6#%&CROSSGF_pFcp-FQ6%7$FTFW-F`p6&F2$\"\"&!\"\"Fb p$\"\"*Fhq-%*THICKNESSG6#\"\"#-FQ6%7$FTFfnFdqF[r-FQ6%7$FTFgnFdqF[r-FQ6 %7$FgnFjnFdqF[r-FQ6%7$FjnF]oFdqF[r-FQ6%7$FjnFboFdqF[r-FQ6%7$FjnFeoFdqF [r-FQ6%7$FjnFhoFdqF[r-FQ6%7#7$F'$!+++++?F,-F06&F2$F7FhqFbpFbp-Fdp6#%%L INEG-FQ6%7#7$$\"+++++]F,FUFjsF]t-FQ6%7#7$$\"+++++qF,FUFjsF]t-F$6&7$$\" #IF)$!+d-G3\"*!\"*Q7b~(a~(a~(a~c~)))~=~~~~F.F/F3-F$6&7$F_u$!+PZ()G%)Fc uQ9~~~~~~~~~~~~~~~~~~~~~~~1F.F/F3-F$6&7$F_u$!+i*)=T')FcuQ<~~~~~~~~~~~~ ~~~~~~~~~~~____F.F/F3-F$6&7$F_u$!+++++5F,Q<~~~~~~~~~~~~~~~~~~~~~~~1680 F.F/F3-F$6&7$F_u$!+Z!\\RM)FcuQ9~~~~~~~~~~~~~4~~~~~~~~~~F.F/F3-FQ6&7*7$ F_u$!+1!Q^)pFcu7$F_u$!+/N&)QFFcu7$F_u$\"+)*4V2:Fcu7$F_u$\"+/br`dFcu7$$ \"++++DEF,$\"+++++5F,7$$\"++++vGF,F_x7$$\"++++DJF,F_x7$$\"++++vLF,F_xF [pF_pFcp-FQ6&F_wFipF_pFcp-FQ6&F_wF^qF_pFcp-FQ6%7$F`wFcwFdqF[r-FQ6%7$Fc wFfwFdqF[r-FQ6%7$FfwFiwFdqF[r-FQ6%7$FiwF\\xFdqF[r-FQ6%7$FiwFaxFdqF[r-F Q6%7$FiwFdxFdqF[r-FQ6%7$FiwFgxFdqF[r-FQ6%7#7$F_uF_xFjsF]t-FQ6%7#7$$\"+ ++++?F,FawFjsF]t-FQ6%7#7$$\"+++++SF,FawFjsF]t-FQ6#-%'LEGENDG6#QB__neve r_display_this_legend_entryF.-F$6&7$FbpF*Q:b~c~~(a~(a~(a~c~)))~=~~~~F. 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66" "Curve 67" "Curve 68" "Curve 69" "Curve 70" "Curve 71 " "Curve 72" "Curve 73" "Curve 74" "Curve 75" "Curve 76" "Curve 77" "C urve 78" "Curve 79" "Curve 80" "Curve 81" "Curve 82" "Curve 83" "Curve 84" "Curve 85" "Curve 86" "Curve 87" "Curve 88" "Curve 89" "Curve 90 " "Curve 91" "Curve 92" "Curve 93" "Curve 94" "Curve 95" "Curve 96" "C urve 97" "Curve 98" "Curve 99" "Curve 100" "Curve 101" "Curve 102" "Cu rve 103" "Curve 104" "Curve 105" "Curve 106" "Curve 107" "Curve 108" " Curve 109" "Curve 110" "Curve 111" "Curve 112" "Curve 113" "Curve 114 " "Curve 115" "Curve 116" "Curve 117" "Curve 118" "Curve 119" "Curve 1 20" "Curve 121" "Curve 122" "Curve 123" "Curve 124" "Curve 125" "Curve 126" "Curve 127" "Curve 128" "Curve 129" "Curve 130" "Curve 131" "Cur ve 132" "Curve 133" "Curve 134" "Curve 135" "Curve 136" "Curve 137" "C urve 138" "Curve 139" "Curve 140" "Curve 141" "Curve 142" "Curve 143" "Curve 144" "Curve 145" "Curve 146" "Curve 147" "Curve 148" "Curve 149 " "Curve 150" "Curve 151" "Curve 152" "Curve 153" "Curve 154" "Curve 1 55" "Curve 156" "Curve 157" "Curve 158" "Curve 159" "Curve 160" "Curve 161" "Curve 162" "Curve 163" "Curve 164" "Curve 165" "Curve 166" "Cur ve 167" "Curve 168" "Curve 169" }}}}{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 49 "SO9_15 := SimpleOr derConditions(9,15,'expanded'):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 485 "SOeqs2 := [seq(add(a[i,j],j= 1..i-1)=c[i],i=12..15),op(StageO rderConditions(2,12..15,'expanded')),\n op(StageOrderCondition s(3,12..15,'expanded')),op(StageOrderConditions(4,12..15,'expanded')), \n op(StageOrderConditions(5,12..15,'expanded'))]:\nord_ cdns := [seq(SO9_15[i],i=[102,106,125,212,223,239,245,251,253])]:\nsim p_eqs := [add(b[i]*a[i,1],i=1+1..15)=b[1],seq(add(b[i]*a[i,j],i=j+1..1 5)=b[j]*(1-c[j]),j=[$7..13])]:\ncdns := [op(SOeqs2),op(ord_cdns),op(si mp_eqs)]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 127 "It is po ssible to manage with fewer equations, but the computation may be less efficient if the number of equations is reduced." }}{PARA 0 "" 0 "" {TEXT -1 50 "For example, the simplifying conditions given by " } {XPPEDIT 18 0 "j=8" "6#/%\"jG\"\")" }{TEXT -1 3 ", " }{XPPEDIT 18 0 " j=10" "6#/%\"jG\"#5" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "j=12" "6#/% \"jG\"#7" }{TEXT -1 17 " may be omitted." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "e10 :=\{seq(seq(a[i,j ]=0,i=12..15),j=2..5)\}:\ne11 := `union`(e9,e10):\neqns2 := subs(e11,c dns):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "There are 37 equations and 34 unknowns." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "nops(eqns2);\nindets(eqns2);\nnops(%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#\"#P" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#F5&F%6$F>F8&F%6$F.F+&F%6$F>F;&F%6$F2 F5&F%6$F2F?&F%6$F>F/&F%6$F>F2&F%6$F2F8&F%6$F2F;&F%6$F.F'&F%6$F'F?&F%6$ F2F/&F%6$F>F'&F%6$F>F.&F%6$F.F?&F%6$F'F;&F%6$F'F/&F%6$F'F5&F%6$F'F8&F% 6$F>F(&F%6$F>F+&F%6$F'F2&F%6$F.F(&F%6$F2F(&F%6$F2F+" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#\"#M" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "e12 := solve(\{op(eqns2)\}):\ninfol evel[solve] := 0:\ne13 := `union`(e11,e12):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "e13 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8075 "e13 := \{a[13,1] = 4507784 6760256141387004276823/110315894143992133591739924480+1493491403898138 129099/13100021190238236835840*6^(1/2), a[11,5] = 0, b[8] = -722631631 41715044860361/169939769455665013040000, b[15] = -369769046476619/6555 2230622340, c[2] = 1/49, c[5] = 3/7, c[8] = 40/63, c[10] = 19/18, c[11 ] = 7/9, a[11,6] = 19/25, a[8,5] = 0, a[8,4] = 0, b[14] = 121254571224 2913280000000/130535954653501897388343, a[10,1] = 52918819/138240000, \+ a[4,2] = 0, a[2,1] = 1/49, a[8,1] = 40/567, b[13] = -10678264099993989 152768/2396652442219114419375, b[12] = 1710999041788984993956022337692 5306674323804078983341334325755071278367152457480027/71381427125808828 146076534703173195056090466069876919551824450642795947233961810560, c[ 6] = 8/21+4/63*6^(1/2), c[7] = 8/21-4/63*6^(1/2), c[9] = 10/21, c[4] = 32/235-8/4935*6^(1/2), b[11] = 1883570537693211021/187227575505495910 0, a[12,3] = 0, c[3] = 64/705-16/14805*6^(1/2), a[10,3] = 0, a[11,3] = 0, a[6,3] = 0, a[7,3] = 0, a[8,3] = 0, a[10,5] = 0, a[12,5] = 0, a[9, 5] = 0, a[10,4] = 0, a[10,8] = 44599023/5120000, a[11,4] = 0, a[12,4] \+ = 0, a[9,4] = 0, a[4,3] = 24/235-2/1645*6^(1/2), a[7,1] = 4336/127575+ 479/127575*6^(1/2), a[5,4] = 65097/22400+477/5600*6^(1/2), a[9,7] = 29 5/1344+115/2688*6^(1/2), a[6,5] = 1940224/15069159+779264/15069159*6^( 1/2), a[7,4] = 90731944/400648275-170142739/8413613775*6^(1/2), a[10,7 ] = -1453047743/103680000+4153586941/829440000*6^(1/2), a[11,7] = 1180 508473123/443296800000-136404911099/147765600000*6^(1/2), a[9,6] = 295 /1344-115/2688*6^(1/2), a[3,1] = -165952/1491075+38896/10437525*6^(1/2 ), a[11,1] = 258780283/8618400000+585428803/51710400000*6^(1/2), a[11, 10] = 13167297224/792049782825-9366860848/2376149348475*6^(1/2), a[7,5 ] = 8245504/62429373-22187008/437005611*6^(1/2), a[3,2] = 301312/14910 75-7168/1491075*6^(1/2), a[8,7] = 10/567*6^(1/2)+160/567, a[4,1] = 8/2 35-2/4935*6^(1/2), a[5,3] = -149931/44800-81/700*6^(1/2), a[7,6] = -39 36/340025+11464/3060225*6^(1/2), a[6,4] = 364520/1674351+87715/5023053 *6^(1/2), a[5,1] = 38937/44800+171/5600*6^(1/2), a[6,1] = 176/5103-29/ 5103*6^(1/2), a[8,6] = 160/567-10/567*6^(1/2), a[11,9] = -1260561943/5 91300000+585428803/886950000*6^(1/2), a[11,8] = -106856621/190800000+5 85428803/2289600000*6^(1/2), a[10,6] = -1453047743/103680000-415358694 1/829440000*6^(1/2), a[10,9] = 518179039/25920000, a[14,5] = 0, a[15,5 ] = 0, b[1] = 173734691637390647/4182794002754640000, b[4] = 0, b[5] = 0, b[2] = 0, b[3] = 0, a[15,1] = 1564746779443331677794753119798867/2 864199537049451295689544004000000+5933645037523445166666379/5232675730 4999307928000000*6^(1/2), b[6] = 0, b[7] = 0, a[13,2] = 0, a[15,4] = 0 , a[13,4] = 0, a[14,4] = 0, c[12] = 319999786/2170712113, a[13,9] = -4 93074073683718697930133408597/27602712116408194083051274240+1493491403 898138129099/224694912332563742720*6^(1/2), a[8,2] = 0, a[10,2] = 0, a [11,2] = 0, a[12,2] = 0, c[15] = 1, c[14] = 39/40, c[13] = 15/16, a[7, 2] = 0, a[9,2] = 0, a[9,3] = 0, a[5,2] = 0, a[6,2] = 0, a[15,2] = 0, a [14,2] = 0, a[15,3] = 0, a[14,3] = 0, a[13,3] = 0, b[9] = 145866978918 49999254003/29700462390576849520000, a[9,1] = 95/1344, a[13,5] = 0, a[ 13,6] = 28530732123103900185/9849639992660328448-15062887306567756845/ 5628365710091616256*6^(1/2), a[14,7] = 397408075485926915758262202639/ 18359970961922048000000000000-6493922587539771225254133441201/10281583 73867634688000000000000*6^(1/2), a[13,7] = 530875502237315716994493/24 064781139210466754560-8920823473649531766699837/1347627743795786138255 360*6^(1/2), a[12,7] = 54237115726095689129801119753077769917434333557 0401188235055068891/71570321592203079939677872792265704339576185318351 0539934169000000-25571039199675674205714455677983806004177793498080540 92568810687111/1001984502290843119155490219091719860754066594456914755 9078366000000*6^(1/2), a[12,6] = 1035482118210023049302666700037918495 5505622050895245676387169/14646754893887488951256983176315156567081809 5959204028052600000-32684637880879071688859023194047542236847633606098 54606186699/4184787112539282557501995193232901876309088455977257944360 0000*6^(1/2), a[14,6] = 8283471074731862302286097/75146789494784000000 00000-15668946773152185221466849/4294102256844800000000000*6^(1/2), a[ 15,6] = -57628625604267458078211/39343426545112261600000-6047516944575 480929793/1405122376611152200000*6^(1/2), a[14,8] = 243527849390304790 9370803905780425755549361787/69519418749609616606570949777735680000000 00000+10993151135374962117764000463/3982779843223552000000000000*6^(1/ 2), a[15,10] = -854444885403376103999958576/21640244757394101989947629 875*6^(1/2)+247970161301643841123535377085814726576/117201570827760940 0708896218214309893875, a[13,11] = -1259978731825102407292471875/94764 2075600343143202947072, a[12,8] = -89222557800951915467623899590157884 1509244882985862325637048827/38813900468801845720831005417235164902766 795429189067433939000000+203673881988860870493193973923424476338897710 33202442752063/2206137485722185665289139540357582581323139973012292392 34375*6^(1/2), a[14,13] = -235412270220829707518634576/100049213779824 63725322265625, a[15,9] = 5933645037523445166666379/897521918060373467 750000*6^(1/2)-253787748130868165717697094592697839427/218398243200420 45467560266343639250000, a[15,8] = 53402805337711006499997411/20852016 068909498648000000*6^(1/2)+4336304776662958741869045159571925984937/22 71064142891395647108842129832843500000, a[9,8] = -15/448, a[15,13] = 2 5469705993361596208461643776/2398429987672964863327432743875, a[12,1] \+ = 307213395328582867964430765847473084972824867512957518186088963/5126 364212860621132939944111710304798478633358572140981841000000+119107533 326819222510639750832411974467191643469020133053/291376649057647163340 07503363213354847664112851105748578125*6^(1/2), a[15,12] = -4438603995 3127381302291127198665043342539469696141788200181019629996219153064322 715843/926096283618263046743578868285140736430174204759597450521208350 7724183850275361816608, a[14,9] = 1221461237263884679751555607/1714286 13534976000000000000*6^(1/2)-55812323906910341634712692997508691214893 8889/34184771406932232290260692794368000000000000, b[10] = 10220931799 7264953344/225042304099487188475, a[12,9] = -5506886052357703380348636 42917595195825050633798073448796494914/8353227400420208542513748217742 23772966384453517335472487484375+7241738026270608728646896850610648047 6052519229164240896224/30386421973154632748322110650208212912564003401 8674235171875*6^(1/2), a[15,14] = -10513898964163619809241937920000000 /255387066048362821838491552709590753, a[15,11] = -5296663573769760545 5117615341/39091452325689672618552125990, a[13,10] = -3360355658770810 79047275/8465066424794973677551616*6^(1/2)+200609996314078300148532240 828075/1019933691979646265167106381709312, a[14,11] = -163845778835264 660255510638493965671483/114196173990354810149157741209600000000, a[14 ,1] = 1221461237263884679751555607/9994523002806272000000000000*6^(1/2 )+36716621212098036093935018687105425505961/72248275402215258274603114 496000000000000, a[14,12] = -37972709869158045130412933766281771978445 1899678250630685021894233169654109523405323/74980943726337976053062781 324716052947047103330285761853079535691020475760640000000, a[12,10] = \+ 123396895115495738434549229715587040998178289648593721222723693824/169 60466422214315122555088502288155618001490986047888958907461648625-8781 560217119727637264447549372070053517105489683916369731584/616968585748 0653009296139869875647732994358307038155314262445125*6^(1/2), a[14,10] = -10993151135374962117764000463/258333325951995046312000000000*6^(1/ 2)+306734586161727173704146823378382330889519/140626680062621422584691 4661737584000000000, a[15,7] = -6650904765188943408799448943/134574190 4975564908028000000*6^(1/2)+1691425739887134646566682083/9612442178396 8922002000000, a[12,11] = -2418901294262986471384856103775514061656722 25318019246672/3661688723471872237814245794078789141770452398980100701 315, a[13,8] = -155850251753928802974915857362119/17401508660534001647 7040216637440+13441422635083243161891/5220309196109974077440*6^(1/2), \+ a[13,12] = -1939162142353174689879923915990531880493671334862071208893 11375/42748092349455088111344007455417233641020280098816132254793728\} :" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "printed coefficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 283 "subs(e13,matrix([seq([c[i],seq(a[i ,j],j=1..i-1),``$(6-i)],i=2..6)]));\nfor ii from 7 to 15 do\n print( ``);\n print(c[ii]=subs(e13,c[ii])); \n for jj to ii-1 do\n p rint(a[ii,jj]=subs(e13,a[ii,jj]));\n end do:\nend do:\n``;\nfor ii t o 15 do\n print(b[ii]=subs(e13,b[ii]));\nend do:" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#K%'matrixG6#7'7(#\"\"\"\"#\\F(%!GF+F+F+7(,&#\"#k\"$0( F)*(\"#;F)\"&0[\"!\"\"\"\"'#F)\"\"#F4,&#\"'_f;\"(v5\\\"F4*(\"&'*)QF)\" )DvV5F4F5F6F),&#\"'78IF;F)*(\"%orF)F;F4F5F6F4F+F+F+7(,&#\"#K\"$N#F)*( \"\")F)\"%N\\F4F5F6F4,&#FJFHF)*(F7F)FKF4F5F6F4\"\"!,&#\"#CFHF)*(F7F)\" %X;F4F5F6F4F+F+7(#\"\"$\"\"(,&#\"&P*Q\"&+[%F)*(\"$r\"F)\"%+cF4F5F6F)FO ,&#\"'J*\\\"FfnF4*(\"#\")F)\"$+(F4F5F6F4,&#\"&(4l\"&+C#F)*(\"$x%F)FinF 4F5F6F)F+7(,&#FJ\"#@F)*(\"\"%F)\"#jF4F5F6F),&#\"$w\"\"%.^F)*(\"#HF)F`p F4F5F6F4FOFO,&#\"'?XO\"(^Vn\"F)*(\"&:x)F)\"(`I-&F4F5F6F),&#\"(C-%>\")f \"p]\"F)*(\"'k#z(F)F]qF4F5F6F)Q(pprint66\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"(,& #\"\")\"#@\"\"\"*(\"\"%F,\"#j!\"\"\"\"'#F,\"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"\",&#\"%OV\"'vv7F(*(\"$z%F(F,!\"\"\" \"'#F(\"\"#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"#\" \"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"$\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"%,&#\")W>t!*\"*v#[1S \"\"\"*(\"*RF9q\"F-\"+vPh8%)!\"\"\"\"'#F-\"\"#F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"&,&#\"(/bC)\")t$HC'\"\"\"*(\")3q=AF- \"*6c+P%!\"\"\"\"'#F-\"\"#F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"a G6$\"\"(\"\"',&#\"%OR\"'D+M!\"\"*(\"&k9\"\"\"\"\"(D-1$F-F(#F0\"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\")#\"#S\"#j" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6 $\"\")\"\"\"#\"#S\"$n&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\" \")\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"$\" \"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"%\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"&\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"',&#\"$g\"\"$n&\"\"\"*(\"#5F-F ,!\"\"F(#F-\"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\" \"(,&*(\"#5\"\"\"\"$n&!\"\"\"\"'#F,\"\"#F,#\"$g\"F-F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\" \"*#\"#5\"#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"\"# \"#&*\"%W8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"#\"\"! " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"%\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"',&#\"$&H\"%W8\"\"\"*(\"$:\"F-\"%)o#! \"\"F(#F-\"\"#F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\" (,&#\"$&H\"%W8\"\"\"*(\"$:\"F-\"%)o#!\"\"\"\"'#F-\"\"#F-" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\")#!#:\"$[%" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#5# \"#>\"#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"\"#\")>)= H&\"*++CQ\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"#\"\"! " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"',&#\"+Vx/`9\"*++o.\"!\"\"*(\"+Tpe`T\"\"\"\"*++ WH)F-F(#F0\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\" (,&#\"+Vx/`9\"*++o.\"!\"\"*(\"+Tpe`T\"\"\"\"*++WH)F-\"\"'#F0\"\"#F0" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\")#\")B!*fW\"(++7&" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"*#\"*R!z\"=&\")++#f# " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#6#\"\"(\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"#6\"\"\",&#\"*$G!ye#\"+++S=')F(*(\"*.)GaeF(\",++S5<&!\"\"\"\"' #F(\"\"#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"#\"\"! 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H\"*o&4Ed6PU&\"]o+++pT$*R0^$=`=w&RVqlAzsynR*zI?#f@.dr\"\"\"*(\"^o6ro5) oD4a!3)\\$zxT+1Q)znbWr0Unv'*>R5dDF-\"_o+++m$y!fv9pX%fmSvg)><4>-\\b\">J %3H-X)>+\"!\"\"\"\"'#F-\"\"#F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"#7\"\"),&#\"jnF)[qjDB'e)H)[C4:%)y:!f**Qina\">&4!ybA#*)\"\\o+++ RRVn!*=HazmF!\\;NsT05$3sX=!)o/!R\")Q!\"\"*(\"fnj?vUC?L5x*)QjZCM#R(R>$ \\q3')))>)Qn.#\"\"\"\"gnvVBR#H7I(*RJK\"e#ed.aR\"*Glc=Ad[Ph?#F-\"\"'#F0 \"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"*,&#\"jn9 \\\\'z[M2)zL10De>&fD0w/[1h]o*okG(31FE!QU'QIF-\"\"'#F0\"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%\"aG6$\"#7\"#5,&#\"]oCQpBFA@Pf['*Gy\")*4/(e:(H#\\XVQd\\:^*oRB\"\" _oD'[;Y2*e*))y/')4\\,!=c:)G-&)3bD7:V@AkYgp\"\"\"\"*(\"hn%eJ(pj\"Ro*[0r ^`+2s$\\vWksjF(>r@g:y)F-\"[oD^WiUJb\"QqIeV*HtZc()p)RhH4Il![deoph!\"\" \"\"'#F-\"\"#F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"#6#! ZsmC>!=`Asc;19bx.h&[Qrk)HE%H,*=C\"en:8q+,)*)R_/xT\"*yySzXU\"yBs=ZB()oh O" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#8#\"#:\"#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"a G6$\"#8\"\"\",&#\">BoF/qQThDgn%y2X\"?![C*R@+58!\"\"\"\"'#F(\"\"#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"#8\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\" \"',&#\"5&=+R5B@tI&G\"4[%G.m#**R'\\)*\"\"\"*(\"5XovnlI()G1:F-\"4cih\"4 5dOGc!\"\"F(#F-\"\"#F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"# 8\"\"(,&#\"9$\\%*pr:tB-b(3`\"8gXvm/@R6ykS#\"\"\"*(\":P)*pm<`\\OZB3#*)F -\":g`DQhy&zVxiZ8!\"\"\"\"'#F-\"\"#F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"),&#\"B>@Ode\"\\(H!)GRv^-&e:\"BSuj;-/xk,S`g'3:S4.A&F-\"\"'#F0\"\"#F0" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"*,&#\"?(f3M8Izp=PotS2 $\\\">SUF^I3%>3k67FgF!\"\"*(\"7*4H\"Q\")*QS\"\\$\\\"\"\"\"\"6?FujDL7\\ pC#F-\"\"'#F0\"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8 \"#5,&*(\"9vs/z53xecNgL\"\"\"\":;;bxO(\\zCk1l%)!\"\"\"\"'#F,\"\"#F.#\" Bv!G3CK&[,IySJ'**41?\"C7$4pL*>5F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"#6#!=v=Z#H2C5D=ty*f7\"\"inGPzaA8;))4!G?5kLsTbu+W86)3b%\\B4[F%" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"cG6#\"#9#\"#R\"#S" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\" #9\"\"\",&*(\"=2cb^(zY)QEP7Y@7F(\"=++++++si!G+BX***!\"\"\"\"'#F(\"\"#F (#\"Jhf]Da5(o=]$R4O!)477i;n$\"J++++++'\\9Jgu#e_@-aF[A(F(" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#9\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#9\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#9\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"#9\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#9\" \"',&#\":(4'G-B'=tu5Z$G)\":+++++Sy%\\*yY^(\"\"\"*(\";\\oY@_=_JxY*oc\"F -\":+++++![%oD-TH%!\"\"F(#F-\"\"#F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"aG6$\"#9\"\"(,&#\"?RE?i#ed\"p#f[v!3uR\">++++++[?#>'4(*f$=\"\"\"* (\"@,7WLTDD7xRveAR\\'F-\"@++++++)oMw'QPe\"G5!\"\"\"\"'#F-\"\"#F1" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#9\"\"),&#\"O(yh$\\bvD/y0R! 3P4z/.R\\y_V#\"O++++++oNxx\\4d1mh4'\\(=%>&p\"\"\"*(\">j/+kx6i\\PN6:$*4 \"F-\"=++++++_NAV)zF)R!\"\"\"\"'#F-\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#9\"\"*,&*(\"=2cb^(zY)QEP7Y@7\"\"\"\"<++++++ w\\`8'G9j/+kx6i\\PN6:$*4\"\"\"\"\"?++++?JY]*>&fKL$e#!\"\"\"\"'#F,\" \"#F.#\"K>&*)3L#QyL#o9/P9\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"#9\"#7#!_pB`SB&4TlpJB%*=-&oI1Dy'**=X%y>x\"GmP$HTI^/e\"p)4F(z$ \"^p+++S1wv/-\"pN&zI&=w&GIL5Zq%H0;ZK\"yiI0wzLEP%4)\\(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#9\"#8#!DcEA`sjC)zP@ \\+5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#:\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%\"aG6$\"#:\"\"\",&#\"Cn))z>Jv%zx;LV%znuk:\"C+++/Sa*o&H^%\\q`*>kGF(* (\":zjmm^WBv.XO$fF(\";+++GzI**\\IdnK_!\"\"\"\"'#F(\"\"#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#:\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#:\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#:\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"#:\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#:\" \"',&#\"86#y!euE/ciGw&\"8++ghA6XlUV$R!\"\"*(\"7$zH4[vX%p^Zg\"\"\"\"7++ ?_6hwB709F-F(#F0\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\" #:\"\"(,&*(\"=V*[%*z3M%*)=lZ!4l'\"\"\"\"=+++G!3\\cv\\!>uX8!\"\"\"\"'#F ,\"\"#F.#\"=$3#omlkMr))RdU\"p\"\";+++-?#*oRy@W7'*F," }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"#:\"\"),&*(\";6u***\\15rP`!GS`\"\"\"\";+++ [')\\4*og,_3#!\"\"\"\"'#F,\"\"#F,#\"IP\\)f#>df^/p=ueHmwZIOV\"I++]VG$)H @%)3rk&R\"*G9k5F#F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#:\" \"*,&*(\":zjmm^WBv.XO$f\"\"\"\"9++vnMPg!=>_(*)!\"\"\"\"'#F,\"\"#F,#\"H F%Ryp#f%4(pv)Hw% **)>5%RdZCS;#!\"\"\"\"'#F,\"\"#F.#\"Hwls9e3x``B6%Qk,8;qzC\"IvQ*)4V@=i* )32S4wF3d,s6F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#:\"#6#!> T`h<^X0wpPdjmH&\">!*f7_&=En*oDBX\"4R" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%\"aG6$\"#:\"#7#!apVerAV1`\">i**H'>5=+#)yThpp%RDMV]m)>F6H-8QFJ&*Rg QW\"`p3m\"=Ov-&Q=Cx]$37_]uffZ?u,VO29&Go)yNuYIE=OG'4E*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#:\"#8#\">wPkh%3ifhL*fqpa#\"@vQuKuKj['H n()*H%)R#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#:\"#9#!D+++?z $>C4)>O;k*)*Q^5\"E`2f4Fb\"\\Q=#GO[g1(Qb#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"\"#\"3Z1RP;p MP<\"4++kaF+%z#=%" }}{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#\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%\"bG6#\"\"'\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"( \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\")#!8h.'[/:<9jJEs \"9++/8]mb%p(R*p\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"*# \"8.SD***\\=*yp'e9\"8++_\\od!Ri/qH" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%\"bG6#\"#5#\"6WL&\\E(*zJ4A5\"6v%)=([*4/B/D#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#6#\"4@5@$pP0d$)=\"4+\"f\\0bdFs=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#7#\"^pF+[dC:n$y72bdKM8M$)*yS!QKu mIDpPB-cR*\\)*)yT!**4r\"\"^pg0\"='RBZfzU1XC=b>p()pgY!4c]>tJqMl2Y\"G)3e 7F9Qr" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#8#!8oF:*)R***4k#y 1\"\"7v$>W6>AW_mR#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#9#\" :+++!G8HC7da77\"9V$)Q(*=]`Y&f`I\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ &%\"bG6#\"#:#!0>mZY!p(p$\"/SBiIAbl" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "RK9_15eqs := [op(RowSumConditions(15,'expanded')),op(OrderConditions(9,15,'expande d'))]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Check:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "expand(subs(e13,R K9_15eqs)):\nmap(u_->`if`(lhs(u_)=rhs(u_),0,1),%);\nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7`jl\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$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 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 34 "Appendix: related order conditions" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "#-------- -----------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 145 "u nrelated 129, 130, 134, 135, 136, 139, 140, 141, 143, 151, 152, 156, \+ 157, 166, 168, 173, 175, 180, 185, 186, 187, 212, 221, 223\n\nrelated \+ groups" }}{PARA 0 "" 0 "" {TEXT -1 19 "132, 137, 148, 153," }}{PARA 0 "" 0 "" {TEXT -1 19 "133, 138, 150, 163," }}{PARA 0 "" 0 "" {TEXT -1 59 "131, 147, 149, 192,\n142, 164, 170, 222,\n162, 184, 188, 227," }} {PARA 0 "" 0 "" {TEXT -1 18 "181, 197, 202, 239" }}{PARA 0 "" 0 "" {TEXT -1 38 "144, 158, 167, 177, 193, 207, 213, 244" }}{PARA 0 "" 0 " " {TEXT -1 38 "145, 159, 169, 178, 194, 209, 215, 245" }}{PARA 0 "" 0 "" {TEXT -1 38 "154, 165, 172, 189, 206, 208, 224, 248" }}{PARA 0 "" 0 "" {TEXT -1 38 "183, 200, 204, 205, 228, 231, 235, 251" }}{PARA 0 " " 0 "" {TEXT -1 78 "146, 160, 161, 174, 182, 196, 198, 201, 203, 211, \+ 218, 230, 232, 238, 241, 253" }}{PARA 0 "" 0 "" {TEXT -1 78 "155, 171, 176, 179, 190, 195, 199, 210, 214, 216, 225, 229, 237, 240, 246, 255 " }}{PARA 0 "" 0 "" {TEXT -1 79 "191, 217, 219, 220, 226, 233, 234, 23 6, 242, 243, 247, 249, 250, 252, 254, 256 " }}{PARA 0 "" 0 "" {TEXT -1 43 "#------------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 44 "#----------------------- --------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 43 "construction of the order 8 embedded scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "e13 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8075 "e13 := \{a[13,1] = 4507784 6760256141387004276823/110315894143992133591739924480+1493491403898138 129099/13100021190238236835840*6^(1/2), a[11,5] = 0, b[8] = -722631631 41715044860361/169939769455665013040000, b[15] = -369769046476619/6555 2230622340, c[2] = 1/49, c[5] = 3/7, c[8] = 40/63, c[10] = 19/18, c[11 ] = 7/9, a[11,6] = 19/25, a[8,5] = 0, a[8,4] = 0, b[14] = 121254571224 2913280000000/130535954653501897388343, a[10,1] = 52918819/138240000, \+ a[4,2] = 0, a[2,1] = 1/49, a[8,1] = 40/567, b[13] = -10678264099993989 152768/2396652442219114419375, b[12] = 1710999041788984993956022337692 5306674323804078983341334325755071278367152457480027/71381427125808828 146076534703173195056090466069876919551824450642795947233961810560, c[ 6] = 8/21+4/63*6^(1/2), c[7] = 8/21-4/63*6^(1/2), c[9] = 10/21, c[4] = 32/235-8/4935*6^(1/2), b[11] = 1883570537693211021/187227575505495910 0, a[12,3] = 0, c[3] = 64/705-16/14805*6^(1/2), a[10,3] = 0, a[11,3] = 0, a[6,3] = 0, a[7,3] = 0, a[8,3] = 0, a[10,5] = 0, a[12,5] = 0, a[9, 5] = 0, a[10,4] = 0, a[10,8] = 44599023/5120000, a[11,4] = 0, a[12,4] \+ = 0, a[9,4] = 0, a[4,3] = 24/235-2/1645*6^(1/2), a[7,1] = 4336/127575+ 479/127575*6^(1/2), a[5,4] = 65097/22400+477/5600*6^(1/2), a[9,7] = 29 5/1344+115/2688*6^(1/2), a[6,5] = 1940224/15069159+779264/15069159*6^( 1/2), a[7,4] = 90731944/400648275-170142739/8413613775*6^(1/2), a[10,7 ] = -1453047743/103680000+4153586941/829440000*6^(1/2), a[11,7] = 1180 508473123/443296800000-136404911099/147765600000*6^(1/2), a[9,6] = 295 /1344-115/2688*6^(1/2), a[3,1] = -165952/1491075+38896/10437525*6^(1/2 ), a[11,1] = 258780283/8618400000+585428803/51710400000*6^(1/2), a[11, 10] = 13167297224/792049782825-9366860848/2376149348475*6^(1/2), a[7,5 ] = 8245504/62429373-22187008/437005611*6^(1/2), a[3,2] = 301312/14910 75-7168/1491075*6^(1/2), a[8,7] = 10/567*6^(1/2)+160/567, a[4,1] = 8/2 35-2/4935*6^(1/2), a[5,3] = -149931/44800-81/700*6^(1/2), a[7,6] = -39 36/340025+11464/3060225*6^(1/2), a[6,4] = 364520/1674351+87715/5023053 *6^(1/2), a[5,1] = 38937/44800+171/5600*6^(1/2), a[6,1] = 176/5103-29/ 5103*6^(1/2), a[8,6] = 160/567-10/567*6^(1/2), a[11,9] = -1260561943/5 91300000+585428803/886950000*6^(1/2), a[11,8] = -106856621/190800000+5 85428803/2289600000*6^(1/2), a[10,6] = -1453047743/103680000-415358694 1/829440000*6^(1/2), a[10,9] = 518179039/25920000, a[14,5] = 0, a[15,5 ] = 0, b[1] = 173734691637390647/4182794002754640000, b[4] = 0, b[5] = 0, b[2] = 0, b[3] = 0, a[15,1] = 1564746779443331677794753119798867/2 864199537049451295689544004000000+5933645037523445166666379/5232675730 4999307928000000*6^(1/2), b[6] = 0, b[7] = 0, a[13,2] = 0, a[15,4] = 0 , a[13,4] = 0, a[14,4] = 0, c[12] = 319999786/2170712113, a[13,9] = -4 93074073683718697930133408597/27602712116408194083051274240+1493491403 898138129099/224694912332563742720*6^(1/2), a[8,2] = 0, a[10,2] = 0, a [11,2] = 0, a[12,2] = 0, c[15] = 1, c[14] = 39/40, c[13] = 15/16, a[7, 2] = 0, a[9,2] = 0, a[9,3] = 0, a[5,2] = 0, a[6,2] = 0, a[15,2] = 0, a [14,2] = 0, a[15,3] = 0, a[14,3] = 0, a[13,3] = 0, b[9] = 145866978918 49999254003/29700462390576849520000, a[9,1] = 95/1344, a[13,5] = 0, a[ 13,6] = 28530732123103900185/9849639992660328448-15062887306567756845/ 5628365710091616256*6^(1/2), a[14,7] = 397408075485926915758262202639/ 18359970961922048000000000000-6493922587539771225254133441201/10281583 73867634688000000000000*6^(1/2), a[13,7] = 530875502237315716994493/24 064781139210466754560-8920823473649531766699837/1347627743795786138255 360*6^(1/2), a[12,7] = 54237115726095689129801119753077769917434333557 0401188235055068891/71570321592203079939677872792265704339576185318351 0539934169000000-25571039199675674205714455677983806004177793498080540 92568810687111/1001984502290843119155490219091719860754066594456914755 9078366000000*6^(1/2), a[12,6] = 1035482118210023049302666700037918495 5505622050895245676387169/14646754893887488951256983176315156567081809 5959204028052600000-32684637880879071688859023194047542236847633606098 54606186699/4184787112539282557501995193232901876309088455977257944360 0000*6^(1/2), a[14,6] = 8283471074731862302286097/75146789494784000000 00000-15668946773152185221466849/4294102256844800000000000*6^(1/2), a[ 15,6] = -57628625604267458078211/39343426545112261600000-6047516944575 480929793/1405122376611152200000*6^(1/2), a[14,8] = 243527849390304790 9370803905780425755549361787/69519418749609616606570949777735680000000 00000+10993151135374962117764000463/3982779843223552000000000000*6^(1/ 2), a[15,10] = -854444885403376103999958576/21640244757394101989947629 875*6^(1/2)+247970161301643841123535377085814726576/117201570827760940 0708896218214309893875, a[13,11] = -1259978731825102407292471875/94764 2075600343143202947072, a[12,8] = -89222557800951915467623899590157884 1509244882985862325637048827/38813900468801845720831005417235164902766 795429189067433939000000+203673881988860870493193973923424476338897710 33202442752063/2206137485722185665289139540357582581323139973012292392 34375*6^(1/2), a[14,13] = -235412270220829707518634576/100049213779824 63725322265625, a[15,9] = 5933645037523445166666379/897521918060373467 750000*6^(1/2)-253787748130868165717697094592697839427/218398243200420 45467560266343639250000, a[15,8] = 53402805337711006499997411/20852016 068909498648000000*6^(1/2)+4336304776662958741869045159571925984937/22 71064142891395647108842129832843500000, a[9,8] = -15/448, a[15,13] = 2 5469705993361596208461643776/2398429987672964863327432743875, a[12,1] \+ = 307213395328582867964430765847473084972824867512957518186088963/5126 364212860621132939944111710304798478633358572140981841000000+119107533 326819222510639750832411974467191643469020133053/291376649057647163340 07503363213354847664112851105748578125*6^(1/2), a[15,12] = -4438603995 3127381302291127198665043342539469696141788200181019629996219153064322 715843/926096283618263046743578868285140736430174204759597450521208350 7724183850275361816608, a[14,9] = 1221461237263884679751555607/1714286 13534976000000000000*6^(1/2)-55812323906910341634712692997508691214893 8889/34184771406932232290260692794368000000000000, b[10] = 10220931799 7264953344/225042304099487188475, a[12,9] = -5506886052357703380348636 42917595195825050633798073448796494914/8353227400420208542513748217742 23772966384453517335472487484375+7241738026270608728646896850610648047 6052519229164240896224/30386421973154632748322110650208212912564003401 8674235171875*6^(1/2), a[15,14] = -10513898964163619809241937920000000 /255387066048362821838491552709590753, a[15,11] = -5296663573769760545 5117615341/39091452325689672618552125990, a[13,10] = -3360355658770810 79047275/8465066424794973677551616*6^(1/2)+200609996314078300148532240 828075/1019933691979646265167106381709312, a[14,11] = -163845778835264 660255510638493965671483/114196173990354810149157741209600000000, a[14 ,1] = 1221461237263884679751555607/9994523002806272000000000000*6^(1/2 )+36716621212098036093935018687105425505961/72248275402215258274603114 496000000000000, a[14,12] = -37972709869158045130412933766281771978445 1899678250630685021894233169654109523405323/74980943726337976053062781 324716052947047103330285761853079535691020475760640000000, a[12,10] = \+ 123396895115495738434549229715587040998178289648593721222723693824/169 60466422214315122555088502288155618001490986047888958907461648625-8781 560217119727637264447549372070053517105489683916369731584/616968585748 0653009296139869875647732994358307038155314262445125*6^(1/2), a[14,10] = -10993151135374962117764000463/258333325951995046312000000000*6^(1/ 2)+306734586161727173704146823378382330889519/140626680062621422584691 4661737584000000000, a[15,7] = -6650904765188943408799448943/134574190 4975564908028000000*6^(1/2)+1691425739887134646566682083/9612442178396 8922002000000, a[12,11] = -2418901294262986471384856103775514061656722 25318019246672/3661688723471872237814245794078789141770452398980100701 315, a[13,8] = -155850251753928802974915857362119/17401508660534001647 7040216637440+13441422635083243161891/5220309196109974077440*6^(1/2), \+ a[13,12] = -1939162142353174689879923915990531880493671334862071208893 11375/42748092349455088111344007455417233641020280098816132254793728\} :" }{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 61 "We can obtain an embedded 14 stage order 8 sche me as follows." }}{PARA 0 "" 0 "" {TEXT -1 94 "We remove stages 14 and 15 from the 15 stage order 9 scheme and introduce a new stage 14 with :" }}{PARA 256 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "c[14] = 1;" " 6#/&%\"cG6#\"#9\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[14,2] = 0; " "6#/&%\"aG6$\"#9\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[14,3 ] = 0;" "6#/&%\"aG6$\"#9\"\"$\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 " a[14,4]=0" "6#/&%\"aG6$\"#9\"\"%\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[14,5]=0" "6#/&%\"aG6$\"#9\"\"&\"\"!" }{TEXT -1 1 "," }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`b*`[2]=0" "6#/&%#b*G6#\"\"#\" \"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[3]=0" "6#/&%#b*G6#\"\"$\" \"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[4]=0" "6#/&%#b*G6#\"\"%\" \"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[5]=0" "6#/&%#b*G6#\"\"&\" \"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[6]=0" "6#/&%#b*G6#\"\"'\" \"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[7] = 0;" "6#/&%#b*G6#\"\"( \"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 56 "where the weigh ts of the order 8 scheme are denoted by " }{XPPEDIT 18 0 "`b*`" "6#%# b*G" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 156 "We incorporate the order 8 quadrature conditions, the \+ row sum conditions for this stage and stage-order conditions so that t his new stage has stage-order 4." }}{PARA 0 "" 0 "" {TEXT -1 48 "We al so incorporate the simplifying conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(`b*`[i]*a[i,j],i = j+1 .. 14) = `b* `[j]*(1-c[j]);" "6#/-%$SumG6$*&&%#b*G6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+ ;,&F0F,F,F,\"#9*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 1 "," }} {PARA 0 "" 0 "" {TEXT -1 5 "for " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\" \"" }{TEXT -1 10 " ( with " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\" \"\"\"!" }{TEXT -1 5 " ), " }{XPPEDIT 18 0 "j = 9;" "6#/%\"jG\"\"*" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "j = 11;" "6#/%\"jG\"#6" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "j = 13;" "6#/%\"jG\"#8" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 451 "`Qeqs*` := subs(b=`b*`,QuadratureConditions(8,14,'expanded')):\nS O_eqs2 := [add(a[14,j],j=1..13)=c[14],add(a[14,j]*c[j],j=2..13)=1/2*c[ 14]^2,\n add(a[14,j]*c[j]^2,j=2..13)=1/3*c[14]^3,add(a[14,j]*c[ j]^3,j=2..13)=1/4*c[14]^4,\n add(a[14,j]*c[j]^4,j=2..13)=1/5*c[ 14]^5]:\n`simp_eqs*` := [add(`b*`[i]*a[i,1],i=2..14)=`b*`[1],seq(add(` b*`[i]*a[i,j],i=j+1..14)=`b*`[j]*(1-c[j]),j=[9,11,13])]:\n`cdns*` := [ op(`simp_eqs*`),op(SO_eqs2),op(`Qeqs*`)]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "e14 := `union`(re move(u_->member(op(1,lhs(u_)),[14,15]) or op(0,lhs(u_))=b,e13),\n \+ \{c[14]=1,seq(a[14,i]=0,i=2..5),seq(`b*`[i]=0,i=2..7)\}):\n" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "We have 1 7 equations for the 17 unknown coefficients." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "eqns3 := subs(e14,`cdns*`):\nnops(%);\nindets(eqns 3);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<3&%\"aG6$\"#9\"#6&F%6$F'\"#7&F%6$F'\"\")&F%6$F'\" \"*&F%6$F'\"#5&F%6$F'\"\"(&F%6$F'\"#8&F%6$F'\"\"\"&F%6$F'\"\"'&%#b*G6# F=&FB6#F.&FB6#F1&FB6#F4&FB6#F(&FB6#F+&FB6#F:&FB6#F'" }}{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 74 "e15 := solve(\{op(eqns3)\}):\ninfol evel[solve] := 0:\ne16 := `union`(e14,e15):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e16" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6464 "e16 := \{a[13,1] = 4507784 6760256141387004276823/110315894143992133591739924480+1493491403898138 129099/13100021190238236835840*6^(1/2), a[11,5] = 0, c[2] = 1/49, c[5] = 3/7, c[8] = 40/63, c[10] = 19/18, c[11] = 7/9, a[11,6] = 19/25, `b* `[1] = 89673698740537/2188798536240000, a[8,5] = 0, a[8,4] = 0, `b*`[9 ] = 32248235262596025897/70884158450064080000, a[10,1] = 52918819/1382 40000, a[4,2] = 0, a[2,1] = 1/49, a[8,1] = 40/567, `b*`[8] = -50462783 649374452713/198296113717228720000, c[6] = 8/21+4/63*6^(1/2), c[7] = 8 /21-4/63*6^(1/2), c[9] = 10/21, c[4] = 32/235-8/4935*6^(1/2), a[12,3] \+ = 0, c[3] = 64/705-16/14805*6^(1/2), a[10,3] = 0, a[11,3] = 0, a[6,3] \+ = 0, a[7,3] = 0, a[8,3] = 0, a[10,5] = 0, a[12,5] = 0, a[9,5] = 0, a[1 0,4] = 0, a[10,8] = 44599023/5120000, a[11,4] = 0, a[12,4] = 0, a[9,4] = 0, `b*`[10] = -3511133566564032/35760734800490575, c[14] = 1, a[4,3 ] = 24/235-2/1645*6^(1/2), a[7,1] = 4336/127575+479/127575*6^(1/2), a[ 5,4] = 65097/22400+477/5600*6^(1/2), a[9,7] = 295/1344+115/2688*6^(1/2 ), a[6,5] = 1940224/15069159+779264/15069159*6^(1/2), a[7,4] = 9073194 4/400648275-170142739/8413613775*6^(1/2), a[10,7] = -1453047743/103680 000+4153586941/829440000*6^(1/2), a[11,7] = 1180508473123/443296800000 -136404911099/147765600000*6^(1/2), a[9,6] = 295/1344-115/2688*6^(1/2) , a[3,1] = -165952/1491075+38896/10437525*6^(1/2), a[11,1] = 258780283 /8618400000+585428803/51710400000*6^(1/2), a[11,10] = 13167297224/7920 49782825-9366860848/2376149348475*6^(1/2), a[7,5] = 8245504/62429373-2 2187008/437005611*6^(1/2), a[3,2] = 301312/1491075-7168/1491075*6^(1/2 ), a[8,7] = 10/567*6^(1/2)+160/567, a[4,1] = 8/235-2/4935*6^(1/2), a[5 ,3] = -149931/44800-81/700*6^(1/2), a[7,6] = -3936/340025+11464/306022 5*6^(1/2), a[6,4] = 364520/1674351+87715/5023053*6^(1/2), a[5,1] = 389 37/44800+171/5600*6^(1/2), a[6,1] = 176/5103-29/5103*6^(1/2), a[8,6] = 160/567-10/567*6^(1/2), a[11,9] = -1260561943/591300000+585428803/886 950000*6^(1/2), a[11,8] = -106856621/190800000+585428803/2289600000*6^ (1/2), a[10,6] = -1453047743/103680000-4153586941/829440000*6^(1/2), a [10,9] = 518179039/25920000, a[14,5] = 0, `b*`[12] = 33263220100518204 10035389555466180704547593970934049395166930777066599/1373323797461522 4189190609759523239412338434762518802069756703639509120, a[14,8] = 101 91868422097140506105655622328061/4787568421535046196059430031281250+10 11994132602976653624021/602746155289116328000000*6^(1/2), a[14,9] = -8 042480997962909835623073494340513/1506681707334195127293078358250000+1 12443792511441850402669/25943672957609370250000*6^(1/2), a[13,2] = 0, \+ a[13,4] = 0, a[14,4] = 0, c[12] = 319999786/2170712113, `b*`[2] = 0, a [14,12] = -62036795119613861352341988130533894349158136534583361187229 9990616582621/19607180039578465698076705097131503568687683950979327148 7889571798793056, `b*`[3] = 0, `b*`[4] = 0, `b*`[14] = 10688504668909/ 28093813123860, a[13,9] = -493074073683718697930133408597/276027121164 08194083051274240+1493491403898138129099/224694912332563742720*6^(1/2) , `b*`[5] = 0, a[8,2] = 0, `b*`[6] = 0, `b*`[7] = 0, a[14,1] = 1414107 54314242261138804477571/335191282257666323640452000000+112443792511441 850402669/1512551672706650408000000*6^(1/2), a[10,2] = 0, a[11,2] = 0, a[12,2] = 0, c[13] = 15/16, a[7,2] = 0, a[9,2] = 0, a[9,3] = 0, a[5,2 ] = 0, a[6,2] = 0, a[14,2] = 0, a[14,3] = 0, a[13,3] = 0, a[9,1] = 95/ 1344, a[13,5] = 0, a[14,13] = -3540712440917044503420862464/6932876173 8078084467429795125, a[14,11] = -11555892521024455643655681/1591511770 1775915252975590, a[14,6] = -2794429334116030797321/113725689677191760 0000-137693983745103858723/40616317741854200000*6^(1/2), a[14,7] = 278 93647118108877562026663/2778562296720245822000000-10391959690445779064 1030873/38899872154083441508000000*6^(1/2), a[13,6] = 2853073212310390 0185/9849639992660328448-15062887306567756845/5628365710091616256*6^(1 /2), a[13,7] = 530875502237315716994493/24064781139210466754560-892082 3473649531766699837/1347627743795786138255360*6^(1/2), a[12,7] = 54237 1157260956891298011197530777699174343335570401188235055068891/71570321 5922030799396778727922657043395761853183510539934169000000-25571039199 67567420571445567798380600417779349808054092568810687111/1001984502290 8431191554902190917198607540665944569147559078366000000*6^(1/2), a[12, 6] = 10354821182100230493026667000379184955505622050895245676387169/14 6467548938874889512569831763151565670818095959204028052600000-32684637 88087907168885902319404754223684763360609854606186699/4184787112539282 5575019951932329018763090884559772579443600000*6^(1/2), a[13,11] = -12 59978731825102407292471875/947642075600343143202947072, a[12,8] = -892 225578009519154676238995901578841509244882985862325637048827/388139004 68801845720831005417235164902766795429189067433939000000+2036738819888 6087049319397392342447633889771033202442752063/22061374857221856652891 3954035758258132313997301229239234375*6^(1/2), a[9,8] = -15/448, a[12, 1] = 307213395328582867964430765847473084972824867512957518186088963/5 126364212860621132939944111710304798478633358572140981841000000+119107 533326819222510639750832411974467191643469020133053/291376649057647163 34007503363213354847664112851105748578125*6^(1/2), a[14,10] = 17127775 4522230541963559693237323184/1168212455792999305788440538350463625-161 91906121647626457984336/625530609794743596276941125*6^(1/2), a[12,9] = -550688605235770338034863642917595195825050633798073448796494914/8353 22740042020854251374821774223772966384453517335472487484375+7241738026 2706087286468968506106480476052519229164240896224/30386421973154632748 3221106502082129125640034018674235171875*6^(1/2), `b*`[11] = 293286057 408279/538164919532900, a[13,10] = -336035565877081079047275/846506642 4794973677551616*6^(1/2)+200609996314078300148532240828075/10199336919 79646265167106381709312, a[12,10] = 1233968951154957384345492297155870 40998178289648593721222723693824/1696046642221431512255508850228815561 8001490986047888958907461648625-87815602171197276372644475493720700535 17105489683916369731584/6169685857480653009296139869875647732994358307 038155314262445125*6^(1/2), `b*`[13] = -35480458347621122048/114126306 772338781875, a[12,11] = -24189012942629864713848561037755140616567222 5318019246672/36616887234718722378142457940787891417704523989801007013 15, a[13,8] = -155850251753928802974915857362119/174015086605340016477 040216637440+13441422635083243161891/5220309196109974077440*6^(1/2), a [13,12] = -19391621423531746898799239159905318804936713348620712088931 1375/42748092349455088111344007455417233641020280098816132254793728\}: " }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "printed coefficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 289 "subs(e16,matrix([seq([c[i],seq(a[i ,j],j=1..i-1),``$(6-i)],i=2..6)]));\nfor ii from 7 to 14 do\n print( ``);\n print(c[ii]=subs(e16,c[ii])); \n for jj to ii-1 do\n p rint(a[ii,jj]=subs(e16,a[ii,jj]));\n end do:\nend do:\n``;\nfor ii t o 14 do\n print(`b*`[ii]=subs(e16,`b*`[ii]));\nend do:" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7(#\"\"\"\"#\\F(%!GF+F+F+7(,&#\"#k \"$0(F)*(\"#;F)\"&0[\"!\"\"\"\"'#F)\"\"#F4,&#\"'_f;\"(v5\\\"F4*(\"&'*) QF)\")DvV5F4F5F6F),&#\"'78IF;F)*(\"%orF)F;F4F5F6F4F+F+F+7(,&#\"#K\"$N# F)*(\"\")F)\"%N\\F4F5F6F4,&#FJFHF)*(F7F)FKF4F5F6F4\"\"!,&#\"#CFHF)*(F7 F)\"%X;F4F5F6F4F+F+7(#\"\"$\"\"(,&#\"&P*Q\"&+[%F)*(\"$r\"F)\"%+cF4F5F6 F)FO,&#\"'J*\\\"FfnF4*(\"#\")F)\"$+(F4F5F6F4,&#\"&(4l\"&+C#F)*(\"$x%F) FinF4F5F6F)F+7(,&#FJ\"#@F)*(\"\"%F)\"#jF4F5F6F),&#\"$w\"\"%.^F)*(\"#HF )F`pF4F5F6F4FOFO,&#\"'?XO\"(^Vn\"F)*(\"&:x)F)\"(`I-&F4F5F6F),&#\"(C-%> \")f\"p]\"F)*(\"'k#z(F)F]qF4F5F6F)Q(pprint96\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"(,& #\"\")\"#@\"\"\"*(\"\"%F,\"#j!\"\"\"\"'#F,\"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"\",&#\"%OV\"'vv7F(*(\"$z%F(F,!\"\"\" \"'#F(\"\"#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"#\" \"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"$\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"%,&#\")W>t!*\"*v#[1S \"\"\"*(\"*RF9q\"F-\"+vPh8%)!\"\"\"\"'#F-\"\"#F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"&,&#\"(/bC)\")t$HC'\"\"\"*(\")3q=AF- \"*6c+P%!\"\"\"\"'#F-\"\"#F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"a G6$\"\"(\"\"',&#\"%OR\"'D+M!\"\"*(\"&k9\"\"\"\"\"(D-1$F-F(#F0\"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\")#\"#S\"#j" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6 $\"\")\"\"\"#\"#S\"$n&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\" \")\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"$\" \"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"%\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"&\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"',&#\"$g\"\"$n&\"\"\"*(\"#5F-F ,!\"\"F(#F-\"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\" \"(,&*(\"#5\"\"\"\"$n&!\"\"\"\"'#F,\"\"#F,#\"$g\"F-F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" 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ZsmC>!=`Asc;19bx.h&[Qrk)HE%H,*=C\"en:8q+,)*)R_/xT\"*yySzXU\"yBs=ZB()oh O" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#8#\"#:\"#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"a G6$\"#8\"\"\",&#\">BoF/qQThDgn%y2X\"?![C*R@+58!\"\"\"\"'#F(\"\"#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"#8\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\" \"',&#\"5&=+R5B@tI&G\"4[%G.m#**R'\\)*\"\"\"*(\"5XovnlI()G1:F-\"4cih\"4 5dOGc!\"\"F(#F-\"\"#F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"# 8\"\"(,&#\"9$\\%*pr:tB-b(3`\"8gXvm/@R6ykS#\"\"\"*(\":P)*pm<`\\OZB3#*)F -\":g`DQhy&zVxiZ8!\"\"\"\"'#F-\"\"#F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"),&#\"B>@Ode\"\\(H!)GRv^-&e:\"BSuj;-/xk,S`g'3:S4.A&F-\"\"'#F0\"\"#F0" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"*,&#\"?(f3M8Izp=PotS2 $\\\">SUF^I3%>3k67FgF!\"\"*(\"7*4H\"Q\")*QS\"\\$\\\"\"\"\"\"6?FujDL7\\ pC#F-\"\"'#F0\"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8 \"#5,&*(\"9vs/z53xecNgL\"\"\"\":;;bxO(\\zCk1l%)!\"\"\"\"'#F,\"\"#F.#\" Bv!G3CK&[,IySJ'**41?\"C7$4pL*>5F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"#6#!=v=Z#H2C5D=ty*f7\"\"inGPzaA8;))4!G?5kLsTbu+W86)3b%\\B4[F%" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"cG6#\"#9\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#9 \"\"\",&#\"?rvZ/)Q6EUUJa2TT\"\"?+++_/kBjmdAG\">N$F(*(\"9pES]=W6DzVC6F( \":+++3/l1Fn^D^\"!\"\"\"\"'#F(\"\"#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#9\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6 $\"#9\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#9\"\"% \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#9\"\"&\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#9\"\"',&#\"7@tzIg6M$HWz#\" 7++g<>x'*oDP6!\"\"*(\"6B(eQ5XP)RpP\"\"\"\"\"5++?a=uR5F-\";+++3:W$3a@()**)Q! \"\"\"\"'#F-\"\"#F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#9\" \"),&#\"Dh!GBibc5109(4A%o=>5\"C]7GJ+Vfg>Y]`@%ovy%\"\"\"*(\":@Si`m(HgKT *>,\"F-\"9+++Gj6*Gbhu-'!\"\"\"\"'#F-\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#9\"\"*,&#\"C80M%\\tIiN)4H'z*4[U!)\"C++De$yI HF^>Mtq\"o1:!\"\"*(\"9pES]=W6DzVC6\"\"\"\"8++Dq$4w&HnVf#F-\"\"'#F0\"\" #F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#9\"#5,&#\"E%=BtB$pf N'>aIA_axFr\"\"FDOY]$Q0W)y0$**HzbC@o6\"\"\"*(\";OV)zXEwk@h!>>;F-\" M_8'Qh>^zO?'\"cocIz)zr&*)y[rKz4&Ro(ooN]Jr40n2)pl%y&R+=2'>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#9\"#8#!=kC'3U.X/<4W72a$\">D^zHuY%3y !Q" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"*#\"5(*e-'fi_B[A$\"5++3k+XeT)3( " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#5#!1KScmN86N\"2v0\\+[t gd$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#6#\"0z#3u0'G$H\"0+H `>\\;Q&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#7#\"ao*fmqxIp;& R\\S$4(RfZXq!=mab*QN+T?=05?KEL\"bo?\"4&ROqc(p?!)=DwM%QBTRK_f(41>*=C_hu zBLP\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#8#!5[?7@wMe/[N\" 6v=yQBx1j79\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#9#\"/4*oY ])o5\"/gQ78Q4G" }}}{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 0 104 "`RK8_14eqs*` := subs(b=`b*`,[op(Ro wSumConditions(14,'expanded')),op(OrderConditions(8,14,'expanded'))]): " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Check :" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "expand(subs(e16,`RK8_14 eqs*`)):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7ax\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$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#\"$8#" }}}{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 princi pal error terms. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 185 "`errte rms8_14*` := subs(b=`b*`,PrincipalErrorTerms(8,14,'expanded')):\nsm := 0:\nfor ct to nops(`errterms8_14*`) do\n sm := sm+(evalf(subs(e16,` errterms8_14*`[ct])))^2;\nend do:\nsqrt(sm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+JPY/;!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 138 "We can include the new stage for the embedded \+ scheme as an additional 16th stage added to the order 9 scheme along w ith the coefficients " }{XPPEDIT 18 0 "a[16,14] = 0;" "6#/&%\"aG6$\"# ;\"#9\"\"!" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[16,15] = 0;" "6#/& %\"aG6$\"#;\"#:\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 13 "T he weights " }{XPPEDIT 18 0 "`b*`[i]" "6#&%#b*G6#%\"iG" }{TEXT -1 7 " for " }{XPPEDIT 18 0 "i=1" "6#/%\"iG\"\"\"" }{TEXT -1 89 " . . 13 o f the 16 stage combined scheme are those of the 14 stage scheme and th e weight " }{XPPEDIT 18 0 "`b*`[14];" "6#&%#b*G6#\"#9" }{TEXT -1 34 " in the 14 stage scheme becomes " }{XPPEDIT 18 0 "`b*`[16];" "6#&%#b *G6#\"#;" }{TEXT -1 25 " in the 16 stage scheme." }}{PARA 0 "" 0 "" {TEXT -1 8 "We set " }{XPPEDIT 18 0 "c[16] = 1;" "6#/&%\"cG6#\"#;\"\" \"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[14] = 0;" "6#/&%#b*G6#\"#9 \"\"!" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "`b*`[15] = 0;" "6#/&%#b*G 6#\"#:\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 66 "We can mak e the order 9 scheme into a 16 stage scheme by setting " }{XPPEDIT 18 0 "b[16] = 0;" "6#/&%\"bG6#\"#;\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 201 "e17 := \{c[16]=1,seq(a[16,i]=subs(e16,a[14,i]),i=1..13),a[16,14]=0,a[16,15]= 0,b[16]=0,\nseq(`b*`[i]=subs(e16,`b*`[i]),i=1..13),`b*`[14]=0,`b*`[15] =0,`b*`[16]=subs(e16,`b*`[14])\}:\ne18 := `union`(e13,e17):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e18" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9875 "e18 := \{a[13,1] = 45077846760256141387004276823/110315894143992133591739924480+149349 1403898138129099/13100021190238236835840*6^(1/2), a[11,5] = 0, b[8] = \+ -72263163141715044860361/169939769455665013040000, b[15] = -3697690464 76619/65552230622340, c[2] = 1/49, c[5] = 3/7, c[8] = 40/63, c[10] = 1 9/18, c[11] = 7/9, a[11,6] = 19/25, `b*`[1] = 89673698740537/218879853 6240000, a[8,5] = 0, a[8,4] = 0, b[14] = 1212545712242913280000000/130 535954653501897388343, `b*`[9] = 32248235262596025897/7088415845006408 0000, a[16,4] = 0, a[10,1] = 52918819/138240000, a[4,2] = 0, a[2,1] = \+ 1/49, a[8,1] = 40/567, b[13] = -10678264099993989152768/23966524422191 14419375, `b*`[8] = -50462783649374452713/198296113717228720000, c[16] = 1, b[12] = 17109990417889849939560223376925306674323804078983341334 325755071278367152457480027/713814271258088281460765347031731950560904 66069876919551824450642795947233961810560, c[6] = 8/21+4/63*6^(1/2), c [7] = 8/21-4/63*6^(1/2), c[9] = 10/21, c[4] = 32/235-8/4935*6^(1/2), b [11] = 1883570537693211021/1872275755054959100, a[12,3] = 0, c[3] = 64 /705-16/14805*6^(1/2), a[10,3] = 0, a[11,3] = 0, a[6,3] = 0, a[7,3] = \+ 0, a[8,3] = 0, a[10,5] = 0, a[16,5] = 0, a[12,5] = 0, a[9,5] = 0, a[10 ,4] = 0, a[10,8] = 44599023/5120000, a[11,4] = 0, a[12,4] = 0, a[9,4] \+ = 0, `b*`[10] = -3511133566564032/35760734800490575, b[16] = 0, `b*`[1 4] = 0, a[16,14] = 0, a[16,15] = 0, `b*`[15] = 0, a[16,1] = 1414107543 14242261138804477571/335191282257666323640452000000+112443792511441850 402669/1512551672706650408000000*6^(1/2), a[16,8] = 101918684220971405 06105655622328061/4787568421535046196059430031281250+10119941326029766 53624021/602746155289116328000000*6^(1/2), a[16,9] = -8042480997962909 835623073494340513/1506681707334195127293078358250000+1124437925114418 50402669/25943672957609370250000*6^(1/2), a[16,10] = 17127775452223054 1963559693237323184/1168212455792999305788440538350463625-161919061216 47626457984336/625530609794743596276941125*6^(1/2), a[16,6] = -2794429 334116030797321/1137256896771917600000-137693983745103858723/406163177 41854200000*6^(1/2), a[16,7] = 27893647118108877562026663/277856229672 0245822000000-103919596904457790641030873/38899872154083441508000000*6 ^(1/2), a[16,13] = -3540712440917044503420862464/693287617380780844674 29795125, a[16,11] = -11555892521024455643655681/159151177017759152529 75590, a[16,12] = -620367951196138613523419881305338943491581365345833 611872299990616582621/196071800395784656980767050971315035686876839509 793271487889571798793056, `b*`[16] = 10688504668909/28093813123860, a[ 4,3] = 24/235-2/1645*6^(1/2), a[7,1] = 4336/127575+479/127575*6^(1/2), a[5,4] = 65097/22400+477/5600*6^(1/2), a[9,7] = 295/1344+115/2688*6^( 1/2), a[6,5] = 1940224/15069159+779264/15069159*6^(1/2), a[7,4] = 9073 1944/400648275-170142739/8413613775*6^(1/2), a[10,7] = -1453047743/103 680000+4153586941/829440000*6^(1/2), a[11,7] = 1180508473123/443296800 000-136404911099/147765600000*6^(1/2), a[9,6] = 295/1344-115/2688*6^(1 /2), a[3,1] = -165952/1491075+38896/10437525*6^(1/2), a[11,1] = 258780 283/8618400000+585428803/51710400000*6^(1/2), a[11,10] = 13167297224/7 92049782825-9366860848/2376149348475*6^(1/2), a[7,5] = 8245504/6242937 3-22187008/437005611*6^(1/2), a[3,2] = 301312/1491075-7168/1491075*6^( 1/2), a[8,7] = 10/567*6^(1/2)+160/567, a[4,1] = 8/235-2/4935*6^(1/2), \+ a[5,3] = -149931/44800-81/700*6^(1/2), a[7,6] = -3936/340025+11464/306 0225*6^(1/2), a[6,4] = 364520/1674351+87715/5023053*6^(1/2), a[5,1] = \+ 38937/44800+171/5600*6^(1/2), a[6,1] = 176/5103-29/5103*6^(1/2), a[8,6 ] = 160/567-10/567*6^(1/2), a[11,9] = -1260561943/591300000+585428803/ 886950000*6^(1/2), a[11,8] = -106856621/190800000+585428803/2289600000 *6^(1/2), a[10,6] = -1453047743/103680000-4153586941/829440000*6^(1/2) , a[10,9] = 518179039/25920000, a[14,5] = 0, a[15,5] = 0, b[1] = 17373 4691637390647/4182794002754640000, b[4] = 0, b[5] = 0, b[2] = 0, b[3] \+ = 0, `b*`[12] = 332632201005182041003538955546618070454759397093404939 5166930777066599/13733237974615224189190609759523239412338434762518802 069756703639509120, a[15,1] = 1564746779443331677794753119798867/28641 99537049451295689544004000000+5933645037523445166666379/52326757304999 307928000000*6^(1/2), b[6] = 0, b[7] = 0, a[13,2] = 0, a[15,4] = 0, a[ 13,4] = 0, a[14,4] = 0, c[12] = 319999786/2170712113, `b*`[2] = 0, `b* `[3] = 0, `b*`[4] = 0, a[13,9] = -493074073683718697930133408597/27602 712116408194083051274240+1493491403898138129099/224694912332563742720* 6^(1/2), `b*`[5] = 0, a[8,2] = 0, `b*`[6] = 0, `b*`[7] = 0, a[10,2] = \+ 0, a[11,2] = 0, a[12,2] = 0, c[15] = 1, c[14] = 39/40, c[13] = 15/16, \+ a[7,2] = 0, a[9,2] = 0, a[9,3] = 0, a[5,2] = 0, a[6,2] = 0, a[15,2] = \+ 0, a[16,2] = 0, a[14,2] = 0, a[15,3] = 0, a[14,3] = 0, a[13,3] = 0, a[ 16,3] = 0, b[9] = 14586697891849999254003/29700462390576849520000, a[9 ,1] = 95/1344, a[13,5] = 0, a[13,6] = 28530732123103900185/98496399926 60328448-15062887306567756845/5628365710091616256*6^(1/2), a[14,7] = 3 97408075485926915758262202639/18359970961922048000000000000-6493922587 539771225254133441201/1028158373867634688000000000000*6^(1/2), a[13,7] = 530875502237315716994493/24064781139210466754560-892082347364953176 6699837/1347627743795786138255360*6^(1/2), a[12,7] = 54237115726095689 1298011197530777699174343335570401188235055068891/71570321592203079939 6778727922657043395761853183510539934169000000-25571039199675674205714 45567798380600417779349808054092568810687111/1001984502290843119155490 2190917198607540665944569147559078366000000*6^(1/2), a[12,6] = 1035482 1182100230493026667000379184955505622050895245676387169/14646754893887 4889512569831763151565670818095959204028052600000-32684637880879071688 85902319404754223684763360609854606186699/4184787112539282557501995193 2329018763090884559772579443600000*6^(1/2), a[14,6] = 8283471074731862 302286097/7514678949478400000000000-15668946773152185221466849/4294102 256844800000000000*6^(1/2), a[15,6] = -57628625604267458078211/3934342 6545112261600000-6047516944575480929793/1405122376611152200000*6^(1/2) , a[14,8] = 2435278493903047909370803905780425755549361787/69519418749 60961660657094977773568000000000000+10993151135374962117764000463/3982 779843223552000000000000*6^(1/2), a[15,10] = -854444885403376103999958 576/21640244757394101989947629875*6^(1/2)+2479701613016438411235353770 85814726576/1172015708277609400708896218214309893875, a[13,11] = -1259 978731825102407292471875/947642075600343143202947072, a[12,8] = -89222 5578009519154676238995901578841509244882985862325637048827/38813900468 801845720831005417235164902766795429189067433939000000+203673881988860 87049319397392342447633889771033202442752063/2206137485722185665289139 54035758258132313997301229239234375*6^(1/2), a[14,13] = -2354122702208 29707518634576/10004921377982463725322265625, a[15,9] = 59336450375234 45166666379/897521918060373467750000*6^(1/2)-2537877481308681657176970 94592697839427/21839824320042045467560266343639250000, a[15,8] = 53402 805337711006499997411/20852016068909498648000000*6^(1/2)+4336304776662 958741869045159571925984937/2271064142891395647108842129832843500000, \+ a[9,8] = -15/448, a[15,13] = 25469705993361596208461643776/23984299876 72964863327432743875, a[12,1] = 30721339532858286796443076584747308497 2824867512957518186088963/51263642128606211329399441117103047984786333 58572140981841000000+1191075333268192225106397508324119744671916434690 20133053/29137664905764716334007503363213354847664112851105748578125*6 ^(1/2), a[15,12] = -44386039953127381302291127198665043342539469696141 788200181019629996219153064322715843/926096283618263046743578868285140 7364301742047595974505212083507724183850275361816608, a[14,9] = 122146 1237263884679751555607/171428613534976000000000000*6^(1/2)-55812323906 9103416347126929975086912148938889/34184771406932232290260692794368000 000000000, b[10] = 102209317997264953344/225042304099487188475, a[12,9 ] = -550688605235770338034863642917595195825050633798073448796494914/8 35322740042020854251374821774223772966384453517335472487484375+7241738 0262706087286468968506106480476052519229164240896224/30386421973154632 7483221106502082129125640034018674235171875*6^(1/2), `b*`[11] = 293286 057408279/538164919532900, a[15,14] = -1051389896416361980924193792000 0000/255387066048362821838491552709590753, a[15,11] = -529666357376976 05455117615341/39091452325689672618552125990, a[13,10] = -336035565877 081079047275/8465066424794973677551616*6^(1/2)+20060999631407830014853 2240828075/1019933691979646265167106381709312, a[14,11] = -16384577883 5264660255510638493965671483/114196173990354810149157741209600000000, \+ a[14,1] = 1221461237263884679751555607/9994523002806272000000000000*6^ (1/2)+36716621212098036093935018687105425505961/7224827540221525827460 3114496000000000000, a[14,12] = -3797270986915804513041293376628177197 84451899678250630685021894233169654109523405323/7498094372633797605306 2781324716052947047103330285761853079535691020475760640000000, a[12,10 ] = 123396895115495738434549229715587040998178289648593721222723693824 /16960466422214315122555088502288155618001490986047888958907461648625- 8781560217119727637264447549372070053517105489683916369731584/61696858 57480653009296139869875647732994358307038155314262445125*6^(1/2), `b*` [13] = -35480458347621122048/114126306772338781875, a[14,10] = -109931 51135374962117764000463/258333325951995046312000000000*6^(1/2)+3067345 86161727173704146823378382330889519/1406266800626214225846914661737584 000000000, a[15,7] = -6650904765188943408799448943/1345741904975564908 028000000*6^(1/2)+1691425739887134646566682083/96124421783968922002000 000, a[12,11] = -24189012942629864713848561037755140616567222531801924 6672/3661688723471872237814245794078789141770452398980100701315, a[13, 8] = -155850251753928802974915857362119/174015086605340016477040216637 440+13441422635083243161891/5220309196109974077440*6^(1/2), a[13,12] = -193916214235317468987992391599053188049367133486207120889311375/4274 8092349455088111344007455417233641020280098816132254793728\}:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "printed coe fficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 362 "subs(e18,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$( 6-i)],i=2..6)]));\nfor ii from 7 to 16 do\n print(``);\n print(c[i i]=subs(e18,c[ii])); \n for jj to ii-1 do\n print(a[ii,jj]=subs (e18,a[ii,jj]));\n end do:\nend do:print(``);\nfor ii to 16 do\n p rint(b[ii]=subs(e18,b[ii]));\nend do:print(``);\nfor ii to 16 do\n p rint(`b*`[ii]=subs(e18,`b*`[ii]));\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7(#\"\"\"\"#\\F(%!GF+F+F+7(,&#\"#k\"$0(F) *(\"#;F)\"&0[\"!\"\"\"\"'#F)\"\"#F4,&#\"'_f;\"(v5\\\"F4*(\"&'*)QF)\")D vV5F4F5F6F),&#\"'78IF;F)*(\"%orF)F;F4F5F6F4F+F+F+7(,&#\"#K\"$N#F)*(\" \")F)\"%N\\F4F5F6F4,&#FJFHF)*(F7F)FKF4F5F6F4\"\"!,&#\"#CFHF)*(F7F)\"%X ;F4F5F6F4F+F+7(#\"\"$\"\"(,&#\"&P*Q\"&+[%F)*(\"$r\"F)\"%+cF4F5F6F)FO,& #\"'J*\\\"FfnF4*(\"#\")F)\"$+(F4F5F6F4,&#\"&(4l\"&+C#F)*(\"$x%F)FinF4F 5F6F)F+7(,&#FJ\"#@F)*(\"\"%F)\"#jF4F5F6F),&#\"$w\"\"%.^F)*(\"#HF)F`pF4 F5F6F4FOFO,&#\"'?XO\"(^Vn\"F)*(\"&:x)F)\"(`I-&F4F5F6F),&#\"(C-%>\")f\" p]\"F)*(\"'k#z(F)F]qF4F5F6F)Q(pprint86\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"(,&#\"\")\"# @\"\"\"*(\"\"%F,\"#j!\"\"\"\"'#F,\"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"\",&#\"%OV\"'vv7F(*(\"$z%F(F,!\"\"\"\"'#F(\"\" #F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"#\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"$\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"%,&#\")W>t!*\"*v#[1S\"\"\"*(\" *RF9q\"F-\"+vPh8%)!\"\"\"\"'#F-\"\"#F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"&,&#\"(/bC)\")t$HC'\"\"\"*(\")3q=AF-\"*6c+P%! \"\"\"\"'#F-\"\"#F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"( \"\"',&#\"%OR\"'D+M!\"\"*(\"&k9\"\"\"\"\"(D-1$F-F(#F0\"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" cG6#\"\")#\"#S\"#j" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\" \"\"#\"#S\"$n&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"# \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"$\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"%\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"',&#\"$g\"\"$n&\"\"\"*(\"#5F-F,!\"\"F (#F-\"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"(,&*( \"#5\"\"\"\"$n&!\"\"\"\"'#F,\"\"#F,#\"$g\"F-F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"*# \"#5\"#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"\"#\"#&* \"%W8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"#\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"$\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"',&#\"$&H\"%W8\"\"\"*(\"$:\"F-\"%)o#!\"\"F(#F- \"\"#F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"(,&#\"$&H \"%W8\"\"\"*(\"$:\"F-\"%)o#!\"\"\"\"'#F-\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\")#!#:\"$[%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#5#\" #>\"#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"\"#\")>)=H& \"*++CQ\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"',&#\"+Vx/`9\"*++o.\"!\"\"*(\"+Tpe`T\"\"\"\"*++ WH)F-F(#F0\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\" (,&#\"+Vx/`9\"*++o.\"!\"\"*(\"+Tpe`T\"\"\"\"*++WH)F-\"\"'#F0\"\"#F0" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\")#\")B!*fW\"(++7&" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"*#\"*R!z\"=&\")++#f# " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#6#\"\"(\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"#6\"\"\",&#\"*$G!ye#\"+++S=')F(*(\"*.)GaeF(\",++S5<&!\"\"\"\"' #F(\"\"#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"#\"\"! " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"'#\"#>\"#D" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ &%\"aG6$\"#6\"\"(,&#\".BJZ30=\"\"-++!oHV%\"\"\"*(\"-*46\\SO\"F-\"-++gl x9!\"\"\"\"'#F-\"\"#F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"# 6\"\"),&#\"*@m&o5\"*++!3>!\"\"*(\"*.)Gae\"\"\"\"+++g*G#F-\"\"'#F0\"\"# F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"*,&#\"+V>cg7\"* ++I\"f!\"\"*(\"*.)Gae\"\"\"\"*++&p))F-\"\"'#F0\"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"#5,&#\",CsHnJ\"\"-DGy\\?z\"\"\"*(\" +[3'oO*F-\".v%[$\\hP#!\"\"\"\"'#F-\"\"#F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#7#\" *'y***>$\"+8@rq@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\" \",&#\"jnj*)3'==v&H^n[#G(\\3tu%ewIW'z'GeG`R8sI\"[o+++T=)49s&eLjy%)z/.r 6T%*RH8@1'G@kj7&F(*(\"Z`I8?!pMk\">nW(>TK3vR1^A#>oKLv5>\"F(\"fnD\"y&[d5 ^G6kw%[N8KO.v+Mjrkd!\\mP\"H!\"\"\"\"'#F(\"\"#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"#7\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\" \"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"',&#\"inp rQwcC&*30Ac]b\\=z.+nm-$\\I-5#=@[N5\"jn++g_!GS?ff4=3nl::j<$)pD^*)[()Q*[ vYY\"\"\"\"*(\"hn*p'=1Y&)41OjZoBUv/%>B!f))or!z3)yj%oKF-\"in++gV%zDxfX) 34j(=!HB$>&*>]dDGRD6(y%=%!\"\"F(#F-\"\"#F1" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"(,&#\"]o\"*)o]0N#)=,/dNLMu\"*px2`(>6!) H\"*o&4Ed6PU&\"]o+++pT$*R0^$=`=w&RVqlAzsynR*zI?#f@.dr\"\"\"*(\"^o6ro5) oD4a!3)\\$zxT+1Q)znbWr0Unv'*>R5dDF-\"_o+++m$y!fv9pX%fmSvg)><4>-\\b\">J %3H-X)>+\"!\"\"\"\"'#F-\"\"#F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"#7\"\"),&#\"jnF)[qjDB'e)H)[C4:%)y:!f**Qina\">&4!ybA#*)\"\\o+++ RRVn!*=HazmF!\\;NsT05$3sX=!)o/!R\")Q!\"\"*(\"fnj?vUC?L5x*)QjZCM#R(R>$ \\q3')))>)Qn.#\"\"\"\"gnvVBR#H7I(*RJK\"e#ed.aR\"*Glc=Ad[Ph?#F-\"\"'#F0 \"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"*,&#\"jn9 \\\\'z[M2)zL10De>&fD0w/[1h]o*okG(31FE!QU'QIF-\"\"'#F0\"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%\"aG6$\"#7\"#5,&#\"]oCQpBFA@Pf['*Gy\")*4/(e:(H#\\XVQd\\:^*oRB\"\" _oD'[;Y2*e*))y/')4\\,!=c:)G-&)3bD7:V@AkYgp\"\"\"\"*(\"hn%eJ(pj\"Ro*[0r ^`+2s$\\vWksjF(>r@g:y)F-\"[oD^WiUJb\"QqIeV*HtZc()p)RhH4Il![deoph!\"\" \"\"'#F-\"\"#F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"#6#! ZsmC>!=`Asc;19bx.h&[Qrk)HE%H,*=C\"en:8q+,)*)R_/xT\"*yySzXU\"yBs=ZB()oh O" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#8#\"#:\"#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"a G6$\"#8\"\"\",&#\">BoF/qQThDgn%y2X\"?![C*R@+58!\"\"\"\"'#F(\"\"#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"#8\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\" \"',&#\"5&=+R5B@tI&G\"4[%G.m#**R'\\)*\"\"\"*(\"5XovnlI()G1:F-\"4cih\"4 5dOGc!\"\"F(#F-\"\"#F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"# 8\"\"(,&#\"9$\\%*pr:tB-b(3`\"8gXvm/@R6ykS#\"\"\"*(\":P)*pm<`\\OZB3#*)F -\":g`DQhy&zVxiZ8!\"\"\"\"'#F-\"\"#F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"),&#\"B>@Ode\"\\(H!)GRv^-&e:\"BSuj;-/xk,S`g'3:S4.A&F-\"\"'#F0\"\"#F0" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"\"*,&#\"?(f3M8Izp=PotS2 $\\\">SUF^I3%>3k67FgF!\"\"*(\"7*4H\"Q\")*QS\"\\$\\\"\"\"\"\"6?FujDL7\\ pC#F-\"\"'#F0\"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8 \"#5,&*(\"9vs/z53xecNgL\"\"\"\":;;bxO(\\zCk1l%)!\"\"\"\"'#F,\"\"#F.#\" Bv!G3CK&[,IySJ'**41?\"C7$4pL*>5F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#8\"#6#!=v=Z#H2C5D=ty*f7\"\"inGPzaA8;))4!G?5kLsTbu+W86)3b%\\B4[F%" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"cG6#\"#9#\"#R\"#S" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\" #9\"\"\",&*(\"=2cb^(zY)QEP7Y@7F(\"=++++++si!G+BX***!\"\"\"\"'#F(\"\"#F (#\"Jhf]Da5(o=]$R4O!)477i;n$\"J++++++'\\9Jgu#e_@-aF[A(F(" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#9\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#9\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#9\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"#9\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#9\" \"',&#\":(4'G-B'=tu5Z$G)\":+++++Sy%\\*yY^(\"\"\"*(\";\\oY@_=_JxY*oc\"F -\":+++++![%oD-TH%!\"\"F(#F-\"\"#F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"aG6$\"#9\"\"(,&#\"?RE?i#ed\"p#f[v!3uR\">++++++[?#>'4(*f$=\"\"\"* (\"@,7WLTDD7xRveAR\\'F-\"@++++++)oMw'QPe\"G5!\"\"\"\"'#F-\"\"#F1" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#9\"\"),&#\"O(yh$\\bvD/y0R! 3P4z/.R\\y_V#\"O++++++oNxx\\4d1mh4'\\(=%>&p\"\"\"*(\">j/+kx6i\\PN6:$*4 \"F-\"=++++++_NAV)zF)R!\"\"\"\"'#F-\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#9\"\"*,&*(\"=2cb^(zY)QEP7Y@7\"\"\"\"<++++++ w\\`8'G9j/+kx6i\\PN6:$*4\"\"\"\"\"?++++?JY]*>&fKL$e#!\"\"\"\"'#F,\" \"#F.#\"K>&*)3L#QyL#o9/P9\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"#9\"#7#!_pB`SB&4TlpJB%*=-&oI1Dy'**=X%y>x\"GmP$HTI^/e\"p)4F(z$ \"^p+++S1wv/-\"pN&zI&=w&GIL5Zq%H0;ZK\"yiI0wzLEP%4)\\(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#9\"#8#!DcEA`sjC)zP@ \\+5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#:\"\"\"" }}{PARA 11 "" 1 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{XPPMATH 20 "6#/&%#b*G6#\"#5#!1KScmN86N\"2v0\\+[t gd$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#6#\"0z#3u0'G$H\"0+H `>\\;Q&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#7#\"ao*fmqxIp;& R\\S$4(RfZXq!=mab*QN+T?=05?KEL\"bo?\"4&ROqc(p?!)=DwM%QBTRK_f(41>*=C_hu zBLP\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#8#!5[?7@wMe/[N\" 6v=yQBx1j79\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#9\"\"!" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#:\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%#b*G6#\"#;#\"/4*oY])o5\"/gQ78Q4G" }}}{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 "RK9_16eqs := [op(RowSumConditions(16,'expanded')),op(OrderConditi ons(9,16,'expanded'))]:\n`RK8_16eqs*` := subs(b=`b*`,OrderConditions(8 ,16,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Check:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "expan d(subs(e18,RK9_16eqs)):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);\nexpand(su bs(e18,`RK8_16eqs*`)):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7ajl\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$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 12 "" 1 "" {XPPMATH 20 "6#7dw\"\" !F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $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 33 "seq(c[i]=subs(e18,c[i]),i=2..16);" }} {PARA 11 "" 1 "" {XPPMATH 20 "61/&%\"cG6#\"\"##\"\"\"\"#\\/&F%6#\"\"$, &#\"#k\"$0(F)*(\"#;F)\"&0[\"!\"\"\"\"'#F)F'F6/&F%6#\"\"%,&#\"#K\"$N#F) *(\"\")F)\"%N\\F6F7F8F6/&F%6#\"\"&#F.\"\"(/&F%6#F7,&#FB\"#@F)*(F\"#=/&F%6#\"#6#FIFin/&F%6#\"#7#\"*'y***>$\"+8@rq@/&F%6 #\"#8#\"#:F4/&F%6#\"#9#\"#RFen/&F%6#FcpF)/&F%6#F4F)" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "#---------------------- ---" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 " " {TEXT -1 52 "#---------------------------------------------------" } }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "characteristics of the embedded \+ scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "coefficient s of the combined scheme " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10011 "ee := \{c[2]=1/49,\nc[3]=64/705- 16/14805*6^(1/2),\nc[4]=32/235-8/4935*6^(1/2),\nc[5]=3/7,\nc[6]=8/21+4 /63*6^(1/2),\nc[7]=8/21-4/63*6^(1/2),\nc[8]=40/63,\nc[9]=10/21,\nc[10] =19/18,\nc[11]=7/9,\nc[12]=319999786/2170712113,\nc[13]=15/16,\nc[14]= 39/40,\nc[15]=1,\nc[16]=1,\n\na[2,1]=1/49,\na[3,1]=-165952/1491075+388 96/10437525*6^(1/2),\na[3,2]=301312/1491075-7168/1491075*6^(1/2),\na[4 ,1]=8/235-2/4935*6^(1/2),\na[4,2]=0,\na[4,3]=24/235-2/1645*6^(1/2),\na [5,1]=38937/44800+171/5600*6^(1/2),\na[5,2]=0,\na[5,3]=-149931/44800-8 1/700*6^(1/2),\na[5,4]=65097/22400+477/5600*6^(1/2),\na[6,1]=176/5103- 29/5103*6^(1/2),\na[6,2]=0,\na[6,3]=0,\na[6,4]=364520/1674351+87715/50 23053*6^(1/2),\na[6,5]=1940224/15069159+779264/15069159*6^(1/2),\na[7, 1]=4336/127575+479/127575*6^(1/2),\na[7,2]=0,\na[7,3]=0,\na[7,4]=90731 944/400648275-170142739/8413613775*6^(1/2),\na[7,5]=8245504/62429373-2 2187008/437005611*6^(1/2),\na[7,6]=-3936/340025+11464/3060225*6^(1/2), \na[8,1]=40/567,\na[8,2]=0,\na[8,3]=0,\na[8,4]=0,\na[8,5]=0,\na[8,6]=1 60/567-10/567*6^(1/2),\na[8,7]=10/567*6^(1/2)+160/567,\na[9,1]=95/1344 ,\na[9,2]=0,\na[9,3]=0,\na[9,4]=0,\na[9,5]=0,\na[9,6]=295/1344-115/268 8*6^(1/2),\na[9,7]=295/1344+115/2688*6^(1/2),\na[9,8]=-15/448,\na[10,1 ]=52918819/138240000,\na[10,2]=0,\na[10,3]=0,\na[10,4]=0,\na[10,5]=0, \na[10,6]=-1453047743/103680000-4153586941/829440000*6^(1/2),\na[10,7] =-1453047743/103680000+4153586941/829440000*6^(1/2),\na[10,8]=44599023 /5120000,\na[10,9]=518179039/25920000,\na[11,1]=258780283/8618400000+5 85428803/51710400000*6^(1/2),\na[11,2]=0,\na[11,3]=0,\na[11,4]=0,\na[1 1,5]=0,\na[11,6]=19/25,\na[11,7]=1180508473123/443296800000-1364049110 99/147765600000*6^(1/2),\na[11,8]=-106856621/190800000+585428803/22896 00000*6^(1/2),\na[11,9]=-1260561943/591300000+585428803/886950000*6^(1 /2),\na[11,10]=13167297224/792049782825-9366860848/2376149348475*6^(1/ 2),\na[12,1]=307213395328582867964430765847473084972824867512957518186 088963/\n 51263642128606211329399441117103047984786333585721409 81841000000+\n 119107533326819222510639750832411974467191643469 020133053/\n 29137664905764716334007503363213354847664112851105 748578125*6^(1/2),\na[12,2]=0,\na[12,3]=0,\na[12,4]=0,\na[12,5]=0,\na[ 12,6]=10354821182100230493026667000379184955505622050895245676387169/ \n 146467548938874889512569831763151565670818095959204028052600 000-\n 32684637880879071688859023194047542236847633606098546061 86699/\n 418478711253928255750199519323290187630908845597725794 43600000*6^(1/2),\na[12,7]=5423711572609568912980111975307776991743433 35570401188235055068891/\n 715703215922030799396778727922657043 395761853183510539934169000000-\n 25571039199675674205714455677 98380600417779349808054092568810687111/\n 100198450229084311915 54902190917198607540665944569147559078366000000*6^(1/2),\na[12,8]=-892 225578009519154676238995901578841509244882985862325637048827/\n \+ 38813900468801845720831005417235164902766795429189067433939000000+\n 20367388198886087049319397392342447633889771033202442752063/ \n 22061374857221856652891395403575825813231399730122923923437 5*6^(1/2),\na[12,9]=-5506886052357703380348636429175951958250506337980 73448796494914/\n 83532274004202085425137482177422377296638445 3517335472487484375+\n 724173802627060872864689685061064804760 52519229164240896224/\n 30386421973154632748322110650208212912 5640034018674235171875*6^(1/2),\na[12,10]=1233968951154957384345492297 15587040998178289648593721222723693824/\n 16960466422214315122 555088502288155618001490986047888958907461648625-\n 8781560217 119727637264447549372070053517105489683916369731584/\n 6169685 857480653009296139869875647732994358307038155314262445125*6^(1/2),\na[ 12,11]=-241890129426298647138485610377551406165672225318019246672/\n \+ 3661688723471872237814245794078789141770452398980100701315,\na [13,1]=45077846760256141387004276823/110315894143992133591739924480+\n 1493491403898138129099/13100021190238236835840*6^(1/2),\na[13, 2]=0,\na[13,3]=0,\na[13,4]=0,\na[13,5]=0,\na[13,6]=2853073212310390018 5/9849639992660328448-\n 15062887306567756845/56283657100916162 56*6^(1/2),\na[13,7]=530875502237315716994493/24064781139210466754560- \n 8920823473649531766699837/1347627743795786138255360*6^(1/2), \na[13,8]=-155850251753928802974915857362119/1740150866053400164770402 16637440+\n 13441422635083243161891/5220309196109974077440*6^( 1/2),\na[13,9]=-493074073683718697930133408597/27602712116408194083051 274240+\n 1493491403898138129099/224694912332563742720*6^(1/2) ,\na[13,10]=200609996314078300148532240828075/101993369197964626516710 6381709312-\n 336035565877081079047275/84650664247949736775516 16*6^(1/2),\na[13,11]=-1259978731825102407292471875/947642075600343143 202947072,\na[13,12]=-193916214235317468987992391599053188049367133486 207120889311375/\n 427480923494550881113440074554172336410202 80098816132254793728,\na[14,1]=367166212120980360939350186871054255059 61/72248275402215258274603114496000000000000+\n 122146123726388 4679751555607/9994523002806272000000000000*6^(1/2),\na[14,2]=0,\na[14, 3]=0,\na[14,4]=0,\na[14,5]=0,\na[14,6]=8283471074731862302286097/75146 78949478400000000000-\n 15668946773152185221466849/429410225684 4800000000000*6^(1/2),\na[14,7]=397408075485926915758262202639/1835997 0961922048000000000000-\n 6493922587539771225254133441201/10281 58373867634688000000000000*6^(1/2),\na[14,8]=2435278493903047909370803 905780425755549361787/6951941874960961660657094977773568000000000000+ \n 10993151135374962117764000463/3982779843223552000000000000*6 ^(1/2),\na[14,9]=-558123239069103416347126929975086912148938889/341847 71406932232290260692794368000000000000+\n 12214612372638846797 51555607/171428613534976000000000000*6^(1/2),\na[14,10]=30673458616172 7173704146823378382330889519/14062668006262142258469146617375840000000 00-\n 10993151135374962117764000463/25833332595199504631200000 0000*6^(1/2),\na[14,11]=-163845778835264660255510638493965671483/11419 6173990354810149157741209600000000,\na[14,12]=-37972709869158045130412 9337662817719784451899678250630685021894233169654109523405323/\n \+ 749809437263379760530627813247160529470471033302857618530795356910 20475760640000000,\na[14,13]=-235412270220829707518634576/100049213779 82463725322265625,\na[15,1]=1564746779443331677794753119798867/2864199 537049451295689544004000000+\n 5933645037523445166666379/523267 57304999307928000000*6^(1/2),\na[15,2]=0,\na[15,3]=0,\na[15,4]=0,\na[1 5,5]=0,\na[15,6]=-57628625604267458078211/39343426545112261600000-\n \+ 6047516944575480929793/1405122376611152200000*6^(1/2),\na[15,7] =1691425739887134646566682083/96124421783968922002000000-\n 665 0904765188943408799448943/1345741904975564908028000000*6^(1/2),\na[15, 8]=4336304776662958741869045159571925984937/22710641428913956471088421 29832843500000+\n 53402805337711006499997411/208520160689094986 48000000*6^(1/2),\na[15,9]=-253787748130868165717697094592697839427/21 839824320042045467560266343639250000+\n 5933645037523445166666 379/897521918060373467750000*6^(1/2),\na[15,10]=2479701613016438411235 35377085814726576/1172015708277609400708896218214309893875-\n \+ 854444885403376103999958576/21640244757394101989947629875*6^(1/2),\na[ 15,11]=-52966635737697605455117615341/39091452325689672618552125990,\n a[15,12]=-443860399531273813022911271986650433425394696961417882001810 19629996219153064322715843/\n 9260962836182630467435788682851 407364301742047595974505212083507724183850275361816608,\na[15,13]=2546 9705993361596208461643776/2398429987672964863327432743875,\na[15,14]=- 10513898964163619809241937920000000/2553870660483628218384915527095907 53,\na[16,1]=141410754314242261138804477571/33519128225766632364045200 0000+\n 112443792511441850402669/1512551672706650408000000*6^(1 /2),\na[16,2]=0,\na[16,3]=0,\na[16,4]=0,\na[16,5]=0,\na[16,6]=-2794429 334116030797321/1137256896771917600000-\n 13769398374510385872 3/40616317741854200000*6^(1/2),\na[16,7]=27893647118108877562026663/27 78562296720245822000000-\n 103919596904457790641030873/38899872 154083441508000000*6^(1/2),\na[16,8]=101918684220971405061056556223280 61/4787568421535046196059430031281250+\n 1011994132602976653624 021/602746155289116328000000*6^(1/2),\na[16,9]=-8042480997962909835623 073494340513/1506681707334195127293078358250000+\n 11244379251 1441850402669/25943672957609370250000*6^(1/2),\na[16,10]=1712777545222 30541963559693237323184/1168212455792999305788440538350463625-\n \+ 16191906121647626457984336/625530609794743596276941125*6^(1/2),\na[ 16,11]=-11555892521024455643655681/15915117701775915252975590,\na[16,1 2]=-620367951196138613523419881305338943491581365345833611872299990616 582621/\n 196071800395784656980767050971315035686876839509793 271487889571798793056,\na[16,13]=-3540712440917044503420862464/6932876 1738078084467429795125,\na[16,14]=0,\na[16,15]=0,\n\nb[1]=173734691637 390647/4182794002754640000,\nb[2]=0,\nb[3]=0,\nb[4]=0,\nb[5]=0,\nb[6]= 0,\nb[7]=0,\nb[8]=-72263163141715044860361/169939769455665013040000,\n b[9]=14586697891849999254003/29700462390576849520000,\nb[10]=102209317 997264953344/225042304099487188475,\nb[11]=1883570537693211021/1872275 755054959100,\nb[12]=1710999041788984993956022337692530667432380407898 3341334325755071278367152457480027/\n 713814271258088281460765347 03173195056090466069876919551824450642795947233961810560,\nb[13]=-1067 8264099993989152768/2396652442219114419375,\nb[14]=1212545712242913280 000000/130535954653501897388343,\nb[15]=-369769046476619/6555223062234 0,\nb[16]=0,\n\n`b*`[1]=89673698740537/2188798536240000,\n`b*`[2]=0,\n `b*`[3]=0,\n`b*`[4]=0,\n`b*`[5]=0,\n`b*`[6]=0,\n`b*`[7]=0,\n`b*`[8]=-5 0462783649374452713/198296113717228720000,\n`b*`[9]=322482352625960258 97/70884158450064080000,\n`b*`[10]=-3511133566564032/35760734800490575 ,\n`b*`[11]=293286057408279/538164919532900,\n`b*`[12]=332632201005182 0410035389555466180704547593970934049395166930777066599/\n 137 33237974615224189190609759523239412338434762518802069756703639509120, \n`b*`[13]=-35480458347621122048/114126306772338781875,\n`b*`[14]=0,\n `b*`[15]=0,\n`b*`[16]=10688504668909/28093813123860\}:" }}}{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[9,16];" "6#&%\"TG6$\"\"*\"#;" }{TEXT -1 129 " denote the vector whose components are the principal error terms of the 16 s tage, order 9 scheme (the error terms of order 10)." }}{PARA 0 "" 0 " " {TEXT -1 5 "Let " }{XPPEDIT 18 0 "`T*`[8,16];" "6#&%#T*G6$\"\")\"#; " }{TEXT -1 146 " denote the vector whose components are the principa l error terms of the embedded 16 stage, order 8 scheme (the error term s of order 9) and let " }{XPPEDIT 18 0 "`T*`[9, 16];" "6#&%#T*G6$\"\" *\"#;" }{TEXT -1 100 " denote the vector whose components are the err or terms of order 10 of the embedded order 8 scheme." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Denote the 2-norm of th ese vectors by " }{XPPEDIT 18 0 "abs(abs(T[9,16]));" "6#-%$absG6#-F$6 #&%\"TG6$\"\"*\"#;" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "abs(abs(`T*`[8,1 6]));" "6#-%$absG6#-F$6#&%#T*G6$\"\")\"#;" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "abs(abs(`T*`[9,16]));" "6#-%$absG6#-F$6#&%#T*G6$\"\"*\" #;" }{TEXT -1 15 " respectively." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 8 "Define: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A[10] = abs(abs(T[9, 16]));" "6#/&%\"AG6#\"#5-%$abs G6#-F)6#&%\"TG6$\"\"*\"#;" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "B[10] \+ = abs(abs(`T*`[9,16]))/abs(abs(`T*`[8,16]));" "6#/&%\"BG6#\"#5*&-%$abs G6#-F*6#&%#T*G6$\"\"*\"#;\"\"\"-F*6#-F*6#&F/6$\"\")F2!\"\"" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "C[10] = abs(abs(`T*`[9,16]-T[9,16]))/abs(ab s(`T*`[8,16]));" "6#/&%\"CG6#\"#5*&-%$absG6#-F*6#,&&%#T*G6$\"\"*\"#;\" \"\"&%\"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[10];" "6#&%\"AG6#\"#5" }{TEXT -1 73 " is a minimum, if the embedded scheme is to be used for error control, " }{XPPEDIT 18 0 "B[10];" "6#&%\"BG6#\"#5" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "C [10];" "6#&%\"CG6#\"#5" }{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 0 168 "errterms9_16 := PrincipalEr rorTerms(9,16,'expanded'):\n`errterms9_16*` :=subs(b=`b*`,errterms9_16 ):\n`errterms8_16*` := subs(b=`b*`,PrincipalErrorTerms(8,16,'expanded' )):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 468 "nmB := 0: \nfor ct to nops(`errterms9_16*`) do\n \+ nmB := nmB+evalf(subs(ee,`errterms9_16*`[ct]))^2;\nend do:\nsnmB := sq rt(nmB):\ndnB := 0:\nfor ct to nops(`errterms8_16*`) do\n dnB := dnB +evalf(subs(ee,`errterms8_16*`[ct]))^2;\nend do:\nsdnB := sqrt(dnB):\n nmC := 0:\nfor ct to nops(errterms9_16) do\n nmC := nmC+(evalf(subs( ee,`errterms9_16*`[ct]))-evalf(subs(ee,errterms9_16[ct])))^2;\nend do: \nsnmC := sqrt(nmC):\n'B[10]'= evalf[8](snmB/sdnB);\n'C[10]'= evalf[8] (snmC/sdnB);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"BG6#\"#5$\")RX@K! \"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"#5$\")S_@K!\"(" }}} {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 36 "co efficients of the combined scheme " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10011 "ee := \{c[2]=1/49,\nc[3]= 64/705-16/14805*6^(1/2),\nc[4]=32/235-8/4935*6^(1/2),\nc[5]=3/7,\nc[6] =8/21+4/63*6^(1/2),\nc[7]=8/21-4/63*6^(1/2),\nc[8]=40/63,\nc[9]=10/21, \nc[10]=19/18,\nc[11]=7/9,\nc[12]=319999786/2170712113,\nc[13]=15/16, \nc[14]=39/40,\nc[15]=1,\nc[16]=1,\n\na[2,1]=1/49,\na[3,1]=-165952/149 1075+38896/10437525*6^(1/2),\na[3,2]=301312/1491075-7168/1491075*6^(1/ 2),\na[4,1]=8/235-2/4935*6^(1/2),\na[4,2]=0,\na[4,3]=24/235-2/1645*6^( 1/2),\na[5,1]=38937/44800+171/5600*6^(1/2),\na[5,2]=0,\na[5,3]=-149931 /44800-81/700*6^(1/2),\na[5,4]=65097/22400+477/5600*6^(1/2),\na[6,1]=1 76/5103-29/5103*6^(1/2),\na[6,2]=0,\na[6,3]=0,\na[6,4]=364520/1674351+ 87715/5023053*6^(1/2),\na[6,5]=1940224/15069159+779264/15069159*6^(1/2 ),\na[7,1]=4336/127575+479/127575*6^(1/2),\na[7,2]=0,\na[7,3]=0,\na[7, 4]=90731944/400648275-170142739/8413613775*6^(1/2),\na[7,5]=8245504/62 429373-22187008/437005611*6^(1/2),\na[7,6]=-3936/340025+11464/3060225* 6^(1/2),\na[8,1]=40/567,\na[8,2]=0,\na[8,3]=0,\na[8,4]=0,\na[8,5]=0,\n a[8,6]=160/567-10/567*6^(1/2),\na[8,7]=10/567*6^(1/2)+160/567,\na[9,1] =95/1344,\na[9,2]=0,\na[9,3]=0,\na[9,4]=0,\na[9,5]=0,\na[9,6]=295/1344 -115/2688*6^(1/2),\na[9,7]=295/1344+115/2688*6^(1/2),\na[9,8]=-15/448, \na[10,1]=52918819/138240000,\na[10,2]=0,\na[10,3]=0,\na[10,4]=0,\na[1 0,5]=0,\na[10,6]=-1453047743/103680000-4153586941/829440000*6^(1/2),\n a[10,7]=-1453047743/103680000+4153586941/829440000*6^(1/2),\na[10,8]=4 4599023/5120000,\na[10,9]=518179039/25920000,\na[11,1]=258780283/86184 00000+585428803/51710400000*6^(1/2),\na[11,2]=0,\na[11,3]=0,\na[11,4]= 0,\na[11,5]=0,\na[11,6]=19/25,\na[11,7]=1180508473123/443296800000-136 404911099/147765600000*6^(1/2),\na[11,8]=-106856621/190800000+58542880 3/2289600000*6^(1/2),\na[11,9]=-1260561943/591300000+585428803/8869500 00*6^(1/2),\na[11,10]=13167297224/792049782825-9366860848/237614934847 5*6^(1/2),\na[12,1]=30721339532858286796443076584747308497282486751295 7518186088963/\n 5126364212860621132939944111710304798478633358 572140981841000000+\n 11910753332681922251063975083241197446719 1643469020133053/\n 2913766490576471633400750336321335484766411 2851105748578125*6^(1/2),\na[12,2]=0,\na[12,3]=0,\na[12,4]=0,\na[12,5] =0,\na[12,6]=103548211821002304930266670003791849555056220508952456763 87169/\n 146467548938874889512569831763151565670818095959204028 052600000-\n 32684637880879071688859023194047542236847633606098 54606186699/\n 418478711253928255750199519323290187630908845597 72579443600000*6^(1/2),\na[12,7]=5423711572609568912980111975307776991 74343335570401188235055068891/\n 715703215922030799396778727922 657043395761853183510539934169000000-\n 25571039199675674205714 45567798380600417779349808054092568810687111/\n 100198450229084 31191554902190917198607540665944569147559078366000000*6^(1/2),\na[12,8 ]=-892225578009519154676238995901578841509244882985862325637048827/\n \+ 38813900468801845720831005417235164902766795429189067433939000 000+\n 2036738819888608704931939739234244763388977103320244275 2063/\n 220613748572218566528913954035758258132313997301229239 234375*6^(1/2),\na[12,9]=-55068860523577033803486364291759519582505063 3798073448796494914/\n 835322740042020854251374821774223772966 384453517335472487484375+\n 7241738026270608728646896850610648 0476052519229164240896224/\n 303864219731546327483221106502082 129125640034018674235171875*6^(1/2),\na[12,10]=12339689511549573843454 9229715587040998178289648593721222723693824/\n 169604664222143 15122555088502288155618001490986047888958907461648625-\n 87815 60217119727637264447549372070053517105489683916369731584/\n 61 69685857480653009296139869875647732994358307038155314262445125*6^(1/2) ,\na[12,11]=-241890129426298647138485610377551406165672225318019246672 /\n 366168872347187223781424579407878914177045239898010070131 5,\na[13,1]=45077846760256141387004276823/1103158941439921335917399244 80+\n 1493491403898138129099/13100021190238236835840*6^(1/2),\n a[13,2]=0,\na[13,3]=0,\na[13,4]=0,\na[13,5]=0,\na[13,6]=28530732123103 900185/9849639992660328448-\n 15062887306567756845/562836571009 1616256*6^(1/2),\na[13,7]=530875502237315716994493/2406478113921046675 4560-\n 8920823473649531766699837/1347627743795786138255360*6^( 1/2),\na[13,8]=-155850251753928802974915857362119/17401508660534001647 7040216637440+\n 13441422635083243161891/522030919610997407744 0*6^(1/2),\na[13,9]=-493074073683718697930133408597/276027121164081940 83051274240+\n 1493491403898138129099/224694912332563742720*6^ (1/2),\na[13,10]=200609996314078300148532240828075/1019933691979646265 167106381709312-\n 336035565877081079047275/846506642479497367 7551616*6^(1/2),\na[13,11]=-1259978731825102407292471875/9476420756003 43143202947072,\na[13,12]=-1939162142353174689879923915990531880493671 33486207120889311375/\n 4274809234945508811134400745541723364 1020280098816132254793728,\na[14,1]=3671662121209803609393501868710542 5505961/72248275402215258274603114496000000000000+\n 1221461237 263884679751555607/9994523002806272000000000000*6^(1/2),\na[14,2]=0,\n a[14,3]=0,\na[14,4]=0,\na[14,5]=0,\na[14,6]=8283471074731862302286097/ 7514678949478400000000000-\n 15668946773152185221466849/4294102 256844800000000000*6^(1/2),\na[14,7]=397408075485926915758262202639/18 359970961922048000000000000-\n 6493922587539771225254133441201/ 1028158373867634688000000000000*6^(1/2),\na[14,8]=24352784939030479093 70803905780425755549361787/6951941874960961660657094977773568000000000 000+\n 10993151135374962117764000463/39827798432235520000000000 00*6^(1/2),\na[14,9]=-558123239069103416347126929975086912148938889/34 184771406932232290260692794368000000000000+\n 1221461237263884 679751555607/171428613534976000000000000*6^(1/2),\na[14,10]=3067345861 61727173704146823378382330889519/1406266800626214225846914661737584000 000000-\n 10993151135374962117764000463/2583333259519950463120 00000000*6^(1/2),\na[14,11]=-163845778835264660255510638493965671483/1 14196173990354810149157741209600000000,\na[14,12]=-3797270986915804513 04129337662817719784451899678250630685021894233169654109523405323/\n \+ 74980943726337976053062781324716052947047103330285761853079535 691020475760640000000,\na[14,13]=-235412270220829707518634576/10004921 377982463725322265625,\na[15,1]=1564746779443331677794753119798867/286 4199537049451295689544004000000+\n 5933645037523445166666379/52 326757304999307928000000*6^(1/2),\na[15,2]=0,\na[15,3]=0,\na[15,4]=0, \na[15,5]=0,\na[15,6]=-57628625604267458078211/39343426545112261600000 -\n 6047516944575480929793/1405122376611152200000*6^(1/2),\na[ 15,7]=1691425739887134646566682083/96124421783968922002000000-\n \+ 6650904765188943408799448943/1345741904975564908028000000*6^(1/2),\n a[15,8]=4336304776662958741869045159571925984937/227106414289139564710 8842129832843500000+\n 53402805337711006499997411/2085201606890 9498648000000*6^(1/2),\na[15,9]=-2537877481308681657176970945926978394 27/21839824320042045467560266343639250000+\n 59336450375234451 66666379/897521918060373467750000*6^(1/2),\na[15,10]=24797016130164384 1123535377085814726576/1172015708277609400708896218214309893875-\n \+ 854444885403376103999958576/21640244757394101989947629875*6^(1/2) ,\na[15,11]=-52966635737697605455117615341/390914523256896726185521259 90,\na[15,12]=-4438603995312738130229112719866504334253946969614178820 0181019629996219153064322715843/\n 92609628361826304674357886 82851407364301742047595974505212083507724183850275361816608,\na[15,13] =25469705993361596208461643776/2398429987672964863327432743875,\na[15, 14]=-10513898964163619809241937920000000/25538706604836282183849155270 9590753,\na[16,1]=141410754314242261138804477571/335191282257666323640 452000000+\n 112443792511441850402669/1512551672706650408000000 *6^(1/2),\na[16,2]=0,\na[16,3]=0,\na[16,4]=0,\na[16,5]=0,\na[16,6]=-27 94429334116030797321/1137256896771917600000-\n 137693983745103 858723/40616317741854200000*6^(1/2),\na[16,7]=278936471181088775620266 63/2778562296720245822000000-\n 103919596904457790641030873/388 99872154083441508000000*6^(1/2),\na[16,8]=1019186842209714050610565562 2328061/4787568421535046196059430031281250+\n 10119941326029766 53624021/602746155289116328000000*6^(1/2),\na[16,9]=-80424809979629098 35623073494340513/1506681707334195127293078358250000+\n 112443 792511441850402669/25943672957609370250000*6^(1/2),\na[16,10]=17127775 4522230541963559693237323184/1168212455792999305788440538350463625-\n \+ 16191906121647626457984336/625530609794743596276941125*6^(1/2) ,\na[16,11]=-11555892521024455643655681/15915117701775915252975590,\na [16,12]=-6203679511961386135234198813053389434915813653458336118722999 90616582621/\n 1960718003957846569807670509713150356868768395 09793271487889571798793056,\na[16,13]=-3540712440917044503420862464/69 328761738078084467429795125,\na[16,14]=0,\na[16,15]=0,\n\nb[1]=1737346 91637390647/4182794002754640000,\nb[2]=0,\nb[3]=0,\nb[4]=0,\nb[5]=0,\n b[6]=0,\nb[7]=0,\nb[8]=-72263163141715044860361/1699397694556650130400 00,\nb[9]=14586697891849999254003/29700462390576849520000,\nb[10]=1022 09317997264953344/225042304099487188475,\nb[11]=1883570537693211021/18 72275755054959100,\nb[12]=17109990417889849939560223376925306674323804 078983341334325755071278367152457480027/\n 7138142712580882814607 6534703173195056090466069876919551824450642795947233961810560,\nb[13]= -10678264099993989152768/2396652442219114419375,\nb[14]=12125457122429 13280000000/130535954653501897388343,\nb[15]=-369769046476619/65552230 622340,\nb[16]=0,\n\n`b*`[1]=89673698740537/2188798536240000,\n`b*`[2] =0,\n`b*`[3]=0,\n`b*`[4]=0,\n`b*`[5]=0,\n`b*`[6]=0,\n`b*`[7]=0,\n`b*`[ 8]=-50462783649374452713/198296113717228720000,\n`b*`[9]=3224823526259 6025897/70884158450064080000,\n`b*`[10]=-3511133566564032/357607348004 90575,\n`b*`[11]=293286057408279/538164919532900,\n`b*`[12]=3326322010 051820410035389555466180704547593970934049395166930777066599/\n \+ 13733237974615224189190609759523239412338434762518802069756703639509 120,\n`b*`[13]=-35480458347621122048/114126306772338781875,\n`b*`[14]= 0,\n`b*`[15]=0,\n`b*`[16]=10688504668909/28093813123860\}:" }}}{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 s tability function R for the 16 stage, order 9 scheme is given (approxi mately) as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "expand(subs(ee,StabilityFunction(9,16,'exp anded'))):\nmap(convert,evalf[28](%),rational,24):\nR := unapply(%,z): \n'R(z)'=R(z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"RG6#%\"zG,B\"\" \"F)F'F)*&#F)\"\"#F)*$)F'F,F)F)F)*&#F)\"\"'F)*$)F'\"\"$F)F)F)*&#F)\"#C F)*$)F'\"\"%F)F)F)*&#F)\"$?\"F)*$)F'\"\"&F)F)F)*&#F)\"$?(F)*$)F'F1F)F) F)*&#F)\"%S]F)*$)F'\"\"(F)F)F)*&#F)\"&?.%F)*$)F'\"\")F)F)F)*&#F)\"'!)G OF)*$)F'\"\"*F)F)F)*&#\"*=e8I$\"1Z^qU#HJL\"F)*$)F'\"#5F)F)F)*&#\")-L(e '\"0xOj%[m*)GF)*$)F'\"#6F)F)F)*&#\")(\\rd$\"0_S5 " 0 "" {MPLTEXT 1 0 26 "z0 := newton(R(z)=1,z=- 4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$!++RzRR!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 302 "z0 \+ := newton(R(z)=1,z=-4):\np1 := plot([R(z),1],z=-4.49..0.49,color=[red, blue]):\np2 := plot([[[z0,1]]$3],style=point,symbol=[circle,cross,diam ond],color=black):\np3 := plot([[z0,0],[z0,1]],linestyle=3,color=COLOR (RGB,0,.5,0)):\nplots[display]([p1,p2,p3],view=[-4.49..0.49,-0.07..1.4 7],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 253 253 {PLOTDATA 2 "6+-%'CURVESG6$7fn7$$!3A++++++!\\%!#<$\"3r1Axh-Y5gF*7$$!3? +D1VD'GY%F*$\"3-zxIADYKbF*7$$!3;+]7'3DdV%F*$\"3)41;p\\]'*3&F*7$$!38+v= Hwe3WF*$\"35d!R_r^'zYF*7$$!35++Ds,X\"Q%F*$\"3#R4#\\T%R-I%F*7$$!3[v=2;! QyN%F*$\"3k\\3`[+P$*RF*7$$!3(*\\P*)feAMVF*$\"3&H-#\\*Hxnq$F*7$$!3ZCcr. Ph5VF*$\"3oL\\wpiBRMF*7$$!3%)*\\Pva,qG%F*$\"33GCD\"Q*e*=$F*7$$!33]iDpG *QB%F*$\"3)H`,-*f!zo#F*7$$!3I+](4>%y!=%F*$\"3oD&e2*f'*fAF*7$$!3l++!RZB t7%F*$\"3QOJy^De$*=F*7$$!35+]#oviQ2%F*$\"3[!>e;=CHe\"F*7$$!3$*\\i?Hhl? SF*$\"3ua;#)f(e8K\"F*7$$!3A+ve,&\\u'RF*$\"3#ow@ct&f+6F*7$$!3?+v8#z!zoQ F*$\"3/k!z\"R'*p*z(!#=7$$!3;+v.8ajmPF*$\"3;BHLD#QxU&Fbp7$$!3C+v$Q<')4m $F*$\"39A;hP?d=PFbp7$$!3G+v3`dnbNF*$\"3-+l*fq$zdDFbp7$$!37++0T5NZMF*$ \"3q]8!4R!ykN$F*$\"3)R=il_A]I\"Fbp7$$!3E++:Qn_WKF*$\" 3KEUhB1B[(*!#>7$$!3#****\\s&QnOJF*$\"3!\\jzhK'e?yFar7$$!3&****\\=iPF.$ F*$\"39!))*zIesfoFar7$$!3Q+v$Gc`$QHF*$\"3)=.(>`fU7lFar7$$!3+++l(y@h#GF *$\"33\"RmXG=za'Far7$$!3=++g_n/JFF*$\"3cGT!)*e%QZoFar7$$!3C+vt%)=X?EF* $\"3<>7OIf4>uFar7$$!3D++5(ocD_#F*$\"3?YK]*pk*z!)Far7$$!3#**\\P(R,::CF* $\"3gg=PQ=YV*)Far7$$!3&)*\\7sqtGJ#F*$\"3yE8x-j'y))*Far7$$!30+]Pz*eh?#F *$\"3u`=](Hb(*4\"Fbp7$$!3l*\\([Y:;3@F*$\"3^gb`%GMK@\"Fbp7$$!3$***\\7w! eC+#F*$\"3Q\")RB[6'*[8Fbp7$$!3%**\\Py(=m#*=F*$\"3X1v#y;rf]\"Fbp7$$!3%) *\\iW'R3(z\"F*$\"3/<5JU!Gtl\"Fbp7$$!3-+]d*>dQp\"F*$\"3%)=e-\\gzP=Fbp7$ $!3)*****p]Q@(e\"F*$\"3'=YwJ&4yW?Fbp7$$!3I+]FRT)G[\"F*$\"3#GP4r$QtpAFb p7$$!3')*\\P*y(R>Q\"F*$\"32T.*[4m3^#Fbp7$$!3++]Kx#e)p7F*$\"3Zk,5ES&H\"QFbp7$$!3](***\\tEbw&)Fbp$\"3t (eIa9a:C%Fbp7$$!39-]P2EBuvFbp$\"3]tIgh-t)o%Fbp7$$!3>)*\\(GL>l_'Fbp$\"3 >TYn$yJm?&Fbp7$$!3z++]-\")=-bFbp$\"3K`RY8eBodFbp7$$!3$*)*\\PM#3)HWFbp$ \"3!)Gh$e?&>@kFbp7$$!3;(****f'*ypR$Fbp$\"39o#*yMQ&)>rFbp7$$!3L)**\\U'= wSBFbp$\"3eS`kr_,8zFbp7$$!3?$*\\Pt2H$H\"Fbp$\"3!)y#4siZoy)Fbp7$$!33R** **pit2LFar$\"3+r)*)36PYn*Fbp7$$\"3Z^+]FkzBxFar$\"3!RZ/%=\"*H!3\"F*7$$ \"35*****z2`!f\"F*7$$\"3l+]i_F06GFbp$\"3j=eZVIfC8F*7 $$\"3\\.]Pt1&z\"QFbp$\"3s2$)3Y=\"\\Y\"F*7$$\"3!***************[Fbp$\"3 *fe>+A;Bj\"F*-%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!F^^lF]^l-F$6$7S7$F( $\"\"\"F^^l7$F=Fc^l7$FQFc^l7$FenFc^l7$F_oFc^l7$FioFc^l7$F^pFc^l7$FdpFc ^l7$FipFc^l7$F^qFc^l7$FcqFc^l7$FhqFc^l7$F]rFc^l7$FcrFc^l7$FhrFc^l7$F]s Fc^l7$FbsFc^l7$FgsFc^l7$F\\tFc^l7$FatFc^l7$FftFc^l7$F[uFc^l7$F`uFc^l7$ FeuFc^l7$FjuFc^l7$F_vFc^l7$FdvFc^l7$FivFc^l7$F^wFc^l7$FcwFc^l7$FhwFc^l 7$F]xFc^l7$FbxFc^l7$FgxFc^l7$F\\yFc^l7$FayFc^l7$FfyFc^l7$F[zFc^l7$F`zF c^l7$FezFc^l7$FjzFc^l7$F_[lFc^l7$Fd[lFc^l7$Fi[lFc^l7$F^\\lFc^l7$Fc\\lF c^l7$Fh\\lFc^l7$F]]lFc^l7$Fb]lFc^l-Fg]l6&Fi]lF]^lF]^lFj]l-F$6&7#7$$!3= ++++RzRRF*Fc^l-%'SYMBOLG6#%'CIRCLEG-Fg]l6&Fi]lF^^lF^^lF^^l-%&STYLEG6#% &POINTG-F$6&Fial-F^bl6#%&CROSSGFablFcbl-F$6&Fial-F^bl6#%(DIAMONDGFablF cbl-F$6%7$7$F[blF]^lFjal-%&COLORG6&Fi]lF]^l$\"\"&!\"\"F]^l-%*LINESTYLE G6#\"\"$-%+AXESLABELSG6%Q\"z6\"Q!Fcdl-%%FONTG6#%(DEFAULTG-Ffdl6$%*HELV ETICAG\"\"*-%%VIEWG6$;$!$\\%!\"#$\"#\\Fcel;$!\"(Fcel$\"$Z\"Fcel" 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 49 "The following picture shows the stability region." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1450 "R := z ->add(z^j/j!,j=0..9)+\n 3301358 18/1333129242705147*z^10+65873302/288966484633677*z^11+\n 35771497 /749767917104052*z^12-20093725/1183463108774183*z^13+\n 2452151/36 97406882441710*z^14-504637/276239108715109746*z^15:\npts := []: z0 := \+ 0:\nfor ct from 0 to 320 do\n zz := newton(R(z)=exp(ct*Pi/20*I),z=z0 ):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np1 := plot(pts,color=COLOR(RGB,.1,.28,.55)):\np2 := plots[polygonplot]([seq ([pts[i-1],pts[i],[-2.3,0]],i=2..nops(pts))],\n style=patchno grid,color=COLOR(RGB,.2,.55,.95)):\npts := []: z0 := 1.5+4*I:\nfor ct \+ from 0 to 40 do\n zz := newton(R(z)=exp(ct*Pi/20*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np3 := plot(pts,co lor=COLOR(RGB,.1,.28,.55)):\np4 := plots[polygonplot]([seq([pts[i-1],p ts[i],[1.42,4.03]],i=2..nops(pts))],\n style=patchnogrid,colo r=COLOR(RGB,.2,.55,.95)):\npts := []: z0 := 1.5-4*I:\nfor ct from 0 to 60 do\n zz := newton(R(z)=exp(ct*Pi/30*I),z=z0):\n z0 := zz:\n \+ pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np5 := plot(pts,color=COLOR (RGB,.1,.28,.55)):\np6 := plots[polygonplot]([seq([pts[i-1],pts[i],[1. 42,-4.03]],i=2..nops(pts))],\n style=patchnogrid,color=COLOR( RGB,.2,.55,.95)):\np7 := plot([[[-4.59,0],[2.09,0]],[[0,-4.59],[0,4.59 ]]],color=black,linestyle=3):\nplots[display]([p||(1..7)],view=[-4.59. .2.09,-4.59..4.59],font=[HELVETICA,9],\n labels=[`Re(z)`, `Im(z)`],axes=boxed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 501 551 551 {PLOTDATA 2 "6/-%'CURVESG6$7]_l7$$\"\"!F)F(7$F($ \"3++++Fjzq:!#=7$F($\"3)******Rl#fTJF-7$F($\"3D+++\")*)Q7ZF-7$$\"3'*** ***\\?*GpQ!#F$\"3%)******>`=$G'F-7$$\"3G+++tb(G1(!#E$\"3i*****\\x\")R& yF-7$$\"3[+++ue()))p!#D$\"3!)*****H$*yZU*F-7$$\"3')*******oWlp%!#C$\"3 -+++1zb*4\"!#<7$$\"3>+++v*30S#!#B$\"3++++d)QmD\"FL7$$\"3m+++zz&*y**FP$ \"3$******\\.APT\"FL7$$\"37+++:HvAN!#A$\"3)******\\!)33d\"FL7$$\"31+++ ;=Y(3\"!#@$\"35+++(o'*ys\"FL7$$\"3#******\\b5d*HF[o$\"3++++]f'\\)=FL7$ $\"33+++*>tlY(F[o$\"34+++%*Q%>/#FL7$$\"35+++&GHvp\"!#?$\"3-+++))Ri)>#F 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" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 148 "We can distort the bou ndary curve horizontally by taking the 11th root of the real part of p oints along the curve. In this way we see that there is " }{TEXT 260 19 "no largest interval" }{TEXT -1 97 " on the nonnegative imaginary a xis that contains the origin and lies inside the stability region." }} {PARA 0 "" 0 "" {TEXT -1 120 "However, the stability region intersects the nonnegative imaginary axis in an interval that does not contain t he origin." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 480 "R := z ->add(z^j/j!,j=0..9)+\n 330135818/13331 29242705147*z^10+65873302/288966484633677*z^11+\n 35771497/7497679 17104052*z^12-20093725/1183463108774183*z^13+\n 2452151/3697406882 441710*z^14-504637/276239108715109746*z^15:\nDigits := 25:\npts := []: z0 := 0:\nfor ct from 0 to 270 do\n zz := newton(R(z)=exp(ct*Pi/200 *I),z=z0):\n z0 := zz:\n pts := [op(pts),[surd(Re(zz),11),Im(zz)]] :\nend do:\nplot(pts,color=COLOR(RGB,0,.4,1),thickness=2,font=[HELVETI CA,9]);\nDigits := 10:" }}{PARA 13 "" 1 "" {GLPLOT2D 287 308 308 {PLOTDATA 2 "6(-%'CURVESG6#7[\\l7$$\"\"!F)F(7$$!:]Ljt.KG#\\$)Q8Z!#F$\" :A8B>m*[zEjzq:!#E7$$!:!H/8!RK)*e=Fu%))F-$\":ALYQKz*e`EfTJF07$$!:Y&GD*4 @`Hh)>y7F0$\":&o_x&)*o%Q!)*)Q7ZF07$$!:'[+_*)Q*yf_U'e;F0$\":`^JykezrI&= $G'F07$$!:)fnE?I5RXS2H?F0$\":<+L7J[uRj\")R&yF07$$!:WvU7*3_C*)p1\"R#F0$ \":?i\\N)z$p2'zxC%*F07$$!:#oj3\\8l4fS]XFF0$\":LS)))pFkvGub*4\"!#D7$$!: -;>(Q)pI**poG4$F0$\":;QRzv\"fVhqjc7FO7$$!:vOx\\`)[F7/QLMF0$\":tS.%*zS: Tp;PT\"FO7$$!:-._Q@***[zEjzq:FO7$$!:1L3E1=2.4qP4%F0 $\":4[?*o&Ru%ff(ys\"FO7$$!:U%>;5(\\eq'y?8WF0$\":zZ$pT+R:#fb\\)=FO7$$!: 6'yx^Smb_l\"\\s%F0$\":'33pcCM$[AN?/#FO7$$!:pj=K.>>r@b#G]F0$\":$Q;kK))H ^d[6*>#FO7$$!:>W@N/lox&3OA`F0$\":0%fsuGE>!\\%>cBFO7$$!:^@tec]]6_$31cF0 $\":Y2[u4TsG7uK^#FO7$$!:e?tS]_zte#)y(eF0$\":G\\(HMXCbbPNqEFO7$$!:vF#Qe z2d<#ec8'F0$\":*HTso7HB)QLu#GFO7$$!:<9P*3`(37WmkP'F0$\":<<9&*3594-8X)H FO7$$!:p))e4H:`,U3gf'F0$\":?&\\jbckf`EfTJFO7$$!:9a;+%RrL:xj(y'F0$\":M- /Cll!G'Gs')H$FO7$$!:Nx!4+[Zpf9?SpF0$\":LtzP`qn*=>vbMFO7$$!:&p&R(R^8+b$ 4D.(F0$\":g*G(fd1f;bJGh$FO7$$!:ml)**G%e^x?P[,(F0$\":OWrm6$oN%=6*pPFO7$ $!:O1Rj>>V@,N4q'F0$\":R'33B]R1<3*p#RFO7$$\":F')QUb/%Rh)ftd'F0$\":F?5iuj(G`K' RR)F0$\":(>_B(4R!H:(H#)R%FO7$$\":B%Q@WSgN-E;l*)F0$\":>4V1IU$4[$4`b%FO7 $$\":^@3(>C-0X)4Q[*F0$\":NmAvXkY4)*)Q7ZFO7$$\":$QT+va\")=H5qs**F0$\":F A,87inQho%p[FO7$$\":u&3+_L))z?pFW5FO$\":)34/*39zoC[l-&FO7$$\":FVk'*Her m&y+!4\"FO$\":>ai\\n]5+)yi$=&FO7$$\":&*)zwwA=()yD$[8\"FO$\":tvNG!)3*H8 vqS`FO7$$\":KeK(**y5Gc>**y6FO$\":A#[=[o@zYry(\\&FO7$$\":#)=Rd%pUY84lA7 FO$\":24e(eN\"\\0ym[l&FO7$$\":u:lr(*z.&*yEfE\"FO$\":j!eB\"z0WYTY>\"eFO 7$$\":wZ%)R)\\0X_c!*38FO$\":#Ry%)>-(o\"\\g-pfFO7$$\":>ISVb&o-;Al^8FO$ \":LX`=$G'FO7$$\": s;x7Vl:5?OmV\"FO$\":!px@ZaX+=LUUanFO7$$\":A#)o'H]<*f;6Lc\" FO$\":9ujZWJ>P(Q]6pFO7$$\":%p8[UaaXy&3ag\"FO$\":T#=EOg'*>;NeoqFO7$$\": Q(>\\%QjA$*3nuk\"FO$\":>1CkP4O5;jcA(FO7$$\":eY4&3SnQ8y\\*o\"FO$\":kGs7 Wx=(3Gu#Q(FO7$$\":&p?B\"\\odh+5:t\"FO$\":%o@0[=T#)fC#)RvFO7$$\":\")Q%G \"pmNhS6Nx\"FO$\":kVehj9F]6-pp(FO7$$\":\"GK!yb)=+.&3b\"=FO$\":!e<,`_^6 v<)R&yFO7$$\":FDr)f0N%*\\n]d=FO$\":CD*R/wF+T916!)FO7$$\":_^*)Q[FO$\":*=7kEpHd%z?_K)FO7 $$\":crdIh+Kgr]N)>FO$\":L1V>@Ay[[+B[)FO7$$\":E$RP$z/FBm#fD?FO$\":()eUD 9\\pi=!QR')FO7$$\":6f$\\`vN(\\S_w1#FO$\":$y.z>/!y0!*fkz)FO7$$\":Q3rA5e ^'4=t4@FO$\":*o/Ooid))H'RN&*)FO7$$\":+9\\KKA))HSK=:#FO$\":\"H#[1])>bw$ >16*FO7$$\":k\\$HI2'*)fTbR>#FO$\":>4serATK9*pn#*FO7$$\":>tFG*Q=j?=5OAF O$\":&[c+hMJ&H$*yZU*FO7$$\":oP+gNqJ3Os#yAFO$\":-5wW'RP0\\(e=e*FO7$$\": 1Q%ea1MxyvY?BFO$\":-([:P$=3`fQ*Q(*FO7$$\":pa'fHuwPPyoiBFO$\":06([q)p9f Z=g*)*FO7$$\":d:E)pE-bRL$\\S#FO$\":@:r(>kRbR)4`+\"!#C7$$\":z&3H&4hGZ9/ sW#FO$\":I>4n0LNf$y,@5Fg_l7$$\":E)Qh`%[_J=+&*[#FO$\":C\"f*[#=mIPesO5Fg _l7$$\":B.Xv[E2#o7#=`#FO$\":O?3,\"GlHWQV_5Fg_l7$$\":ld\"yaR082r;uDFO$ \":nAA'[,]fd=9o5Fg_l7$$\":VtD)pI;K5t`;EFO$\":>KA0+rcz()\\Q3\"Fg_l7$$\" :IQxez#[X*RJ*eEFO$\":z&=*3)zV?1zb*4\"Fg_l7$$\":hC,EU=kX\")[8q#FO$\":Pi Y4N(=BVfE:6Fg_l7$$\":!=,Qod4C?*)yVFFO$\":t/,]3%p+!*R(48\"Fg_l7$$\":OsP VUki5,^iy#FO$\":/F.Ji`tv/#oY6Fg_l7$$\":&Gy-c!=ajJM(GGFO$\":^-7\\oe`q6! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "Digits := 2 0:\nz0 := 0.4*I:\nfor ct from 63 to 66 do\n newton(R(z)=exp(ct*Pi/50 0*I),z=z0);\nend do;\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #^$$!55k<[T/!fS?*!#L$\"5_pKogVnSeR!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!5?y0@#f\"!#L$\"55?cVp'fQ7-%!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"54p(QzN.tq'**!#L$\"5r@;Xy\\/2%3%!#?" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#^$$\"522FQj*> " 0 "" {MPLTEXT 1 0 114 "Digits := 15:\nz0 := 3.35*I:\nfor ct from 113 to 116 do\n newto n(R(z)=exp(ct*Pi/100*I),z=z0);\nend do;\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0-F[z?a*[!#<$\"0;8I+yMK$!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0[HT*H/^=!#<$\"0f(=*==.M$!#9" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#^$$!0(o&H7c3R\"!#<$\"0c>T.6nN$!#9" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#^$$!0v/2g=?#[!#<$\"0I5a)\\msL!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 "Then we apply the bisect ion method to calculate the parameter value associated with each inter section point." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 332 "Digits := 20:\nreal_part := proc(u)\n Re(newton(R(z)=exp(u*Pi*I),z=0.4*I))\ne nd proc:\nu0 := bisect('real_part'(u),u=0.126..0.132);\nnewton(R(z)=ex p(u0*Pi*I),z=0.4*I);``;\nreal_part := proc(u)\n Re(newton(R(z)=exp(u *Pi*I),z=3.35*I))\nend proc:\nu0 := bisect('real_part'(u),u=1.13..1.16 );\nnewton(R(z)=exp(u0*Pi*I),z=3.35*I);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"5***3xZ$RXZ#G\"!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"5j)QXK8f7!HS!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"5t`0.]KCyX6!#>" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"5:\"*fGFb7&)\\L!#>" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 73 "stability region intersects the nonegative imaginary axis in the i nterval" }{TEXT -1 39 " [ 0.4029, 3.3499 ] (approximately)." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "#-------- ----------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 97 "The stability function R* for the 16 stage, o rder 8 scheme is given (approximately) as follows." }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 145 "expand(subs(ee,subs(b=`b*`,StabilityFunctio n(8,16,'expanded')))):\nmap(convert,evalf[28](%),rational,24):\n`R*` : = unapply(%,z):\n'`R*`(z)'=`R*`(z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6 #/-%#R*G6#%\"zG,@\"\"\"F)F'F)*&#F)\"\"#F)*$)F'F,F)F)F)*&#F)\"\"'F)*$)F '\"\"$F)F)F)*&#F)\"#CF)*$)F'\"\"%F)F)F)*&#F)\"$?\"F)*$)F'\"\"&F)F)F)*& #F)\"$?(F)*$)F'F1F)F)F)*&#F)\"%S]F)*$)F'\"\"(F)F)F)*&#F)\"&?.%F)*$)F' \"\")F)F)F)*&#\"*u'pfB\"0\"*eQST$z6F)*$)F'\"\"*F)F)F)*&#\"*P.)yE\"0%y \\V7qHKF)*$)F'\"#5F)F)F)*&#\"*v^A(=\"0Zy\\OhzP)F)*$)F'\"#6F)F)F)*&#\") dm%z'\"1\"4!)[3 " 0 "" {MPLTEXT 1 0 31 "z_0 := newton(`R*`(z)=-1,z=-4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$z_0G$!+T,MDR!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 322 "z_0 := newton(`R*`(z)=-1,z =-4):\np_1 := plot([`R*`(z),-1],z=-4.39..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=[-4.39..0.49,-1.47..1.47], font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 369 264 264 {PLOTDATA 2 "6+-%'CURVESG6$7Y7$$!3o*************Q%!#<$!3>R]:;ogCUF*7$$ !3Amm\">Z2MO%F*$!3%)3;FpW&>\"RF*7$$!3lLL$Q%\\\"oL%F*$!3*\\8bE!*H.i$F*7 $$!3?++v:CA5VF*$!31VxZ^lW[LF*7$$!3ummm())HOG%F*$!3u[AJZ')3&4$F*7$$!3\" )**\\naQNPUF*$!3](4VtJ))\\p#F*7$$!3'GL$o@y2\">%F*$!3MmzXcW%>M#F*7$$!3h **\\s#eN!RTF*$!3[c>V(y[[*>F*7$$!3MmmwVL*p3%F*$!33Z?VE,U%p\"F*7$$!3Vmm1 NhgMSF*$!3>*>i#=FJL9F*7$$!3_mmOE*=A)RF*$!3-jb07hX37F*7$$!3#HL$[$[Uz(QF *$!3UgcBQ./5&)!#=7$$!3Ammhs[E\"y$F*$!3W+.P0PO`gF^o7$$!3U***\\j!3;\"o$F *$!3/A1.j!*epTF^o7$$!3Imm\"y.Lwd$F*$!3KA_s%*)*[`FF^o7$$!3a***\\TGPWZ$F 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^]lF^]lF[]l-F$6&7#7$$!3&******49S`#RF*Fd]l-%'SYMBOLG6#%'CIRCLEG-Fh\\l6 &Fj\\lF_]lF_]lF_]l-%&STYLEG6#%&POINTG-F$6&Fj`l-F_al6#%&CROSSGFbalFdal- F$6&Fj`l-F_al6#%(DIAMONDGFbalFdal-F$6%7$7$F\\alF^]lF[al-%&COLORG6&Fj\\ lF^]l$\"\"&Fe]lF^]l-%*LINESTYLEG6#\"\"$-%+AXESLABELSG6%Q\"z6\"Q!Fccl-% %FONTG6#%(DEFAULTG-Ffcl6$%*HELVETICAG\"\"*-%%VIEWG6$;$!$R%!\"#$\"#\\Fc dl;$!$Z\"Fcdl$\"$Z\"Fcdl" 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 "T he following picture shows the stability region for the 16 stage, orde r 8 scheme. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1465 "`R*` := z -> add(z^j/j!,j=0..8)+\n 235969674 /117934140385891*z^9+267880337/322970124349784*z^10+\n 187225175/8 37796136497847*z^11-67946657/1076917084880091*z^12+\n 15013713/634 5666097879765*z^13-711890/109586961553712731*z^14:\npts := []: z0 := 0 :\nfor ct from 0 to 280 do\n zz := newton(`R*`(z)=exp(ct*Pi/20*I),z= z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np_1 := plot(pts,color=COLOR(RGB,0,.14,.35)):\np_2 := plots[polygonplot]([ seq([pts[i-1],pts[i],[-1.9,0]],i=2..nops(pts))],\n style=patc hnogrid,color=COLOR(RGB,.1,.35,.75)):\npts := []: z0 := 1+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(pts),[Re(zz),Im(zz)]]:\nend do:\np_3 := plot (pts,color=COLOR(RGB,0,.14,.35)):\np_4 := plots[polygonplot]([seq([pts [i-1],pts[i],[.99,4.04]],i=2..nops(pts))],\n style=patchnogri d,color=COLOR(RGB,.1,.35,.75)):\npts := []: z0 := 1-4.2*I:\nfor ct fro m 0 to 60 do\n zz := newton(`R*`(z)=exp(ct*Pi/30*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np_5 := plot(pts,c olor=COLOR(RGB,0,.14,.35)):\np_6 := plots[polygonplot]([seq([pts[i-1], pts[i],[.99,-4.04]],i=2..nops(pts))],\n style=patchnogrid,col or=COLOR(RGB,.1,.35,.75)):\np_7 := plot([[[-4.39,0],[1.79,0]],[[0,-4.4 9],[0,4.49]]],color=black,linestyle=3):\nplots[display]([p_||(1..7)],v iew=[-4.39..1.79,-4.49..4.49],font=[HELVETICA,9],\n labels= [`Re(z)`,`Im(z)`],axes=boxed,scaling=constrained);" }}{PARA 13 "" 1 " " {GLPLOT2D 396 515 515 {PLOTDATA 2 "6/-%'CURVESG6$7e\\l7$$\"\"!F)F(7$ F($\"3++++Fjzq:!#=7$F($\"3)******Rl#fTJF-7$$\"3'******\\zB0*o!#F$\"3?+ ++()*)Q7ZF-7$$\"3#******\\;.o>\"!#D$\"37+++x`=$G'F-7$$\"3/+++2e')z5!#C $\"3q*****R'>)R&yF-7$$\"37+++)QD**R'F@$\"3H+++H%yZU*F-7$$\"3<+++b*zv\" G!#B$\"3#*******Hqb*4\"!#<7$$\"31+++&H`+*)*FK$\"3#******R\\LmD\"FN7$$ \"33+++7)Qg)G!#A$\"3(*******y\"*p89FN7$$\"3S******)zT5;(FW$\"31+++y4tq :FN7$$\"3,+++kl\"*=:!#@$\"3#******\\Gsws\"FN7$$\"3$******4lMDq#F\\o$\" 3%******\\E,W)=FN7$$\"3*)*****R!R$*GPF\\o$\"36+++zXoS?FN7$$\"3#******> 9pzl#F\\o$\"3-+++,O7'>#FN7$$!39+++`,)oG&F\\o$\"3%******>-8,N#FN7$$!3)) *****HwXp&H!#?$\"3'******f\"\\$=]#FN7$$!3))*****Hm-Hf)Ffp$\"37+++y\\F] EFN7$$!3;+++F+8l>!#>$\"3!******>tjTz#FN7$$!3s*****R4C#pQFaq$\"39+++U%3 <$HFN7$$!3++++xUj!y'Faq$\"3)******p;7-1$FN7$$!3++++Cm:OFN7$$!3`*****fqCv6&F-$\"3& )*****H#z\"Qk$FN7$$!3!******4Z$3tbF-$\"3++++I-lnOFN7$$!3,+++4.I;gF-$\" 3'********G!H(o$FN7$$!33+++a\")e[kF-$\"3%******H^*>.PFN7$$!3y*****z+'= roF-$\"3!)*****z>Pdr$FN7$$!3W+++sT;&G(F-$\"3()*****pW&=DPFN7$$!3Y+++*o X9p(F-$\"34+++$*owJPFN7$$!31+++')G$34)F-$\"3'*******4ulNPFN7$$!3s***** H]MS[)F-$\"33+++T`*pt$FN7$$!3/+++[>pr))F-$\"3:+++E#))et$FN7$$!3_+++WYS a#*F-$\"3!)*****>xGh5o$FN7$$!3)**** **\\U\"=[6FN$\"39+++E)[Sm$FN7$$!3-+++.&**[=\"FN$\"37+++B#[Zk$FN7$$!3\" ******Hm7=A\"FN$\"3:+++n]7BOFN7$$!3%******f2.\"f7FN$\"3/+++Id9*f$FN7$$ !3%******RK**pH\"FN$\"3=+++9\")ysNFN7$$!33+++-gyN8FN$\"33+++%4fSa$FN7$ $!3%******f05eP\"FN$\"3-+++^:-8NFN7$$!3$******z@xuT\"FN$\"3)******\\KL )zMFN7$$!3/+++prAh9FN$\"3\")*****f^'zWMFN7$$!3%*******oHZ2:FN$\"37+++Q `S3MFN7$$!3)******p@(\\c:FN$\"33+++bROrLFN7$$!3$******zhC$3;FN$\"34+++ GQ`MLFN7$$!3#********pJEm\"FN$\"3=+++q,z)H$FN7$$!3*)*****\\al(=!>9L=FN$ \"3()******4QU.KFN7$$!3#******foM)*)=FN$\"33+++hF N$\"3'******z$>t\\JFN7$$!3++++.[n**>FN$\"3!******zrd[7$FN7$$!3;+++%3EB 0#FN$\"3/+++Y(f25$FN7$$!3&)******4sL.@FN$\"3=+++Z]3xIFN7$$!3))*****>n1 F:#FN$\"3=+++%[NN0$FN7$$!3-+++DYZ+AFN$\"3%******4al)HIFN7$$!3?+++i]qYA FN$\"35+++#ove+$FN7$$!3=+++pWZ\"H#FN$\"3-+++=YS\")HFN7$$!3.+++R\\'[L#F N$\"3#)*****R.Aj&HFN7$$!3'******HEgpP#FN$\"3')******[B_IHFN7$$!3++++HR %yT#FN$\"3%******pk>R!HFN7$$!3)******RM)fdCFN$\"3=+++*pVk(GFN7$$!3=+++ 4[I'\\#FN$\"3;+++op.[GFN7$$!32+++hR/MDFN$\"3%)*****pW_'=GFN7$$!3;+++wk *3d#FN$\"3/+++[AD)y#FN7$$!33+++tQ%pg#FN$\"31+++Po!ov#FN7$$!3<+++F$pAk# FN$\"39+++#)[HCFFN7$$!3%)*****pLepn#FN$\"3!*******oOq!p#FN7$$!3!)***** z8*46FFN$\"3)******R/Igl#FN7$$!3.+++%o#yWFFN$\"3\")*****>%=G?EFN7$$!33 +++F?5yFFN$\"3%******fpzMe#FN7$$!38+++w3:6GFN$\"3%******p0fca#FN7$$!3 \")*****R=@S%GFN$\"30+++^@(o]#FN7$$!3%******fi*zwGFN$\"35+++'[*=nCFN7$ $!3%)*****H0j&4HFN$\"3=+++&Q+nU#FN7$$!3>+++-KPUHFN$\"39+++D?^&Q#FN7$$! 3!*******e7FvHFN$\"3!)*****\\_YPM#FN7$$!3))*****H8t#3IFN$\"3%******p*e `,BFN7$$!3*)*****4Sm8/$FN$\"3%******p9:!fAFN7$$!3#)******\\(4X2$FN$\"3 )******fD9j@#FN7$$!3;+++!HNw5$FN$\"3%)*****4?]N<#FN7$$!3*)*****4``19$F N$\"3'******4B?38#FN7$$!39+++<-YtJFN$\"33+++\"Q(>)3#FN7$$!3#)*****\\!R %f?$FN$\"39+++i'Gd/#FN7$$!3!******4;$*zB$FN$\"3?+++7eV.?FN7$$!3++++ND] pKFN$\"3'******\\-=8'>FN7$$!3@+++$)oP+LFN$\"3)*******))eN>>FN7$$!3\")* ****4*R`ILFN$\"3-+++yd^x=FN7$$!3?+++Gd!*fLFN$\"3-+++,TvN=FN7$$!35+++t! 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16/14805*6^(1/2),\nc[4]=32/235-8/4935*6^(1/2),\nc[5]=3/7,\nc[6]=8/21+4 /63*6^(1/2),\nc[7]=8/21-4/63*6^(1/2),\nc[8]=40/63,\nc[9]=10/21,\nc[10] =19/18,\nc[11]=7/9,\nc[12]=319999786/2170712113,\nc[13]=15/16,\nc[14]= 39/40,\nc[15]=1,\nc[16]=1,\n\na[2,1]=1/49,\na[3,1]=-165952/1491075+388 96/10437525*6^(1/2),\na[3,2]=301312/1491075-7168/1491075*6^(1/2),\na[4 ,1]=8/235-2/4935*6^(1/2),\na[4,2]=0,\na[4,3]=24/235-2/1645*6^(1/2),\na [5,1]=38937/44800+171/5600*6^(1/2),\na[5,2]=0,\na[5,3]=-149931/44800-8 1/700*6^(1/2),\na[5,4]=65097/22400+477/5600*6^(1/2),\na[6,1]=176/5103- 29/5103*6^(1/2),\na[6,2]=0,\na[6,3]=0,\na[6,4]=364520/1674351+87715/50 23053*6^(1/2),\na[6,5]=1940224/15069159+779264/15069159*6^(1/2),\na[7, 1]=4336/127575+479/127575*6^(1/2),\na[7,2]=0,\na[7,3]=0,\na[7,4]=90731 944/400648275-170142739/8413613775*6^(1/2),\na[7,5]=8245504/62429373-2 2187008/437005611*6^(1/2),\na[7,6]=-3936/340025+11464/3060225*6^(1/2), \na[8,1]=40/567,\na[8,2]=0,\na[8,3]=0,\na[8,4]=0,\na[8,5]=0,\na[8,6]=1 60/567-10/567*6^(1/2),\na[8,7]=10/567*6^(1/2)+160/567,\na[9,1]=95/1344 ,\na[9,2]=0,\na[9,3]=0,\na[9,4]=0,\na[9,5]=0,\na[9,6]=295/1344-115/268 8*6^(1/2),\na[9,7]=295/1344+115/2688*6^(1/2),\na[9,8]=-15/448,\na[10,1 ]=52918819/138240000,\na[10,2]=0,\na[10,3]=0,\na[10,4]=0,\na[10,5]=0, \na[10,6]=-1453047743/103680000-4153586941/829440000*6^(1/2),\na[10,7] =-1453047743/103680000+4153586941/829440000*6^(1/2),\na[10,8]=44599023 /5120000,\na[10,9]=518179039/25920000,\na[11,1]=258780283/8618400000+5 85428803/51710400000*6^(1/2),\na[11,2]=0,\na[11,3]=0,\na[11,4]=0,\na[1 1,5]=0,\na[11,6]=19/25,\na[11,7]=1180508473123/443296800000-1364049110 99/147765600000*6^(1/2),\na[11,8]=-106856621/190800000+585428803/22896 00000*6^(1/2),\na[11,9]=-1260561943/591300000+585428803/886950000*6^(1 /2),\na[11,10]=13167297224/792049782825-9366860848/2376149348475*6^(1/ 2),\na[12,1]=307213395328582867964430765847473084972824867512957518186 088963/\n 51263642128606211329399441117103047984786333585721409 81841000000+\n 119107533326819222510639750832411974467191643469 020133053/\n 29137664905764716334007503363213354847664112851105 748578125*6^(1/2),\na[12,2]=0,\na[12,3]=0,\na[12,4]=0,\na[12,5]=0,\na[ 12,6]=10354821182100230493026667000379184955505622050895245676387169/ \n 146467548938874889512569831763151565670818095959204028052600 000-\n 32684637880879071688859023194047542236847633606098546061 86699/\n 418478711253928255750199519323290187630908845597725794 43600000*6^(1/2),\na[12,7]=5423711572609568912980111975307776991743433 35570401188235055068891/\n 715703215922030799396778727922657043 395761853183510539934169000000-\n 25571039199675674205714455677 98380600417779349808054092568810687111/\n 100198450229084311915 54902190917198607540665944569147559078366000000*6^(1/2),\na[12,8]=-892 225578009519154676238995901578841509244882985862325637048827/\n \+ 38813900468801845720831005417235164902766795429189067433939000000+\n 20367388198886087049319397392342447633889771033202442752063/ \n 22061374857221856652891395403575825813231399730122923923437 5*6^(1/2),\na[12,9]=-5506886052357703380348636429175951958250506337980 73448796494914/\n 83532274004202085425137482177422377296638445 3517335472487484375+\n 724173802627060872864689685061064804760 52519229164240896224/\n 30386421973154632748322110650208212912 5640034018674235171875*6^(1/2),\na[12,10]=1233968951154957384345492297 15587040998178289648593721222723693824/\n 16960466422214315122 555088502288155618001490986047888958907461648625-\n 8781560217 119727637264447549372070053517105489683916369731584/\n 6169685 857480653009296139869875647732994358307038155314262445125*6^(1/2),\na[ 12,11]=-241890129426298647138485610377551406165672225318019246672/\n \+ 3661688723471872237814245794078789141770452398980100701315,\na [13,1]=45077846760256141387004276823/110315894143992133591739924480+\n 1493491403898138129099/13100021190238236835840*6^(1/2),\na[13, 2]=0,\na[13,3]=0,\na[13,4]=0,\na[13,5]=0,\na[13,6]=2853073212310390018 5/9849639992660328448-\n 15062887306567756845/56283657100916162 56*6^(1/2),\na[13,7]=530875502237315716994493/24064781139210466754560- \n 8920823473649531766699837/1347627743795786138255360*6^(1/2), \na[13,8]=-155850251753928802974915857362119/1740150866053400164770402 16637440+\n 13441422635083243161891/5220309196109974077440*6^( 1/2),\na[13,9]=-493074073683718697930133408597/27602712116408194083051 274240+\n 1493491403898138129099/224694912332563742720*6^(1/2) ,\na[13,10]=200609996314078300148532240828075/101993369197964626516710 6381709312-\n 336035565877081079047275/84650664247949736775516 16*6^(1/2),\na[13,11]=-1259978731825102407292471875/947642075600343143 202947072,\na[13,12]=-193916214235317468987992391599053188049367133486 207120889311375/\n 427480923494550881113440074554172336410202 80098816132254793728,\na[14,1]=367166212120980360939350186871054255059 61/72248275402215258274603114496000000000000+\n 122146123726388 4679751555607/9994523002806272000000000000*6^(1/2),\na[14,2]=0,\na[14, 3]=0,\na[14,4]=0,\na[14,5]=0,\na[14,6]=8283471074731862302286097/75146 78949478400000000000-\n 15668946773152185221466849/429410225684 4800000000000*6^(1/2),\na[14,7]=397408075485926915758262202639/1835997 0961922048000000000000-\n 6493922587539771225254133441201/10281 58373867634688000000000000*6^(1/2),\na[14,8]=2435278493903047909370803 905780425755549361787/6951941874960961660657094977773568000000000000+ \n 10993151135374962117764000463/3982779843223552000000000000*6 ^(1/2),\na[14,9]=-558123239069103416347126929975086912148938889/341847 71406932232290260692794368000000000000+\n 12214612372638846797 51555607/171428613534976000000000000*6^(1/2),\na[14,10]=30673458616172 7173704146823378382330889519/14062668006262142258469146617375840000000 00-\n 10993151135374962117764000463/25833332595199504631200000 0000*6^(1/2),\na[14,11]=-163845778835264660255510638493965671483/11419 6173990354810149157741209600000000,\na[14,12]=-37972709869158045130412 9337662817719784451899678250630685021894233169654109523405323/\n \+ 749809437263379760530627813247160529470471033302857618530795356910 20475760640000000,\na[14,13]=-235412270220829707518634576/100049213779 82463725322265625,\na[15,1]=1564746779443331677794753119798867/2864199 537049451295689544004000000+\n 5933645037523445166666379/523267 57304999307928000000*6^(1/2),\na[15,2]=0,\na[15,3]=0,\na[15,4]=0,\na[1 5,5]=0,\na[15,6]=-57628625604267458078211/39343426545112261600000-\n \+ 6047516944575480929793/1405122376611152200000*6^(1/2),\na[15,7] =1691425739887134646566682083/96124421783968922002000000-\n 665 0904765188943408799448943/1345741904975564908028000000*6^(1/2),\na[15, 8]=4336304776662958741869045159571925984937/22710641428913956471088421 29832843500000+\n 53402805337711006499997411/208520160689094986 48000000*6^(1/2),\na[15,9]=-253787748130868165717697094592697839427/21 839824320042045467560266343639250000+\n 5933645037523445166666 379/897521918060373467750000*6^(1/2),\na[15,10]=2479701613016438411235 35377085814726576/1172015708277609400708896218214309893875-\n \+ 854444885403376103999958576/21640244757394101989947629875*6^(1/2),\na[ 15,11]=-52966635737697605455117615341/39091452325689672618552125990,\n a[15,12]=-443860399531273813022911271986650433425394696961417882001810 19629996219153064322715843/\n 9260962836182630467435788682851 407364301742047595974505212083507724183850275361816608,\na[15,13]=2546 9705993361596208461643776/2398429987672964863327432743875,\na[15,14]=- 10513898964163619809241937920000000/2553870660483628218384915527095907 53,\na[16,1]=141410754314242261138804477571/33519128225766632364045200 0000+\n 112443792511441850402669/1512551672706650408000000*6^(1 /2),\na[16,2]=0,\na[16,3]=0,\na[16,4]=0,\na[16,5]=0,\na[16,6]=-2794429 334116030797321/1137256896771917600000-\n 13769398374510385872 3/40616317741854200000*6^(1/2),\na[16,7]=27893647118108877562026663/27 78562296720245822000000-\n 103919596904457790641030873/38899872 154083441508000000*6^(1/2),\na[16,8]=101918684220971405061056556223280 61/4787568421535046196059430031281250+\n 1011994132602976653624 021/602746155289116328000000*6^(1/2),\na[16,9]=-8042480997962909835623 073494340513/1506681707334195127293078358250000+\n 11244379251 1441850402669/25943672957609370250000*6^(1/2),\na[16,10]=1712777545222 30541963559693237323184/1168212455792999305788440538350463625-\n \+ 16191906121647626457984336/625530609794743596276941125*6^(1/2),\na[ 16,11]=-11555892521024455643655681/15915117701775915252975590,\na[16,1 2]=-620367951196138613523419881305338943491581365345833611872299990616 582621/\n 196071800395784656980767050971315035686876839509793 271487889571798793056,\na[16,13]=-3540712440917044503420862464/6932876 1738078084467429795125,\na[16,14]=0,\na[16,15]=0,\n\nb[1]=173734691637 390647/4182794002754640000,\nb[2]=0,\nb[3]=0,\nb[4]=0,\nb[5]=0,\nb[6]= 0,\nb[7]=0,\nb[8]=-72263163141715044860361/169939769455665013040000,\n b[9]=14586697891849999254003/29700462390576849520000,\nb[10]=102209317 997264953344/225042304099487188475,\nb[11]=1883570537693211021/1872275 755054959100,\nb[12]=1710999041788984993956022337692530667432380407898 3341334325755071278367152457480027/\n 713814271258088281460765347 03173195056090466069876919551824450642795947233961810560,\nb[13]=-1067 8264099993989152768/2396652442219114419375,\nb[14]=1212545712242913280 000000/130535954653501897388343,\nb[15]=-369769046476619/6555223062234 0,\nb[16]=0,\n\n`b*`[1]=89673698740537/2188798536240000,\n`b*`[2]=0,\n `b*`[3]=0,\n`b*`[4]=0,\n`b*`[5]=0,\n`b*`[6]=0,\n`b*`[7]=0,\n`b*`[8]=-5 0462783649374452713/198296113717228720000,\n`b*`[9]=322482352625960258 97/70884158450064080000,\n`b*`[10]=-3511133566564032/35760734800490575 ,\n`b*`[11]=293286057408279/538164919532900,\n`b*`[12]=332632201005182 0410035389555466180704547593970934049395166930777066599/\n 137 33237974615224189190609759523239412338434762518802069756703639509120, \n`b*`[13]=-35480458347621122048/114126306772338781875,\n`b*`[14]=0,\n `b*`[15]=0,\n`b*`[16]=10688504668909/28093813123860\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 5 "nodes" } {TEXT -1 2 ": " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "seq(c[i]=s ubs(ee,c[i]),i=2..16);" }}{PARA 11 "" 1 "" {XPPMATH 20 "61/&%\"cG6#\" \"##\"\"\"\"#\\/&F%6#\"\"$,&#\"#k\"$0(F)*(\"#;F)\"&0[\"!\"\"\"\"'#F)F' F6/&F%6#\"\"%,&#\"#K\"$N#F)*(\"\")F)\"%N\\F6F7F8F6/&F%6#\"\"&#F.\"\"(/ &F%6#F7,&#FB\"#@F)*(F\"#=/&F%6#\"#6#FIFin/&F%6# \"#7#\"*'y***>$\"+8@rq@/&F%6#\"#8#\"#:F4/&F%6#\"#9#\"#RFen/&F%6#FcpF)/ &F%6#F4F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 20 "linking coefficients" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "seq(seq(a[i,j]=subs(ee,a[i,j]),j=1..i-1),i=1..16); " }}{PARA 12 "" 1 "" {XPPMATH 20 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A lgorithms, 44 (2007) 291-307." }}{PARA 0 "" 0 "" {TEXT -1 9 " \+ " }{URLLINK 17 "http://users.ntua.gr/tsitoura/Heinzel.pdf" 4 "http://u sers.ntua.gr/tsitoura/Heinzel.pdf" "" }{TEXT -1 8 " and " } {URLLINK 17 "http://users.ntua.gr/tsitoura/NumericalAlgorithms.txt" 4 "http://users.ntua.gr/tsitoura/NumericalAlgorithms.txt" "" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "N ote" }{TEXT -1 180 ": The scheme constructed here has the same nodes a s the scheme given in the preceding paper, but the weights of both the order 9 scheme and the order 8 embedded scheme are changed." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "The nodes are: \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[2] = 1/46" "6#/ &%\"cG6#\"\"#*&\"\"\"F)\"#Y!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[ 3] = 96755252944/718444993695-11256225944/718444993695*sqrt(6)" "6#/&% \"cG6#\"\"$,&*&\",WHDbn*\"\"\"\"-&p$*\\W=(!\"\"F+*(\",WfAc7\"F+\"-&p$* \\W=(F--%%sqrtG6#\"\"'F+F-" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 4 8377626472/239481664565-5628112972/239481664565*sqrt(6)" "6#/&%\"cG6# \"\"%,&*&\",skix$[\"\"\"\"-lXm\"[R#!\"\"F+*(\"+sH6GcF+\"-lXm\"[R#F--%% sqrtG6#\"\"'F+F-" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5] = 71/136" "6# /&%\"cG6#\"\"&*&\"#r\"\"\"\"$O\"!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6] = 276/715-46/715*sqrt(6)" "6#/&%\"cG6#\"\"',&*&\"$w#\"\"\"\"$: (!\"\"F+*(\"#YF+F,F--%%sqrtG6#F'F+F-" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[7] = 276/715+46/715*sqrt(6)" "6# /&%\"cG6#\"\"(,&*&\"$w#\"\"\"\"$:(!\"\"F+*(\"#YF+F,F--%%sqrtG6#\"\"'F+ F+" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[8] = 92/143" "6#/&%\"cG6#\"\") *&\"##*\"\"\"\"$V\"!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[9] = 69/ 143" "6#/&%\"cG6#\"\"**&\"#p\"\"\"\"$V\"!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[10] = 3/44" "6#/&%\"cG6#\"#5*&\"\"$\"\"\"\"#W!\"\"" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "c[11] = 103/411" "6#/&%\"cG6#\"#6*&\" $.\"\"\"\"\"$6%!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[12] = 302582 48819701/45339732981913" "6#/&%\"cG6#\"#7*&\"/,(>)[#e-$\"\"\"\"/8>)HtR `%!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[13] = 59/69" "6#/&%\"cG6# \"#8*&\"#f\"\"\"\"#p!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[14] = 4 4/49" "6#/&%\"cG6#\"#9*&\"#W\"\"\"\"#\\!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[15] = 1" "6#/&%\"cG6#\"#:\"\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[16] = 1" "6#/&%\"cG6#\"#;\"\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "#-------- -------------------------------------------" }}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 28 "checking the combined scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "coefficients of the combined scheme " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16669 "ee := \{c[2]=1/46,\nc[3]=96755252944/718444993695-11256225944/7 18444993695*6^(1/2),\nc[4]=48377626472/239481664565-5628112972/2394816 64565*6^(1/2),\nc[5]=71/136,\nc[6]=276/715-46/715*6^(1/2),\nc[7]=276/7 15+46/715*6^(1/2),\nc[8]=92/143,\nc[9]=69/143,\nc[10]=3/44,\nc[11]=103 /411,\nc[12]=30258248819701/45339732981913,\nc[13]=59/69,\nc[14]=44/49 ,\nc[15]=1,\nc[16]=1,\n\na[2,1]=1/46,\na[3,1]=-16328795117593872453281 6/516163208965408589753025+\n 42011574289334042817176/516163208 965408589753025*6^(1/2),\na[3,2]=232801278267248934720896/516163208965 408589753025-\n 50098553466700618240256/516163208965408589753025 *6^(1/2),\na[4,1]=12094406618/239481664565-1407028243/239481664565*6^( 1/2),\na[4,2]=0,\na[4,3]=36283219854/239481664565-4221084729/239481664 565*6^(1/2),\na[5,1]=450479172821804238979159483/489985471732935255816 699904+\n 65404175703680378526395577/244992735866467627908349952 *6^(1/2),\na[5,2]=0,\na[5,3]=-1663285823745576633021875313/48998547173 2935255816699904-\n 258991054585998425691922779/244992735866467 627908349952*6^(1/2),\na[5,4]=734303944921586208649981787/244992735866 467627908349952+\n 96793439441159023582763601/122496367933233813 954174976*6^(1/2),\na[6,1]=188634486760257/2753187875656075-4045100355 6679/5506375751312150*6^(1/2),\na[6,2]=0,\na[6,3]=0,\na[6,4]=890541395 040155939974909749/3404930508779360011084250045-\n 2354148424451 43790083329443/6809861017558720022168500090*6^(1/2),\na[6,5]=127509164 130554343284736/2278805333809176804299525-\n 51090254569210884 816896/2278805333809176804299525*6^(1/2),\na[7,1]=523150756520001/5294 592068569375+372205675002861/137659393782803750*6^(1/2),\na[7,2]=0,\na [7,3]=0,\na[7,4]=121832502441158811994748302664452173/6319431229672072 722127362725145820625-\n 12054008141355156662680357922224203047/ 164305211971473890775311430853791336250*6^(1/2),\na[7,5]=-734518889112 3909155979140554752/52428978281511938535235507146875+\n 7138219 5182457889488943971467264/681576717659655200958061592909375*6^(1/2),\n a[7,6]=84211752143498940768/206389046233053165625+567839841668979868/1 8762640566641196875*6^(1/2),\na[8,1]=92/1287,\na[8,2]=0,\na[8,3]=0,\na [8,4]=0,\na[8,5]=0,\na[8,6]=368/1287+23/1287*6^(1/2),\na[8,7]=368/1287 -23/1287*6^(1/2),\na[9,1]=1311/18304,\na[9,2]=0,\na[9,3]=0,\na[9,4]=0, \na[9,5]=0,\na[9,6]=4071/18304+1587/36608*6^(1/2),\na[9,7]=4071/18304- 1587/36608*6^(1/2),\na[9,8]=-621/18304,\na[10,1]=2451872601/5043406438 4,\na[10,2]=0,\na[10,3]=0,\na[10,4]=0,\na[10,5]=0,\na[10,6]=84329349/1 146228736-1383050643/100868128768*6^(1/2),\na[10,7]=84329349/114622873 6+1383050643/100868128768*6^(1/2),\na[10,8]=-1098320769/50434064384,\n a[10,9]=-333490521/3152129024,\na[11,1]=-11290810941252792923651/16694 69461414577748900000-\n 76218489460616423924209/10016816768487 466493400000*6^(1/2),\na[11,2]=0,\na[11,3]=0,\na[11,4]=0,\na[11,5]=0, \na[11,6]=1/30,\na[11,7]=-44608220078798131601386867/17783274316806616 26219300000-\n 302663621648107819403033939/5334982295041984878 657900000*6^(1/2),\na[11,8]=4768550623191902657077/3353207892584835649 50000+76218489460616423924209/9388982099237539818600000*6^(1/2),\na[11 ,9]=76371166597983496297729/1268154687036073482337500+\n 762184 89460616423924209/1902232030554110223506250*6^(1/2),\na[11,10]=1283709 2726068800321242176/73489499260117750229428125+\n 224387232972 054752032871296/13889515360162254793361915625*6^(1/2),\na[12,1]=-35584 3792738780589211336013211266011894384892859307606673616682840321037292 934264483298005378753134077/611498050133846699976127131429172460287752 5435623381536957589416556451501095664913528943447961301150000+16342530 0686746737060984103973048211168317072573898840779749222007416544158960 4495769296514008733014761/62508689569237662664226328990537629273859148 897483455711122025147021504233422352449406977468048856200000*6^(1/2), \na[12,2]=0,\na[12,3]=0,\na[12,4]=0,\na[12,5]=0,\na[12,6]=-17594918026 0408634887753230236883672639389038840971696926196578253409432857792421 73824532021369934883/\n 34620197299885474706340736056297763905 521990158606221624621429312196525421587764433517710597688597280-\n \+ 17188002444689712939111901065805768959625842814399469853824584200 29691073118839588510586218092229/\n 134447368154895047403264994393 389374390376660810121249027656036163870001637234036635020235330829504* 6^(1/2),\na[12,7]=-138207307502597872414466248711907117101238289950946 956575872497287994333538943652935288845283125425999567221/449445892634 1069611642228743708681268522187616121807077973900118489313298715689055 81758203549930436500950000+3734226099348227821171135121746934552518699 21393106586844571126171756173532479385325874622983683540733325899/1797 7835705364278446568914974834725074088750464487228311895600473957253194 86275622327032814199721746003800000*6^(1/2),\na[12,8]=1263244583627961 9113294210780168171565726892406599945732333835965116776912265868042131 556830549327654393/112996477298237313277639902405971868302745384545450 862247028276227308103806571175581620305423011393900000-630354731220308 8429495101153246145287920801370707526715790327134571780988988474483681 572268319398771221/225992954596474626555279804811943736605490769090901 724494056552454616207613142351163240610846022787800000*6^(1/2),\na[12, 9]=1115379785277627713874987636432084522404410826010464628546537384975 239192450316138834225701046102632948257/334752063996028040585008210877 6916598468832017158981794068212683234002575269671076605501548156712544 287500-163425300686746737060984103973048211168317072573898840779749222 0074165441589604495769296514008733014761/11870640567235036900177596130 4145978669107518338970985605255768909007183520201102007287288941727395 18750*6^(1/2),\na[12,10]=111953957936197928184195726875956182710949222 53700867106612850891912759357989068567623193453186950145251776/3714676 2745465379216384266737567863919875050348378042013698628944987922182903 823702648363618722076517534375-687320121745403419867910288709505619542 179230939368838936545299414049008577113662220686991034529999350912/123 8225424848845973879475557918928797329168344945934733789954298166264072 7634607900882787872907358839178125*6^(1/2),\na[12,11]=7382510418747476 8875967421005730375156586805528272061538441201406502469378399602291332 635087412/\n 9287127253902959914328914474066891452514620247016 5909630170359355065425993729905138177287271317,\na[13,1]=-647190707441 44335733962214412431202035561004419937320357188840883/\n 10120 9800434111984325751891003355158288493635359529771992774000000+\n \+ 3559372342256314491923633965576963561357143961289/\n 363572 43706101884581258626118802982597639184500000*6^(1/2),\na[13,2]=0,\na[1 3,3]=0,\na[13,4]=0,\na[13,5]=0,\na[13,6]=-6600558885055349219496257716 4587068367312330247/\n 196036127235831406972940284420798499574 08380800-\n 13176568020345600854085799476910283462657777/\n \+ 76130534848866565814734091037203300805469440*6^(1/2),\na[13,7]=- 9318570701703691930191534222789494006237103047579881711/\n 356 2968864769188061196125478115789029665196590638000000+\n 643475 7944786940764537553522803508307647498263055753667/\n 712593772 9538376122392250956231578059330393181276000000*6^(1/2),\na[13,8]=-3761 276943318114593157887595063197569838926771255702620302334773297/\n \+ 14642165429350376228890216175250454997381873206410282367993860000 00-\n 3559372342256314491923633965576963561357143961289/\n \+ 34078442106289230846820763513066043517445275500000*6^(1/2),\na[ 13,9]=4386968817173326931645746836051457829029794774044320649595802272 17/\n 190320793597985643524049406938231176153947641989529318077 977593750-\n 3559372342256314491923633965576963561357143961289/ \n 6904380415341463658460171786984220252917056671875*6^(1/2),\n a[13,10]=3939307739733062346504442284413039354939067076806862872467577 6122752/\n 241046661167498470550788957260179457915876779829574 05550119962671875-\n 91119931961761650993245029518770267170742 8854089984/\n 438379739036741087515091959387714352376018403776 5625*6^(1/2),\na[13,11]=1455014345065512890761705230599774567929341275 785651/\n 437274533681980646175086038705021590800143506216585, \na[13,12]=43491235157516875762138510242422928845963515012067399050743 561004230490804423974878600834753885909/\n 1565007371693347353 2700213239415706508578020603344410725067680963543589421556571669667160 279586515,\na[14,1]=11135230389730673778222991060954644373352582465850 30516055024186482995053/\n 117208182195266626857155482543623815 8838606320428900711229598548800375000-\n 9656355719858106181793 772673915704646989403181/\n 42650123991481333817759653132492492 758879187500*6^(1/2),\na[14,2]=0,\na[14,3]=0,\na[14,4]=0,\na[14,5]=0, \na[14,6]=65156558806314041332627176859309009839731119/\n 11498 348446055040912040697694327646399463400+\n 28939086030349239034 37694020377944093049/\n 446537803730292850953036803663209568911 20*6^(1/2),\na[14,7]=8209640140354849710164186531509948506760631106345 347/\n 2089832016538282470163966831380108687881472614625000-\n \+ 7326425889801057568096943993446608268878122947239059/\n \+ 4179664033076564940327933662760217375762945229250000*6^(1/2),\na[14,8] =358387301461393123075253933751833574944722054371964074460462507967392 441417/\n 60038122073498036770225353795012178236450255253187133 0639897526611625875000+\n 9656355719858106181793772673915704646 989403181/\n 39976896846716122011518307352108347913593312500*6^ (1/2),\na[14,9]=-64285140412531160510172381720022061372098175856836237 6903035228606467992291/\n 274647144865445671171423994203344292 862210985060954852952360191727428234375+\n 7725084575886484945 4350181391325637175915225448/\n 647953806793658725308271653359 02056306758765625*6^(1/2),\na[14,10]=-36996958930176351863976128599782 4520507173970793320519753635225871563412913152/\n 18085087026 8987312974476662632186804296491491034759756207691410652763681078125+\n 454852979828196233587213868032125351691788847437824/\n \+ 946232027174365750372191179393662736769436409578125*6^(1/2),\na[14 ,11]=-7715303458281199041757411869319392732268755939408305903151657039 /\n 148306285785801995659935429289182147283195837962842168065 3323310,\na[14,12]=-94543290861814042087512801900367982103477409806135 4410087876969999388668430388640158852256997513450369741874915618358283 /118333625607368137927111484394931198556993205054781849088795998776610 9714104308279282406274069868573023638087923030436570,\na[14,13]=105691 1827593717127690972016166243945915857152155/\n 735979081412953 7930306952068888958882079118176267,\na[15,1]=-358442884482751721102900 3988869342002889757294838350269774673649/\n 208571634122713325 8652824258948052402773350356190879990897000000+\n 184185415092 25838359706930946702924369912480521/\n 3950819028766747069393 9925841123267455183000000*6^(1/2),\na[15,2]=0,\na[15,3]=0,\na[15,4]=0, \na[15,5]=0,\na[15,6]=-10950654480421820732850785842671218433296327603 /\n 1006547124867343561064069495275386629319739200+\n \+ 207046340886115163444903302653766033657147/\n 3908920873271237 130345900952525773317746560*6^(1/2), a[15,7]=-190787636465921540998883 304987599740985291747006984827/\n 2613436700522753347330289940 1390866713643703491000000+\n 178911278528379700355580290396510 513302568946148710869/\n 5226873401045506694660579880278173342 7287406982000000*6^(1/2), a[15,8]=-11573384193124353728682067589927476 0337634770259165918831723224267751/\n 794160022535984080063167 31727203547297279769620940498280027513000000-\n 49730062074909 7635712087135560978957987636974067/\n 999861123434045989100479 661671504230211939000000*6^(1/2), a[15,9]=2028363041702323309905709572 77958545540280636322218855276965040523111/\n 404834522658668650 51188583915057088814854119415174645941812575437500-\n 184185415 09225838359706930946702924369912480521/\n 750275729020608217505 1091686174851271537156250*6^(1/2), a[15,10]=99854707672539845032167700 901378988710484675887497768961753983967005376/\n 2398926904363 4452040903285829718875782291016518318453275155443387390625-\n \+ 54224186203160868130977204707093409345022342653824/\n 54782834 560687628399797684955475647938658789640625*6^(1/2), a[15,11]=474324561 414678755109830553528867775624956239150831585597139/\n 4515791 1635690366640337614874729721281201275171059286117565, a[15,12]=5838559 5002511207073587386944190632124915223533408383221102363939423894748980 604430394490848526064113167169126734415639048389/224741650740382631378 5622253865369318598611634721135768699232414191791839296674642114874667 0582600045180015758041428626487315,\na[15,13]=-20617842720508682752890 38565787612564079661263925986910/\n 6305818041153225448392022 471538861648368501130406546697, a[15,14]=21447821690294300536862603750 92251514835023987332030/\n 52553908524700142200738164428629074 85166958089934203,\na[16,1]=694779880028538161253441908996808488756491 17294292403621110427/\n 276315727858040840861280510910553807509 95025720848651597000000-\n 185426619462312591171592206632768608 526005483/\n 523404556022605918220275652595264842483000000*6^(1 /2),\na[16,2]=0,\na[16,3]=0,\na[16,4]=0,\na[16,5]=0,\na[16,6]=15540986 8146947995309862276450550758522148769/\n 1333473761189900616265 8099703042439679259200+\n 2948833469555291377668077700549094120 2439/\n 51785388784073810340419804672009474482560*6^(1/2), a[16 ,7]=3093423962117018793580109802227401427197073610480521/\n 346 228127881950258134565897477182544747165591000000-\n 22233492818 00786799168112412879091860919395860086487/\n 692456255763900516 269131794954365089494331182000000*6^(1/2), a[16,8]=8531832724323943375 6228728328071128940131418878118395907092074851/\n 4717952020428 847111777781739872410825294720094032625268331000000+\n 50065187 25482439961632989579084752430202148041/\n 132461614562644113149 59283823372471782839000000*6^(1/2), a[16,9]=-6614008744991438032754810 1344360694447375771589305352069924632043/\n 755387081649282115 3530435425077432706612170298471572545794812500+\n 185426619462 312591171592206632768608526005483/\n 9939653828313910466202350 1334196929221531250*6^(1/2), a[16,10]=-9964789456529458151748570686358 53357841406776256125134079522701632/\n 1776466558483769642410 55783826452426783687047504396906456095890625+\n 2373460729117 6011669963802448994381891328701824/\n 31554915787156112484260 150916445096576431546875*6^(1/2), a[16,11]=-44807476521380792971932033 1516183261898760393629529216/\n 45888783596485008769753055117 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652552648033910994214803877492780746797410373232909048652380685636,\na [12,7]=.20128430845560909506500331018201158854604633245475713718540627 11019851103578044630768,\na[12,8]=.43472227732547894832402079376789062 96377158021512663708497332494465155235838566783081e-1,\na[12,9]=-.4029 9857147530725077534998391017927599688381408200603830350700093676710867 98888005998e-2,\na[12,10]=.1654153572157061277142048209789895237987311 754507351909076354321086346892202059287151,\na[12,11]=.794918624125123 4457332208655151818006701133749850328100205972816224305266505346645467 ,\na[13,1]=-.399649659687948924971577067118614486956785845654183051926 7865896296345008297248542057,\na[13,2]=0.,\na[13,3]=0.,\na[13,4]=0.,\n a[13,5]=0.,\na[13,6]=-3.7909657756839315855474263811624937288735306502 43984061346027140586497794125572509467,\na[13,7]=-.4034932565353010338 751580781549804409247140410717448998803019090880817642871497642309,\na [13,8]=-2.824638795304352633780496682862207150757396843855776265553078 430950310988635063725530,\na[13,9]=1.042268927721859855333742832898214 168171271665967857126556429052907339665873769572504,\na[13,10]=1.12510 9564204366039742370365369240784568404608625876680489767032719107805685 884102981,\na[13,11]=3.32746188718986816186934832571938138898823948848 9197928234000109150397276869698610030,\na[13,12]=2.7789795718635560632 58182192557836277378714516293481181107157295767535371911926683861,\na[ 14,1]=.395453063500852371570982182057569227829863693035751703990197922 0884531037588519241752,\na[14,2]=0.,\na[14,3]=0.,\na[14,4]=0.,\na[14,5 ]=0.,\na[14,6]=5.82534730759650564865380791881446903058807699346898770 5808737112378995526246556899129,\na[14,7]=-.36527452339161313311889856 84697445207362339866197220939093923018677177193362904683069,\na[14,8]= 1.18860324058346533283780076203192232785089866681398483833069094219073 6927343514615809,\na[14,9]=.579704676383579213471102717626879728610392 1675338802551516458553273095993182856938673,\na[14,10]=-.8682486258908 7693262676988867897834489078687404305827255276926859137256616377917414 83,\na[14,11]=-5.20227677296454721392873650976792184255513966123663646 4752994488651123107996835974167,\na[14,12]=-.7989554142075338254321112 105867591561522852647111204098439044902504871017280125986247,\na[14,13 ]=.1436062320636379263279246377888900800674591637171164112471764724772 461548842396945110,\na[15,1]=-.576618924854752802186173070168401944772 6810364864003402086963008193772260796268425145,\na[15,2]=0.,\na[15,3]= 0.,\na[15,4]=0.,\na[15,5]=0.,\na[15,6]=-10.749681815061955661704516716 75389240410663049700076115683642548465951289532708129891,\na[15,7]=1.0 8412935121620890650261947597225472735862245595495311350230382187630614 8675839975592,\na[15,8]=-2.6756133191843819522108664931269629627753050 40498470916918421125605263456424877027114,\na[15,9]=-1.002908794145111 883032876236736495897295857932742193125577670568216125419495558139448, \na[15,10]=1.737962905137535569136158518599296524353829016489861458494 774667809465502081924598406,\na[15,11]=10.5036868232807807992532674656 9485415871491181460722061519204580168136447062527385070,\na[15,12]=2.5 9789828944333771307610231674258427211944092738211878637908977571991127 8563038055136,\na[15,13]=-.3269653926889719551274310441134675954557371 749902466142491975579219682457906850340524,\na[15,14]=.408110876857311 2662937157838902311218594074672839181802221969701351998431717518622006 ,\na[16,1]=1.646660891073387291106101035029422199207259812771676776521 235397749301308187032785786,\na[16,2]=0.,\na[16,3]=0.,\na[16,4]=0.,\na [16,5]=0.,\na[16,6]=13.04933414168316949396923079450422417415407470563 834242714053942047570178357626460722,\na[16,7]=1.069781170539730606447 594518614727754672957465098439041960158866261551333069407248351,\na[16 ,8]=19.009572283156672276207296468203223682768880017588215855970213758 94324645159162489432,\na[16,9]=-4.186205770343479780244976666808376725 365605260392577881420824302025421208499117915685,\na[16,10]=-3.7669027 6244720410015285749569311889498345100794510956420576779282279843123649 9459472,\na[16,11]=-9.764363534973491472406237674333426075243901930871 790217372448046597064314951986150631,\na[16,12]=-16.846469831025290009 52559041176861225199681774536016418027038146657517789700718980516,\na[ 16,13]=.78859341233650569459943943225193613678660394347296774167727416 45906609752704637952792,\na[16,14]=0.,\na[16,15]=0.,\n\nb[1]=.14909020 8197846102248361710238255271413924285803362739156491480302264842147336 1310107e-1,\nb[2]=0.,\nb[3]=0.,\nb[4]=0.,\nb[5]=0.,\nb[6]=0.,\nb[7]=0. ,\nb[8]=-.204080446920541512583491209341347918044934015126336352504928 5194417309050921652444102,\nb[9]=.229014386005704472647724693370664760 3531373802374130416209307539979468662050656748873,\nb[10]=.12800558251 1473756692082115737292022737723618892437240858502252806873896563006659 3081,\nb[11]=.22380626846054143649770066956485937727843730877752263380 50004586203873888538276205345,\nb[12]=.3955316529370005442055238915642 165166696733613795013837137887761225125843284107725787,\nb[13]=.541664 6758806981196568364538360743735174887121333155233519587876315231311599 922270715e-1,\nb[14]=.126914396524459036856433853121680376577665791087 0158628817571359283072191064518669055,\nb[15]=.31732672073507843493506 16957520189993515525495877836337410411517232415270466981438788e-1,\nb[ 16]=0.,\n\n`b*`[1]=.17312993442620729691199256523962107505017980602177 46839783040686135947285216431022474e-1,\n`b*`[2]=0.,\n`b*`[3]=0.,\n`b* `[4]=0.,\n`b*`[5]=0.,\n`b*`[6]=0.,\n`b*`[7]=0.,\n`b*`[8]=.401501833710 6195564009249847409861894215665048750691482133739698770541804983334916 224,\n`b*`[9]=.1698319708498420063111024105747673790482974707524540893 992750950174820584098348378075,\n`b*`[10]=.121959058498074079162948864 9398179871350014052451386889883854163460662246390206728645,\n`b*`[11]= .236646207918179948395735259079611638452026563280301330925416878248439 9679078940590933,\n`b*`[12]=-.2180405319307897885323826201233300337903 709926928467201016312519616654971860092341981,\n`b*`[13]=.228749019444 8336071465421621185843328221983327148307874886008208962533437763456473 381,\n`b*`[14]=0.,\n`b*`[15]=0.,\n`b*`[16]=.42039448066619861423929682 14560039940626273522287520668874866471501024910241621524753e-1\}:" }}} {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 "RK9_16eqs := [op(RowSumConditions(16,'expanded')),op (OrderConditions(9,16,'expanded'))]:\n`RK8_16eqs*` := subs(b=`b*`,Orde rConditions(8,16,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 6 "Check:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "expand(subs(ee,RK9_16eqs)):\nmap(u->lhs(u)-rhs(u),%) ;\nnops(%);\nexpand(subs(ee,`RK8_16eqs*`)):\nmap(u->lhs(u)-rhs(u),%); \nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7ajl\"\"!F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$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 12 "" 1 "" {XPPMATH 20 "6#7dw\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$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 257 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 "Next we set-up stage-orde r conditions to check for stage-orders from 2 to 6 inclusive. " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "for ct from 2 to 6 do\n so ||ct||_16 := StageOrderConditions(ct,16,'expanded');\nend do:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 138 "Stage 3 \+ has stage-order 2, stages 4 and 5 have stage-order 3, stages 6 and 7 h ave stage-order 4, while stages 8 to 16 have stage-order 5. " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 175 "[seq([seq(expand(subs(ee,so ||i||_16[j])),i=2..6)],j=1..14)]:\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,%):\nsim plify(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#70\"\"#\"\"$F%\"\"%F&\" \"&F'F'F'F'F'F'F'F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 27 "The simplifying conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j+1 .. 16) = b[j]*(1-c[j] );" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F, \"#;*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 9 " . . 15, " }}{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 0 151 "[Sum(b[i]*a[i,1],i=1+1..16) =b[1],seq(Sum(b[i]*a[i,j],i=j+1..16)=b[j]*(1-c[j]),j=2..15)]:\nmap(u-> lhs(u)-rhs(u),subs(ee,eval(subs(Sum=add,%))));\nnops(%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#71\"\"!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 27 "The simplifying conditions:" }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(`b*`[i]*a[i,j],i = j+1 .. 16) = ` b*`[j]*(1-c[j]);" "6#/-%$SumG6$*&&%#b*G6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/ F+;,&F0F,F,F,\"#;*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 9 " . . 15, " }} {PARA 0 "" 0 "" {TEXT -1 8 "(where " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\" cG6#\"\"\"\"\"!" }{TEXT -1 17 " ) are satisfied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 163 "[Sum(`b*`[i ]*a[i,1],i=1+1..16)=`b*`[1],seq(Sum(`b*`[i]*a[i,j],i=j+1..16)=`b*`[j]* (1-c[j]),j=2..15)]:\nmap(u->lhs(u)-rhs(u),subs(ee,eval(subs(Sum=add,%) )));\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#71\"\"!F$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 93 "We can calculate t he principal error norm, that is, the 2-norm of the principal error te rms. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "errterms9_16 := Pr incipalErrorTerms(9,16,'expanded'):\nsm := 0:\nfor ct to nops(errterms 9_16) do\n sm := sm+(evalf(subs(ee,errterms9_16[ct])))^2;\nend do:\n sqrt(sm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+>\"*[;O!#;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "We can also cal culate the principal error norm of the order 8 embedded scheme." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "`errterms8_16*` := subs(b=` b*`,PrincipalErrorTerms(8,16,'expanded')):\nsm := 0:\nfor ct to nops(` errterms8_16*`) do\n sm := sm+(evalf(subs(ee,`errterms8_16*`[ct])))^ 2;\nend do:\nsqrt(sm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+$Q$4*f$! #:" }}}{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 34 "construction of the order 9 scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 94 "First we construct a 15 stage orde r 9 scheme starting with a consideration of stages 2 to 11. " }}{PARA 0 "" 0 "" {TEXT -1 21 "We specify the nodes:" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "c[2] = 1/46;" "6#/&%\"cG6#\"\"#*&\"\" \"F)\"#Y!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5] = 71/136;" "6#/& %\"cG6#\"\"&*&\"#r\"\"\"\"$O\"!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[8] = 92/143;" "6#/&%\"cG6#\"\")*&\"##*\"\"\"\"$V\"!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[10] = 3/44;" "6#/&%\"cG6#\"#5*&\"\"$\"\"\"\" #W!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[11] = 103/411;" "6#/&%\"c G6#\"#6*&\"$.\"\"\"\"\"$6%!\"\"" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 35 "and the zero linking coefficients: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[4,2]=0" "6#/&%\"aG6$\"\"%\"\"#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[5,2]=0" "6#/&%\"aG6$\"\"&\"\"#\"\"! " }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[6,2]=0" "6#/&%\"aG6$\"\"'\"\"#\" \"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[6,3]=0" "6#/&%\"aG6$\"\"'\"\"$ \"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[7,2]=0" "6#/&%\"aG6$\"\"(\" \"#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[7,3]=0" "6#/&%\"aG6$\"\"( \"\"$\"\"!" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "a[8,2]=0" "6#/&%\"aG6$\"\")\"\"#\"\"!" }{TEXT -1 2 ", \+ " }{XPPEDIT 18 0 "a[8,3]=0" "6#/&%\"aG6$\"\")\"\"$\"\"!" }{TEXT -1 2 " , " }{XPPEDIT 18 0 "a[8,4]=0" "6#/&%\"aG6$\"\")\"\"%\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[8,5]=0" "6#/&%\"aG6$\"\")\"\"&\"\"!" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[9,2]=0 " "6#/&%\"aG6$\"\"*\"\"#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[9,3] =0" "6#/&%\"aG6$\"\"*\"\"$\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[9, 4]=0" "6#/&%\"aG6$\"\"*\"\"%\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[ 9,5]=0" "6#/&%\"aG6$\"\"*\"\"&\"\"!" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[10,2]=0" "6#/&%\"aG6$\"#5\"\"#\" \"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[10,3]=0" "6#/&%\"aG6$\"#5\"\"$ \"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[10,4]=0" "6#/&%\"aG6$\"#5\" \"%\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[10,5]=0" "6#/&%\"aG6$\"#5 \"\"&\"\"!" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "a[11,2]=0" "6#/&%\"aG6$\"#6\"\"#\"\"!" }{TEXT -1 2 ", \+ " }{XPPEDIT 18 0 "a[11,3]=0" "6#/&%\"aG6$\"#6\"\"$\"\"!" }{TEXT -1 2 " , " }{XPPEDIT 18 0 "a[11,4]=0" "6#/&%\"aG6$\"#6\"\"%\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[11,5]=0" "6#/&%\"aG6$\"#6\"\"&\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 20 " in stages 2 to 11. " }}{PARA 0 "" 0 "" {TEXT -1 17 "We also specify " }{XPPEDIT 18 0 "a[11,6] = 1/ 30;" "6#/&%\"aG6$\"#6\"\"'*&\"\"\"F*\"#I!\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "The stage -order equations relating to these stages, such that stages " } {XPPEDIT 18 0 "3,4,5,6,7,8,9,10,11;" "6+\"\"$\"\"%\"\"&\"\"'\"\"(\"\") \"\"*\"#5\"#6" }{TEXT -1 19 " have stage orders " }{XPPEDIT 18 0 "2,3, 3,4,4,5,5,5,5;" "6+\"\"#\"\"$F$\"\"%F%\"\"&F&F&F&" }{TEXT -1 96 " resp ectively taken together with the row-sum conditions can then be solved to obtain the nodes " }{XPPEDIT 18 0 "c[3],c[4],c[6],c[7],c[9],c[11]; " "6(&%\"cG6#\"\"$&F$6#\"\"%&F$6#\"\"'&F$6#\"\"(&F$6#\"\"*&F$6#\"#6" } {TEXT -1 67 " and the remaining non-zero linking coefficients for thes e stages. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "The computation is made more efficient by " }{TEXT 260 48 "incl uding explicitly relations between the nodes" }{TEXT -1 42 " arising f rom the stage-order conditions: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[3]=2/3" "6#/&%\"cG6 #\"\"$*&\"\"#\"\"\"F'!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[4]" "6#& %\"cG6#\"\"%" }{TEXT -1 8 ", " }{XPPEDIT 18 0 "c[4] = (6-sqrt(6) )*(9*c[8]-24*c[5]-4*c[5]*sqrt(6))*c[8]/(60*(2*c[8]-6*c[5]-c[5]*sqrt(6) ));" "6#/&%\"cG6#\"\"%**,&\"\"'\"\"\"-%%sqrtG6#F*!\"\"F+,(*&\"\"*F+&F% 6#\"\")F+F+*&\"#CF+&F%6#\"\"&F+F/*(F'F+&F%6#F:F+-F-6#F*F+F/F+&F%6#F5F+ *&\"#gF+,(*&\"\"#F+&F%6#F5F+F+*&F*F+&F%6#F:F+F/*&&F%6#F:F+-F-6#F*F+F/F +F/" }{TEXT -1 2 ", " }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[6]=((6-sqrt(6))*c[8])/10" "6#/ &%\"cG6#\"\"'*(,&F'\"\"\"-%%sqrtG6#F'!\"\"F*&F%6#\"\")F*\"#5F." } {TEXT -1 8 ", " }{XPPEDIT 18 0 "c[7]=((6+sqrt(6))*c[8])/10" "6#/ &%\"cG6#\"\"(*(,&\"\"'\"\"\"-%%sqrtG6#F*F+F+&F%6#\"\")F+\"#5!\"\"" } {TEXT -1 8 ", " }{XPPEDIT 18 0 "c[9]=3/4" "6#/&%\"cG6#\"\"**&\" \"$\"\"\"\"\"%!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "c[8]" "6#&%\"cG6# \"\")" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "The equations that lead to " }{XPPEDIT 18 0 "c[6] < c[7]" "6#2&%\"cG6#\"\"'&F%6#\"\"(" }{TEXT -1 19 ", have been chosen. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 818 "RSeqs := RowSumConditions(11,'expanded'):\nSOeqs := [op(Stage OrderConditions(2,11,'expanded')),\n op(StageOrderCondit ions(3,4..11,'expanded')),\n op(StageOrderConditions(4,6 ..11,'expanded')),\n op(StageOrderConditions(5,8..11,'ex panded'))]:\nnode_eqsA := [c[3]=2/3*c[4],c[9]=3/4*c[8],c[6]=1/10*(6-6^ (1/2))*c[8],c[7]=1/10*(6+6^(1/2))*c[8],\n c[4]=1/60*(6-6^ (1/2))*(9*c[8]-24*c[5]-4*c[5]*6^(1/2))*c[8]/(2*c[8]-6*c[5]-c[5]*6^(1/2 ))]:\n\ne1 := \{c[2]=1/46,c[5]=71/136,c[8]=92/143,c[10]=3/44,c[11]=103 /411,\n seq(a[i,2]=0,i=4..11),seq(a[i,3]=0,i=6..11),\n \+ seq(a[i,4]=0,i=8..11),seq(a[i,5]=0,i=8..11),a[11,6]=1/30\}:\neq ns := expand(rationalize(subs(e1,[op(RSeqs),op(SOeqs),op(node_eqsA)])) ):\nconvert(ListTools[Enumerate](%),matrix);\n``;\nindets(eqns);\nnops (%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7L7$\"\"\"/&%\"aG6 $\"\"#F(#F(\"#Y7$F-/,&&F+6$\"\"$F(F(&F+6$F5F-F(&%\"cG6#F57$F5/,&&F+6$ \"\"%F(F(&F+6$F@F5F(&F96#F@7$F@/,(&F+6$\"\"&F(F(&F+6$FJF5F(&F+6$FJF@F( #\"#r\"$O\"7$FJ/,(&F+6$\"\"'F(F(&F+6$FWF@F(&F+6$FWFJF(&F96#FW7$FW/,*&F +6$\"\"(F(F(&F+6$F]oF@F(&F+6$F]oFJF(&F+6$F]oFWF(&F96#F]o7$F]o/,(&F+6$ \"\")F(F(&F+6$F[pFWF(&F+6$F[pF]oF(#\"##*\"$V\"7$F[p/,*&F+6$\"\"*F(F(&F +6$FhpFWF(&F+6$FhpF]oF(&F+6$FhpF[pF(&F96#Fhp7$Fhp/,,&F+6$\"#5F(F(&F+6$ FfqFWF(&F+6$FfqF]oF(&F+6$FfqF[pF(&F+6$FfqFhpF(#F5\"#W7$Ffq/,.&F+6$\"#6 F(F(#F(\"#IF(&F+6$FfrF]oF(&F+6$FfrF[pF(&F+6$FfrFhpF(&F+6$FfrFfqF(#\"$. \"\"$6%7$Ffr/,$*&F.F(F6F(F(,$*&#F(F-F(*$)F8F-F(F(F(7$\"#7/*&FAF(F8F(,$ *&FjsF(*$)FCF-F(F(F(7$\"#8/,&*&FKF(F8F(F(*&FMF(FCF(F(#\"%T]\"&#*p$7$\" #9/,&*&FXF(FCF(F(*&FOF(FZF(F(,$*&FjsF(*$)FfnF-F(F(F(7$\"#:/,(*&F^oF(FC F(F(*&FOF(F`oF(F(*&FboF(FfnF(F(,$*&FjsF(*$)FdoF-F(F(F(7$\"#;/,&*&F\\pF (FfnF(F(*&F^pF(FdoF(F(#\"%KU\"&\\/#7$\"#/,,*&FgrF(FfnF(F(*&Fi rF(FdoF(F(*&F`pF(F[sF(F(*&F]sF(F_qF(F(*&F_rF(F_sF(F(#\"&41\"\"'UyL7$\" #?/*&FAF(F\\tF(,$*&#F(F5F(*$)FCF5F(F(F(7$\"#@/,&*&FKF(F\\tF(F(*&FMF(Fd tF(F(#\"'6zN\"(oja(7$\"#A/,&*&FXF(FdtF(F(*&#F\\u\"&'\\=F(FZF(F(,$*&Fcy F(*$)FfnF5F(F(F(7$\"#B/,(*&F^oF(FdtF(F(*&FezF(F`oF(F(*&FboF(FguF(F(,$* &FcyF(*$)FdoF5F(F(F(7$\"#C/,&*&F\\pF(FguF(F(*&F^pF(FbvF(F(#\"')oy(\"(@ Ex)7$\"#D/,(*&FipF(FguF(F(*&F[qF(FbvF(F(*&#\"%k%)F[wF(F]qF(F(,$*&FcyF( *$)F_qF5F(F(F(7$\"#E/,**&FgqF(FguF(F(*&FiqF(FbvF(F(*&Ff\\lF(F[rF(F(*&F ]rF(FfwF(F(#Fhp\"&%=&)7$\"#F/,,*&FgrF(FfuF(F(*&FirF(FbvF(F(*&Ff\\lF(F[ sF(F(*&F]sF(FfwF(F(*&#Fhp\"%O>F(F_sF(F(#\"(FF4\"\"*$fz#3#7$\"#G/,&*&FX F(FeyF(F(*&#F]z\"(ca^#F(FZF(F(,$*&#F(F@F(*$)FfnF@F(F(F(7$\"#H/,(*&F^oF (FeyF(F(*&Fj^lF(F`oF(F(*&FboF(FjzF(F(,$*&F^_lF(*$)FdoF@F(F(F(7$Fhr/,&* &F\\pF(FjzF(F(*&F^pF(Fe[lF(F(#\")C)4z\"\"*,;;=%7$\"#J/,(*&FipF(FjzF(F( *&F[qF(Fe[lF(F(*&#F]\\l\"(2U#HF(F]qF(F(,$*&F^_lF(*$)F_qF@F(F(F(7$\"#K/ ,**&FgqF(FjzF(F(*&FiqF(Fe[lF(F(*&F[alF(F[rF(F(*&F]rF(F[]lF(F(#\"#\")\" )%Q#*\\\"7$\"#L/,,*&FgrF(FizF(F(*&FirF(Fe[lF(F(*&F[alF(F[sF(F(*&F]sF(F []lF(F(*&#Fg]lFe]lF(F_sF(F(#\"*\")3b7\"\"-kp@PT67$\"#M/,&*&F\\pF(F`_lF (F(*&F^pF(F[`lF(F(#\"+K_\"3f'\"-:Za&)*)H7$\"#N/,(*&FipF(F`_lF(F(*&F[qF (F[`lF(F(*&#\")'HR;(Fc`lF(F]qF(F(,$*&#F(FJF(*$)F_qFJF(F(F(7$\"#O/,**&F gqF(F`_lF(F(*&FiqF(F[`lF(F(*&FiclF(F[rF(F(*&F]rF(F`alF(F(#\"$V#\"*?6eC )7$\"#P/,,*&FgrF(F__lF(F(*&FirF(F[`lF(F(*&FiclF(F[sF(F(*&F]sF(F`alF(F( *&#Fjal\"('4[PF(F_sF(F(#\",V2u#f6\"/b_@&*zje7$\"#Q/F8,$*&#F-F5F(FCF(F( 7$\"#R/F_q#\"#pFbp7$\"#S/Ffn,&#\"$w#\"$:(F(*(F/F(Fjfl!\"\"FWFjsF\\gl7$ \"#T/Fdo,&FhflF(*(F/F(FjflF\\glFWFjsF(7$\"#U/FC,&#\",skix$[\"-lXm\"[R# F(*(\"+sH6GcF(FhglF\\glFWFjsF\\glQ(pprint86\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "e2 := expand(rationalize(solve(\{op(eqns) \}))):\ne3 := `union`(e1,e2):\ninfolevel[solve] := 0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 106 " If the equations giving the relati ons between the nodes are omitted we need to select the solution with \+ " }{XPPEDIT 18 0 "c[6] < c[7]" "6#2&%\"cG6#\"\"'&F%6#\"\"(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 23 " Thus we require that " } {XPPEDIT 18 0 "c[6]=276/715-46/715*sqrt(6)" "6#/&%\"cG6#\"\"',&*&\"$w# \"\"\"\"$:(!\"\"F+*(\"#YF+F,F--%%sqrtG6#F'F+F-" }{TEXT -1 1 " " } {TEXT 270 1 "~" }{TEXT -1 20 " 0.2284244361 and " }{XPPEDIT 18 0 "c[ 7]=276/715+46/715*sqrt(6)" "6#/&%\"cG6#\"\"(,&*&\"$w#\"\"\"\"$:(!\"\"F +*(\"#YF+F,F--%%sqrtG6#\"\"'F+F+" }{TEXT -1 1 " " }{TEXT 271 1 "~" } {TEXT -1 48 " 0.5436035359, rather than the other way round. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "The follo wing commands achieve this." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 185 "eqns := subs(e1,[op(RSeqs),op(SOeq s)]):\nsol := solve(\{op(eqns)\}):\ne2 := op(select(u_->evalf(subs(u_, c[6]) " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e3 \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3255 "e3 := \{a[7,6] = 84211752143498940768/206389046233053165625+ 567839841668979868/18762640566641196875*6^(1/2), a[6,2] = 0, a[6,3] = \+ 0, c[2] = 1/46, c[5] = 71/136, c[8] = 92/143, c[11] = 103/411, c[10] = 3/44, a[4,2] = 0, a[5,2] = 0, a[11,1] = -11290810941252792923651/1669 469461414577748900000-76218489460616423924209/100168167684874664934000 00*6^(1/2), a[6,4] = 890541395040155939974909749/340493050877936001108 4250045-235414842445143790083329443/6809861017558720022168500090*6^(1/ 2), a[10,6] = 84329349/1146228736-1383050643/100868128768*6^(1/2), a[9 ,7] = 4071/18304-1587/36608*6^(1/2), c[7] = 276/715+46/715*6^(1/2), a[ 7,2] = 0, a[7,3] = 0, a[8,2] = 0, a[8,3] = 0, a[8,4] = 0, a[8,5] = 0, \+ a[9,2] = 0, a[9,3] = 0, a[9,4] = 0, a[9,5] = 0, a[10,2] = 0, a[10,3] = 0, a[10,4] = 0, a[10,5] = 0, a[11,2] = 0, a[11,3] = 0, a[11,4] = 0, a [11,5] = 0, a[11,6] = 1/30, a[8,6] = 368/1287+23/1287*6^(1/2), a[7,4] \+ = 121832502441158811994748302664452173/6319431229672072722127362725145 820625-12054008141355156662680357922224203047/164305211971473890775311 430853791336250*6^(1/2), c[6] = 276/715-46/715*6^(1/2), a[11,7] = -446 08220078798131601386867/1778327431680661626219300000-30266362164810781 9403033939/5334982295041984878657900000*6^(1/2), a[8,7] = 368/1287-23/ 1287*6^(1/2), a[10,7] = 84329349/1146228736+1383050643/100868128768*6^ (1/2), a[10,9] = -333490521/3152129024, a[11,9] = 76371166597983496297 729/1268154687036073482337500+76218489460616423924209/1902232030554110 223506250*6^(1/2), a[4,3] = 36283219854/239481664565-4221084729/239481 664565*6^(1/2), a[7,5] = -7345188891123909155979140554752/524289782815 11938535235507146875+71382195182457889488943971467264/6815767176596552 00958061592909375*6^(1/2), a[11,10] = 12837092726068800321242176/73489 499260117750229428125+224387232972054752032871296/13889515360162254793 361915625*6^(1/2), a[5,1] = 450479172821804238979159483/48998547173293 5255816699904+65404175703680378526395577/244992735866467627908349952*6 ^(1/2), a[2,1] = 1/46, a[8,1] = 92/1287, a[9,1] = 1311/18304, a[10,8] \+ = -1098320769/50434064384, a[9,8] = -621/18304, a[10,1] = 2451872601/5 0434064384, c[9] = 69/143, c[4] = 48377626472/239481664565-5628112972/ 239481664565*6^(1/2), c[3] = 96755252944/718444993695-11256225944/7184 44993695*6^(1/2), a[7,1] = 523150756520001/5294592068569375+3722056750 02861/137659393782803750*6^(1/2), a[3,1] = -163287951175938724532816/5 16163208965408589753025+42011574289334042817176/5161632089654085897530 25*6^(1/2), a[4,1] = 12094406618/239481664565-1407028243/239481664565* 6^(1/2), a[3,2] = 232801278267248934720896/516163208965408589753025-50 098553466700618240256/516163208965408589753025*6^(1/2), a[6,1] = 18863 4486760257/2753187875656075-40451003556679/5506375751312150*6^(1/2), a [6,5] = 127509164130554343284736/2278805333809176804299525-51090254569 210884816896/2278805333809176804299525*6^(1/2), a[11,8] = 476855062319 1902657077/335320789258483564950000+76218489460616423924209/9388982099 237539818600000*6^(1/2), a[9,6] = 4071/18304+1587/36608*6^(1/2), a[5,3 ] = -1663285823745576633021875313/489985471732935255816699904-25899105 4585998425691922779/244992735866467627908349952*6^(1/2), a[5,4] = 7343 03944921586208649981787/244992735866467627908349952+967934394411590235 82763601/122496367933233813954174976*6^(1/2)\}:" }{TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "printed coe fficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 150 "for ii from 2 to 11 do\n print(``);\n print(c[ii ]=subs(e3,c[ii])); \n for jj to ii-1 do\n print(a[ii,jj]=subs(e 3,a[ii,jj]));\n end do:\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"##\"\"\"\"#Y" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"#\"\"\"#F(\"#Y" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" cG6#\"\"$,&#\",WHDbn*\"-&p$*\\W=(\"\"\"*(\",WfAc7\"F,F+!\"\"\"\"'#F,\" \"#F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"$\"\"\",&#\"9;G` C(Qf<^zGj\"\"9DIv*e3a'*3K;;&!\"\"*(\"8wr\"G/M$*Gu:,UF(F,F-\"\"'#F(\"\" #F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"$\"\"#,&#\"9'*3sM* [sEy7!GB\"9DIv*e3a'*3K;;&\"\"\"*(\"8c-C=1qmMb)4]F-F,!\"\"\"\"'#F-F(F0 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"%,&#\",skix$[\"-lXm\"[R#\"\"\"*(\"+sH6GcF,F+!\"\"\" \"'#F,\"\"#F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"%\"\"\", &#\",=mS%47\"-lXm\"[R#F(*(\"+V#GqS\"F(F,!\"\"\"\"'#F(\"\"#F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"%\"\"#\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"aG6$\"\"%\"\"$,&#\",a)>KGO\"-lXm\"[R#\"\"\"*(\"+ HZ3@UF-F,!\"\"\"\"'#F-\"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"&#\"#r\"$O\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"\",&#\"<$[f\"z*QU!=#GpD%)*fea5**e#\"\"\"\"<_*\\$3 zink'et#*\\CF-\"\"'#F0\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"\"&\"\"%,&#\"<(y\")*\\'3ie@\\%RIM(\"<_*\\$3zink'et#*\\C\"\"\"*( \";,Ow#eB!f6WRMz'*F-\"\")e6W-D$=7\"FD1#e9D FOF@ss?nH7V>j\"\"\"*(\"GZI?CA#zN!oim:b893S07F-\"H]iL\"z`3V6`x!*QZr>@0V ;!\"\"\"\"'#F-\"\"#F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\" (\"\"&,&#\"@_ZbS\"zf:4R7\"*))=Xt\"Avo92bBN&Q>^\"Gy*GC&!\"\"*(\"AksYrR% *)[*)yX#=&>#Qr\"\"\"\"Bv$4Hfh!e4?b'fwrw:oF-\"\"'#F0\"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"',&#\"5o2%*)\\V@v6U)\"6Dc; `IBY!*Q1#\"\"\"*(\"3o)z*o;%)RycF-\"5vo>TmcSEw=!\"\"F(#F-\"\"#F-" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"cG6#\"\")#\"##*\"$V\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG 6$\"\")\"\"\"#\"##*\"%(G\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6 $\"\")\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\" $\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"%\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"&\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"',&#\"$o$\"%(G\"\"\"\"*(\"#BF- F,!\"\"F(#F-\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\") \"\"(,&#\"$o$\"%(G\"\"\"\"*(\"#BF-F,!\"\"\"\"'#F-\"\"#F0" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6# \"\"*#\"#p\"$V\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\" \"#\"%68\"&/$=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"# \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"$\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"%\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"',&#\"%rS\"&/$=\"\"\"*(\"%(e\"F-\"&3m $!\"\"F(#F-\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\" \"(,&#\"%rS\"&/$=\"\"\"*(\"%(e\"F-\"&3m$!\"\"\"\"'#F-\"\"#F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\")#!$@'\"&/$=" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6# \"#5#\"\"$\"#W" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"\"# \"+,E(=X#\",%QkSV]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\" \"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"$\"\"!" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"%\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"',&#\")\\$HV)\"+O(Gi9\"\"\"\"*(\"+V10$ Q\"F-\"-o(G\"o35!\"\"F(#F-\"\"#F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ &%\"aG6$\"#5\"\"(,&#\")\\$HV)\"+O(Gi9\"\"\"\"*(\"+V10$Q\"F-\"-o(G\"o35 !\"\"\"\"'#F-\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5 \"\")#!+p2K)4\"\",%QkSV]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"#5\"\"*#!*@0\\L$\"+C!H@:$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#6#\"$.\"\"$6%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"\",&#\"8^O#Hz_7%4\"3H6\":++!*[ xd99Yp%p;!\"\"*(\"84U#RU;1Y*[=i(F(\";++S$\\mu[on\"o,5F-\"\"'#F(\"\"#F- " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"#6\"\"'#\"\"\"\"#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"#6\"\"(,&#\";noQ,;8)zy+A3Y%\"=++I>iih1oJuKyy5[;ijEI \"\"\"\"=++!zly[)>/&H#)\\L&F-\"\"'#F0\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"),&#\"7xql->>B1boZ\"9++&\\c$[e#*y?`L\" \"\"*(\"84U#RU;1Y*[=i(F-\":++g=)RvB*4#)*)Q*!\"\"\"\"'#F-\"\"#F-" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"*,&#\"8HxH'\\$)zfm6Pw \":+vL#[tg.(oa\"o7\"\"\"*(\"84U#RU;1Y*[=i(F-\":]i]B-6a0.KA!>!\"\"\"\"' #F-\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"#5,&#\";w @C@.!)ogs#4PG\"\";D\"G%H-v<,E*\\*[t\"\"\"*(\"<'HrG._Z0sHB(QC#F-\">Dc\" >O$zaA;g`^*)Q\"!\"\"\"\"'#F-\"\"#F-" }}}{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 27 "We do not need to specify " }{XPPEDIT 18 0 "c[12]" "6#&%\"cG6#\"#7" }{TEXT -1 43 ", because, a ccording to Verner, the nodes " }{XPPEDIT 18 0 "c[8]" "6#&%\"cG6#\"\" )" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "c[9]" "6#&%\"cG6#\"\"*" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "c[10]" "6#&%\"cG6#\"#5" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "c[11]" "6#&%\"cG6#\"#6" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "c[12]" "6#&%\"cG6#\"#7" }{TEXT -1 30 " 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),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-F-F,F3F-/F,;F:F-F-*&-F&6$*&-F*6#F,F--F /6#*&,&F-F-F,F3\"\"#-F66#FNF3F-/F,;F:F-F--F&6$*&-F?6#F,F--F/6#*&,&F-F- F,F3FN-F66#FNF3F-/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[8])*(x-c[9])*(x-c[10])*(x-c[11])" "6#/-%\"pG6#%\"xG*,F'\"\"\",&F' F)&%\"cG6#\"\")!\"\"F),&F'F)&F,6#\"\"*F/F),&F'F)&F,6#\"#5F/F),&F'F)&F, 6#\"#6F/F)" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "q(x)=(x-c[12])*p(x) " "6#/-%\"qG6#%\"xG*&,&F'\"\"\"&%\"cG6#\"#7!\"\"F*-%\"pG6#F'F*" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 155 "See: J.H. Verner, SIAM Journal of Numerical Analysis 197 8, 772-790, \"Explicit Runge-Kutta methods with estimates of the Local Truncation Error.\" (page 780)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "p := x -> x*(x-c[8])*(x-c[9 ])*(x-c[10])*(x-c[11]):\n'p(x)'=p(x);\nq := x -> (x-c[12])*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),&F'F)&F,6#\"#5F/F),&F'F)& F,6#\"#6F/F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"qG6#%\"xG*.,&F'\" \"\"&%\"cG6#\"#7!\"\"F*F'F*,&F'F*&F,6#\"\")F/F*,&F'F*&F,6#\"\"*F/F*,&F 'F*&F,6#\"#5F/F*,&F'F*&F,6#\"#6F/F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "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);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ IeqG/*&-%$IntG6$,$*&#\"\"\"\"\"'F-*&-%\"pG6#%\"xGF-),&F-F-F3!\"\"\"\"$ F-F-F-/F3;\"\"!F-F--F(6$*&-%\"qGF2F-F5F-F8F-*&-F(6$,$*&#F-\"\"#F-*&F0F -)F5FFF-F-F-F8F--F(6$,$*&FEF-*&F>F-FHF-F-F-F8F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "e4 := solve( subs(e3,value(Ieq)),\{c[12]\}):\nc[12]=subs(e4,c[12]);\ne5 := `union`( e3,e4):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#7#\"/,(>)[#e-$ \"/8>)HtR`%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Thus " }{XPPEDIT 18 0 "c[12] = 30258248819701/45339732981913" " 6#/&%\"cG6#\"#7*&\"/,(>)[#e-$\"\"\"\"/8>)HtR`%!\"\"" }{TEXT -1 1 "." } }{PARA 0 "" 0 "" {TEXT -1 99 "Now we can use the quadrature equations \+ to find the weights once the remaining nodes once the nodes" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[13] = 59/69;" "6#/&%\"cG6 #\"#8*&\"#f\"\"\"\"#p!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[14] = \+ 44/49;" "6#/&%\"cG6#\"#9*&\"#W\"\"\"\"#\\!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[15]=1" "6#/&%\"cG6#\"#:\"\"\"" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 38 "are specified along with the weights " } {XPPEDIT 18 0 "b[2]=0" "6#/&%\"bG6#\"\"#\"\"!" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "b[3]=0" "6#/&%\"bG6#\"\"$\"\"!" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "b[4]=0" "6#/&%\"bG6#\"\"%\"\"!" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "b[5]=0" "6#/&%\"bG6#\"\"&\"\"!" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "b[6]=0" "6#/&%\"bG6#\"\"'\"\"!" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "b[7]=0" "6#/&%\"bG6#\"\"(\"\"!" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 288 "Qeqs := QuadratureConditions(9,15,'expanded'):\ne6 := \{seq(b[i]= 0,i=2..7),c[13]=59/69,c[14]=44/49,c[15]=1\}:\ne7 := `union`(e5,e6):\nq uadeqns := subs(e7,Qeqs):\nfor ct to nops(quadeqns) do\n print(`equa tion `||ct); print(``);print(quadeqns[ct]);print(``);\nend do:\ninde ts(quadeqns);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~ ~~1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,4&%\"bG6#\"\"\"F(&F&6#\"\")F(&F&6#\"\"*F(&F&6#\"#5F(& F&6#\"#6F(&F&6#\"#7F(&F&6#\"#8F(&F&6#\"#9F(&F&6#\"#:F(F(" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equatio n~~~2G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,2*&#\"##*\"$V\"\"\"\"&%\"bG6#\"\")F)F)*&#\"#pF(F)&F+6 #\"\"*F)F)*&#\"\"$\"#WF)&F+6#\"#5F)F)*&#\"$.\"\"$6%F)&F+6#\"#6F)F)*&# \"/,(>)[#e-$\"/8>)HtR`%F)&F+6#\"#7F)F)*&#\"#fF0F)&F+6#\"#8F)F)*&#F7\"# \\F)&F+6#\"#9F)F)&F+6#\"#:F)#F)\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~3G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,2*&#\"% k%)\"&\\/#\"\"\"&%\"bG6#\"\")F)F)*&#\"%hZF(F)&F+6#\"\"*F)F)*&#F3\"%O>F )&F+6#\"#5F)F)*&#\"&41\"\"'@*o\"F)&F+6#\"#6F)F)*&#\"<,%Hdgfp$\\j@;c:* \"=p&R^y)\\p6(oQ\"pb?F)&F+6#\"#7F)F)*&#\"%\"[$F0F)&F+6#\"#8F)F)*&#F6\" %,CF)&F+6#\"#9F)F)&F+6#\"#:F)#F)\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~4G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,2*&#\" ')oy(\"(2U#H\"\"\"&%\"bG6#\"\")F)F)*&#\"'4&G$F(F)&F+6#\"\"*F)F)*&#\"#F \"&%=&)F)&F+6#\"#5F)F)*&#\"(FF4\"\")JlUpF)&F+6#\"#6F)F)*&#\"J,\"HPWD@; .c*Q!['))>d8H.x#\"J(\\:wDo;q4KKe.CdRd)\\/K*F)&F+6#\"#7F)F)*&#\"'z`?F0F )&F+6#\"#8F)F)*&#F7\"'\\w6F)&F+6#\"#9F)F)&F+6#\"#:F)#F)\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equ ation~~~5G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,2*&#\")'HR;(\"*,;;=%\"\"\"&%\"bG6#\"\")F)F)*&#\")@rmA F(F)&F+6#\"\"*F)F)*&#\"#\")\"('4[PF)&F+6#\"#5F)F)*&#\"*\")3b7\"\",TUIM &GF)&F+6#\"#6F)F)*&#\"W,)=eBD7di5p0s?07,U`Y&z5I3`#Q)\"Xhd]4o'ox()4ccc@ \\S.cbE7j0yq'eA%F)&F+6#\"#7F)F)*&#\")ht67F0F)&F+6#\"#8F)F)*&#F7\"(,[w& F)&F+6#\"#9F)F)&F+6#\"#:F)#F)\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~6G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,2*&#\"+K_ \"3f'\",V*3rzf\"\"\"&%\"bG6#\"\")F)F)*&#\"+\\8.k:F(F)&F+6#\"\"*F)F)*&# \"$V#\"*Ci\"\\;F)&F+6#\"#5F)F)*&#\",V2u#f6\"/^I/*fF<\"F)&F+6#\"#6F)F)* &#\"_o,&)*\\bY2mf*H0#ePMn5[/D-O.EYEE@if.2k`#\"`o$z+BG=d(GjLh=R/sY+uMKQ g68W#*4Ih$\\o*f\">F)&F+6#\"#7F)F)*&#\"**HC\\rF0F)&F+6#\"#8F)F)*&#F7\"* \\_Z#GF)&F+6#\"#9F)F)&F+6#\"#:F)#F)\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~7G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,2*& #\"-W8+bjg\".\\)yl)4b)\"\"\"&%\"bG6#\"\")F)F)*&#\"-\"3j\"=z5F(F)&F+6# \"\"*F)F)*&#\"$H(\"+cQJcsF)&F+6#\"#5F)F)*&#\".HlH_S>\"\"1hRp1K/?[F)&F+ 6#\"#6F)F)*&#\"\\p,#oU2lZ*Q;!f$44`gt5CxP*Hs'y^MBtY)\\l?$\\=-_BZn(\"]p4 qb%HOPY8\\3!\\XLj.q6&\\X6>#\\SusBNi_:D(zAWbyqo)F)&F+6#\"#7F)F)*&#\",TO `!=UF0F)&F+6#\"#8F)F)*&#F7\",,sGTQ\"F)&F+6#\"#9F)F)&F+6#\"#:F)#F)\"\"( " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equation~~~8G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,2*&#\"/[O7gYyb\"12ax!3\"zA7\"\"\"&%\"bG6# \"\")F)F)*&#\".*e_KNYuF(F)&F+6#\"\"*F)F)*&#\"%(=#\"-k'4yF>$F)&F+6#\"#5 F)F)*&#\"0([UlQ()H7\"4rz@^zvP5)>F)&F+6#\"#6F)F)*&#\"jp,zimX^q%[OX)R8)e =&e%>)R8$)=%G/;6%GpP$3\"p'e>n(es#=(*QpBAB\"[q<#yV%*yHa0y%Hu%*Q!y[sJtD* *\\'\\Ib!pn:],*[E@l%o%o1CV]?#)pQRF)&F+6#\"#7F)F)*&#\".>[[^')[#F0F)&F+6 #\"#8F)F)*&#F7\"-\\G2B#y'F)&F+6#\"#9F)F)&F+6#\"#:F)#F)F-" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-equatio n~~~9G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,2*&#\"1;cPJ()=K^\"3,K)3bC\"f[<\"\"\"&%\"bG6#\"\")F)F) *&#\"0T'GWP)z8&F(F)&F+6#\"\"*F)F)*&#\"%hl\"/;_iB#[S\"F)&F+6#\"#5F)F)*& #\"2hh(Q\"3qnE\"\"6\"3'e]z^=l?9)F)&F+6#\"#6F)F)*&#\"gq,w20p&H\")o&\\X] ^M^a1^L-D;>Rx&)3')Rk!)*Gy]/Q@AHWEw$3@`N5x6BoEq\"iq@\"*=#4H'\\V(QS)))Q? 4[%=KXQKov;BlG+2pPY[?#Q%>+9c_z&y\"F)&F+6#\"#7F)F)*&#\"0@Vg P/$o9F0F)&F+6#\"#8F)F)*&#F7\"/,'p0$HBLF)&F+6#\"#9F)F)&F+6#\"#:F)#F)F3 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<+&%\"bG6#\"#:&F%6#\"\"\"&F%6#\"\")&F%6#\"\"*&F%6#\"#5&F%6#\"#6& F%6#\"#7&F%6#\"#9&F%6#\"#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"*" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "e8 := solve(\{op(quadeqns)\}):\ne9 := `union`(e7,e8):\ninfolevel [solve] := 0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "for ii to 15 do\n wt_val := subs(e9,b[ii]);\n if wt_val<>0 then print(b[ii]=wt_val) end if;\nend do:\n``;\nevalf[ 8](subs(e9,[seq(b[i],i=1..15)]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ &%\"bG6#\"\"\"#\"98*f9Y)>biVCfQ\";'H>th]Z;#H(H&)e#" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"bG6#\"\")#!Dt'f)>+3y<2UHy))=z3:)\"Eg(4N)*>An(y_& **zx+aR*R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"*#\"E,5O@=di mdy*\\W_:j$f$*\"F+7hup\"=&f_qYW-&49!o3%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#5#\"Cs-([nEX&[')eQCCqas@\"DDX@[v'3J\"oFST3MGsp\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#6#\"L>Kct'=>#z26[4Ut*RrCnu $>\"\"L+_f%[V`-PPbzxR1Y7V&RQL&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"bG6#\"#7#\"br$yEbo%*[VhU4PV)z=UhNJBqRgw!*o-8Napg=\"pXF$ow$*G!3#4b(G \">czg?P%3!f)))**y\"cr+/zI0x2$4C5Qg!)3(=*[C3EUCx_McWFQ#f'GYC,R/Md11oO] qNdVM$Q$p0%fPNG(*>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#8#\" D$=8Fi=L*p.mw$*zY**49\"\"E+SmHyD)Hok5#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#9#\"D\"*f*>sQ1,1FfBR#*3'[+\"\"D+gLi\"oNz'pb >_MJFw\"z" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#:#\"9hdq " 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 4271 "e9 := \{b[8] = -81508791888782942071778080019859673/399395400777 999552787672219983509760, a[7,6] = 84211752143498940768/20638904623305 3165625+567839841668979868/18762640566641196875*6^(1/2), a[6,2] = 0, a [6,3] = 0, c[2] = 1/46, c[5] = 71/136, c[8] = 92/143, c[11] = 103/411, c[10] = 3/44, c[13] = 59/69, c[14] = 44/49, c[15] = 1, a[4,2] = 0, a[ 5,2] = 0, a[11,1] = -11290810941252792923651/1669469461414577748900000 -76218489460616423924209/10016816768487466493400000*6^(1/2), a[6,4] = \+ 890541395040155939974909749/3404930508779360011084250045-2354148424451 43790083329443/6809861017558720022168500090*6^(1/2), a[10,6] = 8432934 9/1146228736-1383050643/100868128768*6^(1/2), a[9,7] = 4071/18304-1587 /36608*6^(1/2), c[7] = 276/715+46/715*6^(1/2), a[7,2] = 0, a[7,3] = 0, a[8,2] = 0, a[8,3] = 0, a[8,4] = 0, a[8,5] = 0, a[9,2] = 0, a[9,3] = \+ 0, a[9,4] = 0, a[9,5] = 0, a[10,2] = 0, a[10,3] = 0, a[10,4] = 0, a[10 ,5] = 0, a[11,2] = 0, a[11,3] = 0, a[11,4] = 0, a[11,5] = 0, b[2] = 0, b[3] = 0, b[4] = 0, b[5] = 0, b[6] = 0, b[7] = 0, b[10] = 21725470242 43858864854526674870272/16972283408414027681310867548214525, a[11,6] = 1/30, a[8,6] = 368/1287+23/1287*6^(1/2), a[7,4] = 1218325024411588119 94748302664452173/6319431229672072722127362725145820625-12054008141355 156662680357922224203047/164305211971473890775311430853791336250*6^(1/ 2), c[6] = 276/715-46/715*6^(1/2), a[11,7] = -446082200787981316013868 67/1778327431680661626219300000-302663621648107819403033939/5334982295 041984878657900000*6^(1/2), a[8,7] = 368/1287-23/1287*6^(1/2), a[10,7] = 84329349/1146228736+1383050643/100868128768*6^(1/2), a[10,9] = -333 490521/3152129024, a[11,9] = 76371166597983496297729/12681546870360734 82337500+76218489460616423924209/1902232030554110223506250*6^(1/2), c[ 12] = 30258248819701/45339732981913, a[4,3] = 36283219854/239481664565 -4221084729/239481664565*6^(1/2), a[7,5] = -73451888911239091559791405 54752/52428978281511938535235507146875+7138219518245788948894397146726 4/681576717659655200958061592909375*6^(1/2), a[11,10] = 12837092726068 800321242176/73489499260117750229428125+224387232972054752032871296/13 889515360162254793361915625*6^(1/2), a[5,1] = 450479172821804238979159 483/489985471732935255816699904+65404175703680378526395577/24499273586 6467627908349952*6^(1/2), a[2,1] = 1/46, a[8,1] = 92/1287, a[9,1] = 13 11/18304, b[11] = 1193746724713997342094811077921918673563219/53338395 43124606397779553737025343484595200, a[10,8] = -1098320769/50434064384 , b[1] = 385924436255198461459913/25885297292164750617319296, b[13] = \+ 11409994679937666036993318622713183/2106468298193165235233172578296640 00, a[9,8] = -621/18304, a[10,1] = 2451872601/50434064384, b[9] = 9359 36315524449978576662571821361001/4086801409502444670525951816974611200 , b[14] = 10048608923923592706010638721995991/791762731345219556967935 68162336000, c[9] = 69/143, c[4] = 48377626472/239481664565-5628112972 /239481664565*6^(1/2), c[3] = 96755252944/718444993695-11256225944/718 444993695*6^(1/2), a[7,1] = 523150756520001/5294592068569375+372205675 002861/137659393782803750*6^(1/2), a[3,1] = -163287951175938724532816/ 516163208965408589753025+42011574289334042817176/516163208965408589753 025*6^(1/2), a[4,1] = 12094406618/239481664565-1407028243/239481664565 *6^(1/2), b[15] = 957935979810312917705761/30187687238921485361088000, b[12] = 7899888590084372060795619128755092080289376683274569118606954 3513026890766039702331356142187984337094261434894685526783/19972835375 9405693383344357357050366806065734043901244628659238274456345277244226 082448918708806038102409307770530790400, a[3,2] = 23280127826724893472 0896/516163208965408589753025-50098553466700618240256/5161632089654085 89753025*6^(1/2), a[6,1] = 188634486760257/2753187875656075-4045100355 6679/5506375751312150*6^(1/2), a[6,5] = 127509164130554343284736/22788 05333809176804299525-51090254569210884816896/2278805333809176804299525 *6^(1/2), a[11,8] = 4768550623191902657077/335320789258483564950000+76 218489460616423924209/9388982099237539818600000*6^(1/2), a[9,6] = 4071 /18304+1587/36608*6^(1/2), a[5,3] = -1663285823745576633021875313/4899 85471732935255816699904-258991054585998425691922779/244992735866467627 908349952*6^(1/2), a[5,4] = 734303944921586208649981787/24499273586646 7627908349952+96793439441159023582763601/122496367933233813954174976*6 ^(1/2)\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 109 "It remains to determine the linking \+ coefficients in stages 12 to 15. We have the following zero coefficien ts." }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "a[12,2]=0" "6 #/&%\"aG6$\"#7\"\"#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[12,3]=0" "6#/&%\"aG6$\"#7\"\"$\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[12,4]=0 " "6#/&%\"aG6$\"#7\"\"%\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[12,5 ]=0" "6#/&%\"aG6$\"#7\"\"&\"\"!" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "a[13,2]=0" "6#/&%\"aG6$\"#8\"\"#\"\"! " }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[13,3]=0" "6#/&%\"aG6$\"#8\"\"$\" \"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[13,4]=0" "6#/&%\"aG6$\"#8\"\"% \"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[13,5]=0" "6#/&%\"aG6$\"#8\" \"&\"\"!" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "a[14,2]=0" "6#/&%\"aG6$\"#9\"\"#\"\"!" }{TEXT -1 2 ", \+ " }{XPPEDIT 18 0 "a[14,3]=0" "6#/&%\"aG6$\"#9\"\"$\"\"!" }{TEXT -1 2 " , " }{XPPEDIT 18 0 "a[14,4]=0" "6#/&%\"aG6$\"#9\"\"%\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[14,5]=0" "6#/&%\"aG6$\"#9\"\"&\"\"!" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[15,2]=0 " "6#/&%\"aG6$\"#:\"\"#\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[15,3] =0" "6#/&%\"aG6$\"#:\"\"$\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[15, 4]=0" "6#/&%\"aG6$\"#:\"\"%\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[1 5,5]=0" "6#/&%\"aG6$\"#:\"\"&\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 154 "We make use of the stage -order conditions for stages 12 to 15 so that all these stages all hav e stage-order 5 and incorporate the simplifying conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j+1 .. \+ 15) = b[j]*(1-c[j]);" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+%\"j GF,/F+;,&F0F,F,F,\"#:*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "for " }{XPPEDIT 18 0 "j=1" "6#/%\"jG \"\"\"" }{TEXT -1 10 " ( with " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6# \"\"\"\"\"!" }{TEXT -1 5 " ), " }{XPPEDIT 18 0 "j=7" "6#/%\"jG\"\"(" }{TEXT -1 8 " . . 13." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 185 "Then it turns out that the following collection of \" simple\" order conditions (given in abreviated form) is sufficient to \+ determine the remaining linking coefficients in stages 11 to 15." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 194 "SO9 := SimpleOrderConditions(9):\n[seq([i,SO9[i]],i=[102,106,125, 212,223,239,245,251,253])]:\nlinalg[augment](linalg[delcols](%,2..2),m atrix([[` `]$(linalg[rowdim](%))]),linalg[delcols](%,1..1));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7+7%\"$-\"%#~~G/*(%\"bG\" \"\"%\"cGF--%!G6#*&%\"aGF--F06#*&F3F--F06#*&)F.\"\"$F-F3F-F-F-F-#F-\"$ g*7%\"$1\"F)/*&F,F--F06#*&F3F--F06#*&F3F--F06#*&)F.\"\"%F-F3F-F-F-F-#F -\"%!o\"7%\"$D\"F)/*(F,F-F.F--F06#*&)F.\"\"&F-F3F-F-#F-\"#[7%\"$7#F)/* (F,F-)F.\"\"#F-F/F-#F-\"%!3\"7%\"$B#F)/*(F,F-F.F-FBF-#F-\"%!*=7%\"$R#F )/*(F,F-FhnF-FEF-#F-\"$q#7%\"$X#F)/*(F,F-F.F--F06#*&F3F-FSF-F-#F-\"$y$ 7%\"$^#F)/*(F,F-FhnF-FSF-#F-\"#a7%\"$`#F)/*(F,F-F.F--F06#*&)F.\"\"'F-F 3F-F-#F-\"#jQ(pprint16\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "The associated trees" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 456 "AST9 := AllSimpleTrees(9):\nwhch : = [102,106,125,212,223,239,245,251,253]:\nm := 3: ## number of trees p er row\nnn := nops(whch): q := iquo(nn,m,'r'):\nfor i to nn do \n p|| i := DrawTree(AST9[whch[i]],height=4,width=2,show_ordercondition=true, \n color=COLOR(RGB,.5,0,.9),font_color=black);\nend do:\npp := plot ([[1,1]],style=line,axes=none):\nplots[display](convert([seq([p||((k-1 )*m+1..m*k)],k=1..q),\n `if`(r>0,[p||(m*q+1..nn),pp$(m*(q+1)-nn)],NUL L)],array));" }}{PARA 13 "" 1 "" {GLPLOT2D 782 948 948 {PLOTDATA 2 "6f u-%%TEXTG6&7$$\"#g\"\"!$!+E!G3\"R!\")Q5b~c~~(a~(a~c~))~=~~~6\"-%'COLOU RG6&%$RGBGF)F)F)-%%FONTG6$%(COURIERG\"#5-F$6&7$F'$!+uu)G%QF,Q7~~~~~~~~ ~~~~~~~~~~~~~1F.F/F3-F$6&7$F'$!+(*)=T'QF,Q9~~~~~~~~~~~~~~~~~~~~~___F.F /F3-F$6&7$F'$!+,+++SF,Q9~~~~~~~~~~~~~~~~~~~~~270F.F/F3-F$6&7$F'$!+0\\R MQF,Q7~~~~2~~~~~~~~4~~~~~~~~F.F/F3-%'CURVESG6&7+7$F'$!+,Q^)p$F,7$$\"++ +++bF,$!+^`)QF$F,7$F'FZ7$$\"+++++lF,FZ7$Fhn$!+,pD\\GF,7$$\"+LLLLjF,$!+ ]%GYU#F,7$$\"+WWWWkF,F`o7$$\"+cbbblF,F`o7$$\"+nmmmmF,F`o-%'SYMBOLG6#%' CIRCLEG-%&COLORG6&F2$F)F)FbpFbp-%&STYLEG6#%&POINTG-FQ6&FS-F\\p6#%(DIAM ONDGF_pFcp-FQ6&FS-F\\p6#%&CROSSGF_pFcp-FQ6%7$FTFW-F`p6&F2$\"\"&!\"\"Fb p$\"\"*Fhq-%*THICKNESSG6#\"\"#-FQ6%7$FTFfnFdqF[r-FQ6%7$FTFgnFdqF[r-FQ6 %7$FgnFjnFdqF[r-FQ6%7$FjnF]oFdqF[r-FQ6%7$FjnFboFdqF[r-FQ6%7$FjnFeoFdqF [r-FQ6%7$FjnFhoFdqF[r-FQ6%7#7$F'$!+++++?F,-F06&F2$F7FhqFbpFbp-Fdp6#%%L INEG-FQ6%7#7$$\"+++++]F,FUFjsF]t-FQ6%7#7$$\"+++++qF,FUFjsF]t-F$6&7$$\" #IF)$!+d-G3\"*!\"*Q7b~(a~(a~(a~c~)))~=~~~~F.F/F3-F$6&7$F_u$!+PZ()G%)Fc uQ9~~~~~~~~~~~~~~~~~~~~~~~1F.F/F3-F$6&7$F_u$!+i*)=T')FcuQ<~~~~~~~~~~~~ ~~~~~~~~~~~____F.F/F3-F$6&7$F_u$!+++++5F,Q<~~~~~~~~~~~~~~~~~~~~~~~1680 F.F/F3-F$6&7$F_u$!+Z!\\RM)FcuQ9~~~~~~~~~~~~~4~~~~~~~~~~F.F/F3-FQ6&7*7$ F_u$!+1!Q^)pFcu7$F_u$!+/N&)QFFcu7$F_u$\"+)*4V2:Fcu7$F_u$\"+/br`dFcu7$$ \"++++DEF,$\"+++++5F,7$$\"++++vGF,F_x7$$\"++++DJF,F_x7$$\"++++vLF,F_xF [pF_pFcp-FQ6&F_wFipF_pFcp-FQ6&F_wF^qF_pFcp-FQ6%7$F`wFcwFdqF[r-FQ6%7$Fc wFfwFdqF[r-FQ6%7$FfwFiwFdqF[r-FQ6%7$FiwF\\xFdqF[r-FQ6%7$FiwFaxFdqF[r-F Q6%7$FiwFdxFdqF[r-FQ6%7$FiwFgxFdqF[r-FQ6%7#7$F_uF_xFjsF]t-FQ6%7#7$$\"+ ++++?F,FawFjsF]t-FQ6%7#7$$\"+++++SF,FawFjsF]t-FQ6#-%'LEGENDG6#QB__neve r_display_this_legend_entryF.-F$6&7$FbpF*Q:b~c~~(a~(a~(a~c~)))~=~~~~F. 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66" "Curve 67" "Curve 68" "Curve 69" "Curve 70" "Curve 71 " "Curve 72" "Curve 73" "Curve 74" "Curve 75" "Curve 76" "Curve 77" "C urve 78" "Curve 79" "Curve 80" "Curve 81" "Curve 82" "Curve 83" "Curve 84" "Curve 85" "Curve 86" "Curve 87" "Curve 88" "Curve 89" "Curve 90 " "Curve 91" "Curve 92" "Curve 93" "Curve 94" "Curve 95" "Curve 96" "C urve 97" "Curve 98" "Curve 99" "Curve 100" "Curve 101" "Curve 102" "Cu rve 103" "Curve 104" "Curve 105" "Curve 106" "Curve 107" "Curve 108" " Curve 109" "Curve 110" "Curve 111" "Curve 112" "Curve 113" "Curve 114 " "Curve 115" "Curve 116" "Curve 117" "Curve 118" "Curve 119" "Curve 1 20" "Curve 121" "Curve 122" "Curve 123" "Curve 124" "Curve 125" "Curve 126" "Curve 127" "Curve 128" "Curve 129" "Curve 130" "Curve 131" "Cur ve 132" "Curve 133" "Curve 134" "Curve 135" "Curve 136" "Curve 137" "C urve 138" "Curve 139" "Curve 140" "Curve 141" "Curve 142" "Curve 143" "Curve 144" "Curve 145" "Curve 146" "Curve 147" "Curve 148" "Curve 149 " "Curve 150" "Curve 151" "Curve 152" "Curve 153" "Curve 154" "Curve 1 55" "Curve 156" "Curve 157" "Curve 158" "Curve 159" "Curve 160" "Curve 161" "Curve 162" "Curve 163" "Curve 164" "Curve 165" "Curve 166" "Cur ve 167" "Curve 168" "Curve 169" }}}}{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 49 "SO9_15 := SimpleOr derConditions(9,15,'expanded'):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 485 "SOeqs2 := [seq(add(a[i,j],j= 1..i-1)=c[i],i=12..15),op(StageO rderConditions(2,12..15,'expanded')),\n op(StageOrderCondition s(3,12..15,'expanded')),op(StageOrderConditions(4,12..15,'expanded')), \n op(StageOrderConditions(5,12..15,'expanded'))]:\nord_ cdns := [seq(SO9_15[i],i=[102,106,125,212,223,239,245,251,253])]:\nsim p_eqs := [add(b[i]*a[i,1],i=1+1..15)=b[1],seq(add(b[i]*a[i,j],i=j+1..1 5)=b[j]*(1-c[j]),j=[$7..13])]:\ncdns := [op(SOeqs2),op(ord_cdns),op(si mp_eqs)]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 127 "It is po ssible to manage with fewer equations, but the computation may be less efficient if the number of equations is reduced." }}{PARA 0 "" 0 "" {TEXT -1 50 "For example, the simplifying conditions given by " } {XPPEDIT 18 0 "j=8" "6#/%\"jG\"\")" }{TEXT -1 3 ", " }{XPPEDIT 18 0 " j=10" "6#/%\"jG\"#5" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "j=12" "6#/% \"jG\"#7" }{TEXT -1 17 " may be omitted." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "e10 :=\{seq(seq(a[i,j ]=0,i=12..15),j=2..5)\}:\ne11 := `union`(e9,e10):\neqns2 := subs(e11,c dns):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "There are 37 equations and 34 unknowns." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "nops(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 74 "e12 := solve(\{op(eqns2) \}):\ninfolevel[solve] := 0:\ne13 := `union`(e11,e12):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "e13 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13251 "e13 := \{b[8] = -8 1508791888782942071778080019859673/39939540077799955278767221998350976 0, a[7,6] = 84211752143498940768/206389046233053165625+567839841668979 868/18762640566641196875*6^(1/2), a[6,2] = 0, a[6,3] = 0, c[2] = 1/46, c[5] = 71/136, c[8] = 92/143, c[11] = 103/411, c[10] = 3/44, c[13] = \+ 59/69, c[14] = 44/49, c[15] = 1, a[4,2] = 0, a[5,2] = 0, a[11,1] = -11 290810941252792923651/1669469461414577748900000-7621848946061642392420 9/10016816768487466493400000*6^(1/2), a[6,4] = 89054139504015593997490 9749/3404930508779360011084250045-235414842445143790083329443/68098610 17558720022168500090*6^(1/2), a[10,6] = 84329349/1146228736-1383050643 /100868128768*6^(1/2), a[9,7] = 4071/18304-1587/36608*6^(1/2), c[7] = \+ 276/715+46/715*6^(1/2), a[7,2] = 0, a[7,3] = 0, a[8,2] = 0, a[8,3] = 0 , a[8,4] = 0, a[8,5] = 0, a[9,2] = 0, a[9,3] = 0, a[9,4] = 0, a[9,5] = 0, a[10,2] = 0, a[10,3] = 0, a[10,4] = 0, a[10,5] = 0, a[11,2] = 0, a [11,3] = 0, a[11,4] = 0, a[11,5] = 0, a[12,2] = 0, a[12,3] = 0, a[12,4 ] = 0, a[12,5] = 0, a[13,5] = 0, a[13,4] = 0, a[13,3] = 0, a[13,2] = 0 , a[14,2] = 0, a[14,3] = 0, a[14,4] = 0, a[14,5] = 0, a[15,2] = 0, a[1 5,3] = 0, a[15,4] = 0, a[15,5] = 0, b[2] = 0, b[3] = 0, b[4] = 0, b[5] = 0, b[6] = 0, b[7] = 0, b[10] = 2172547024243858864854526674870272/1 6972283408414027681310867548214525, a[11,6] = 1/30, a[8,6] = 368/1287+ 23/1287*6^(1/2), a[7,4] = 121832502441158811994748302664452173/6319431 229672072722127362725145820625-12054008141355156662680357922224203047/ 164305211971473890775311430853791336250*6^(1/2), c[6] = 276/715-46/715 *6^(1/2), a[11,7] = -44608220078798131601386867/1778327431680661626219 300000-302663621648107819403033939/5334982295041984878657900000*6^(1/2 ), a[8,7] = 368/1287-23/1287*6^(1/2), a[10,7] = 84329349/1146228736+13 83050643/100868128768*6^(1/2), a[10,9] = -333490521/3152129024, a[11,9 ] = 76371166597983496297729/1268154687036073482337500+7621848946061642 3924209/1902232030554110223506250*6^(1/2), c[12] = 30258248819701/4533 9732981913, a[4,3] = 36283219854/239481664565-4221084729/239481664565* 6^(1/2), a[7,5] = -7345188891123909155979140554752/5242897828151193853 5235507146875+71382195182457889488943971467264/68157671765965520095806 1592909375*6^(1/2), a[11,10] = 12837092726068800321242176/734894992601 17750229428125+224387232972054752032871296/138895153601622547933619156 25*6^(1/2), a[5,1] = 450479172821804238979159483/489985471732935255816 699904+65404175703680378526395577/244992735866467627908349952*6^(1/2), a[2,1] = 1/46, a[8,1] = 92/1287, a[9,1] = 1311/18304, b[11] = 1193746 724713997342094811077921918673563219/533383954312460639777955373702534 3484595200, a[10,8] = -1098320769/50434064384, b[1] = 3859244362551984 61459913/25885297292164750617319296, b[13] = 1140999467993766603699331 8622713183/210646829819316523523317257829664000, a[12,6] = -1718800244 4689712939111901065805768959625842814399469853824584200296910731188395 88510586218092229/1344473681548950474032649943933893743903766608101212 49027656036163870001637234036635020235330829504*6^(1/2)-17594918026040 8634887753230236883672639389038840971696926196578253409432857792421738 24532021369934883/3462019729988547470634073605629776390552199015860622 1624621429312196525421587764433517710597688597280, a[12,7] = -13820730 7502597872414466248711907117101238289950946956575872497287994333538943 652935288845283125425999567221/449445892634106961164222874370868126852 218761612180707797390011848931329871568905581758203549930436500950000+ 3734226099348227821171135121746934552518699213931065868445711261717561 73532479385325874622983683540733325899/1797783570536427844656891497483 4725074088750464487228311895600473957253194862756223270328141997217460 03800000*6^(1/2), a[14,7] = 820964014035484971016418653150994850676063 1106345347/2089832016538282470163966831380108687881472614625000-732642 5889801057568096943993446608268878122947239059/41796640330765649403279 33662760217375762945229250000*6^(1/2), a[9,8] = -621/18304, a[10,1] = \+ 2451872601/50434064384, a[15,12] = 58385595002511207073587386944190632 1249152235334083832211023639394238947489806044303944908485260641131671 69126734415639048389/2247416507403826313785622253865369318598611634721 1357686992324141917918392966746421148746670582600045180015758041428626 487315, a[12,10] = -68732012174540341986791028870950561954217923093936 8838936545299414049008577113662220686991034529999350912/12382254248488 4597387947555791892879732916834494593473378995429816626407276346079008 82787872907358839178125*6^(1/2)+11195395793619792818419572687595618271 094922253700867106612850891912759357989068567623193453186950145251776/ 3714676274546537921638426673756786391987505034837804201369862894498792 2182903823702648363618722076517534375, b[9] = 935936315524449978576662 571821361001/4086801409502444670525951816974611200, b[14] = 1004860892 3923592706010638721995991/79176273134521955696793568162336000, c[9] = \+ 69/143, a[14,10] = -36996958930176351863976128599782452050717397079332 0519753635225871563412913152/18085087026898731297447666263218680429649 1491034759756207691410652763681078125+45485297982819623358721386803212 5351691788847437824/94623202717436575037219117939366273676943640957812 5*6^(1/2), c[4] = 48377626472/239481664565-5628112972/239481664565*6^( 1/2), c[3] = 96755252944/718444993695-11256225944/718444993695*6^(1/2) , a[7,1] = 523150756520001/5294592068569375+372205675002861/1376593937 82803750*6^(1/2), a[3,1] = -163287951175938724532816/51616320896540858 9753025+42011574289334042817176/516163208965408589753025*6^(1/2), a[4, 1] = 12094406618/239481664565-1407028243/239481664565*6^(1/2), b[15] = 957935979810312917705761/30187687238921485361088000, b[12] = 78998885 9008437206079561912875509208028937668327456911860695435130268907660397 02331356142187984337094261434894685526783/1997283537594056933833443573 5705036680606573404390124462865923827445634527724422608244891870880603 8102409307770530790400, a[3,2] = 232801278267248934720896/516163208965 408589753025-50098553466700618240256/516163208965408589753025*6^(1/2), a[6,1] = 188634486760257/2753187875656075-40451003556679/550637575131 2150*6^(1/2), a[6,5] = 127509164130554343284736/2278805333809176804299 525-51090254569210884816896/2278805333809176804299525*6^(1/2), a[11,8] = 4768550623191902657077/335320789258483564950000+7621848946061642392 4209/9388982099237539818600000*6^(1/2), a[9,6] = 4071/18304+1587/36608 *6^(1/2), a[5,3] = -1663285823745576633021875313/489985471732935255816 699904-258991054585998425691922779/244992735866467627908349952*6^(1/2) , a[5,4] = 734303944921586208649981787/244992735866467627908349952+967 93439441159023582763601/122496367933233813954174976*6^(1/2), a[15,13] \+ = -2061784272050868275289038565787612564079661263925986910/63058180411 53225448392022471538861648368501130406546697, a[13,9] = 43869688171733 2693164574683605145782902979477404432064959580227217/19032079359798564 3524049406938231176153947641989529318077977593750-35593723422563144919 23633965576963561357143961289/6904380415341463658460171786984220252917 056671875*6^(1/2), a[13,11] = 1455014345065512890761705230599774567929 341275785651/437274533681980646175086038705021590800143506216585, a[15 ,1] = 18418541509225838359706930946702924369912480521/3950819028766747 0693939925841123267455183000000*6^(1/2)-358442884482751721102900398886 9342002889757294838350269774673649/20857163412271332586528242589480524 02773350356190879990897000000, a[12,9] = -1634253006867467370609841039 7304821116831707257389884077974922200741654415896044957692965140087330 14761/1187064056723503690017759613041459786691075183389709856052557689 0900718352020110200728728894172739518750*6^(1/2)+111537978527762771387 4987636432084522404410826010464628546537384975239192450316138834225701 046102632948257/334752063996028040585008210877691659846883201715898179 4068212683234002575269671076605501548156712544287500, a[15,7] = -19078 7636465921540998883304987599740985291747006984827/26134367005227533473 302899401390866713643703491000000+178911278528379700355580290396510513 302568946148710869/522687340104550669466057988027817334272874069820000 00*6^(1/2), a[15,10] = 99854707672539845032167700901378988710484675887 497768961753983967005376/239892690436344520409032858297188757822910165 18318453275155443387390625-5422418620316086813097720470709340934502234 2653824/54782834560687628399797684955475647938658789640625*6^(1/2), a[ 14,1] = 11135230389730673778222991060954644373352582465850305160550241 86482995053/1172081821952666268571554825436238158838606320428900711229 598548800375000-9656355719858106181793772673915704646989403181/4265012 3991481333817759653132492492758879187500*6^(1/2), a[14,11] = -77153034 58281199041757411869319392732268755939408305903151657039/1483062857858 019956599354292891821472831958379628421680653323310, a[14,9] = -642851 404125311605101723817200220613720981758568362376903035228606467992291/ 2746471448654456711714239942033442928622109850609548529523601917274282 34375+77250845758864849454350181391325637175915225448/6479538067936587 2530827165335902056306758765625*6^(1/2), a[12,8] = -630354731220308842 9495101153246145287920801370707526715790327134571780988988474483681572 268319398771221/225992954596474626555279804811943736605490769090901724 494056552454616207613142351163240610846022787800000*6^(1/2)+1263244583 6279619113294210780168171565726892406599945732333835965116776912265868 042131556830549327654393/112996477298237313277639902405971868302745384 545450862247028276227308103806571175581620305423011393900000, a[14,12] = -945432908618140420875128019003679821034774098061354410087876969999 388668430388640158852256997513450369741874915618358283/118333625607368 1379271114843949311985569932050547818490887959987766109714104308279282 406274069868573023638087923030436570, a[13,12] = 434912351575168757621 3851024242292884596351501206739905074356100423049080442397487860083475 3885909/15650073716933473532700213239415706508578020603344410725067680 963543589421556571669667160279586515, a[12,1] = 1634253006867467370609 8410397304821116831707257389884077974922200741654415896044957692965140 08733014761/6250868956923766266422632899053762927385914889748345571112 2025147021504233422352449406977468048856200000*6^(1/2)-355843792738780 5892113360132112660118943848928593076066736166828403210372929342644832 98005378753134077/6114980501338466999761271314291724602877525435623381 536957589416556451501095664913528943447961301150000, a[15,8] = -115733 841931243537286820675899274760337634770259165918831723224267751/794160 02253598408006316731727203547297279769620940498280027513000000-4973006 20749097635712087135560978957987636974067/9998611234340459891004796616 71504230211939000000*6^(1/2), a[13,1] = -64719070744144335733962214412 431202035561004419937320357188840883/101209800434111984325751891003355 158288493635359529771992774000000+355937234225631449192363396557696356 1357143961289/36357243706101884581258626118802982597639184500000*6^(1/ 2), a[12,11] = 7382510418747476887596742100573037515658680552827206153 8441201406502469378399602291332635087412/92871272539029599143289144740 668914525146202470165909630170359355065425993729905138177287271317, a[ 14,13] = 1056911827593717127690972016166243945915857152155/73597908141 29537930306952068888958882079118176267, a[13,8] = -3761276943318114593 157887595063197569838926771255702620302334773297/146421654293503762288 9021617525045499738187320641028236799386000000-35593723422563144919236 33965576963561357143961289/3407844210628923084682076351306604351744527 5500000*6^(1/2), a[13,10] = 393930773973306234650444228441303935493906 70768068628724675776122752/2410466611674984705507889572601794579158767 7982957405550119962671875-91119931961761650993245029518770267170742885 4089984/4383797390367410875150919593877143523760184037765625*6^(1/2), \+ a[15,9] = 202836304170232330990570957277958545540280636322218855276965 040523111/404834522658668650511885839150570888148541194151746459418125 75437500-18418541509225838359706930946702924369912480521/7502757290206 082175051091686174851271537156250*6^(1/2), a[15,14] = 2144782169029430 053686260375092251514835023987332030/525539085247001422007381644286290 7485166958089934203, a[15,11] = 47432456141467875510983055352886777562 4956239150831585597139/45157911635690366640337614874729721281201275171 059286117565, a[14,8] = 3583873014613931230752539337518335749447220543 71964074460462507967392441417/6003812207349803677022535379501217823645 02552531871330639897526611625875000+9656355719858106181793772673915704 646989403181/39976896846716122011518307352108347913593312500*6^(1/2), \+ a[15,6] = -10950654480421820732850785842671218433296327603/10065471248 67343561064069495275386629319739200+2070463408861151634449033026537660 33657147/3908920873271237130345900952525773317746560*6^(1/2), a[13,7] \+ = -9318570701703691930191534222789494006237103047579881711/35629688647 69188061196125478115789029665196590638000000+6434757944786940764537553 522803508307647498263055753667/712593772953837612239225095623157805933 0393181276000000*6^(1/2), a[13,6] = -660055888505534921949625771645870 68367312330247/19603612723583140697294028442079849957408380800-1317656 8020345600854085799476910283462657777/76130534848866565814734091037203 300805469440*6^(1/2), a[14,6] = 65156558806314041332627176859309009839 731119/11498348446055040912040697694327646399463400+289390860303492390 3437694020377944093049/44653780373029285095303680366320956891120*6^(1/ 2)\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 20 "printed coefficients" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 213 "for ii from 2 to 15 do\n \+ print(``);\n print(c[ii]=subs(e13,c[ii])); \n for jj to ii-1 do\n print(a[ii,jj]=subs(e13,a[ii,jj]));\n end do:\nend do:\n``;\nf or ii to 15 do\n print(b[ii]=subs(e13,b[ii]));\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6 #\"\"##\"\"\"\"#Y" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"#\" \"\"#F(\"#Y" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"cG6#\"\"$,&#\",WHDbn*\"-&p$*\\W=(\"\"\"*(\",WfAc 7\"F,F+!\"\"\"\"'#F,\"\"#F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG 6$\"\"$\"\"\",&#\"9;G`C(Qf<^zGj\"\"9DIv*e3a'*3K;;&!\"\"*(\"8wr\"G/M$*G u:,UF(F,F-\"\"'#F(\"\"#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"\"$\"\"#,&#\"9'*3sM*[sEy7!GB\"9DIv*e3a'*3K;;&\"\"\"*(\"8c-C=1qmMb)4] F-F,!\"\"\"\"'#F-F(F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"%,&#\",skix$[\"-lXm\"[R#\"\"\"* (\"+sH6GcF,F+!\"\"\"\"'#F,\"\"#F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ &%\"aG6$\"\"%\"\"\",&#\",=mS%47\"-lXm\"[R#F(*(\"+V#GqS\"F(F,!\"\"\"\"' #F(\"\"#F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"%\"\"#\"\"! " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"%\"\"$,&#\",a)>KGO\"- lXm\"[R#\"\"\"*(\"+HZ3@UF-F,!\"\"\"\"'#F-\"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"&# \"#r\"$O\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"\",&# \"<$[f\"z*QU!=#GpD%)*fea 5**e#\"\"\"\"<_*\\$3zink'et#*\\CF-\"\"'#F0\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"%,&#\"<(y\")*\\'3ie@\\%RIM(\"<_*\\$3z ink'et#*\\C\"\"\"*(\";,Ow#eB!f6WRMz'*F-\"\")e6W-D$=7\"FD1#e9D FOF@ss?nH7V>j\"\"\"*(\"GZI?CA#zN!oim:b893S07F-\"H]iL\"z`3V6`x!*QZr>@0V ;!\"\"\"\"'#F-\"\"#F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\" (\"\"&,&#\"@_ZbS\"zf:4R7\"*))=Xt\"Avo92bBN&Q>^\"Gy*GC&!\"\"*(\"AksYrR% *)[*)yX#=&>#Qr\"\"\"\"Bv$4Hfh!e4?b'fwrw:oF-\"\"'#F0\"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"',&#\"5o2%*)\\V@v6U)\"6Dc; `IBY!*Q1#\"\"\"*(\"3o)z*o;%)RycF-\"5vo>TmcSEw=!\"\"F(#F-\"\"#F-" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"cG6#\"\")#\"##*\"$V\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG 6$\"\")\"\"\"#\"##*\"%(G\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6 $\"\")\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\" $\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"%\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"&\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"',&#\"$o$\"%(G\"\"\"\"*(\"#BF- F,!\"\"F(#F-\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\") \"\"(,&#\"$o$\"%(G\"\"\"\"*(\"#BF-F,!\"\"\"\"'#F-\"\"#F0" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6# \"\"*#\"#p\"$V\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\" \"#\"%68\"&/$=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"# \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"$\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"%\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\"',&#\"%rS\"&/$=\"\"\"*(\"%(e\"F-\"&3m $!\"\"F(#F-\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\" \"(,&#\"%rS\"&/$=\"\"\"*(\"%(e\"F-\"&3m$!\"\"\"\"'#F-\"\"#F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"*\"\")#!$@'\"&/$=" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6# \"#5#\"\"$\"#W" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"\"# \"+,E(=X#\",%QkSV]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\" \"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"$\"\"!" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"%\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"',&#\")\\$HV)\"+O(Gi9\"\"\"\"*(\"+V10$ Q\"F-\"-o(G\"o35!\"\"F(#F-\"\"#F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ &%\"aG6$\"#5\"\"(,&#\")\\$HV)\"+O(Gi9\"\"\"\"*(\"+V10$Q\"F-\"-o(G\"o35 !\"\"\"\"'#F-\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5 \"\")#!+p2K)4\"\",%QkSV]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"#5\"\"*#!*@0\\L$\"+C!H@:$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#6#\"$.\"\"$6%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"\",&#\"8^O#Hz_7%4\"3H6\":++!*[ xd99Yp%p;!\"\"*(\"84U#RU;1Y*[=i(F(\";++S$\\mu[on\"o,5F-\"\"'#F(\"\"#F- " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"#6\"\"'#\"\"\"\"#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"#6\"\"(,&#\";noQ,;8)zy+A3Y%\"=++I>iih1oJuKyy5[;ijEI \"\"\"\"=++!zly[)>/&H#)\\L&F-\"\"'#F0\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"),&#\"7xql->>B1boZ\"9++&\\c$[e#*y?`L\" \"\"*(\"84U#RU;1Y*[=i(F-\":++g=)RvB*4#)*)Q*!\"\"\"\"'#F-\"\"#F-" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"*,&#\"8HxH'\\$)zfm6Pw \":+vL#[tg.(oa\"o7\"\"\"*(\"84U#RU;1Y*[=i(F-\":]i]B-6a0.KA!>!\"\"\"\"' #F-\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"#5,&#\";w @C@.!)ogs#4PG\"\";D\"G%H-v<,E*\\*[t\"\"\"*(\"<'HrG._Z0sHB(QC#F-\">Dc\" >O$zaA;g`^*)Q\"!\"\"\"\"'#F-\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#7#\"/,(>)[#e-$\"/8 >)HtR`%" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"\",&*(\"bq hZ,L(3S^'Hpd\\/'*eTa;u+A#\\(z2%))*QdsqJo6@[I(R5%)41Pnu'o+`Uj\"F(\"cq++ ?c)[!ou(pS\\CNAMB/:-Z^-A6rbM[(*)[\"fQFHw`!**GjAkEmP#p&*o3D'!\"\"\"\"'# F(\"\"#F(#\"aqxS8`(y`+)H$[kU$HHP5KSGo;On1wIfG*[Q%*=,m7@8gL6#*e!yQFzVeN \"bq++:,8'zWV*GN\"\\m&4,:XclT*edp`\"QBcVDv(GgC6RHr *oWC+)=<\"\"\"\"^q/&H3LN--Nm.Msj,+(Q;OglF!\\7753mw.Ru$*Q$R%*\\E.u/&*[: otWM\"!\"\"F(#F,\"\"#F.#\"`q$)[$*p8-KX#QEppr4%)Q!*QREn$)o BIKv()[j3/E!=\\f<\"`q!G(f)o(f5x^LWw(e@a_'>7$H9iC;A1'e,*>_0Rw(HcgtSjqua ))*H(>?Y$F." }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"(,&#\" gq@sc**fUDJGX))GNHlV*QNL%*zG(\\sedcp%4&**GQ75r[iY9C(yf-vI2#Q\"\"gq ++&4]O/$*\\N?e9G.FBivi[>`s&RZ+c*=JGs[k/v)3u]sM[(\\\"*olWyUO0d$y(z\"F- \"\"'#F0\"\"#F0" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"), &*(\"bq@7x)R>$oAd\"o$[u%))*))4yrX8F.z:n_22P,3#zGXhC`65&\\H%)3.AJZNI'\" \"\"\"dq++!yyAg%3hSK;^B98w?;YX_l0%\\C\"[!)z_bEYZ'faH*fA! \"\"\"\"'#F,\"\"#F.#\"cq$RawK\\0$obJ@/oeE7px;^'f$QLKd%**f1C*osl:'zi$eWKE\"\"dq++!RR6IU0.i\"ev6d1Q53tAw#GqCi3XXXQXFIo=(fS-*RwF8tB) Hxk*H6F," }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"\"*,&*(\"bq hZ,L(3S^'Hpd\\/'*eTa;u+A#\\(z2%))*QdsqJo6@[I(R5%)41Pnu'o+`Uj\"\"\"\"\" cq](=&RF<%*)G(G2?5,-_$=2!4*odD0c)4(*Q$=v5p'yf9/8'fx,!p.NscS1(=\"!\"\" \"\"'#F,\"\"#F.#\"eqd#[Hj-h/,dAM)QhJ]C>R_(\\QPlaGYY5g#3T/C_%3Kkj()\\(Q rFwF&yz`6\"\"eq+vGWDrc\"[:]0m2r'p_d-SB$o7#oSz\")*er,K)o%)f;px3@3]eS!Gg *R1_ZLF," }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#7\"#5,&*(\"dq7 4N***HX.\"*po?Am8rd3!\\ST*HXl$*Q)o$R4Bz@a>c]4()G5z')>MSX<7?to\"\"\"\"f 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f%)GHUU-^Q@wvo^d^B\"\\V\"]q:lez-;n'p;dc:U*eVN'4on]s5WM.1-y&3lq:%RK@+F` tM$prt+l:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#9#\"#W\"#\\" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#9\"\"\",&#\"do`]*H['=C]0;0.&eY#e_LPWY&41\"*HAyPnI(*Q I_86\"do+]P+)[&)fH7r+*G/K1'Q)e\"QiVD[br&oim_>#=3s6F(*(\"O\"=.%*)pk/d\" RnsPz\"=1\"e)>dNc'*F(\"P+v=z)eF\\#\\KJlfx\"QL\"[\"*R7]E%!\"\"\"\"'#F( \"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#9\"\"#\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#9\"\"$\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#9\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#9\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#9\"\"',&#\"M>6tR)4!4$foPa?sW\\dL=vLRDvI7$Rh9I(Qe$\"fo+] (ei6m_(*)R1Lr=`_D]kBy@,&z``Aqn.)\\t?7Q+'\"\"\"*(\"O\"=.%*)pk/d\"RnsPz \"=1\"e)>dNc'*F-\"P+DJ$f8zM3@N2$=:,AhrYo*o(*R!\"\"\"\"'#F-\"\"#F-" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#9\"\"*,&#\"fo\"H#*zY1'G_.. 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" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#:\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#:\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#:\"\"',&#\"P.wK'HL%=7nUey]Gt?=U![a1&4\"\"O+ #R(>$HmQv_\\pS1hNMn[7Zl+\"!\"\"*(\"KZrlLgw`EI.\\Wj^6')3MYq?\"\"\"\"Lgl u\"V+++\"\\ .Pk8n'3R,%**GItM`F_+nV8E!\"\"*(\"Wp3r[h%*oDI80^'R!H!eb.qz$G&y7\"*y\"\" \"\"\"V+++#)pS(GFMt\"y-))z0m%p1b/,M(oA&F-\"\"'#F0\"\"#F0" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#:\"\"),&#\"`o^xECKsJ)=f;f-xMwLgZF**e n?oGPNCJ>%Qt:\"\"_o+++8v-!G)\\S4ip(zsHZN?F@IU]r;mz/5*)f/MM7h)***F-\"\"'# F0\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#:\"\"*,&#\"`o 6J_S]'pFb)=AKO1GSbaezFd4d!*4LK-T&[\"))3d]\"Re )=^]'o'eE_M[S\"\"\"*(\"P@0[7*pV#HqY4$pqf$QeA4:a=%=F-\"O]i:P:F^[<'o\"4^ ]<#31-HdF](!\"\"\"\"'#F-\"\"#F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"#:\"#5,&#\"bow`+nR)Rvh*ox\\()en%[5())*y8!4qn@.X)RDn2Z&)**\"boD 1R(QVa:vKX=$=l,\"H#yv)=(HeG.4/_WjV!p#*)R#\"\"\"*(\"SCQlUB-X$4M42Z?x48o 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"6#/&%\"bG6#\"\")#!Dt'f)>+3y<2UHy))=z3:)\"Eg(4N)*>An(y_& **zx+aR*R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"\"*#\"E,5O@=di mdy*\\W_:j$f$*\"F+7hup\"=&f_qYW-&49!o3%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#5#\"Cs-([nEX&[')eQCCqas@\"DDX@[v'3J\"oFST3MGsp\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#6#\"L>Kct'=>#z26[4Ut*RrCnu $>\"\"L+_f%[V`-PPbzxR1Y7V&RQL&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"bG6#\"#7#\"br$yEbo%*[VhU4PV)z=UhNJBqRgw!*o-8Napg=\"pXF$ow$*G!3#4b(G \">czg?P%3!f)))**y\"cr+/zI0x2$4C5Qg!)3(=*[C3EUCx_McWFQ#f'GYC,R/Md11oO] qNdVM$Q$p0%fPNG(*>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#8#\" D$=8Fi=L*p.mw$*zY**49\"\"E+SmHyD)Hok5#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#9#\"D\"*f*>sQ1,1FfBR#*3'[+\"\"D+gLi\"oNz'pb >_MJFw\"z" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"bG6#\"#:#\"9hdq " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "RK9_15eqs := [op(RowSumC onditions(15,'expanded')),op(OrderConditions(9,15,'expanded'))]:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Check:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "expand(subs(e13,RK9_15eqs)): \nmap(u_->`if`(lhs(u_)=rhs(u_),0,1),%);\nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7`jl\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$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 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 34 "Appendix: related order conditions" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "#-------- -----------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 145 "u nrelated 129, 130, 134, 135, 136, 139, 140, 141, 143, 151, 152, 156, \+ 157, 166, 168, 173, 175, 180, 185, 186, 187, 212, 221, 223\n\nrelated \+ groups" }}{PARA 0 "" 0 "" {TEXT -1 19 "132, 137, 148, 153," }}{PARA 0 "" 0 "" {TEXT -1 19 "133, 138, 150, 163," }}{PARA 0 "" 0 "" {TEXT -1 59 "131, 147, 149, 192,\n142, 164, 170, 222,\n162, 184, 188, 227," }} {PARA 0 "" 0 "" {TEXT -1 18 "181, 197, 202, 239" }}{PARA 0 "" 0 "" {TEXT -1 38 "144, 158, 167, 177, 193, 207, 213, 244" }}{PARA 0 "" 0 " " {TEXT -1 38 "145, 159, 169, 178, 194, 209, 215, 245" }}{PARA 0 "" 0 "" {TEXT -1 38 "154, 165, 172, 189, 206, 208, 224, 248" }}{PARA 0 "" 0 "" {TEXT -1 38 "183, 200, 204, 205, 228, 231, 235, 251" }}{PARA 0 " " 0 "" {TEXT -1 78 "146, 160, 161, 174, 182, 196, 198, 201, 203, 211, \+ 218, 230, 232, 238, 241, 253" }}{PARA 0 "" 0 "" {TEXT -1 78 "155, 171, 176, 179, 190, 195, 199, 210, 214, 216, 225, 229, 237, 240, 246, 255 " }}{PARA 0 "" 0 "" {TEXT -1 78 "191, 217, 219, 220, 226, 233, 234, 23 6, 242, 243, 247, 249, 250, 252, 254, 256" }}{PARA 0 "" 0 "" {TEXT -1 43 "#------------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 44 "#----------------------- --------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 43 "construction of the order 8 embedded scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "e13 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13251 "e13 := \{b[8] = -81508791 888782942071778080019859673/399395400777999552787672219983509760, a[7, 6] = 84211752143498940768/206389046233053165625+567839841668979868/187 62640566641196875*6^(1/2), a[6,2] = 0, a[6,3] = 0, c[2] = 1/46, c[5] = 71/136, c[8] = 92/143, c[11] = 103/411, c[10] = 3/44, c[13] = 59/69, \+ c[14] = 44/49, c[15] = 1, a[4,2] = 0, a[5,2] = 0, a[11,1] = -112908109 41252792923651/1669469461414577748900000-76218489460616423924209/10016 816768487466493400000*6^(1/2), a[6,4] = 890541395040155939974909749/34 04930508779360011084250045-235414842445143790083329443/680986101755872 0022168500090*6^(1/2), a[10,6] = 84329349/1146228736-1383050643/100868 128768*6^(1/2), a[9,7] = 4071/18304-1587/36608*6^(1/2), c[7] = 276/715 +46/715*6^(1/2), a[7,2] = 0, a[7,3] = 0, a[8,2] = 0, a[8,3] = 0, a[8,4 ] = 0, a[8,5] = 0, a[9,2] = 0, a[9,3] = 0, a[9,4] = 0, a[9,5] = 0, a[1 0,2] = 0, a[10,3] = 0, a[10,4] = 0, a[10,5] = 0, a[11,2] = 0, a[11,3] \+ = 0, a[11,4] = 0, a[11,5] = 0, a[12,2] = 0, a[12,3] = 0, a[12,4] = 0, \+ a[12,5] = 0, a[13,5] = 0, a[13,4] = 0, a[13,3] = 0, a[13,2] = 0, a[14, 2] = 0, a[14,3] = 0, a[14,4] = 0, a[14,5] = 0, a[15,2] = 0, a[15,3] = \+ 0, a[15,4] = 0, a[15,5] = 0, b[2] = 0, b[3] = 0, b[4] = 0, b[5] = 0, b [6] = 0, b[7] = 0, b[10] = 2172547024243858864854526674870272/16972283 408414027681310867548214525, a[11,6] = 1/30, a[8,6] = 368/1287+23/1287 *6^(1/2), a[7,4] = 121832502441158811994748302664452173/63194312296720 72722127362725145820625-12054008141355156662680357922224203047/1643052 11971473890775311430853791336250*6^(1/2), c[6] = 276/715-46/715*6^(1/2 ), a[11,7] = -44608220078798131601386867/1778327431680661626219300000- 302663621648107819403033939/5334982295041984878657900000*6^(1/2), a[8, 7] = 368/1287-23/1287*6^(1/2), a[10,7] = 84329349/1146228736+138305064 3/100868128768*6^(1/2), a[10,9] = -333490521/3152129024, a[11,9] = 763 71166597983496297729/1268154687036073482337500+76218489460616423924209 /1902232030554110223506250*6^(1/2), c[12] = 30258248819701/45339732981 913, a[4,3] = 36283219854/239481664565-4221084729/239481664565*6^(1/2) , a[7,5] = -7345188891123909155979140554752/52428978281511938535235507 146875+71382195182457889488943971467264/681576717659655200958061592909 375*6^(1/2), a[11,10] = 12837092726068800321242176/7348949926011775022 9428125+224387232972054752032871296/13889515360162254793361915625*6^(1 /2), a[5,1] = 450479172821804238979159483/489985471732935255816699904+ 65404175703680378526395577/244992735866467627908349952*6^(1/2), a[2,1] = 1/46, a[8,1] = 92/1287, a[9,1] = 1311/18304, b[11] = 11937467247139 97342094811077921918673563219/5333839543124606397779553737025343484595 200, a[10,8] = -1098320769/50434064384, b[1] = 38592443625519846145991 3/25885297292164750617319296, b[13] = 11409994679937666036993318622713 183/210646829819316523523317257829664000, a[12,6] = -17188002444689712 9391119010658057689596258428143994698538245842002969107311883958851058 6218092229/13444736815489504740326499439338937439037666081012124902765 6036163870001637234036635020235330829504*6^(1/2)-175949180260408634887 7532302368836726393890388409716969261965782534094328577924217382453202 1369934883/34620197299885474706340736056297763905521990158606221624621 429312196525421587764433517710597688597280, a[12,7] = -138207307502597 8724144662487119071171012382899509469565758724972879943335389436529352 88845283125425999567221/4494458926341069611642228743708681268522187616 12180707797390011848931329871568905581758203549930436500950000+3734226 0993482278211711351217469345525186992139310658684457112617175617353247 9385325874622983683540733325899/17977835705364278446568914974834725074 0887504644872283118956004739572531948627562232703281419972174600380000 0*6^(1/2), a[14,7] = 8209640140354849710164186531509948506760631106345 347/2089832016538282470163966831380108687881472614625000-7326425889801 057568096943993446608268878122947239059/417966403307656494032793366276 0217375762945229250000*6^(1/2), a[9,8] = -621/18304, a[10,1] = 2451872 601/50434064384, a[15,12] = 583855950025112070735873869441906321249152 2353340838322110236393942389474898060443039449084852606411316716912673 4415639048389/22474165074038263137856222538653693185986116347211357686 992324141917918392966746421148746670582600045180015758041428626487315, a[12,10] = -687320121745403419867910288709505619542179230939368838936 545299414049008577113662220686991034529999350912/123822542484884597387 9475557918928797329168344945934733789954298166264072763460790088278787 2907358839178125*6^(1/2)+111953957936197928184195726875956182710949222 53700867106612850891912759357989068567623193453186950145251776/3714676 2745465379216384266737567863919875050348378042013698628944987922182903 823702648363618722076517534375, b[9] = 9359363155244499785766625718213 61001/4086801409502444670525951816974611200, b[14] = 10048608923923592 706010638721995991/79176273134521955696793568162336000, c[9] = 69/143, a[14,10] = -369969589301763518639761285997824520507173970793320519753 635225871563412913152/180850870268987312974476662632186804296491491034 759756207691410652763681078125+454852979828196233587213868032125351691 788847437824/946232027174365750372191179393662736769436409578125*6^(1/ 2), c[4] = 48377626472/239481664565-5628112972/239481664565*6^(1/2), c [3] = 96755252944/718444993695-11256225944/718444993695*6^(1/2), a[7,1 ] = 523150756520001/5294592068569375+372205675002861/13765939378280375 0*6^(1/2), a[3,1] = -163287951175938724532816/516163208965408589753025 +42011574289334042817176/516163208965408589753025*6^(1/2), a[4,1] = 12 094406618/239481664565-1407028243/239481664565*6^(1/2), b[15] = 957935 979810312917705761/30187687238921485361088000, b[12] = 789988859008437 2060795619128755092080289376683274569118606954351302689076603970233135 6142187984337094261434894685526783/19972835375940569338334435735705036 6806065734043901244628659238274456345277244226082448918708806038102409 307770530790400, a[3,2] = 232801278267248934720896/5161632089654085897 53025-50098553466700618240256/516163208965408589753025*6^(1/2), a[6,1] = 188634486760257/2753187875656075-40451003556679/5506375751312150*6^ (1/2), a[6,5] = 127509164130554343284736/2278805333809176804299525-510 90254569210884816896/2278805333809176804299525*6^(1/2), a[11,8] = 4768 550623191902657077/335320789258483564950000+76218489460616423924209/93 88982099237539818600000*6^(1/2), a[9,6] = 4071/18304+1587/36608*6^(1/2 ), a[5,3] = -1663285823745576633021875313/489985471732935255816699904- 258991054585998425691922779/244992735866467627908349952*6^(1/2), a[5,4 ] = 734303944921586208649981787/244992735866467627908349952+9679343944 1159023582763601/122496367933233813954174976*6^(1/2), a[15,13] = -2061 784272050868275289038565787612564079661263925986910/630581804115322544 8392022471538861648368501130406546697, a[13,9] = 438696881717332693164 574683605145782902979477404432064959580227217/190320793597985643524049 406938231176153947641989529318077977593750-355937234225631449192363396 5576963561357143961289/69043804153414636584601717869842202529170566718 75*6^(1/2), a[13,11] = 14550143450655128907617052305997745679293412757 85651/437274533681980646175086038705021590800143506216585, a[15,1] = 1 8418541509225838359706930946702924369912480521/39508190287667470693939 925841123267455183000000*6^(1/2)-3584428844827517211029003988869342002 889757294838350269774673649/208571634122713325865282425894805240277335 0356190879990897000000, a[12,9] = -16342530068674673706098410397304821 11683170725738988407797492220074165441589604495769296514008733014761/1 1870640567235036900177596130414597866910751833897098560525576890900718 352020110200728728894172739518750*6^(1/2)+1115379785277627713874987636 4320845224044108260104646285465373849752391924503161388342257010461026 32948257/3347520639960280405850082108776916598468832017158981794068212 683234002575269671076605501548156712544287500, a[15,7] = -190787636465 921540998883304987599740985291747006984827/261343670052275334733028994 01390866713643703491000000+1789112785283797003555802903965105133025689 46148710869/52268734010455066946605798802781733427287406982000000*6^(1 /2), a[15,10] = 998547076725398450321677009013789887104846758874977689 61753983967005376/2398926904363445204090328582971887578229101651831845 3275155443387390625-54224186203160868130977204707093409345022342653824 /54782834560687628399797684955475647938658789640625*6^(1/2), a[14,1] = 111352303897306737782229910609546443733525824658503051605502418648299 5053/11720818219526662685715548254362381588386063204289007112295985488 00375000-9656355719858106181793772673915704646989403181/42650123991481 333817759653132492492758879187500*6^(1/2), a[14,11] = -771530345828119 9041757411869319392732268755939408305903151657039/14830628578580199565 99354292891821472831958379628421680653323310, a[14,9] = -6428514041253 11605101723817200220613720981758568362376903035228606467992291/2746471 44865445671171423994203344292862210985060954852952360191727428234375+7 7250845758864849454350181391325637175915225448/64795380679365872530827 165335902056306758765625*6^(1/2), a[12,8] = -6303547312203088429495101 1532461452879208013707075267157903271345717809889884744836815722683193 98771221/2259929545964746265552798048119437366054907690909017244940565 52454616207613142351163240610846022787800000*6^(1/2)+12632445836279619 1132942107801681715657268924065999457323338359651167769122658680421315 56830549327654393/1129964772982373132776399024059718683027453845454508 62247028276227308103806571175581620305423011393900000, a[14,12] = -945 4329086181404208751280190036798210347740980613544100878769699993886684 30388640158852256997513450369741874915618358283/1183336256073681379271 1148439493119855699320505478184908879599877661097141043082792824062740 69868573023638087923030436570, a[13,12] = 4349123515751687576213851024 2422928845963515012067399050743561004230490804423974878600834753885909 /156500737169334735327002132394157065085780206033444107250676809635435 89421556571669667160279586515, a[12,1] = 16342530068674673706098410397 3048211168317072573898840779749222007416544158960449576929651400873301 4761/62508689569237662664226328990537629273859148897483455711122025147 021504233422352449406977468048856200000*6^(1/2)-3558437927387805892113 3601321126601189438489285930760667361668284032103729293426448329800537 8753134077/61149805013384669997612713142917246028775254356233815369575 89416556451501095664913528943447961301150000, a[15,8] = -1157338419312 43537286820675899274760337634770259165918831723224267751/7941600225359 8408006316731727203547297279769620940498280027513000000-49730062074909 7635712087135560978957987636974067/99986112343404598910047966167150423 0211939000000*6^(1/2), a[13,1] = -647190707441443357339622144124312020 35561004419937320357188840883/1012098004341119843257518910033551582884 93635359529771992774000000+3559372342256314491923633965576963561357143 961289/36357243706101884581258626118802982597639184500000*6^(1/2), a[1 2,11] = 73825104187474768875967421005730375156586805528272061538441201 406502469378399602291332635087412/928712725390295991432891447406689145 25146202470165909630170359355065425993729905138177287271317, a[14,13] \+ = 1056911827593717127690972016166243945915857152155/735979081412953793 0306952068888958882079118176267, a[13,8] = -37612769433181145931578875 95063197569838926771255702620302334773297/1464216542935037622889021617 525045499738187320641028236799386000000-355937234225631449192363396557 6963561357143961289/34078442106289230846820763513066043517445275500000 *6^(1/2), a[13,10] = 3939307739733062346504442284413039354939067076806 8628724675776122752/24104666116749847055078895726017945791587677982957 405550119962671875-911199319617616509932450295187702671707428854089984 /4383797390367410875150919593877143523760184037765625*6^(1/2), a[15,9] = 2028363041702323309905709572779585455402806363222188552769650405231 11/4048345226586686505118858391505708881485411941517464594181257543750 0-18418541509225838359706930946702924369912480521/75027572902060821750 51091686174851271537156250*6^(1/2), a[15,14] = 21447821690294300536862 60375092251514835023987332030/5255390852470014220073816442862907485166 958089934203, a[15,11] = 474324561414678755109830553528867775624956239 150831585597139/451579116356903666403376148747297212812012751710592861 17565, a[14,8] = 35838730146139312307525393375183357494472205437196407 4460462507967392441417/60038122073498036770225353795012178236450255253 1871330639897526611625875000+96563557198581061817937726739157046469894 03181/39976896846716122011518307352108347913593312500*6^(1/2), a[15,6] = -10950654480421820732850785842671218433296327603/100654712486734356 1064069495275386629319739200+20704634088611516344490330265376603365714 7/3908920873271237130345900952525773317746560*6^(1/2), a[13,7] = -9318 570701703691930191534222789494006237103047579881711/356296886476918806 1196125478115789029665196590638000000+64347579447869407645375535228035 08307647498263055753667/7125937729538376122392250956231578059330393181 276000000*6^(1/2), a[13,6] = -6600558885055349219496257716458706836731 2330247/19603612723583140697294028442079849957408380800-13176568020345 600854085799476910283462657777/761305348488665658147340910372033008054 69440*6^(1/2), a[14,6] = 65156558806314041332627176859309009839731119/ 11498348446055040912040697694327646399463400+2893908603034923903437694 020377944093049/44653780373029285095303680366320956891120*6^(1/2)\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 61 "We can obtain an embedded 14 stage order 8 scheme \+ as follows." }}{PARA 0 "" 0 "" {TEXT -1 94 "We remove stages 14 and 15 from the 15 stage order 9 scheme and introduce a new stage 14 with:" }}{PARA 256 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "c[14] = 1;" "6#/ &%\"cG6#\"#9\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[14,2] = 0;" "6 #/&%\"aG6$\"#9\"\"#\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[14,3] = \+ 0;" "6#/&%\"aG6$\"#9\"\"$\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a[14 ,4]=0" "6#/&%\"aG6$\"#9\"\"%\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "a [14,5]=0" "6#/&%\"aG6$\"#9\"\"&\"\"!" }{TEXT -1 1 "," }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`b*`[2]=0" "6#/&%#b*G6#\"\"#\"\"! " }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[3]=0" "6#/&%#b*G6#\"\"$\"\"! " }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[4]=0" "6#/&%#b*G6#\"\"%\"\"! " }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[5]=0" "6#/&%#b*G6#\"\"&\"\"! " }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[6]=0" "6#/&%#b*G6#\"\"'\"\"! " }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[7] = 0;" "6#/&%#b*G6#\"\"(\" \"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 56 "where the weights of the order 8 scheme are denoted by " }{XPPEDIT 18 0 "`b*`" "6#%#b* G" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 156 "We incorporate the order 8 quadrature conditions, the ro w sum conditions for this stage and stage-order conditions so that thi s new stage has stage-order 4." }}{PARA 0 "" 0 "" {TEXT -1 48 "We also incorporate the simplifying conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(`b*`[i]*a[i,j],i = j+1 .. 14) = `b*`[j]*( 1-c[j]);" "6#/-%$SumG6$*&&%#b*G6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F ,F,F,\"#9*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 5 "for " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" } {TEXT -1 10 " ( with " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\" \"!" }{TEXT -1 5 " ), " }{XPPEDIT 18 0 "j = 9;" "6#/%\"jG\"\"*" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "j = 11;" "6#/%\"jG\"#6" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "j = 13;" "6#/%\"jG\"#8" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 451 "`Qeqs*` := subs(b=`b*`,QuadratureConditions(8,14,'expanded')):\nS O_eqs2 := [add(a[14,j],j=1..13)=c[14],add(a[14,j]*c[j],j=2..13)=1/2*c[ 14]^2,\n add(a[14,j]*c[j]^2,j=2..13)=1/3*c[14]^3,add(a[14,j]*c[ j]^3,j=2..13)=1/4*c[14]^4,\n add(a[14,j]*c[j]^4,j=2..13)=1/5*c[ 14]^5]:\n`simp_eqs*` := [add(`b*`[i]*a[i,1],i=2..14)=`b*`[1],seq(add(` b*`[i]*a[i,j],i=j+1..14)=`b*`[j]*(1-c[j]),j=[9,11,13])]:\n`cdns*` := [ op(`simp_eqs*`),op(SO_eqs2),op(`Qeqs*`)]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "e14 := `union`(re move(u_->member(op(1,lhs(u_)),[14,15]) or op(0,lhs(u_))=b,e13),\n \+ \{c[14]=1,seq(a[14,i]=0,i=2..5),seq(`b*`[i]=0,i=2..7)\}):\n" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "We have 1 7 equations for the 17 unknown coefficients." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "eqns3 := subs(e14,`cdns*`):\nnops(%);\nindets(eqns 3);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<3&%#b*G6#\"\"*&F%6#\"#5&F%6#\"\")&F%6#\"\"\"&%\"a G6$\"#9F0&F26$F4\"\"'&F26$F4\"\"(&F26$F4F-&F26$F4F'&F26$F4F*&F26$F4\"# 6&F26$F4\"#7&F26$F4\"#8&F%6#FC&F%6#FF&F%6#FI&F%6#F4" }}{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 74 "e15 := solve(\{op(eqns3)\}):\ninfol evel[solve] := 0:\ne16 := `union`(e14,e15):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e16" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10811 "e16 := \{a[7,6] = 8421175 2143498940768/206389046233053165625+567839841668979868/187626405666411 96875*6^(1/2), a[6,2] = 0, a[6,3] = 0, `b*`[4] = 0, `b*`[5] = 0, `b*`[ 6] = 0, `b*`[2] = 0, `b*`[3] = 0, `b*`[7] = 0, c[2] = 1/46, c[5] = 71/ 136, c[8] = 92/143, c[11] = 103/411, c[10] = 3/44, c[13] = 59/69, a[4, 2] = 0, a[5,2] = 0, a[11,1] = -11290810941252792923651/166946946141457 7748900000-76218489460616423924209/10016816768487466493400000*6^(1/2), a[6,4] = 890541395040155939974909749/3404930508779360011084250045-235 414842445143790083329443/6809861017558720022168500090*6^(1/2), a[10,6] = 84329349/1146228736-1383050643/100868128768*6^(1/2), a[9,7] = 4071/ 18304-1587/36608*6^(1/2), c[7] = 276/715+46/715*6^(1/2), a[7,2] = 0, a [7,3] = 0, a[8,2] = 0, a[8,3] = 0, a[8,4] = 0, a[8,5] = 0, a[9,2] = 0, a[9,3] = 0, a[9,4] = 0, a[9,5] = 0, a[10,2] = 0, a[10,3] = 0, a[10,4] = 0, a[10,5] = 0, a[11,2] = 0, a[11,3] = 0, a[11,4] = 0, a[11,5] = 0, a[12,2] = 0, a[12,3] = 0, a[12,4] = 0, a[12,5] = 0, a[13,5] = 0, a[13 ,4] = 0, a[13,3] = 0, a[13,2] = 0, a[14,2] = 0, a[14,3] = 0, a[14,4] = 0, a[14,5] = 0, a[11,6] = 1/30, a[8,6] = 368/1287+23/1287*6^(1/2), a[ 7,4] = 121832502441158811994748302664452173/63194312296720727221273627 25145820625-12054008141355156662680357922224203047/1643052119714738907 75311430853791336250*6^(1/2), c[6] = 276/715-46/715*6^(1/2), a[11,7] = -44608220078798131601386867/1778327431680661626219300000-302663621648 107819403033939/5334982295041984878657900000*6^(1/2), a[8,7] = 368/128 7-23/1287*6^(1/2), a[10,7] = 84329349/1146228736+1383050643/1008681287 68*6^(1/2), a[10,9] = -333490521/3152129024, a[11,9] = 763711665979834 96297729/1268154687036073482337500+76218489460616423924209/19022320305 54110223506250*6^(1/2), c[12] = 30258248819701/45339732981913, `b*`[12 ] = -42503788329176706927962313423474925759201096452815072640470622305 29640315652520199442309428858040080393/1949352625073773791223065241302 3567235380476570031599740950002149019485955879156117360614282462351914 240, a[4,3] = 36283219854/239481664565-4221084729/239481664565*6^(1/2) , a[7,5] = -7345188891123909155979140554752/52428978281511938535235507 146875+71382195182457889488943971467264/681576717659655200958061592909 375*6^(1/2), a[11,10] = 12837092726068800321242176/7348949926011775022 9428125+224387232972054752032871296/13889515360162254793361915625*6^(1 /2), a[5,1] = 450479172821804238979159483/489985471732935255816699904+ 65404175703680378526395577/244992735866467627908349952*6^(1/2), a[2,1] = 1/46, a[8,1] = 92/1287, a[9,1] = 1311/18304, a[14,10] = -9964789456 52945815174857068635853357841406776256125134079522701632/1776466558483 76964241055783826452426783687047504396906456095890625+2373460729117601 1669963802448994381891328701824/31554915787156112484260150916445096576 431546875*6^(1/2), `b*`[13] = 8307802719022653646372347529833/36318418 934364917848847803074080, a[10,8] = -1098320769/50434064384, `b*`[9] = 596074834977423915714550106443/3509791660513951108318405888848, `b*`[ 1] = 6365795202834324687727/367688881990976571268740, a[12,6] = -17188 0024446897129391119010658057689596258428143994698538245842002969107311 8839588510586218092229/13444736815489504740326499439338937439037666081 0121249027656036163870001637234036635020235330829504*6^(1/2)-175949180 2604086348877532302368836726393890388409716969261965782534094328577924 2173824532021369934883/34620197299885474706340736056297763905521990158 606221624621429312196525421587764433517710597688597280, a[12,7] = -138 2073075025978724144662487119071171012382899509469565758724972879943335 38943652935288845283125425999567221/4494458926341069611642228743708681 2685221876161218070779739001184893132987156890558175820354993043650095 0000+37342260993482278211711351217469345525186992139310658684457112617 1756173532479385325874622983683540733325899/17977835705364278446568914 9748347250740887504644872283118956004739572531948627562232703281419972 1746003800000*6^(1/2), a[9,8] = -621/18304, a[10,1] = 2451872601/50434 064384, a[12,10] = -68732012174540341986791028870950561954217923093936 8838936545299414049008577113662220686991034529999350912/12382254248488 4597387947555791892879732916834494593473378995429816626407276346079008 82787872907358839178125*6^(1/2)+11195395793619792818419572687595618271 094922253700867106612850891912759357989068567623193453186950145251776/ 3714676274546537921638426673756786391987505034837804201369862894498792 2182903823702648363618722076517534375, a[14,9] = -66140087449914380327 548101344360694447375771589305352069924632043/755387081649282115353043 5425077432706612170298471572545794812500+18542661946231259117159220663 2768608526005483/99396538283139104662023501334196929221531250*6^(1/2), a[14,1] = 69477988002853816125344190899680848875649117294292403621110 427/27631572785804084086128051091055380750995025720848651597000000-185 426619462312591171592206632768608526005483/523404556022605918220275652 595264842483000000*6^(1/2), c[9] = 69/143, a[14,11] = -448074765213807 929719320331516183261898760393629529216/458887835964850087697530551177 40509726182346135675245, c[4] = 48377626472/239481664565-5628112972/23 9481664565*6^(1/2), c[3] = 96755252944/718444993695-11256225944/718444 993695*6^(1/2), a[7,1] = 523150756520001/5294592068569375+372205675002 861/137659393782803750*6^(1/2), a[3,1] = -163287951175938724532816/516 163208965408589753025+42011574289334042817176/516163208965408589753025 *6^(1/2), a[4,1] = 12094406618/239481664565-1407028243/239481664565*6^ (1/2), `b*`[10] = 46281133707158379428440684544/3794809034860598699007 46060329, a[3,2] = 232801278267248934720896/516163208965408589753025-5 0098553466700618240256/516163208965408589753025*6^(1/2), a[6,1] = 1886 34486760257/2753187875656075-40451003556679/5506375751312150*6^(1/2), \+ a[6,5] = 127509164130554343284736/2278805333809176804299525-5109025456 9210884816896/2278805333809176804299525*6^(1/2), a[11,8] = 47685506231 91902657077/335320789258483564950000+76218489460616423924209/938898209 9237539818600000*6^(1/2), a[9,6] = 4071/18304+1587/36608*6^(1/2), a[5, 3] = -1663285823745576633021875313/489985471732935255816699904-2589910 54585998425691922779/244992735866467627908349952*6^(1/2), a[5,4] = 734 303944921586208649981787/244992735866467627908349952+96793439441159023 582763601/122496367933233813954174976*6^(1/2), c[14] = 1, `b*`[11] = 4 840963801198418151644350108694156829/204565449993273237622902268045767 56480, `b*`[14] = 12690737099320029004861/301876872389214853610880, a[ 13,9] = 43869688171733269316457468360514578290297947740443206495958022 7217/19032079359798564352404940693823117615394764198952931807797759375 0-3559372342256314491923633965576963561357143961289/690438041534146365 8460171786984220252917056671875*6^(1/2), a[13,11] = 145501434506551289 0761705230599774567929341275785651/43727453368198064617508603870502159 0800143506216585, a[12,9] = -16342530068674673706098410397304821116831 70725738988407797492220074165441589604495769296514008733014761/1187064 0567235036900177596130414597866910751833897098560525576890900718352020 110200728728894172739518750*6^(1/2)+1115379785277627713874987636432084 5224044108260104646285465373849752391924503161388342257010461026329482 57/3347520639960280405850082108776916598468832017158981794068212683234 002575269671076605501548156712544287500, a[14,13] = 987686476545830891 23296762257490493421581752858120/1252466050432043088092798352203994784 04120630180191, a[14,8] = 85318327243239433756228728328071128940131418 878118395907092074851/471795202042884711177778173987241082529472009403 2625268331000000+5006518725482439961632989579084752430202148041/132461 61456264411314959283823372471782839000000*6^(1/2), a[12,8] = -63035473 1220308842949510115324614528792080137070752671579032713457178098898847 4483681572268319398771221/22599295459647462655527980481194373660549076 9090901724494056552454616207613142351163240610846022787800000*6^(1/2)+ 1263244583627961911329421078016817156572689240659994573233383596511677 6912265868042131556830549327654393/11299647729823731327763990240597186 8302745384545450862247028276227308103806571175581620305423011393900000 , a[13,12] = 434912351575168757621385102424229288459635150120673990507 43561004230490804423974878600834753885909/1565007371693347353270021323 9415706508578020603344410725067680963543589421556571669667160279586515 , a[12,1] = 1634253006867467370609841039730482111683170725738988407797 492220074165441589604495769296514008733014761/625086895692376626642263 2899053762927385914889748345571112202514702150423342235244940697746804 8856200000*6^(1/2)-355843792738780589211336013211266011894384892859307 606673616682840321037292934264483298005378753134077/611498050133846699 9761271314291724602877525435623381536957589416556451501095664913528943 447961301150000, a[13,1] = -647190707441443357339622144124312020355610 04419937320357188840883/1012098004341119843257518910033551582884936353 59529771992774000000+3559372342256314491923633965576963561357143961289 /36357243706101884581258626118802982597639184500000*6^(1/2), a[12,11] \+ = 73825104187474768875967421005730375156586805528272061538441201406502 469378399602291332635087412/928712725390295991432891447406689145251462 02470165909630170359355065425993729905138177287271317, a[13,8] = -3761 276943318114593157887595063197569838926771255702620302334773297/146421 6542935037622889021617525045499738187320641028236799386000000-35593723 42256314491923633965576963561357143961289/3407844210628923084682076351 3066043517445275500000*6^(1/2), a[13,10] = 393930773973306234650444228 44130393549390670768068628724675776122752/2410466611674984705507889572 6017945791587677982957405550119962671875-91119931961761650993245029518 7702671707428854089984/43837973903674108751509195938771435237601840377 65625*6^(1/2), a[13,7] = -93185707017036919301915342227894940062371030 47579881711/3562968864769188061196125478115789029665196590638000000+64 34757944786940764537553522803508307647498263055753667/7125937729538376 122392250956231578059330393181276000000*6^(1/2), a[13,6] = -6600558885 0553492194962577164587068367312330247/19603612723583140697294028442079 849957408380800-13176568020345600854085799476910283462657777/761305348 48866565814734091037203300805469440*6^(1/2), `b*`[8] = 112358454167569 10667521184687449/27984543215947278082095867431580, a[14,12] = -979092 0345585957334747823787770618292014719367043272653602430061699291961954 865703160143382924460700238382314/581185283551484202541630621570268158 5402800853944266329705352071827921109964617590149496131182633206079239 05, a[14,6] = 155409868146947995309862276450550758522148769/1333473761 1899006162658099703042439679259200+29488334695552913776680777005490941 202439/51785388784073810340419804672009474482560*6^(1/2), a[14,7] = 30 93423962117018793580109802227401427197073610480521/3462281278819502581 34565897477182544747165591000000-2223349281800786799168112412879091860 919395860086487/692456255763900516269131794954365089494331182000000*6^ (1/2)\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "printed coefficients" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 219 "for ii from 2 to 14 do \n print(``);\n print(c[ii]=subs(e16,c[ii])); \n for jj to ii-1 \+ do\n print(a[ii,jj]=subs(e16,a[ii,jj]));\n end do:\nend do:\n`` ;\nfor ii to 14 do\n print(`b*`[ii]=subs(e16,`b*`[ii]));\nend do:" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%\"cG6#\"\"##\"\"\"\"#Y" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG 6$\"\"#\"\"\"#F(\"#Y" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"$,&#\",WHDbn*\"-&p$*\\W=(\"\"\" *(\",WfAc7\"F,F+!\"\"\"\"'#F,\"\"#F/" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%\"aG6$\"\"$\"\"\",&#\"9;G`C(Qf<^zGj\"\"9DIv*e3a'*3K;;&!\"\"*(\"8w r\"G/M$*Gu:,UF(F,F-\"\"'#F(\"\"#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /&%\"aG6$\"\"$\"\"#,&#\"9'*3sM*[sEy7!GB\"9DIv*e3a'*3K;;&\"\"\"*(\"8c-C =1qmMb)4]F-F,!\"\"\"\"'#F-F(F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"%,&#\",skix$[\"-lXm\" [R#\"\"\"*(\"+sH6GcF,F+!\"\"\"\"'#F,\"\"#F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"%\"\"\",&#\",=mS%47\"-lXm\"[R#F(*(\"+V#GqS \"F(F,!\"\"\"\"'#F(\"\"#F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6 $\"\"%\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"%\"\" $,&#\",a)>KGO\"-lXm\"[R#\"\"\"*(\"+HZ3@UF-F,!\"\"\"\"'#F-\"\"#F0" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"cG6#\"\"&#\"#r\"$O\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6 $\"\"&\"\"\",&#\"<$[f\"z*QU!=#GpD%)*fea5**e#\"\"\"\"<_*\\$3zink'et#*\\CF-\"\"'#F0\"\"#F-" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"%,&#\"<(y\")*\\'3ie@ \\%RIM(\"<_*\\$3zink'et#*\\C\"\"\"*(\";,Ow#eB!f6WRMz'*F-\"\")e6W-D$=7\" FD1#e9DFOF@ss?nH7V>j\"\"\"*(\"GZI?CA#zN!oim:b893S07F-\"H]iL\"z`3V6`x!* QZr>@0V;!\"\"\"\"'#F-\"\"#F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"a G6$\"\"(\"\"&,&#\"@_ZbS\"zf:4R7\"*))=Xt\"Avo92bBN&Q>^\"Gy*GC&!\"\"*(\" AksYrR%*)[*)yX#=&>#Qr\"\"\"\"Bv$4Hfh!e4?b'fwrw:oF-\"\"'#F0\"\"#F0" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"(\"\"',&#\"5o2%*)\\V@v6U) \"6Dc;`IBY!*Q1#\"\"\"*(\"3o)z*o;%)RycF-\"5vo>TmcSEw=!\"\"F(#F-\"\"#F- " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\")#\"##*\"$V\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/& %\"aG6$\"\")\"\"\"#\"##*\"%(G\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"\")\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\" )\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"%\"\" !" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"&\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"',&#\"$o$\"%(G\"\"\" \"*(\"#BF-F,!\"\"F(#F-\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\" aG6$\"\")\"\"(,&#\"$o$\"%(G\"\"\"\"*(\"#BF-F,!\"\"\"\"'#F-\"\"#F0" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"cG6#\"\"*#\"#p\"$V\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6 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"6#/&%#b*G6#\"\"#\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%#b*G6#\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/&%#b*G6#\"\"%\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\" &\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"'\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"(\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%#b*G6#\"\")#\"A\\uo%=@vm5pv;a%eB6\"A!eJu'e4#3ys%f@ VX)z#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"\"*#\"?Vk5]Xr:RUx \\$[2'f\"@[)))eS=$36&R^g;z4N" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b* G6#\"#5#\">WXoS%G%z$erqL6GY\"?H.1Y2!*p)fg[.4[z$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#6#\"FHo:%p3,NW;:=%)>,Q'4%[\"G![cnd/oA!HiPKF $**\\ac/#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#7#!bq$R!3S!e) G%4BW*>?Dl:.kH0Biq/ks]\"GX'4,#fd#\\ZBMJiz#pqw\"H$)y.D%\"cqSU\">NiCG91O !\\@+]4u*fJ+dw/QNscBIT_1B7ztP2DEN\\>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#8#\"@L)HvMsjk`E->F!yI)\"A!3uI!y%)[y\"\\OM*= %=j$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%#b*G6#\"#9#\"8h[+H+K*4P2p7 \"9!)3h`[@*Qso(=I" }}}{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 0 104 "`RK8_14eqs*` := subs(b=`b*` ,[op(RowSumConditions(14,'expanded')),op(OrderConditions(8,14,'expande d'))]):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Check:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "expand(subs(e16 ,`RK8_14eqs*`)):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7ax\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$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#\"$8 #" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "We \+ can calculate the principal error norm, that is, the 2-norm of the pri ncipal error terms. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 185 "`er rterms8_14*` := subs(b=`b*`,PrincipalErrorTerms(8,14,'expanded')):\nsm := 0:\nfor ct to nops(`errterms8_14*`) do\n sm := sm+(evalf(subs(e1 6,`errterms8_14*`[ct])))^2;\nend do:\nsqrt(sm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+_$)4*f$!#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 138 "We can include the new stage for the emb edded scheme as an additional 16th stage added to the order 9 scheme a long with the coefficients " }{XPPEDIT 18 0 "a[16,14] = 0;" "6#/&%\"a G6$\"#;\"#9\"\"!" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[16,15] = 0; " "6#/&%\"aG6$\"#;\"#:\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 13 "The weights " }{XPPEDIT 18 0 "`b*`[i]" "6#&%#b*G6#%\"iG" } {TEXT -1 7 " for " }{XPPEDIT 18 0 "i=1" "6#/%\"iG\"\"\"" }{TEXT -1 89 " . . 13 of the 16 stage combined scheme are those of the 14 stage \+ scheme and the weight " }{XPPEDIT 18 0 "`b*`[14];" "6#&%#b*G6#\"#9" } {TEXT -1 34 " in the 14 stage scheme becomes " }{XPPEDIT 18 0 "`b*`[ 16];" "6#&%#b*G6#\"#;" }{TEXT -1 25 " in the 16 stage scheme." }} {PARA 0 "" 0 "" {TEXT -1 8 "We set " }{XPPEDIT 18 0 "c[16] = 1;" "6#/ &%\"cG6#\"#;\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`b*`[14] = 0;" " 6#/&%#b*G6#\"#9\"\"!" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "`b*`[15] = 0;" "6#/&%#b*G6#\"#:\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 66 "We can make the order 9 scheme into a 16 stage scheme by settin g " }{XPPEDIT 18 0 "b[16] = 0;" "6#/&%\"bG6#\"#;\"\"!" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 201 "e17 := \{c[16]=1,seq(a[16,i]=subs(e16,a[14,i]),i=1..13),a[16, 14]=0,a[16,15]=0,b[16]=0,\nseq(`b*`[i]=subs(e16,`b*`[i]),i=1..13),`b*` [14]=0,`b*`[15]=0,`b*`[16]=subs(e16,`b*`[14])\}:\ne18 := `union`(e13,e 17):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "e18" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16058 "e18 := \{b[8] = -81508791888782942071778080019859673/3993954007 77999552787672219983509760, a[7,6] = 84211752143498940768/206389046233 053165625+567839841668979868/18762640566641196875*6^(1/2), a[6,2] = 0, a[6,3] = 0, `b*`[4] = 0, `b*`[5] = 0, `b*`[6] = 0, `b*`[2] = 0, `b*`[ 3] = 0, `b*`[7] = 0, b[16] = 0, c[2] = 1/46, c[5] = 71/136, c[8] = 92/ 143, c[11] = 103/411, c[10] = 3/44, c[13] = 59/69, c[14] = 44/49, c[15 ] = 1, c[16] = 1, a[4,2] = 0, a[5,2] = 0, a[11,1] = -11290810941252792 923651/1669469461414577748900000-76218489460616423924209/1001681676848 7466493400000*6^(1/2), a[6,4] = 890541395040155939974909749/3404930508 779360011084250045-235414842445143790083329443/68098610175587200221685 00090*6^(1/2), a[10,6] = 84329349/1146228736-1383050643/100868128768*6 ^(1/2), a[9,7] = 4071/18304-1587/36608*6^(1/2), c[7] = 276/715+46/715* 6^(1/2), a[7,2] = 0, a[7,3] = 0, a[8,2] = 0, a[8,3] = 0, a[8,4] = 0, a [8,5] = 0, a[9,2] = 0, a[9,3] = 0, a[9,4] = 0, a[9,5] = 0, a[10,2] = 0 , a[10,3] = 0, a[10,4] = 0, a[10,5] = 0, a[11,2] = 0, a[11,3] = 0, a[1 1,4] = 0, a[11,5] = 0, a[12,2] = 0, a[12,3] = 0, a[12,4] = 0, a[12,5] \+ = 0, a[13,5] = 0, a[13,4] = 0, a[13,3] = 0, a[13,2] = 0, a[14,2] = 0, \+ a[14,3] = 0, a[14,4] = 0, a[14,5] = 0, a[15,2] = 0, a[15,3] = 0, a[15, 4] = 0, a[15,5] = 0, a[16,2] = 0, a[16,3] = 0, a[16,4] = 0, a[16,5] = \+ 0, a[16,14] = 0, a[16,15] = 0, b[2] = 0, b[3] = 0, b[4] = 0, b[5] = 0, b[6] = 0, b[7] = 0, `b*`[14] = 0, `b*`[15] = 0, b[10] = 2172547024243 858864854526674870272/16972283408414027681310867548214525, a[11,6] = 1 /30, a[8,6] = 368/1287+23/1287*6^(1/2), a[7,4] = 121832502441158811994 748302664452173/6319431229672072722127362725145820625-1205400814135515 6662680357922224203047/164305211971473890775311430853791336250*6^(1/2) , c[6] = 276/715-46/715*6^(1/2), a[11,7] = -44608220078798131601386867 /1778327431680661626219300000-302663621648107819403033939/533498229504 1984878657900000*6^(1/2), a[8,7] = 368/1287-23/1287*6^(1/2), a[10,7] = 84329349/1146228736+1383050643/100868128768*6^(1/2), a[10,9] = -33349 0521/3152129024, a[11,9] = 76371166597983496297729/1268154687036073482 337500+76218489460616423924209/1902232030554110223506250*6^(1/2), c[12 ] = 30258248819701/45339732981913, `b*`[12] = -42503788329176706927962 3134234749257592010964528150726404706223052964031565252019944230942885 8040080393/19493526250737737912230652413023567235380476570031599740950 002149019485955879156117360614282462351914240, a[4,3] = 36283219854/23 9481664565-4221084729/239481664565*6^(1/2), a[7,5] = -7345188891123909 155979140554752/52428978281511938535235507146875+713821951824578894889 43971467264/681576717659655200958061592909375*6^(1/2), a[11,10] = 1283 7092726068800321242176/73489499260117750229428125+22438723297205475203 2871296/13889515360162254793361915625*6^(1/2), a[5,1] = 45047917282180 4238979159483/489985471732935255816699904+65404175703680378526395577/2 44992735866467627908349952*6^(1/2), a[2,1] = 1/46, a[8,1] = 92/1287, a [9,1] = 1311/18304, b[11] = 119374672471399734209481107792191867356321 9/5333839543124606397779553737025343484595200, `b*`[13] = 830780271902 2653646372347529833/36318418934364917848847803074080, a[10,8] = -10983 20769/50434064384, `b*`[9] = 596074834977423915714550106443/3509791660 513951108318405888848, b[1] = 385924436255198461459913/258852972921647 50617319296, `b*`[1] = 6365795202834324687727/367688881990976571268740 , b[13] = 11409994679937666036993318622713183/210646829819316523523317 257829664000, a[12,6] = -171880024446897129391119010658057689596258428 1439946985382458420029691073118839588510586218092229/13444736815489504 7403264994393389374390376660810121249027656036163870001637234036635020 235330829504*6^(1/2)-1759491802604086348877532302368836726393890388409 7169692619657825340943285779242173824532021369934883/34620197299885474 7063407360562977639055219901586062216246214293121965254215877644335177 10597688597280, a[12,7] = -1382073075025978724144662487119071171012382 89950946956575872497287994333538943652935288845283125425999567221/4494 4589263410696116422287437086812685221876161218070779739001184893132987 1568905581758203549930436500950000+37342260993482278211711351217469345 5251869921393106586844571126171756173532479385325874622983683540733325 899/179778357053642784465689149748347250740887504644872283118956004739 5725319486275622327032814199721746003800000*6^(1/2), a[14,7] = 8209640 140354849710164186531509948506760631106345347/208983201653828247016396 6831380108687881472614625000-73264258898010575680969439934466082688781 22947239059/4179664033076564940327933662760217375762945229250000*6^(1/ 2), a[9,8] = -621/18304, a[10,1] = 2451872601/50434064384, a[15,12] = \+ 5838559500251120707358738694419063212491522353340838322110236393942389 4748980604430394490848526064113167169126734415639048389/22474165074038 2631378562225386536931859861163472113576869923241419179183929667464211 48746670582600045180015758041428626487315, a[12,10] = -687320121745403 4198679102887095056195421792309393688389365452994140490085771136622206 86991034529999350912/1238225424848845973879475557918928797329168344945 9347337899542981662640727634607900882787872907358839178125*6^(1/2)+111 9539579361979281841957268759561827109492225370086710661285089191275935 7989068567623193453186950145251776/37146762745465379216384266737567863 9198750503483780420136986289449879221829038237026483636187220765175343 75, b[9] = 935936315524449978576662571821361001/4086801409502444670525 951816974611200, b[14] = 10048608923923592706010638721995991/791762731 34521955696793568162336000, c[9] = 69/143, a[14,10] = -369969589301763 518639761285997824520507173970793320519753635225871563412913152/180850 8702689873129744766626321868042964914910347597562076914106527636810781 25+454852979828196233587213868032125351691788847437824/946232027174365 750372191179393662736769436409578125*6^(1/2), c[4] = 48377626472/23948 1664565-5628112972/239481664565*6^(1/2), c[3] = 96755252944/7184449936 95-11256225944/718444993695*6^(1/2), a[7,1] = 523150756520001/52945920 68569375+372205675002861/137659393782803750*6^(1/2), a[3,1] = -1632879 51175938724532816/516163208965408589753025+42011574289334042817176/516 163208965408589753025*6^(1/2), a[4,1] = 12094406618/239481664565-14070 28243/239481664565*6^(1/2), b[15] = 957935979810312917705761/301876872 38921485361088000, b[12] = 7899888590084372060795619128755092080289376 6832745691186069543513026890766039702331356142187984337094261434894685 526783/199728353759405693383344357357050366806065734043901244628659238 274456345277244226082448918708806038102409307770530790400, `b*`[10] = \+ 46281133707158379428440684544/379480903486059869900746060329, a[3,2] = 232801278267248934720896/516163208965408589753025-5009855346670061824 0256/516163208965408589753025*6^(1/2), a[6,1] = 188634486760257/275318 7875656075-40451003556679/5506375751312150*6^(1/2), a[6,5] = 127509164 130554343284736/2278805333809176804299525-51090254569210884816896/2278 805333809176804299525*6^(1/2), a[11,8] = 4768550623191902657077/335320 789258483564950000+76218489460616423924209/9388982099237539818600000*6 ^(1/2), a[9,6] = 4071/18304+1587/36608*6^(1/2), a[5,3] = -166328582374 5576633021875313/489985471732935255816699904-2589910545859984256919227 79/244992735866467627908349952*6^(1/2), a[5,4] = 734303944921586208649 981787/244992735866467627908349952+96793439441159023582763601/12249636 7933233813954174976*6^(1/2), a[15,13] = -20617842720508682752890385657 87612564079661263925986910/6305818041153225448392022471538861648368501 130406546697, `b*`[11] = 4840963801198418151644350108694156829/2045654 4999327323762290226804576756480, a[13,9] = 438696881717332693164574683 605145782902979477404432064959580227217/190320793597985643524049406938 231176153947641989529318077977593750-355937234225631449192363396557696 3561357143961289/6904380415341463658460171786984220252917056671875*6^( 1/2), a[13,11] = 1455014345065512890761705230599774567929341275785651/ 437274533681980646175086038705021590800143506216585, a[15,1] = 1841854 1509225838359706930946702924369912480521/39508190287667470693939925841 123267455183000000*6^(1/2)-3584428844827517211029003988869342002889757 294838350269774673649/208571634122713325865282425894805240277335035619 0879990897000000, a[12,9] = -16342530068674673706098410397304821116831 70725738988407797492220074165441589604495769296514008733014761/1187064 0567235036900177596130414597866910751833897098560525576890900718352020 110200728728894172739518750*6^(1/2)+1115379785277627713874987636432084 5224044108260104646285465373849752391924503161388342257010461026329482 57/3347520639960280405850082108776916598468832017158981794068212683234 002575269671076605501548156712544287500, a[15,7] = -190787636465921540 998883304987599740985291747006984827/261343670052275334733028994013908 66713643703491000000+1789112785283797003555802903965105133025689461487 10869/52268734010455066946605798802781733427287406982000000*6^(1/2), a [15,10] = 998547076725398450321677009013789887104846758874977689617539 83967005376/2398926904363445204090328582971887578229101651831845327515 5443387390625-54224186203160868130977204707093409345022342653824/54782 834560687628399797684955475647938658789640625*6^(1/2), `b*`[16] = 1269 0737099320029004861/301876872389214853610880, a[16,1] = 69477988002853 816125344190899680848875649117294292403621110427/276315727858040840861 28051091055380750995025720848651597000000-1854266194623125911715922066 32768608526005483/523404556022605918220275652595264842483000000*6^(1/2 ), a[14,1] = 111352303897306737782229910609546443733525824658503051605 5024186482995053/11720818219526662685715548254362381588386063204289007 11229598548800375000-9656355719858106181793772673915704646989403181/42 650123991481333817759653132492492758879187500*6^(1/2), a[14,11] = -771 5303458281199041757411869319392732268755939408305903151657039/14830628 57858019956599354292891821472831958379628421680653323310, a[14,9] = -6 4285140412531160510172381720022061372098175856836237690303522860646799 2291/27464714486544567117142399420334429286221098506095485295236019172 7428234375+77250845758864849454350181391325637175915225448/64795380679 365872530827165335902056306758765625*6^(1/2), a[12,8] = -6303547312203 0884294951011532461452879208013707075267157903271345717809889884744836 81572268319398771221/2259929545964746265552798048119437366054907690909 01724494056552454616207613142351163240610846022787800000*6^(1/2)+12632 4458362796191132942107801681715657268924065999457323338359651167769122 65868042131556830549327654393/1129964772982373132776399024059718683027 45384545450862247028276227308103806571175581620305423011393900000, a[1 6,6] = 155409868146947995309862276450550758522148769/13334737611899006 162658099703042439679259200+29488334695552913776680777005490941202439/ 51785388784073810340419804672009474482560*6^(1/2), a[16,7] = 309342396 2117018793580109802227401427197073610480521/34622812788195025813456589 7477182544747165591000000-22233492818007867991681124128790918609193958 60086487/692456255763900516269131794954365089494331182000000*6^(1/2), \+ a[16,8] = 853183272432394337562287283280711289401314188781183959070920 74851/4717952020428847111777781739872410825294720094032625268331000000 +5006518725482439961632989579084752430202148041/1324616145626441131495 9283823372471782839000000*6^(1/2), a[16,9] = -661400874499143803275481 01344360694447375771589305352069924632043/7553870816492821153530435425 077432706612170298471572545794812500+185426619462312591171592206632768 608526005483/99396538283139104662023501334196929221531250*6^(1/2), a[1 4,12] = -9454329086181404208751280190036798210347740980613544100878769 69999388668430388640158852256997513450369741874915618358283/1183336256 0736813792711148439493119855699320505478184908879599877661097141043082 79282406274069868573023638087923030436570, a[13,12] = 4349123515751687 5762138510242422928845963515012067399050743561004230490804423974878600 834753885909/156500737169334735327002132394157065085780206033444107250 67680963543589421556571669667160279586515, a[12,1] = 16342530068674673 7060984103973048211168317072573898840779749222007416544158960449576929 6514008733014761/62508689569237662664226328990537629273859148897483455 711122025147021504233422352449406977468048856200000*6^(1/2)-3558437927 3878058921133601321126601189438489285930760667361668284032103729293426 4483298005378753134077/61149805013384669997612713142917246028775254356 23381536957589416556451501095664913528943447961301150000, a[15,8] = -1 15733841931243537286820675899274760337634770259165918831723224267751/7 9416002253598408006316731727203547297279769620940498280027513000000-49 7300620749097635712087135560978957987636974067/99986112343404598910047 9661671504230211939000000*6^(1/2), a[13,1] = -647190707441443357339622 14412431202035561004419937320357188840883/1012098004341119843257518910 03355158288493635359529771992774000000+3559372342256314491923633965576 963561357143961289/36357243706101884581258626118802982597639184500000* 6^(1/2), a[12,11] = 73825104187474768875967421005730375156586805528272 061538441201406502469378399602291332635087412/928712725390295991432891 4474066891452514620247016590963017035935506542599372990513817728727131 7, a[14,13] = 1056911827593717127690972016166243945915857152155/735979 0814129537930306952068888958882079118176267, a[13,8] = -37612769433181 14593157887595063197569838926771255702620302334773297/1464216542935037 622889021617525045499738187320641028236799386000000-355937234225631449 1923633965576963561357143961289/34078442106289230846820763513066043517 445275500000*6^(1/2), a[13,10] = 3939307739733062346504442284413039354 9390670768068628724675776122752/24104666116749847055078895726017945791 587677982957405550119962671875-911199319617616509932450295187702671707 428854089984/4383797390367410875150919593877143523760184037765625*6^(1 /2), a[15,9] = 2028363041702323309905709572779585455402806363222188552 76965040523111/4048345226586686505118858391505708881485411941517464594 1812575437500-18418541509225838359706930946702924369912480521/75027572 90206082175051091686174851271537156250*6^(1/2), a[15,14] = 21447821690 29430053686260375092251514835023987332030/5255390852470014220073816442 862907485166958089934203, a[15,11] = 474324561414678755109830553528867 775624956239150831585597139/451579116356903666403376148747297212812012 75171059286117565, a[14,8] = 35838730146139312307525393375183357494472 2054371964074460462507967392441417/60038122073498036770225353795012178 2364502552531871330639897526611625875000+96563557198581061817937726739 15704646989403181/39976896846716122011518307352108347913593312500*6^(1 /2), a[15,6] = -10950654480421820732850785842671218433296327603/100654 7124867343561064069495275386629319739200+20704634088611516344490330265 3766033657147/3908920873271237130345900952525773317746560*6^(1/2), a[1 3,7] = -9318570701703691930191534222789494006237103047579881711/356296 8864769188061196125478115789029665196590638000000+64347579447869407645 37553522803508307647498263055753667/7125937729538376122392250956231578 059330393181276000000*6^(1/2), a[13,6] = -6600558885055349219496257716 4587068367312330247/19603612723583140697294028442079849957408380800-13 176568020345600854085799476910283462657777/761305348488665658147340910 37203300805469440*6^(1/2), a[14,6] = 651565588063140413326271768593090 09839731119/11498348446055040912040697694327646399463400+2893908603034 923903437694020377944093049/44653780373029285095303680366320956891120* 6^(1/2), `b*`[8] = 11235845416756910667521184687449/279845432159472780 82095867431580, a[16,10] = -996478945652945815174857068635853357841406 776256125134079522701632/177646655848376964241055783826452426783687047 504396906456095890625+23734607291176011669963802448994381891328701824/ 31554915787156112484260150916445096576431546875*6^(1/2), a[16,11] = -4 48074765213807929719320331516183261898760393629529216/4588878359648500 8769753055117740509726182346135675245, a[16,12] = -9790920345585957334 7478237877706182920147193670432726536024300616992919619548657031601433 82924460700238382314/5811852835514842025416306215702681585402800853944 26632970535207182792110996461759014949613118263320607923905, a[16,13] \+ = 98768647654583089123296762257490493421581752858120/12524660504320430 8809279835220399478404120630180191\}:" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "printed coefficients" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 292 "for ii from 2 to 16 do\n print(``);\n print(c[ii]=subs(e18,c[ ii])); \n for jj to ii-1 do\n print(a[ii,jj]=subs(e18,a[ii,jj]) );\n end do:\nend do:print(``);\nfor ii to 16 do\n print(b[ii]=sub s(e18,b[ii]));\nend do:print(``);\nfor ii to 16 do\n print(`b*`[ii]= subs(e18,`b*`[ii]));\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"##\"\"\"\"#Y" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"#\"\"\"#F(\"#Y" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\" \"$,&#\",WHDbn*\"-&p$*\\W=(\"\"\"*(\",WfAc7\"F,F+!\"\"\"\"'#F,\"\"#F/ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"$\"\"\",&#\"9;G`C(Qf< ^zGj\"\"9DIv*e3a'*3K;;&!\"\"*(\"8wr\"G/M$*Gu:,UF(F,F-\"\"'#F(\"\"#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"$\"\"#,&#\"9'*3sM*[sEy7 !GB\"9DIv*e3a'*3K;;&\"\"\"*(\"8c-C=1qmMb)4]F-F,!\"\"\"\"'#F-F(F0" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/&%\"cG6#\"\"%,&#\",skix$[\"-lXm\"[R#\"\"\"*(\"+sH6GcF,F+!\"\"\"\"'#F ,\"\"#F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"%\"\"\",&#\", =mS%47\"-lXm\"[R#F(*(\"+V#GqS\"F(F,!\"\"\"\"'#F(\"\"#F/" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"%\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"%\"\"$,&#\",a)>KGO\"-lXm\"[R#\"\"\"*(\"+HZ 3@UF-F,!\"\"\"\"'#F-\"\"#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"\"&#\"#r\"$O\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"&\"\"\",&#\"<$[f\"z*QU!=#GpD%)*fea5**e#\"\"\"\"<_*\\$3zin k'et#*\\CF-\"\"'#F0\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6 $\"\"&\"\"%,&#\"<(y\")*\\'3ie@\\%RIM(\"<_*\\$3zink'et#*\\C\"\"\"*(\";, Ow#eB!f6WRMz'*F-\"\")e6W-D$=7\"FD1#e9DFOF@ss?nH7V>j\"\" 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"" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"$\"\"!" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"%\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5\"\"',&#\")\\$HV)\"+O(Gi9\"\"\"\"*(\"+V10$ Q\"F-\"-o(G\"o35!\"\"F(#F-\"\"#F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ &%\"aG6$\"#5\"\"(,&#\")\\$HV)\"+O(Gi9\"\"\"\"*(\"+V10$Q\"F-\"-o(G\"o35 !\"\"\"\"'#F-\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#5 \"\")#!+p2K)4\"\",%QkSV]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"#5\"\"*#!*@0\\L$\"+C!H@:$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"cG6#\"#6#\"$.\"\"$6%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"\",&#\"8^O#Hz_7%4\"3H6\":++!*[ xd99Yp%p;!\"\"*(\"84U#RU;1Y*[=i(F(\";++S$\\mu[on\"o,5F-\"\"'#F(\"\"#F- " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"#6\"\"%\"\"!" 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$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$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#\"$,&" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7dw\"\"!F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$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 33 "seq(c[i]=subs(e18,c[i]),i=2..16);" }} {PARA 12 "" 1 "" {XPPMATH 20 "61/&%\"cG6#\"\"##\"\"\"\"#Y/&F%6#\"\"$,& #\",WHDbn*\"-&p$*\\W=(F)*(\",WfAc7\"F)F2!\"\"\"\"'#F)F'F5/&F%6#\"\"%,& #\",skix$[\"-lXm\"[R#F)*(\"+sH6GcF)F?F5F6F7F5/&F%6#\"\"&#\"#r\"$O\"/&F %6#F6,&#\"$w#\"$:(F)*(F*F)FOF5F6F7F5/&F%6#\"\"(,&FMF)*(F*F)FOF5F6F7F)/ &F%6#\"\")#\"##*\"$V\"/&F%6#\"\"*#\"#pFgn/&F%6#\"#5#F.\"#W/&F%6#\"#6# \"$.\"\"$6%/&F%6#\"#7#\"/,(>)[#e-$\"/8>)HtR`%/&F%6#\"#8#\"#fF]o/&F%6# \"#9#Fco\"#\\/&F%6#\"#:F)/&F%6#\"#;F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 26 "#-------------------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 " #---------------------------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "characteristics of the embedded scheme" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "coefficients of the comb ined scheme " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16669 "ee := \{c[2]=1/46,\nc[3]=96755252944/7184449936 95-11256225944/718444993695*6^(1/2),\nc[4]=48377626472/239481664565-56 28112972/239481664565*6^(1/2),\nc[5]=71/136,\nc[6]=276/715-46/715*6^(1 /2),\nc[7]=276/715+46/715*6^(1/2),\nc[8]=92/143,\nc[9]=69/143,\nc[10]= 3/44,\nc[11]=103/411,\nc[12]=30258248819701/45339732981913,\nc[13]=59/ 69,\nc[14]=44/49,\nc[15]=1,\nc[16]=1,\n\na[2,1]=1/46,\na[3,1]=-1632879 51175938724532816/516163208965408589753025+\n 42011574289334042 817176/516163208965408589753025*6^(1/2),\na[3,2]=232801278267248934720 896/516163208965408589753025-\n 50098553466700618240256/51616320 8965408589753025*6^(1/2),\na[4,1]=12094406618/239481664565-1407028243/ 239481664565*6^(1/2),\na[4,2]=0,\na[4,3]=36283219854/239481664565-4221 084729/239481664565*6^(1/2),\na[5,1]=450479172821804238979159483/48998 5471732935255816699904+\n 65404175703680378526395577/24499273586 6467627908349952*6^(1/2),\na[5,2]=0,\na[5,3]=-166328582374557663302187 5313/489985471732935255816699904-\n 258991054585998425691922779 /244992735866467627908349952*6^(1/2),\na[5,4]=734303944921586208649981 787/244992735866467627908349952+\n 96793439441159023582763601/12 2496367933233813954174976*6^(1/2),\na[6,1]=188634486760257/27531878756 56075-40451003556679/5506375751312150*6^(1/2),\na[6,2]=0,\na[6,3]=0,\n a[6,4]=890541395040155939974909749/3404930508779360011084250045-\n \+ 235414842445143790083329443/6809861017558720022168500090*6^(1/2),\n a[6,5]=127509164130554343284736/2278805333809176804299525-\n 5 1090254569210884816896/2278805333809176804299525*6^(1/2),\na[7,1]=5231 50756520001/5294592068569375+372205675002861/137659393782803750*6^(1/2 ),\na[7,2]=0,\na[7,3]=0,\na[7,4]=121832502441158811994748302664452173/ 6319431229672072722127362725145820625-\n 12054008141355156662680 357922224203047/164305211971473890775311430853791336250*6^(1/2),\na[7, 5]=-7345188891123909155979140554752/52428978281511938535235507146875+ \n 71382195182457889488943971467264/681576717659655200958061592 909375*6^(1/2),\na[7,6]=84211752143498940768/206389046233053165625+567 839841668979868/18762640566641196875*6^(1/2),\na[8,1]=92/1287,\na[8,2] =0,\na[8,3]=0,\na[8,4]=0,\na[8,5]=0,\na[8,6]=368/1287+23/1287*6^(1/2), \na[8,7]=368/1287-23/1287*6^(1/2),\na[9,1]=1311/18304,\na[9,2]=0,\na[9 ,3]=0,\na[9,4]=0,\na[9,5]=0,\na[9,6]=4071/18304+1587/36608*6^(1/2),\na [9,7]=4071/18304-1587/36608*6^(1/2),\na[9,8]=-621/18304,\na[10,1]=2451 872601/50434064384,\na[10,2]=0,\na[10,3]=0,\na[10,4]=0,\na[10,5]=0,\na [10,6]=84329349/1146228736-1383050643/100868128768*6^(1/2),\na[10,7]=8 4329349/1146228736+1383050643/100868128768*6^(1/2),\na[10,8]=-10983207 69/50434064384,\na[10,9]=-333490521/3152129024,\na[11,1]=-112908109412 52792923651/1669469461414577748900000-\n 762184894606164239242 09/10016816768487466493400000*6^(1/2),\na[11,2]=0,\na[11,3]=0,\na[11,4 ]=0,\na[11,5]=0,\na[11,6]=1/30,\na[11,7]=-44608220078798131601386867/1 778327431680661626219300000-\n 302663621648107819403033939/533 4982295041984878657900000*6^(1/2),\na[11,8]=4768550623191902657077/335 320789258483564950000+76218489460616423924209/938898209923753981860000 0*6^(1/2),\na[11,9]=76371166597983496297729/1268154687036073482337500+ \n 76218489460616423924209/1902232030554110223506250*6^(1/2),\n a[11,10]=12837092726068800321242176/73489499260117750229428125+\n \+ 224387232972054752032871296/13889515360162254793361915625*6^(1/2), \na[12,1]=-35584379273878058921133601321126601189438489285930760667361 6682840321037292934264483298005378753134077/61149805013384669997612713 1429172460287752543562338153695758941655645150109566491352894344796130 1150000+16342530068674673706098410397304821116831707257389884077974922 20074165441589604495769296514008733014761/6250868956923766266422632899 0537629273859148897483455711122025147021504233422352449406977468048856 200000*6^(1/2),\na[12,2]=0,\na[12,3]=0,\na[12,4]=0,\na[12,5]=0,\na[12, 6]=-175949180260408634887753230236883672639389038840971696926196578253 40943285779242173824532021369934883/\n 34620197299885474706340 7360562977639055219901586062216246214293121965254215877644335177105976 88597280-\n 17188002444689712939111901065805768959625842814399 46985382458420029691073118839588510586218092229/\n 134447368154895 0474032649943933893743903766608101212490276560361638700016372340366350 20235330829504*6^(1/2),\na[12,7]=-138207307502597872414466248711907117 1012382899509469565758724972879943335389436529352888452831254259995672 21/4494458926341069611642228743708681268522187616121807077973900118489 31329871568905581758203549930436500950000+3734226099348227821171135121 7469345525186992139310658684457112617175617353247938532587462298368354 0733325899/17977835705364278446568914974834725074088750464487228311895 60047395725319486275622327032814199721746003800000*6^(1/2),\na[12,8]=1 2632445836279619113294210780168171565726892406599945732333835965116776 912265868042131556830549327654393/112996477298237313277639902405971868 302745384545450862247028276227308103806571175581620305423011393900000- 6303547312203088429495101153246145287920801370707526715790327134571780 988988474483681572268319398771221/225992954596474626555279804811943736 605490769090901724494056552454616207613142351163240610846022787800000* 6^(1/2),\na[12,9]=1115379785277627713874987636432084522404410826010464 628546537384975239192450316138834225701046102632948257/334752063996028 0405850082108776916598468832017158981794068212683234002575269671076605 501548156712544287500-163425300686746737060984103973048211168317072573 8988407797492220074165441589604495769296514008733014761/11870640567235 0369001775961304145978669107518338970985605255768909007183520201102007 28728894172739518750*6^(1/2),\na[12,10]=111953957936197928184195726875 9561827109492225370086710661285089191275935798906856762319345318695014 5251776/37146762745465379216384266737567863919875050348378042013698628 944987922182903823702648363618722076517534375-687320121745403419867910 2887095056195421792309393688389365452994140490085771136622206869910345 29999350912/1238225424848845973879475557918928797329168344945934733789 9542981662640727634607900882787872907358839178125*6^(1/2),\na[12,11]=7 3825104187474768875967421005730375156586805528272061538441201406502469 378399602291332635087412/\n 9287127253902959914328914474066891 4525146202470165909630170359355065425993729905138177287271317,\na[13,1 ]=-64719070744144335733962214412431202035561004419937320357188840883/ \n 10120980043411198432575189100335515828849363535952977199277 4000000+\n 3559372342256314491923633965576963561357143961289/ \n 36357243706101884581258626118802982597639184500000*6^(1/2), \na[13,2]=0,\na[13,3]=0,\na[13,4]=0,\na[13,5]=0,\na[13,6]=-66005588850 553492194962577164587068367312330247/\n 1960361272358314069729 4028442079849957408380800-\n 131765680203456008540857994769102 83462657777/\n 76130534848866565814734091037203300805469440*6^ (1/2),\na[13,7]=-93185707017036919301915342227894940062371030475798817 11/\n 3562968864769188061196125478115789029665196590638000000+ \n 6434757944786940764537553522803508307647498263055753667/\n \+ 7125937729538376122392250956231578059330393181276000000*6^(1/2 ),\na[13,8]=-376127694331811459315788759506319756983892677125570262030 2334773297/\n 146421654293503762288902161752504549973818732064 1028236799386000000-\n 355937234225631449192363396557696356135 7143961289/\n 3407844210628923084682076351306604351744527550 0000*6^(1/2),\na[13,9]=43869688171733269316457468360514578290297947740 4432064959580227217/\n 1903207935979856435240494069382311761539 47641989529318077977593750-\n 355937234225631449192363396557696 3561357143961289/\n 6904380415341463658460171786984220252917056 671875*6^(1/2),\na[13,10]=39393077397330623465044422844130393549390670 768068628724675776122752/\n 2410466611674984705507889572601794 5791587677982957405550119962671875-\n 911199319617616509932450 295187702671707428854089984/\n 4383797390367410875150919593877 143523760184037765625*6^(1/2),\na[13,11]=14550143450655128907617052305 99774567929341275785651/\n 43727453368198064617508603870502159 0800143506216585,\na[13,12]=434912351575168757621385102424229288459635 15012067399050743561004230490804423974878600834753885909/\n 15 6500737169334735327002132394157065085780206033444107250676809635435894 21556571669667160279586515,\na[14,1]=111352303897306737782229910609546 4437335258246585030516055024186482995053/\n 1172081821952666268 571554825436238158838606320428900711229598548800375000-\n 96563 55719858106181793772673915704646989403181/\n 426501239914813338 17759653132492492758879187500*6^(1/2),\na[14,2]=0,\na[14,3]=0,\na[14,4 ]=0,\na[14,5]=0,\na[14,6]=65156558806314041332627176859309009839731119 /\n 11498348446055040912040697694327646399463400+\n 2893 908603034923903437694020377944093049/\n 44653780373029285095303 680366320956891120*6^(1/2),\na[14,7]=820964014035484971016418653150994 8506760631106345347/\n 2089832016538282470163966831380108687881 472614625000-\n 73264258898010575680969439934466082688781229472 39059/\n 4179664033076564940327933662760217375762945229250000*6 ^(1/2),\na[14,8]=35838730146139312307525393375183357494472205437196407 4460462507967392441417/\n 6003812207349803677022535379501217823 64502552531871330639897526611625875000+\n 965635571985810618179 3772673915704646989403181/\n 3997689684671612201151830735210834 7913593312500*6^(1/2),\na[14,9]=-6428514041253116051017238172002206137 20981758568362376903035228606467992291/\n 27464714486544567117 1423994203344292862210985060954852952360191727428234375+\n 772 50845758864849454350181391325637175915225448/\n 64795380679365 872530827165335902056306758765625*6^(1/2),\na[14,10]=-3699695893017635 18639761285997824520507173970793320519753635225871563412913152/\n \+ 18085087026898731297447666263218680429649149103475975620769141065 2763681078125+\n 45485297982819623358721386803212535169178884 7437824/\n 94623202717436575037219117939366273676943640957812 5*6^(1/2),\na[14,11]=-771530345828119904175741186931939273226875593940 8305903151657039/\n 14830628578580199565993542928918214728319 58379628421680653323310,\na[14,12]=-9454329086181404208751280190036798 2103477409806135441008787696999938866843038864015885225699751345036974 1874915618358283/11833362560736813792711148439493119855699320505478184 90887959987766109714104308279282406274069868573023638087923030436570, \na[14,13]=1056911827593717127690972016166243945915857152155/\n \+ 7359790814129537930306952068888958882079118176267,\na[15,1]=-3584428 844827517211029003988869342002889757294838350269774673649/\n 2 085716341227133258652824258948052402773350356190879990897000000+\n \+ 18418541509225838359706930946702924369912480521/\n 39508 190287667470693939925841123267455183000000*6^(1/2),\na[15,2]=0,\na[15, 3]=0,\na[15,4]=0,\na[15,5]=0,\na[15,6]=-109506544804218207328507858426 71218433296327603/\n 10065471248673435610640694952753866293197 39200+\n 207046340886115163444903302653766033657147/\n \+ 3908920873271237130345900952525773317746560*6^(1/2), a[15,7]=-1907876 36465921540998883304987599740985291747006984827/\n 26134367005 227533473302899401390866713643703491000000+\n 1789112785283797 00355580290396510513302568946148710869/\n 52268734010455066946 605798802781733427287406982000000*6^(1/2), a[15,8]=-115733841931243537 286820675899274760337634770259165918831723224267751/\n 7941600 2253598408006316731727203547297279769620940498280027513000000-\n \+ 497300620749097635712087135560978957987636974067/\n 9998611 23434045989100479661671504230211939000000*6^(1/2), a[15,9]=20283630417 0232330990570957277958545540280636322218855276965040523111/\n 4 0483452265866865051188583915057088814854119415174645941812575437500-\n 18418541509225838359706930946702924369912480521/\n 7502 757290206082175051091686174851271537156250*6^(1/2), a[15,10]=998547076 72539845032167700901378988710484675887497768961753983967005376/\n \+ 239892690436344520409032858297188757822910165183184532751554433873 90625-\n 54224186203160868130977204707093409345022342653824/\n 54782834560687628399797684955475647938658789640625*6^(1/2), a [15,11]=474324561414678755109830553528867775624956239150831585597139/ \n 45157911635690366640337614874729721281201275171059286117565 , a[15,12]=58385595002511207073587386944190632124915223533408383221102 363939423894748980604430394490848526064113167169126734415639048389/224 7416507403826313785622253865369318598611634721135768699232414191791839 2966746421148746670582600045180015758041428626487315,\na[15,13]=-20617 84272050868275289038565787612564079661263925986910/\n 6305818 041153225448392022471538861648368501130406546697, a[15,14]=21447821690 29430053686260375092251514835023987332030/\n 52553908524700142 20073816442862907485166958089934203,\na[16,1]=694779880028538161253441 90899680848875649117294292403621110427/\n 276315727858040840861 28051091055380750995025720848651597000000-\n 185426619462312591 171592206632768608526005483/\n 52340455602260591822027565259526 4842483000000*6^(1/2),\na[16,2]=0,\na[16,3]=0,\na[16,4]=0,\na[16,5]=0, \na[16,6]=155409868146947995309862276450550758522148769/\n 1333 4737611899006162658099703042439679259200+\n 2948833469555291377 6680777005490941202439/\n 5178538878407381034041980467200947448 2560*6^(1/2), a[16,7]=309342396211701879358010980222740142719707361048 0521/\n 346228127881950258134565897477182544747165591000000-\n \+ 2223349281800786799168112412879091860919395860086487/\n \+ 692456255763900516269131794954365089494331182000000*6^(1/2), a[16,8]=8 5318327243239433756228728328071128940131418878118395907092074851/\n \+ 4717952020428847111777781739872410825294720094032625268331000000+ \n 5006518725482439961632989579084752430202148041/\n 132 46161456264411314959283823372471782839000000*6^(1/2), a[16,9]=-6614008 7449914380327548101344360694447375771589305352069924632043/\n \+ 7553870816492821153530435425077432706612170298471572545794812500+\n \+ 185426619462312591171592206632768608526005483/\n 9939653 8283139104662023501334196929221531250*6^(1/2), a[16,10]=-9964789456529 45815174857068635853357841406776256125134079522701632/\n 1776 46655848376964241055783826452426783687047504396906456095890625+\n \+ 23734607291176011669963802448994381891328701824/\n 31554 915787156112484260150916445096576431546875*6^(1/2), a[16,11]=-44807476 5213807929719320331516183261898760393629529216/\n 45888783596 485008769753055117740509726182346135675245, a[16,12]=-9790920345585957 3347478237877706182920147193670432726536024300616992919619548657031601 43382924460700238382314/5811852835514842025416306215702681585402800853 94426632970535207182792110996461759014949613118263320607923905, a[16,1 3]=98768647654583089123296762257490493421581752858120/\n 12524 6605043204308809279835220399478404120630180191,\na[16,14]=0,\na[16,15] =0,\n\nb[1]=385924436255198461459913/25885297292164750617319296,\nb[2] =0,\nb[3]=0,\nb[4]=0,\nb[5]=0,\nb[6]=0,\nb[7]=0,\nb[8]=-81508791888782 942071778080019859673/399395400777999552787672219983509760,\nb[9]=9359 36315524449978576662571821361001/4086801409502444670525951816974611200 ,\nb[10]=2172547024243858864854526674870272/16972283408414027681310867 548214525,\nb[11]=1193746724713997342094811077921918673563219/53338395 43124606397779553737025343484595200,\nb[12]=78998885900843720607956191 2875509208028937668327456911860695435130268907660397023313561421879843 37094261434894685526783/1997283537594056933833443573570503668060657340 4390124462865923827445634527724422608244891870880603810240930777053079 0400,\nb[13]=11409994679937666036993318622713183/210646829819316523523 317257829664000,\nb[14]=10048608923923592706010638721995991/7917627313 4521955696793568162336000, b[15]=957935979810312917705761/301876872389 21485361088000,\nb[16]=0,\n\n`b*`[1]=6365795202834324687727/3676888819 90976571268740,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=0,\n`b*`[5]=0,\n`b*`[ 6]=0,\n`b*`[7]=0,\n`b*`[8]=11235845416756910667521184687449/2798454321 5947278082095867431580,\n`b*`[9]=596074834977423915714550106443/350979 1660513951108318405888848,\n`b*`[10]=46281133707158379428440684544/379 480903486059869900746060329,\n`b*`[11]=4840963801198418151644350108694 156829/20456544999327323762290226804576756480,\n`b*`[12]=-425037883291 7670692796231342347492575920109645281507264047062230529640315652520199 442309428858040080393/194935262507377379122306524130235672353804765700 31599740950002149019485955879156117360614282462351914240,\n`b*`[13]=83 07802719022653646372347529833/36318418934364917848847803074080,\n`b*`[ 14]=0,\n`b*`[15]=0,\n`b*`[16]=12690737099320029004861/3018768723892148 53610880\}:" }}}{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[9,16];" "6#&%\"TG6$\"\"*\"#; " }{TEXT -1 129 " denote the vector whose components are the principa l error terms of the 16 stage, order 9 scheme (the error terms of orde r 10)." }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "`T*`[8,16 ];" "6#&%#T*G6$\"\")\"#;" }{TEXT -1 146 " denote the vector whose com ponents are the principal error terms of the embedded 16 stage, order \+ 8 scheme (the error terms of order 9) and let " }{XPPEDIT 18 0 "`T*`[ 9, 16];" "6#&%#T*G6$\"\"*\"#;" }{TEXT -1 100 " denote the vector whos e components are the error terms of order 10 of the embedded order 8 s cheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Denote the 2-norm of these vectors by " }{XPPEDIT 18 0 "abs(abs(T[9, 16]));" "6#-%$absG6#-F$6#&%\"TG6$\"\"*\"#;" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "abs(abs(`T*`[8,16]));" "6#-%$absG6#-F$6#&%#T*G6$\"\")\" #;" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "abs(abs(`T*`[9,16]));" "6#-% $absG6#-F$6#&%#T*G6$\"\"*\"#;" }{TEXT -1 15 " respectively." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Define: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A[10] = abs(abs(T[9, \+ 16]));" "6#/&%\"AG6#\"#5-%$absG6#-F)6#&%\"TG6$\"\"*\"#;" }{TEXT -1 6 " , " }{XPPEDIT 18 0 "B[10] = abs(abs(`T*`[9,16]))/abs(abs(`T*`[8,16 ]));" "6#/&%\"BG6#\"#5*&-%$absG6#-F*6#&%#T*G6$\"\"*\"#;\"\"\"-F*6#-F*6 #&F/6$\"\")F2!\"\"" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "C[10] = abs(a bs(`T*`[9,16]-T[9,16]))/abs(abs(`T*`[8,16]));" "6#/&%\"CG6#\"#5*&-%$ab sG6#-F*6#,&&%#T*G6$\"\"*\"#;\"\"\"&%\"TG6$F2F3!\"\"F4-F*6#-F*6#&F06$\" \")F3F8" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Dormand and Prince have suggested that as well as at tempting to ensure that " }{XPPEDIT 18 0 "A[10];" "6#&%\"AG6#\"#5" } {TEXT -1 73 " is a minimum, if the embedded scheme is to be used for \+ error control, " }{XPPEDIT 18 0 "B[10];" "6#&%\"BG6#\"#5" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "C[10];" "6#&%\"CG6#\"#5" }{TEXT -1 27 " s hould be chosen so that " }}{PARA 0 "" 0 "" {TEXT -1 68 "they are \"si milar in magnitude\" and also not differ too much from 1." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 168 "errt erms9_16 := PrincipalErrorTerms(9,16,'expanded'):\n`errterms9_16*` :=s ubs(b=`b*`,errterms9_16):\n`errterms8_16*` := subs(b=`b*`,PrincipalErr orTerms(8,16,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 468 "nmB := 0: \nfor ct to nops(`errte rms9_16*`) do\n nmB := nmB+evalf(subs(ee,`errterms9_16*`[ct]))^2;\ne nd do:\nsnmB := sqrt(nmB):\ndnB := 0:\nfor ct to nops(`errterms8_16*`) do\n dnB := dnB+evalf(subs(ee,`errterms8_16*`[ct]))^2;\nend do:\nsd nB := sqrt(dnB):\nnmC := 0:\nfor ct to nops(errterms9_16) do\n nmC : = nmC+(evalf(subs(ee,`errterms9_16*`[ct]))-evalf(subs(ee,errterms9_16[ ct])))^2;\nend do:\nsnmC := sqrt(nmC):\n'B[10]'= evalf[8](snmB/sdnB); \n'C[10]'= evalf[8](snmC/sdnB);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"BG6#\"#5$\")4,,I!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"# 5$\")Pt4I!\"(" }}}{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 36 "coefficients of the combined scheme " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16669 "ee := \{c[2]=1/46,\nc[3]=96755252944/718444993695-11256225944/7184449936 95*6^(1/2),\nc[4]=48377626472/239481664565-5628112972/239481664565*6^( 1/2),\nc[5]=71/136,\nc[6]=276/715-46/715*6^(1/2),\nc[7]=276/715+46/715 *6^(1/2),\nc[8]=92/143,\nc[9]=69/143,\nc[10]=3/44,\nc[11]=103/411,\nc[ 12]=30258248819701/45339732981913,\nc[13]=59/69,\nc[14]=44/49,\nc[15]= 1,\nc[16]=1,\n\na[2,1]=1/46,\na[3,1]=-163287951175938724532816/5161632 08965408589753025+\n 42011574289334042817176/516163208965408589 753025*6^(1/2),\na[3,2]=232801278267248934720896/516163208965408589753 025-\n 50098553466700618240256/516163208965408589753025*6^(1/2), \na[4,1]=12094406618/239481664565-1407028243/239481664565*6^(1/2),\na[ 4,2]=0,\na[4,3]=36283219854/239481664565-4221084729/239481664565*6^(1/ 2),\na[5,1]=450479172821804238979159483/489985471732935255816699904+\n 65404175703680378526395577/244992735866467627908349952*6^(1/2), \na[5,2]=0,\na[5,3]=-1663285823745576633021875313/48998547173293525581 6699904-\n 258991054585998425691922779/244992735866467627908349 952*6^(1/2),\na[5,4]=734303944921586208649981787/244992735866467627908 349952+\n 96793439441159023582763601/122496367933233813954174976 *6^(1/2),\na[6,1]=188634486760257/2753187875656075-40451003556679/5506 375751312150*6^(1/2),\na[6,2]=0,\na[6,3]=0,\na[6,4]=890541395040155939 974909749/3404930508779360011084250045-\n 2354148424451437900833 29443/6809861017558720022168500090*6^(1/2),\na[6,5]=127509164130554343 284736/2278805333809176804299525-\n 51090254569210884816896/22 78805333809176804299525*6^(1/2),\na[7,1]=523150756520001/5294592068569 375+372205675002861/137659393782803750*6^(1/2),\na[7,2]=0,\na[7,3]=0, \na[7,4]=121832502441158811994748302664452173/631943122967207272212736 2725145820625-\n 12054008141355156662680357922224203047/16430521 1971473890775311430853791336250*6^(1/2),\na[7,5]=-73451888911239091559 79140554752/52428978281511938535235507146875+\n 713821951824578 89488943971467264/681576717659655200958061592909375*6^(1/2),\na[7,6]=8 4211752143498940768/206389046233053165625+567839841668979868/187626405 66641196875*6^(1/2),\na[8,1]=92/1287,\na[8,2]=0,\na[8,3]=0,\na[8,4]=0, \na[8,5]=0,\na[8,6]=368/1287+23/1287*6^(1/2),\na[8,7]=368/1287-23/1287 *6^(1/2),\na[9,1]=1311/18304,\na[9,2]=0,\na[9,3]=0,\na[9,4]=0,\na[9,5] =0,\na[9,6]=4071/18304+1587/36608*6^(1/2),\na[9,7]=4071/18304-1587/366 08*6^(1/2),\na[9,8]=-621/18304,\na[10,1]=2451872601/50434064384,\na[10 ,2]=0,\na[10,3]=0,\na[10,4]=0,\na[10,5]=0,\na[10,6]=84329349/114622873 6-1383050643/100868128768*6^(1/2),\na[10,7]=84329349/1146228736+138305 0643/100868128768*6^(1/2),\na[10,8]=-1098320769/50434064384,\na[10,9]= -333490521/3152129024,\na[11,1]=-11290810941252792923651/1669469461414 577748900000-\n 76218489460616423924209/1001681676848746649340 0000*6^(1/2),\na[11,2]=0,\na[11,3]=0,\na[11,4]=0,\na[11,5]=0,\na[11,6] =1/30,\na[11,7]=-44608220078798131601386867/17783274316806616262193000 00-\n 302663621648107819403033939/5334982295041984878657900000 *6^(1/2),\na[11,8]=4768550623191902657077/335320789258483564950000+762 18489460616423924209/9388982099237539818600000*6^(1/2),\na[11,9]=76371 166597983496297729/1268154687036073482337500+\n 762184894606164 23924209/1902232030554110223506250*6^(1/2),\na[11,10]=1283709272606880 0321242176/73489499260117750229428125+\n 224387232972054752032 871296/13889515360162254793361915625*6^(1/2),\na[12,1]=-35584379273878 0589211336013211266011894384892859307606673616682840321037292934264483 298005378753134077/611498050133846699976127131429172460287752543562338 1536957589416556451501095664913528943447961301150000+16342530068674673 7060984103973048211168317072573898840779749222007416544158960449576929 6514008733014761/62508689569237662664226328990537629273859148897483455 711122025147021504233422352449406977468048856200000*6^(1/2),\na[12,2]= 0,\na[12,3]=0,\na[12,4]=0,\na[12,5]=0,\na[12,6]=-175949180260408634887 7532302368836726393890388409716969261965782534094328577924217382453202 1369934883/\n 346201972998854747063407360562977639055219901586 06221624621429312196525421587764433517710597688597280-\n 17188 0024446897129391119010658057689596258428143994698538245842002969107311 8839588510586218092229/\n 1344473681548950474032649943933893743903 76660810121249027656036163870001637234036635020235330829504*6^(1/2),\n a[12,7]=-1382073075025978724144662487119071171012382899509469565758724 97287994333538943652935288845283125425999567221/4494458926341069611642 2287437086812685221876161218070779739001184893132987156890558175820354 9930436500950000+37342260993482278211711351217469345525186992139310658 6844571126171756173532479385325874622983683540733325899/17977835705364 2784465689149748347250740887504644872283118956004739572531948627562232 7032814199721746003800000*6^(1/2),\na[12,8]=12632445836279619113294210 7801681715657268924065999457323338359651167769122658680421315568305493 27654393/1129964772982373132776399024059718683027453845454508622470282 76227308103806571175581620305423011393900000-6303547312203088429495101 1532461452879208013707075267157903271345717809889884744836815722683193 98771221/2259929545964746265552798048119437366054907690909017244940565 52454616207613142351163240610846022787800000*6^(1/2),\na[12,9]=1115379 7852776277138749876364320845224044108260104646285465373849752391924503 16138834225701046102632948257/3347520639960280405850082108776916598468 832017158981794068212683234002575269671076605501548156712544287500-163 4253006867467370609841039730482111683170725738988407797492220074165441 589604495769296514008733014761/118706405672350369001775961304145978669 10751833897098560525576890900718352020110200728728894172739518750*6^(1 /2),\na[12,10]=1119539579361979281841957268759561827109492225370086710 6612850891912759357989068567623193453186950145251776/37146762745465379 2163842667375678639198750503483780420136986289449879221829038237026483 63618722076517534375-6873201217454034198679102887095056195421792309393 68838936545299414049008577113662220686991034529999350912/1238225424848 8459738794755579189287973291683449459347337899542981662640727634607900 882787872907358839178125*6^(1/2),\na[12,11]=73825104187474768875967421 005730375156586805528272061538441201406502469378399602291332635087412/ \n 92871272539029599143289144740668914525146202470165909630170 359355065425993729905138177287271317,\na[13,1]=-6471907074414433573396 2214412431202035561004419937320357188840883/\n 101209800434111 984325751891003355158288493635359529771992774000000+\n 3559372 342256314491923633965576963561357143961289/\n 3635724370610188 4581258626118802982597639184500000*6^(1/2),\na[13,2]=0,\na[13,3]=0,\na [13,4]=0,\na[13,5]=0,\na[13,6]=-66005588850553492194962577164587068367 312330247/\n 19603612723583140697294028442079849957408380800- \n 13176568020345600854085799476910283462657777/\n 761 30534848866565814734091037203300805469440*6^(1/2),\na[13,7]=-931857070 1703691930191534222789494006237103047579881711/\n 356296886476 9188061196125478115789029665196590638000000+\n 643475794478694 0764537553522803508307647498263055753667/\n 712593772953837612 2392250956231578059330393181276000000*6^(1/2),\na[13,8]=-3761276943318 114593157887595063197569838926771255702620302334773297/\n 1464 216542935037622889021617525045499738187320641028236799386000000-\n \+ 3559372342256314491923633965576963561357143961289/\n 34 078442106289230846820763513066043517445275500000*6^(1/2),\na[13,9]=438 696881717332693164574683605145782902979477404432064959580227217/\n \+ 190320793597985643524049406938231176153947641989529318077977593750 -\n 3559372342256314491923633965576963561357143961289/\n \+ 6904380415341463658460171786984220252917056671875*6^(1/2),\na[13,10]= 39393077397330623465044422844130393549390670768068628724675776122752/ \n 24104666116749847055078895726017945791587677982957405550119 962671875-\n 9111993196176165099324502951877026717074288540899 84/\n 4383797390367410875150919593877143523760184037765625*6^( 1/2),\na[13,11]=1455014345065512890761705230599774567929341275785651/ \n 437274533681980646175086038705021590800143506216585,\na[13, 12]=434912351575168757621385102424229288459635150120673990507435610042 30490804423974878600834753885909/\n 15650073716933473532700213 2394157065085780206033444107250676809635435894215565716696671602795865 15,\na[14,1]=111352303897306737782229910609546443733525824658503051605 5024186482995053/\n 1172081821952666268571554825436238158838606 320428900711229598548800375000-\n 96563557198581061817937726739 15704646989403181/\n 426501239914813338177596531324924927588791 87500*6^(1/2),\na[14,2]=0,\na[14,3]=0,\na[14,4]=0,\na[14,5]=0,\na[14,6 ]=65156558806314041332627176859309009839731119/\n 1149834844605 5040912040697694327646399463400+\n 2893908603034923903437694020 377944093049/\n 44653780373029285095303680366320956891120*6^(1/ 2),\na[14,7]=8209640140354849710164186531509948506760631106345347/\n \+ 2089832016538282470163966831380108687881472614625000-\n 7 326425889801057568096943993446608268878122947239059/\n 41796640 33076564940327933662760217375762945229250000*6^(1/2),\na[14,8]=3583873 01461393123075253933751833574944722054371964074460462507967392441417/ \n 600381220734980367702253537950121782364502552531871330639897 526611625875000+\n 96563557198581061817937726739157046469894031 81/\n 39976896846716122011518307352108347913593312500*6^(1/2), \na[14,9]=-64285140412531160510172381720022061372098175856836237690303 5228606467992291/\n 274647144865445671171423994203344292862210 985060954852952360191727428234375+\n 7725084575886484945435018 1391325637175915225448/\n 647953806793658725308271653359020563 06758765625*6^(1/2),\na[14,10]=-36996958930176351863976128599782452050 7173970793320519753635225871563412913152/\n 18085087026898731 2974476662632186804296491491034759756207691410652763681078125+\n \+ 454852979828196233587213868032125351691788847437824/\n 94 6232027174365750372191179393662736769436409578125*6^(1/2),\na[14,11]=- 7715303458281199041757411869319392732268755939408305903151657039/\n \+ 148306285785801995659935429289182147283195837962842168065332331 0,\na[14,12]=-94543290861814042087512801900367982103477409806135441008 7876969999388668430388640158852256997513450369741874915618358283/11833 3625607368137927111484394931198556993205054781849088795998776610971410 4308279282406274069868573023638087923030436570,\na[14,13]=105691182759 3717127690972016166243945915857152155/\n 735979081412953793030 6952068888958882079118176267,\na[15,1]=-358442884482751721102900398886 9342002889757294838350269774673649/\n 208571634122713325865282 4258948052402773350356190879990897000000+\n 184185415092258383 59706930946702924369912480521/\n 3950819028766747069393992584 1123267455183000000*6^(1/2),\na[15,2]=0,\na[15,3]=0,\na[15,4]=0,\na[15 ,5]=0,\na[15,6]=-10950654480421820732850785842671218433296327603/\n \+ 1006547124867343561064069495275386629319739200+\n 207046 340886115163444903302653766033657147/\n 3908920873271237130345 900952525773317746560*6^(1/2), a[15,7]=-190787636465921540998883304987 599740985291747006984827/\n 2613436700522753347330289940139086 6713643703491000000+\n 178911278528379700355580290396510513302 568946148710869/\n 5226873401045506694660579880278173342728740 6982000000*6^(1/2), a[15,8]=-11573384193124353728682067589927476033763 4770259165918831723224267751/\n 794160022535984080063167317272 03547297279769620940498280027513000000-\n 49730062074909763571 2087135560978957987636974067/\n 999861123434045989100479661671 504230211939000000*6^(1/2), a[15,9]=2028363041702323309905709572779585 45540280636322218855276965040523111/\n 404834522658668650511885 83915057088814854119415174645941812575437500-\n 184185415092258 38359706930946702924369912480521/\n 750275729020608217505109168 6174851271537156250*6^(1/2), a[15,10]=99854707672539845032167700901378 988710484675887497768961753983967005376/\n 2398926904363445204 0903285829718875782291016518318453275155443387390625-\n 542241 86203160868130977204707093409345022342653824/\n 54782834560687 628399797684955475647938658789640625*6^(1/2), a[15,11]=474324561414678 755109830553528867775624956239150831585597139/\n 4515791163569 0366640337614874729721281201275171059286117565, a[15,12]=5838559500251 1207073587386944190632124915223533408383221102363939423894748980604430 394490848526064113167169126734415639048389/224741650740382631378562225 3865369318598611634721135768699232414191791839296674642114874667058260 0045180015758041428626487315,\na[15,13]=-20617842720508682752890385657 87612564079661263925986910/\n 6305818041153225448392022471538 861648368501130406546697, a[15,14]=21447821690294300536862603750922515 14835023987332030/\n 52553908524700142200738164428629074851669 58089934203,\na[16,1]=694779880028538161253441908996808488756491172942 92403621110427/\n 276315727858040840861280510910553807509950257 20848651597000000-\n 185426619462312591171592206632768608526005 483/\n 523404556022605918220275652595264842483000000*6^(1/2),\n a[16,2]=0,\na[16,3]=0,\na[16,4]=0,\na[16,5]=0,\na[16,6]=15540986814694 7995309862276450550758522148769/\n 1333473761189900616265809970 3042439679259200+\n 29488334695552913776680777005490941202439/ \n 51785388784073810340419804672009474482560*6^(1/2), a[16,7]=3 093423962117018793580109802227401427197073610480521/\n 34622812 7881950258134565897477182544747165591000000-\n 2223349281800786 799168112412879091860919395860086487/\n 69245625576390051626913 1794954365089494331182000000*6^(1/2), a[16,8]=853183272432394337562287 28328071128940131418878118395907092074851/\n 471795202042884711 1777781739872410825294720094032625268331000000+\n 5006518725482 439961632989579084752430202148041/\n 13246161456264411314959283 823372471782839000000*6^(1/2), a[16,9]=-661400874499143803275481013443 60694447375771589305352069924632043/\n 75538708164928211535304 35425077432706612170298471572545794812500+\n 18542661946231259 1171592206632768608526005483/\n 993965382831391046620235013341 96929221531250*6^(1/2), a[16,10]=-996478945652945815174857068635853357 841406776256125134079522701632/\n 177646655848376964241055783 826452426783687047504396906456095890625+\n 237346072911760116 69963802448994381891328701824/\n 3155491578715611248426015091 6445096576431546875*6^(1/2), a[16,11]=-4480747652138079297193203315161 83261898760393629529216/\n 4588878359648500876975305511774050 9726182346135675245, a[16,12]=-979092034558595733474782378777061829201 4719367043272653602430061699291961954865703160143382924460700238382314 /581185283551484202541630621570268158540280085394426632970535207182792 110996461759014949613118263320607923905, a[16,13]=98768647654583089123 296762257490493421581752858120/\n 1252466050432043088092798352 20399478404120630180191,\na[16,14]=0,\na[16,15]=0,\n\nb[1]=38592443625 5198461459913/25885297292164750617319296,\nb[2]=0,\nb[3]=0,\nb[4]=0,\n b[5]=0,\nb[6]=0,\nb[7]=0,\nb[8]=-81508791888782942071778080019859673/3 99395400777999552787672219983509760,\nb[9]=935936315524449978576662571 821361001/4086801409502444670525951816974611200,\nb[10]=21725470242438 58864854526674870272/16972283408414027681310867548214525,\nb[11]=11937 46724713997342094811077921918673563219/5333839543124606397779553737025 343484595200,\nb[12]=7899888590084372060795619128755092080289376683274 5691186069543513026890766039702331356142187984337094261434894685526783 /199728353759405693383344357357050366806065734043901244628659238274456 345277244226082448918708806038102409307770530790400,\nb[13]=1140999467 9937666036993318622713183/210646829819316523523317257829664000,\nb[14] =10048608923923592706010638721995991/791762731345219556967935681623360 00, b[15]=957935979810312917705761/30187687238921485361088000,\nb[16]= 0,\n\n`b*`[1]=6365795202834324687727/367688881990976571268740,\n`b*`[2 ]=0,\n`b*`[3]=0,\n`b*`[4]=0,\n`b*`[5]=0,\n`b*`[6]=0,\n`b*`[7]=0,\n`b*` [8]=11235845416756910667521184687449/27984543215947278082095867431580, \n`b*`[9]=596074834977423915714550106443/35097916605139511083184058888 48,\n`b*`[10]=46281133707158379428440684544/37948090348605986990074606 0329,\n`b*`[11]=4840963801198418151644350108694156829/2045654499932732 3762290226804576756480,\n`b*`[12]=-42503788329176706927962313423474925 75920109645281507264047062230529640315652520199442309428858040080393/1 9493526250737737912230652413023567235380476570031599740950002149019485 955879156117360614282462351914240,\n`b*`[13]=8307802719022653646372347 529833/36318418934364917848847803074080,\n`b*`[14]=0,\n`b*`[15]=0,\n`b *`[16]=12690737099320029004861/301876872389214853610880\}:" }}}{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 s tability function R for the 16 stage, order 9 scheme is given (approxi mately) as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "expand(subs(ee,StabilityFunction(9,16,'exp anded'))):\nmap(convert,evalf[28](%),rational,24):\nR := unapply(%,z): \n'R(z)'=R(z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"RG6#%\"zG,B\"\" \"F)F'F)*&#F)\"\"#F)*$)F'F,F)F)F)*&#F)\"\"'F)*$)F'\"\"$F)F)F)*&#F)\"#C F)*$)F'\"\"%F)F)F)*&#F)\"$?\"F)*$)F'\"\"&F)F)F)*&#F)\"$?(F)*$)F'F1F)F) F)*&#F)\"%S]F)*$)F'\"\"(F)F)F)*&#F)\"&?.%F)*$)F'\"\")F)F)F)*&#F)\"'!)G OF)*$)F'\"\"*F)F)F)*&#\"*v8HE\"\"0MsA7U%*\\%F)*$)F'\"#5F)F)F)*&#\"*'HJ vA\"1haJ=u<5qF)*$)F'\"#6F)F)F)*&#\")0%[R\"\"2#Go(f!))GI:F)*$)F'\"#7F)F )!\"\"*&#\")SSoM\"2F^KZdLa<\"F)*$)F'\"#8F)F)Fgo*&#\"()Q-E\"284?G'z#p1 \"F)*$)F'\"#9F)F)F)*&#\"'V$[$\"3tCdg%)*)ptOF)*$)F'\"#:F)F)F)" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can f ind the point where the boundary of the stability region intersects th e negative real axis by solving the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R(z) = 1;" "6#/-%\"RG6#%\"zG\"\"\"" } {TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "z0 := newt on(R(z)=1,z=-4.5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$!+J8M,X! \"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 304 "z0 := newton(R(z)=1,z=-4.5):\np1 := plot([R(z),1],z= -5.19..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= [-5.19..0.49,-0.07..1.47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 253 253 {PLOTDATA 2 "6+-%'CURVESG6$7Z7$$!3Q++++++!>&!#<$ \"3tsN;,zsEpF*7$$!3QML3T![!f^F*$\"3wuh:pG&\\Q'F*7$$!3Ynm;#3'4G^F*$\"3_ sNJPtj#)eF*7$$!3a++DBT9(4&F*$\"3UJ*Ru$y7m]F*$\"37FV_&e Mf)\\F*7$$!3E+]U'*)HB,&F*$\"3fBJ#p=L0J%F*7$$!3!pm;&GwYe\\F*$\"3DnCpztx ?PF*7$$!3s+](\\(Q*y*[F*$\"3m!H,VLqt9$F*7$$!3nLLV@,KP[F*$\"3MI%\\(o&**o l#F*7$$!3'RLLd%[MwZF*$\"3*z;ud)znNAF*7$$!3NLL.q&p`r%F*$\"3[)\\?5rys(=F *7$$!3E+]<*4%oaYF*$\"33yZP=ZMu:F*7$$!3;nmJG')*Rf%F*$\"3'HN+\\uevJ\"F*7 $$!3uLLyGAZ\"[%F*$\"3qxYs&eb,U*!#=7$$!3%3+])fw&\\O%F*$\"37Ja%*))=X5mFh o7$$!3$QL$)f7eWC%F*$\"3k2u'edffb%Fho7$$!3A++lN]MCTF*$\"3B$)pv#z/O8$Fho 7$$!3ummYeRz+SF*$\"3dTU_(esk8#Fho7$$!3_LLV-,(>*QF*$\"3$\\xj^Z!HQ:Fho7$ $!35++S:-YpPF*$\"3')3P$e'e?(3\"Fho7$$!3K+++\"HZkk$F*$\"3%HC8%>e35!)!#> 7$$!3;++gW:!z_$F*$\"3O`8C*p!HWjF\\r7$$!3hLL)*\\1D?MF*$\"3/,9g<5czaF\\r 7$$!3'ommSKVAH$F*$\"3*\\5/c$3/5]F\\r7$$!3/nmEGV!Q=$F*$\"3aAhi;8!*[\\F \\r7$$!39++0(*RmdIF*$\"3wB%y'o24a^F\\r7$$!39nmEI%3g%HF*$\"3q5e=&Rx?_&F \\r7$$!3-++0xX]BGF*$\"3uo6![$\\z%3'F\\r7$$!3*)***\\\"R>&oq#F*$\"3A:vtb vG`nF\\r7$$!3gmm;\\r8&e#F*$\"3y)46Sj%fyvF\\r7$$!3ymmrw\\OtCF*$\"3)3+af 6-3X)F\\r7$$!3SLL$))e.GN#F*$\"3**[J\"ywT*>&*F\\r7$$!3nLL)**=uvA#F*$\"3 ]-!QF'>Jy5Fho7$$!3K++:I;c=@F*$\"3O@T&=4IA?\"Fho7$$!31LL.z]#3+#F*$\"3O2 XpmyI_8Fho7$$!3M++?,<>z=F*$\"3%39/jodr_\"Fho7$$!3;++!4<(>gFho7$$!3H++q9zA<:F*$\"3cm!prb* =$>#Fho7$$!3EnmEY;O-9F*$\"3m4V4kA:gCFho7$$!3#)*****pQ<(z7F*$\"3O#))GNj e6y#Fho7$$!3)RL$efMeo6F*$\"3C*)fq%*)o!3JFho7$$!3I****fAZ3Z5F*$\"3I&=J! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 485 "R := z ->add(z^j/j!,j=0..9)+\n 126291375/44 9944212227234*z^10+227531296/7010177418315461*z^11-\n 13948405/153 02888059768282*z^12-34684040/11754335747325127*z^13+\n 2602388/106 69279628200913*z^14+348343/367369898460572473*z^15:\nDigits := 25:\npt s := []: z0 := 0:\nfor ct from 0 to 93 do\n zz := newton(R(z)=exp(ct *Pi/100*I),z=z0):\n z0 := zz:\n pts := [op(pts),[surd(Re(zz),11),I m(zz)]]:\nend do:\nplot(pts,color=COLOR(RGB,.95,0,0),thickness=2,font= [HELVETICA,9]);\nDigits := 10:" }}{PARA 13 "" 1 "" {GLPLOT2D 287 308 308 {PLOTDATA 2 "6(-%'CURVESG6#7jp7$$\"\"!F)F(7$$\":FrJ8'f\"3;Rsle(!#F $\":]EYQKz*e`EfTJ!#E7$$\":()>?Hd=Yg%3CD9F0$\":c!RpZ'ezrI&=$G'F07$$\":z 82\\gu]C6->1#F0$\":e=d;(z$p2'zxC%*F07$$\":g^*)3iJV)=#z2o#F0$\":xY;)Hj\"yWF0$\":Hf>Cl&G^d[6*>#F@7$$ \":h2_+N:**4%=*e1&F0$\":S<%f%*R=(G7uK^#F@7$$\":VR$3#=T[y!z]]cF0$\":Far JI$3B)QLu#GF@7$$\":tw-VONJyXmIB'F0$\":'pxwa_)*e`EfTJF@7$$\":Ub)GQ,JR_E U9oF0$\":]m&QjO*[*=>vbMF@7$$\":0M;)H9'**yxy_R(F0$\":PyuxU;3V=6*pPF@7$$ \":!HV]5b8$[!e@wzF0$\":[n8'H'on'\\/2%3%F@7$$\":N\")H2+w-bl:xb)F0$\":y4 &=Tqx-:(H#)R%F@7$$\":h'z]$G(Gb;wmliW;vX#[l-&F@7$$\":_C\\'e'*\\RcW#4.\"F@$\":j(o%[ ^6<6^22M&F@7$$\":P9A\\IQ.fbL'*3\"F@$\":E=%[E(e'[wn'[l&F@7$$\":(4[2$)>. !z`P&[6F@$\":_-Xk9hh=/E!pfF@7$$\":p+iJb+5_c\\w?\"F@$\":.&HM+sOC2`=$G'F @7$$\":!))>m)Q#4$*)GzpE\"F@$\":brF@$\":RB\\Qz\\#oZ'4`+\"!#C7$$\":,bs'pzaP-t.'*>F@$\":dfuUka- KdDn.\"Fgu7$$\":wb[Ssh=&)*H-e?F@$\":3hQq9z`\")\\T\"o5Fgu7$$\":/N;q+(R[ qX9?@F@$\":W3\"ePJ'oAUd&*4\"Fgu7$$\":\")\\S)Qg)*H:lQ#=#F@$\":`hhV(>A\"Fgu7$$\":cnF4NBc<^l'pBF@$\":6y$He$z5/5@_A\"Fgu 7$$\":tfL^^y$)o@:AV#F@$\":C&emM+)\\5,PmD\"Fgu7$$\":hfL&p(f44u$z%\\#F@$ \":IRwS19!G:H0)G\"Fgu7$$\":y52;,i*e+*ztb#F@$\":af\"o:R*Q3\")o%>8Fgu7$$ \":gugG*os=)[^*>EF@$\":$*\\YyW,mZp%)3N\"Fgu7$$\":GwI)QNjQA_[#o#F@$\":` y'4!3)z@j0I#Q\"Fgu7$$\":Age+I50!)oc\\u#F@$\":+*>ZXX,C6kr89Fgu7$$\":*4I X'>^!fj-M2GF@$\":k'*y6LF)[KA8X9Fgu7$$\":SSVq*4trN!4'pGF@$\":ULumu7\"*) =![lZ\"Fgu7$$\":'oq7cpf_)pM<$HF@$\":$\\-C\">UV-wjz]\"Fgu7$$\":x.nF6\"R pOuo$*HF@$\":mH\")4aV`kP+G`]zq: Fgu7$$\":ee?LtAvA(GI%=yJF@$ \":K.8Q-%4BmfiL;Fgu7$$\":;&R-#3[VFx6\"RKF@$\":6K;v9\"ek97/l;Fgu7$$\":+ %*G6wWP;-*o*H$F@$\":#>wJ-fA6xiX'p\"Fgu7$$\":[Ca]oejUWt)fLF@$\":XR0Ol8x $36(ys\"Fgu7$$\":-4c'>MF@$\":02PoF!=a`cGfFG7A\"pg8@#=Fgu 7$$\":^hxmc()o?I:df$F@$\":8m\"pZ^>))Ro_`=Fgu7$$\":mTzO+wB$>Y<`OF@$\":' RVf\\&\\58VR\\)=Fgu7$$\":Y!eDcz1Zx^!*4PF@$\":bw][_D*)eC^j\">Fgu7$$\":> VTY$R;Yr>$ew$F@$\":VSN`db+N7ix%>Fgu7$$\":7Zifx5E5>s3#QF@$\":Q`Fgu7$$\":e%[mJ^/k&*Q$\\(QF@$\":`K_T*Q%z)z-e5?Fgu7$$\":Km`x19e**Q9z# RF@$\":Ok*4`I\\dxq)>/#Fgu7$$\":M[s./J?P@)pzRF@$\":Un1Nk(\\0m>Rt?Fgu7$$ \":yXNhV--&HY:ISF@$\":'3_qLc$*)Rf%z/@Fgu7$$\":RZ=y&RgsZW8zSF@$\":H!*p! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 118 "Digits := 15:\nz0 := 2.7*I:\nfor ct from 84 to 87 \+ do\n print(newton(R(z)=exp(ct*Pi/100*I),z=z0));\nend do;\nDigits := \+ 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0o%zldYVq!#>$\"0n*Re7bPE!# 9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0m38F,)pK!#>$\"0Q00`Y(oE!#9 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0Seu8+#o=!#>$\"0j)H\"f6**p#!#9 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0Z=T@&Gk')!#>$\"0J)p\"4V5t#!#9 " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "The n we apply the bisection method to calculate the parameter value assoc iated with the intersection point." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 " real_part := proc(u)\n Re(newton(R(z)=exp(u*Pi*I ),z=2.7*I))\nend proc:\nDigits := 15:\nu0 := bisect('real_part'(u),u=0 .84..0.87);\nnewton(R(z)=exp(u0*Pi*I),z=2.7*I);\nDigits := 10:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"0/HVd8pc)!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0O>\"Q'f`'z!#H$\"0$*)*>t.'*o#!#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, 2.6896];" "6#7$\"\"!-%&FloatG6$\"&'*o#!\"%" }{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 97 "The stability func tion R* for the 16 stage, order 8 scheme is given (approximately) as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "expand(subs(ee ,subs(b=`b*`,StabilityFunction(8,16,'expanded')))):\nmap(convert,evalf [28](%),rational,24):\n`R*` := unapply(%,z):\n'`R*`(z)'=`R*`(z);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#/-%#R*G6#%\"zG,@\"\"\"F)F'F)*&#F)\"\"# F)*$)F'F,F)F)F)*&#F)\"\"'F)*$)F'\"\"$F)F)F)*&#F)\"#CF)*$)F'\"\"%F)F)F) *&#F)\"$?\"F)*$)F'\"\"&F)F)F)*&#F)\"$?(F)*$)F'F1F)F)F)*&#F)\"%S]F)*$)F '\"\"(F)F)F)*&#F)\"&?.%F)*$)F'\"\")F)F)F)*&#\"*(R/9M\"0EOu(*f>:\"F)*$) F'\"\"*F)F)F)*&#\"*-jop#\"06m&)oln\"*\"2j'>DDF^BaF)*$)F'\"#9F) 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 inte rsects 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 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "z_0 := newton(`R*`(z)=-1,z=- 4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$z_0G$!+ " 0 "" {MPLTEXT 1 0 322 "z _0 := newton(`R*`(z)=-1,z=-4):\np_1 := plot([`R*`(z),-1],z=-4.39..0.49 ,color=[red,blue]):\np_2 := plot([[[z_0,-1]]$3],style=point,symbol=[ci rcle,cross,diamond],color=black):\np_3 := plot([[z_0,0],[z_0,-1]],line style=3,color=COLOR(RGB,0,.5,0)):\nplots[display]([p_1,p_2,p_3],view=[ -4.39..0.49,-1.47..1.47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 369 264 264 {PLOTDATA 2 "6+-%'CURVESG6$7Y7$$!3o************* Q%!#<$!3;>uVN:tXWF*7$$!3Amm\">Z2MO%F*$!3WbMMf1%47%F*7$$!3lLL$Q%\\\"oL% F*$!3\\Ru>C%*)y\"QF*7$$!3?++v:CA5VF*$!3tH3O%*\\CNNF*7$$!3ummm())HOG%F* $!3]XHx!\\[%F*$ !3uv&[;NBw[#F*7$$!3h**\\s#eN!RTF*$!3`i*3nsfc7#F*7$$!3MmmwVL*p3%F*$!3Pz @8\\k#>\"=F*7$$!3Vmm1NhgMSF*$!3e'>EZ8>)Q:F*7$$!3_mmOE*=A)RF*$!3#G)3pww >.8F*7$$!3#HL$[$[Uz(QF*$!3Yd2&=XCXF*!#=7$$!3Ammhs[E\"y$F*$!3!p5fj2S\"z mF^o7$$!3U***\\j!3;\"o$F*$!3mQ(4J!4'pn%F^o7$$!3Imm\"y.Lwd$F*$!3A7$$!3W****RTi`pJF*$!337!RQ,u$oJF] q7$$!3')*****43\\Q1$F*$!3!H9fGqx-Z*!#@7$$!3m****f=***>'HF*$\"3KEZji5D% 4#F]q7$$!3Amm\"=76&pGF*$\"38x%RDSQmj$F]q7$$!3'GLLn*H`fFF*$\"3p_\\z5018 ^F]q7$$!3yKL$*)4njm#F*$\"3%*p&z`a8q<'F]q7$$!3!)***\\:,$*zb#F*$\"3[(4*) [?sCI(F]q7$$!35LL$4ej?Y#F*$\"37Q%zq)*\\?E)F]q7$$!3g***\\:z8oN#F*$\"3D6 Do4\"fzK*F]q7$$!3S***\\E5\"fcAF*$\"38D))zs*p#R5F^o7$$!3'HLL3C>?:#F*$\" 3'GJ.LD(ee6F^o7$$!3sKL)yi*)f0#F*$\"3s9^,1!4xF\"F^o7$$!3?mm;<(3C&>F*$\" 3pDn!)y$p%=9F^o7$$!3\\mm\"=E<[%=F*$\"3S(zUsHM-e\"F^o7$$!3m***\\Oee6v\" F*$\"3$p_G;*ziNF^o7$$!3))****>HF]X: F*$\"3iqjWEW1K@F^o7$$!3b*****=*zEV9F*$\"3OX&*>4jchBF^o7$$!3$)***\\ni]V M\"F*$\"3ukV,!)G52EF^o7$$!3y****pd(>XB\"F*$\"39Ptk%*>v4HF^o7$$!3MLL$p$ =$e8\"F*$\"3#oT;5H_:@$F^o7$$!3r*****pTh/.\"F*$\"3M^]\"[@C%oNF^o7$$!3ql m;a:!)\\$*F^o$\"3>&3jn:Pf#RF^o7$$!3r$****f;RfI)F^o$\"3[Ig(*RS!zN%F^o7$ $!3$RLL)3guBtF^o$\"3d-z'o5gw![F^o7$$!3o&***\\`62(H'F^o$\"34\"yec7yuK&F ^o7$$!3=HLLB)3LH&F^o$\"3M4g8v%*)**)eF^o7$$!36(***\\(oiCC%F^o$\"3rS5#fP FEa'F^o7$$!3jcmmiHPIKF^o$\"3dRCZL&o%RsF^o7$$!3_jmmR]O&>#F^o$\"3k6>d%[3 *G!)F^o7$$!3kBL$[]F*o6F^o$\"3IN!='eg!o*))F^o7$$!3%)=****>,QdAF]q$\"3YO QT13zw(*F^o7$$\"3Ozmmcej_&)F]q$\"3%o%f4t-H*3\"F*7$$\"3-TLL,V7A=F^o$\"3 1RA:b!p)*>\"F*7$$\"330+]%[**H&GF^o$\"30$))4![4;I8F*7$$\"3C4+]r&y'RQF^o $\"3SBn8_#)4o9F*7$$\"3!***************[F^o$\"3\"G&es?iJK;F*-%'COLOURG6 &%$RGBG$\"*++++\"!\")$\"\"!F`]lF_]l-F$6$7S7$F($!\"\"F`]l7$F=Fe]l7$FGFe ]l7$FQFe]l7$FenFe]l7$FjnFe]l7$F`oFe]l7$FeoFe]l7$FjoFe]l7$F_pFe]l7$FdpF e]l7$FipFe]l7$F_qFe]l7$FdqFe]l7$FjqFe]l7$F_rFe]l7$FdrFe]l7$FirFe]l7$F^ sFe]l7$FcsFe]l7$FhsFe]l7$F]tFe]l7$FbtFe]l7$FgtFe]l7$F\\uFe]l7$FauFe]l7 $FfuFe]l7$F[vFe]l7$F`vFe]l7$FevFe]l7$FjvFe]l7$F_wFe]l7$FdwFe]l7$FiwFe] l7$F^xFe]l7$FcxFe]l7$FhxFe]l7$F]yFe]l7$FbyFe]l7$FgyFe]l7$F\\zFe]l7$Faz Fe]l7$FfzFe]l7$F[[lFe]l7$F`[lFe]l7$Fe[lFe]l7$Fj[lFe]l7$F_\\lFe]l7$Fd\\ lFe]l-Fi\\l6&F[]lF_]lF_]lF\\]l-F$6&7#7$$!3z*****pJ#o+RF*Fe]l-%'SYMBOLG 6#%'CIRCLEG-Fi\\l6&F[]lF`]lF`]lF`]l-%&STYLEG6#%&POINTG-F$6&F[al-F`al6# %&CROSSGFcalFeal-F$6&F[al-F`al6#%(DIAMONDGFcalFeal-F$6%7$7$F]alF_]lF\\ al-%&COLORG6&F[]lF_]l$\"\"&Ff]lF_]l-%*LINESTYLEG6#\"\"$-%%FONTG6$%*HEL VETICAG\"\"*-%+AXESLABELSG6%Q\"z6\"Q!Ficl-Facl6#%(DEFAULTG-%%VIEWG6$;$ !$R%!\"#$\"#\\Fddl;$!$Z\"Fddl$\"$Z\"Fddl" 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 16 stage, order 8 scheme. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1445 "`R*` := z -> add(z^j/j!,j= 0..8)+\n 341404397/115195997743626*z^9+269686302/346176568856611*z ^10+\n 75093175/559161413283247*z^11-61363155/933968332588954*z^12 +\n 27805501/6470639401477803*z^13+916719/54235127252519663*z^14: \npts := []: z0 := 0:\nfor ct from 0 to 280 do\n zz := newton(`R*`(z )=exp(ct*Pi/20*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im( zz)]]:\nend do:\np_1 := plot(pts,color=COLOR(RGB,.2,0,0)):\np_2 := plo ts[polygonplot]([seq([pts[i-1],pts[i],[-1.9,0]],i=2..nops(pts))],\n \+ style=patchnogrid,color=COLOR(RGB,.7,0,0)):\npts := []: z0 := 0 .9+4.2*I:\nfor ct from 0 to 60 do\n zz := newton(`R*`(z)=exp(ct*Pi/3 0*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend d o:\np_3 := plot(pts,color=COLOR(RGB,.2,0,0)):\np_4 := plots[polygonplo t]([seq([pts[i-1],pts[i],[.81,4.02]],i=2..nops(pts))],\n styl e=patchnogrid,color=COLOR(RGB,.7,0,0)):\npts := []: z0 := 0.9-4.2*I:\n for ct from 0 to 60 do\n zz := newton(`R*`(z)=exp(ct*Pi/30*I),z=z0): \n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np_5 := \+ plot(pts,color=COLOR(RGB,.2,0,0)):\np_6 := plots[polygonplot]([seq([pt s[i-1],pts[i],[.81,-4.02]],i=2..nops(pts))],\n style=patchnog rid,color=COLOR(RGB,.7,0,0)):\np_7 := plot([[[-4.59,0],[1.49,0]],[[0,- 4.79],[0,4.79]]],color=black,linestyle=3):\nplots[display]([p_||(1..7) ],view=[-4.59..1.49,-4.79..4.79],font=[HELVETICA,9],\n labe ls=[`Re(z)`,`Im(z)`],axes=boxed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 396 515 515 {PLOTDATA 2 "6/-%'CURVESG6$7e\\l7$$\"\"!F)F (7$F($\"3++++Fjzq:!#=7$F($\"3)******Rl#fTJF-7$$\"30+++E`@Q:!#F$\"3#*** ***p(*)Q7ZF-7$$\"3++++]R?HE!#E$\"3=+++d_=$G'F-7$$\"3;+++i7o+B!#D$\"3#* *****\\<\")R&yF-7$$\"33+++e6?!H\"!#C$\"32+++X]xC%*F-7$$\"3i*****>g=5:& FF$\"3#******R*fb*4\"!#<7$$\"3%******\\&fU7:!#B$\"3++++\\7jc7FN7$$\"3) ******f;&o\\IFR$\"3\"******4b'p89FN7$$\"30+++M#e'=@FR$\"3)******z+N2d 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FbeoFh`gl7%Ffchl7$$\"+&[\"R-6Fbeo$!+Dk%)\\ZFbeoFh`gl7%F\\dhlF``glFh`gl F^cdlF][u-F$6%7$7$$!3/++++++!H&FNF(7$$\"3!**************)=FNF(-%'COLOU RG6&FaboF)F)F)-%*LINESTYLEG6#\"\"$-F$6%7$7$F(Ffdhl7$F($\"3/++++++!H&FN F[ehlF^ehl-%%FONTG6$%*HELVETICAG\"\"*-%*AXESSTYLEG6#%$BOXG-%(SCALINGG6 #%,CONSTRAINEDG-%+AXESLABELSG6%%&Re(z)G%&Im(z)G-Fjehl6#%(DEFAULTG-%%VI EWG6$;$!$H&F`_v$\"$*=F`_v;Fbghl$\"$H&F`_v" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 52 "#---------------------------------------------------" }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "coefficients of the combined sche me" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "coefficient s of the combined scheme " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16669 "ee := \{c[2]=1/46,\nc[3]=9675525 2944/718444993695-11256225944/718444993695*6^(1/2),\nc[4]=48377626472/ 239481664565-5628112972/239481664565*6^(1/2),\nc[5]=71/136,\nc[6]=276/ 715-46/715*6^(1/2),\nc[7]=276/715+46/715*6^(1/2),\nc[8]=92/143,\nc[9]= 69/143,\nc[10]=3/44,\nc[11]=103/411,\nc[12]=30258248819701/45339732981 913,\nc[13]=59/69,\nc[14]=44/49,\nc[15]=1,\nc[16]=1,\n\na[2,1]=1/46,\n a[3,1]=-163287951175938724532816/516163208965408589753025+\n 42 011574289334042817176/516163208965408589753025*6^(1/2),\na[3,2]=232801 278267248934720896/516163208965408589753025-\n 50098553466700618 240256/516163208965408589753025*6^(1/2),\na[4,1]=12094406618/239481664 565-1407028243/239481664565*6^(1/2),\na[4,2]=0,\na[4,3]=36283219854/23 9481664565-4221084729/239481664565*6^(1/2),\na[5,1]=450479172821804238 979159483/489985471732935255816699904+\n 65404175703680378526395 577/244992735866467627908349952*6^(1/2),\na[5,2]=0,\na[5,3]=-166328582 3745576633021875313/489985471732935255816699904-\n 258991054585 998425691922779/244992735866467627908349952*6^(1/2),\na[5,4]=734303944 921586208649981787/244992735866467627908349952+\n 96793439441159 023582763601/122496367933233813954174976*6^(1/2),\na[6,1]=188634486760 257/2753187875656075-40451003556679/5506375751312150*6^(1/2),\na[6,2]= 0,\na[6,3]=0,\na[6,4]=890541395040155939974909749/34049305087793600110 84250045-\n 235414842445143790083329443/680986101755872002216850 0090*6^(1/2),\na[6,5]=127509164130554343284736/22788053338091768042995 25-\n 51090254569210884816896/2278805333809176804299525*6^(1/2 ),\na[7,1]=523150756520001/5294592068569375+372205675002861/1376593937 82803750*6^(1/2),\na[7,2]=0,\na[7,3]=0,\na[7,4]=1218325024411588119947 48302664452173/6319431229672072722127362725145820625-\n 12054008 141355156662680357922224203047/164305211971473890775311430853791336250 *6^(1/2),\na[7,5]=-7345188891123909155979140554752/5242897828151193853 5235507146875+\n 71382195182457889488943971467264/6815767176596 55200958061592909375*6^(1/2),\na[7,6]=84211752143498940768/20638904623 3053165625+567839841668979868/18762640566641196875*6^(1/2),\na[8,1]=92 /1287,\na[8,2]=0,\na[8,3]=0,\na[8,4]=0,\na[8,5]=0,\na[8,6]=368/1287+23 /1287*6^(1/2),\na[8,7]=368/1287-23/1287*6^(1/2),\na[9,1]=1311/18304,\n a[9,2]=0,\na[9,3]=0,\na[9,4]=0,\na[9,5]=0,\na[9,6]=4071/18304+1587/366 08*6^(1/2),\na[9,7]=4071/18304-1587/36608*6^(1/2),\na[9,8]=-621/18304, \na[10,1]=2451872601/50434064384,\na[10,2]=0,\na[10,3]=0,\na[10,4]=0, \na[10,5]=0,\na[10,6]=84329349/1146228736-1383050643/100868128768*6^(1 /2),\na[10,7]=84329349/1146228736+1383050643/100868128768*6^(1/2),\na[ 10,8]=-1098320769/50434064384,\na[10,9]=-333490521/3152129024,\na[11,1 ]=-11290810941252792923651/1669469461414577748900000-\n 762184 89460616423924209/10016816768487466493400000*6^(1/2),\na[11,2]=0,\na[1 1,3]=0,\na[11,4]=0,\na[11,5]=0,\na[11,6]=1/30,\na[11,7]=-4460822007879 8131601386867/1778327431680661626219300000-\n 3026636216481078 19403033939/5334982295041984878657900000*6^(1/2),\na[11,8]=47685506231 91902657077/335320789258483564950000+76218489460616423924209/938898209 9237539818600000*6^(1/2),\na[11,9]=76371166597983496297729/12681546870 36073482337500+\n 76218489460616423924209/190223203055411022350 6250*6^(1/2),\na[11,10]=12837092726068800321242176/7348949926011775022 9428125+\n 224387232972054752032871296/13889515360162254793361 915625*6^(1/2),\na[12,1]=-35584379273878058921133601321126601189438489 2859307606673616682840321037292934264483298005378753134077/61149805013 3846699976127131429172460287752543562338153695758941655645150109566491 3528943447961301150000+16342530068674673706098410397304821116831707257 38988407797492220074165441589604495769296514008733014761/6250868956923 7662664226328990537629273859148897483455711122025147021504233422352449 406977468048856200000*6^(1/2),\na[12,2]=0,\na[12,3]=0,\na[12,4]=0,\na[ 12,5]=0,\na[12,6]=-175949180260408634887753230236883672639389038840971 69692619657825340943285779242173824532021369934883/\n 34620197 2998854747063407360562977639055219901586062216246214293121965254215877 64433517710597688597280-\n 17188002444689712939111901065805768 95962584281439946985382458420029691073118839588510586218092229/\n \+ 1344473681548950474032649943933893743903766608101212490276560361638700 01637234036635020235330829504*6^(1/2),\na[12,7]=-138207307502597872414 4662487119071171012382899509469565758724972879943335389436529352888452 83125425999567221/4494458926341069611642228743708681268522187616121807 07797390011848931329871568905581758203549930436500950000+3734226099348 2278211711351217469345525186992139310658684457112617175617353247938532 5874622983683540733325899/17977835705364278446568914974834725074088750 46448722831189560047395725319486275622327032814199721746003800000*6^(1 /2),\na[12,8]=12632445836279619113294210780168171565726892406599945732 333835965116776912265868042131556830549327654393/112996477298237313277 6399024059718683027453845454508622470282762273081038065711755816203054 23011393900000-6303547312203088429495101153246145287920801370707526715 790327134571780988988474483681572268319398771221/225992954596474626555 2798048119437366054907690909017244940565524546162076131423511632406108 46022787800000*6^(1/2),\na[12,9]=1115379785277627713874987636432084522 404410826010464628546537384975239192450316138834225701046102632948257/ 3347520639960280405850082108776916598468832017158981794068212683234002 575269671076605501548156712544287500-163425300686746737060984103973048 2111683170725738988407797492220074165441589604495769296514008733014761 /118706405672350369001775961304145978669107518338970985605255768909007 18352020110200728728894172739518750*6^(1/2),\na[12,10]=111953957936197 9281841957268759561827109492225370086710661285089191275935798906856762 3193453186950145251776/37146762745465379216384266737567863919875050348 378042013698628944987922182903823702648363618722076517534375-687320121 7454034198679102887095056195421792309393688389365452994140490085771136 62220686991034529999350912/1238225424848845973879475557918928797329168 3449459347337899542981662640727634607900882787872907358839178125*6^(1/ 2),\na[12,11]=73825104187474768875967421005730375156586805528272061538 441201406502469378399602291332635087412/\n 9287127253902959914 3289144740668914525146202470165909630170359355065425993729905138177287 271317,\na[13,1]=-6471907074414433573396221441243120203556100441993732 0357188840883/\n 101209800434111984325751891003355158288493635 359529771992774000000+\n 3559372342256314491923633965576963561 357143961289/\n 3635724370610188458125862611880298259763918450 0000*6^(1/2),\na[13,2]=0,\na[13,3]=0,\na[13,4]=0,\na[13,5]=0,\na[13,6] =-66005588850553492194962577164587068367312330247/\n 196036127 23583140697294028442079849957408380800-\n 13176568020345600854 085799476910283462657777/\n 7613053484886656581473409103720330 0805469440*6^(1/2),\na[13,7]=-9318570701703691930191534222789494006237 103047579881711/\n 3562968864769188061196125478115789029665196 590638000000+\n 6434757944786940764537553522803508307647498263 055753667/\n 7125937729538376122392250956231578059330393181276 000000*6^(1/2),\na[13,8]=-37612769433181145931578875950631975698389267 71255702620302334773297/\n 14642165429350376228890216175250454 99738187320641028236799386000000-\n 35593723422563144919236339 65576963561357143961289/\n 340784421062892308468207635130660 43517445275500000*6^(1/2),\na[13,9]=4386968817173326931645746836051457 82902979477404432064959580227217/\n 190320793597985643524049406 938231176153947641989529318077977593750-\n 35593723422563144919 23633965576963561357143961289/\n 690438041534146365846017178698 4220252917056671875*6^(1/2),\na[13,10]=3939307739733062346504442284413 0393549390670768068628724675776122752/\n 241046661167498470550 78895726017945791587677982957405550119962671875-\n 91119931961 7616509932450295187702671707428854089984/\n 438379739036741087 5150919593877143523760184037765625*6^(1/2),\na[13,11]=1455014345065512 890761705230599774567929341275785651/\n 4372745336819806461750 86038705021590800143506216585,\na[13,12]=43491235157516875762138510242 422928845963515012067399050743561004230490804423974878600834753885909/ \n 15650073716933473532700213239415706508578020603344410725067 680963543589421556571669667160279586515,\na[14,1]=11135230389730673778 22299106095464437335258246585030516055024186482995053/\n 117208 1821952666268571554825436238158838606320428900711229598548800375000-\n 9656355719858106181793772673915704646989403181/\n 42650 123991481333817759653132492492758879187500*6^(1/2),\na[14,2]=0,\na[14, 3]=0,\na[14,4]=0,\na[14,5]=0,\na[14,6]=6515655880631404133262717685930 9009839731119/\n 11498348446055040912040697694327646399463400+ \n 2893908603034923903437694020377944093049/\n 446537803 73029285095303680366320956891120*6^(1/2),\na[14,7]=8209640140354849710 164186531509948506760631106345347/\n 20898320165382824701639668 31380108687881472614625000-\n 732642588980105756809694399344660 8268878122947239059/\n 4179664033076564940327933662760217375762 945229250000*6^(1/2),\na[14,8]=358387301461393123075253933751833574944 722054371964074460462507967392441417/\n 60038122073498036770225 3537950121782364502552531871330639897526611625875000+\n 9656355 719858106181793772673915704646989403181/\n 39976896846716122011 518307352108347913593312500*6^(1/2),\na[14,9]=-64285140412531160510172 3817200220613720981758568362376903035228606467992291/\n 274647 144865445671171423994203344292862210985060954852952360191727428234375+ \n 77250845758864849454350181391325637175915225448/\n \+ 64795380679365872530827165335902056306758765625*6^(1/2),\na[14,10]=-36 9969589301763518639761285997824520507173970793320519753635225871563412 913152/\n 180850870268987312974476662632186804296491491034759 756207691410652763681078125+\n 454852979828196233587213868032 125351691788847437824/\n 946232027174365750372191179393662736 769436409578125*6^(1/2),\na[14,11]=-7715303458281199041757411869319392 732268755939408305903151657039/\n 148306285785801995659935429 2891821472831958379628421680653323310,\na[14,12]=-94543290861814042087 5128019003679821034774098061354410087876969999388668430388640158852256 997513450369741874915618358283/118333625607368137927111484394931198556 9932050547818490887959987766109714104308279282406274069868573023638087 923030436570,\na[14,13]=1056911827593717127690972016166243945915857152 155/\n 7359790814129537930306952068888958882079118176267,\na[1 5,1]=-3584428844827517211029003988869342002889757294838350269774673649 /\n 2085716341227133258652824258948052402773350356190879990897 000000+\n 18418541509225838359706930946702924369912480521/\n \+ 39508190287667470693939925841123267455183000000*6^(1/2),\na[15 ,2]=0,\na[15,3]=0,\na[15,4]=0,\na[15,5]=0,\na[15,6]=-10950654480421820 732850785842671218433296327603/\n 1006547124867343561064069495 275386629319739200+\n 2070463408861151634449033026537660336571 47/\n 3908920873271237130345900952525773317746560*6^(1/2), a[1 5,7]=-190787636465921540998883304987599740985291747006984827/\n \+ 26134367005227533473302899401390866713643703491000000+\n 178 911278528379700355580290396510513302568946148710869/\n 5226873 4010455066946605798802781733427287406982000000*6^(1/2), a[15,8]=-11573 3841931243537286820675899274760337634770259165918831723224267751/\n \+ 7941600225359840800631673172720354729727976962094049828002751300 0000-\n 497300620749097635712087135560978957987636974067/\n \+ 999861123434045989100479661671504230211939000000*6^(1/2), a[15,9 ]=20283630417023233099057095727795854554028063632221885527696504052311 1/\n 4048345226586686505118858391505708881485411941517464594181 2575437500-\n 18418541509225838359706930946702924369912480521/ \n 7502757290206082175051091686174851271537156250*6^(1/2), a[15 ,10]=99854707672539845032167700901378988710484675887497768961753983967 005376/\n 2398926904363445204090328582971887578229101651831845 3275155443387390625-\n 542241862031608681309772047070934093450 22342653824/\n 54782834560687628399797684955475647938658789640 625*6^(1/2), a[15,11]=474324561414678755109830553528867775624956239150 831585597139/\n 4515791163569036664033761487472972128120127517 1059286117565, a[15,12]=5838559500251120707358738694419063212491522353 3408383221102363939423894748980604430394490848526064113167169126734415 639048389/224741650740382631378562225386536931859861163472113576869923 24141917918392966746421148746670582600045180015758041428626487315,\na[ 15,13]=-2061784272050868275289038565787612564079661263925986910/\n \+ 6305818041153225448392022471538861648368501130406546697, a[15,14 ]=2144782169029430053686260375092251514835023987332030/\n 5255 390852470014220073816442862907485166958089934203,\na[16,1]=69477988002 853816125344190899680848875649117294292403621110427/\n 27631572 785804084086128051091055380750995025720848651597000000-\n 18542 6619462312591171592206632768608526005483/\n 5234045560226059182 20275652595264842483000000*6^(1/2),\na[16,2]=0,\na[16,3]=0,\na[16,4]=0 ,\na[16,5]=0,\na[16,6]=155409868146947995309862276450550758522148769/ \n 13334737611899006162658099703042439679259200+\n 29488 334695552913776680777005490941202439/\n 51785388784073810340419 804672009474482560*6^(1/2), a[16,7]=3093423962117018793580109802227401 427197073610480521/\n 34622812788195025813456589747718254474716 5591000000-\n 2223349281800786799168112412879091860919395860086 487/\n 692456255763900516269131794954365089494331182000000*6^(1 /2), a[16,8]=853183272432394337562287283280711289401314188781183959070 92074851/\n 471795202042884711177778173987241082529472009403262 5268331000000+\n 5006518725482439961632989579084752430202148041 /\n 13246161456264411314959283823372471782839000000*6^(1/2), a[ 16,9]=-661400874499143803275481013443606944473757715893053520699246320 43/\n 75538708164928211535304354250774327066121702984715725457 94812500+\n 185426619462312591171592206632768608526005483/\n \+ 99396538283139104662023501334196929221531250*6^(1/2), a[16,10]= -996478945652945815174857068635853357841406776256125134079522701632/\n 177646655848376964241055783826452426783687047504396906456095 890625+\n 23734607291176011669963802448994381891328701824/\n \+ 31554915787156112484260150916445096576431546875*6^(1/2), a[16 ,11]=-448074765213807929719320331516183261898760393629529216/\n \+ 45888783596485008769753055117740509726182346135675245, a[16,12]=-97 9092034558595733474782378777061829201471936704327265360243006169929196 1954865703160143382924460700238382314/58118528355148420254163062157026 8158540280085394426632970535207182792110996461759014949613118263320607 923905, a[16,13]=98768647654583089123296762257490493421581752858120/\n 125246605043204308809279835220399478404120630180191,\na[16,14 ]=0,\na[16,15]=0,\n\nb[1]=385924436255198461459913/2588529729216475061 7319296,\nb[2]=0,\nb[3]=0,\nb[4]=0,\nb[5]=0,\nb[6]=0,\nb[7]=0,\nb[8]=- 81508791888782942071778080019859673/3993954007779995527876722199835097 60,\nb[9]=935936315524449978576662571821361001/40868014095024446705259 51816974611200,\nb[10]=2172547024243858864854526674870272/169722834084 14027681310867548214525,\nb[11]=11937467247139973420948110779219186735 63219/5333839543124606397779553737025343484595200,\nb[12]=789988859008 4372060795619128755092080289376683274569118606954351302689076603970233 1356142187984337094261434894685526783/19972835375940569338334435735705 0366806065734043901244628659238274456345277244226082448918708806038102 409307770530790400,\nb[13]=11409994679937666036993318622713183/2106468 29819316523523317257829664000,\nb[14]=10048608923923592706010638721995 991/79176273134521955696793568162336000, b[15]=95793597981031291770576 1/30187687238921485361088000,\nb[16]=0,\n\n`b*`[1]=6365795202834324687 727/367688881990976571268740,\n`b*`[2]=0,\n`b*`[3]=0,\n`b*`[4]=0,\n`b* `[5]=0,\n`b*`[6]=0,\n`b*`[7]=0,\n`b*`[8]=11235845416756910667521184687 449/27984543215947278082095867431580,\n`b*`[9]=59607483497742391571455 0106443/3509791660513951108318405888848,\n`b*`[10]=4628113370715837942 8440684544/379480903486059869900746060329,\n`b*`[11]=48409638011984181 51644350108694156829/20456544999327323762290226804576756480,\n`b*`[12] =-42503788329176706927962313423474925759201096452815072640470622305296 40315652520199442309428858040080393/1949352625073773791223065241302356 7235380476570031599740950002149019485955879156117360614282462351914240 ,\n`b*`[13]=8307802719022653646372347529833/36318418934364917848847803 074080,\n`b*`[14]=0,\n`b*`[15]=0,\n`b*`[16]=12690737099320029004861/30 1876872389214853610880\}:" }}}{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..16);" } }{PARA 12 "" 1 "" {XPPMATH 20 "61/&%\"cG6#\"\"##\"\"\"\"#Y/&F%6#\"\"$, &#\",WHDbn*\"-&p$*\\W=(F)*(\",WfAc7\"F)F2!\"\"\"\"'#F)F'F5/&F%6#\"\"%, &#\",skix$[\"-lXm\"[R#F)*(\"+sH6GcF)F?F5F6F7F5/&F%6#\"\"&#\"#r\"$O\"/& F%6#F6,&#\"$w#\"$:(F)*(F*F)FOF5F6F7F5/&F%6#\"\"(,&FMF)*(F*F)FOF5F6F7F) /&F%6#\"\")#\"##*\"$V\"/&F%6#\"\"*#\"#pFgn/&F%6#\"#5#F.\"#W/&F%6#\"#6# \"$.\"\"$6%/&F%6#\"#7#\"/,(>)[#e-$\"/8>)HtR`%/&F%6#\"#8#\"#fF]o/&F%6# \"#9#Fco\"#\\/&F%6#\"#:F)/&F%6#\"#;F)" }}}{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..16);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6dr/&%\"aG 6$\"\"#\"\"\"#F(\"#Y/&F%6$\"\"$F(,&#\"9;G`C(Qf<^zGj\"\"9DIv*e3a'*3K;;& !\"\"*(\"8wr\"G/M$*Gu:,UF(F2F3\"\"'#F(F'F(/&F%6$F.F',&#\"9'*3sM*[sEy7! 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HiPKF$**\\ac/#/&F%6#\"#7#!bq$R!3S!e)G%4BW*>?Dl:.kH0Biq/ks]\"GX'4,#fd# \\ZBMJiz#pqw\"H$)y.D%\"cqSU\">NiCG91O!\\@+]4u*fJ+dw/QNscBIT_1 B7ztP2DEN\\>/&F%6#\"#8#\"@L)HvMsjk`E->F!yI)\"A!3uI!y%)[y\"\\OM*=%=j$/& F%6#\"#9F//&F%6#\"#:F//&F%6#\"#;#\"8h[+H+K*4P2p7\"9!)3h`[@*Qso(=I" }}} {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 30 "#=============================" }} {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 3453 "SO9_15 := SimpleOrderConditions(9,15,'expan ded'):\nQeqs := QuadratureConditions(9,15,'expanded'):\n\nRSeqs := Row SumConditions(11,'expanded'):\nSOeqs := [op(StageOrderConditions(2,11, 'expanded')),\n op(StageOrderConditions(3,4..11,'expande d')),\n op(StageOrderConditions(4,6..11,'expanded')),\n \+ op(StageOrderConditions(5,8..11,'expanded'))]:\nnode_eqs A := [c[3]=2/3*c[4],c[9]=3/4*c[8],c[6]=1/10*(6-6^(1/2))*c[8],c[7]=1/10 *(6+6^(1/2))*c[8],\n c[4]=1/40*(6-6^(1/2))*(40*c[5]-18*c[8]+3*c[8 ]*6^(1/2))*c[8]/(15*c[5]-6*c[8]+c[8]*6^(1/2))]:\n\nnode_eqsB := [c[3]= 2/3*c[4],c[9]=3/4*c[8],c[6]=1/10*(6+6^(1/2))*c[8],c[7]=1/10*(6-6^(1/2) )*c[8],\n c[4]=1/40*(6+6^(1/2))*(40*c[5]-18*c[8]-3*c[8]*6^(1/2))* c[8]/(15*c[5]-6*c[8]-c[8]*6^(1/2))]:\n\n`Qeqs*` := subs(b=`b*`,Quadrat ureConditions(8,14,'expanded')):\n\nCeq := -1/12700800*(-9*c[8]-9*c[9] -9*c[10]-9*c[11]+5+126*c[8]*c[9]*c[10]*c[11]+18*c[11]*c[8]+18*c[11]*c[ 9]+18*c[11]*c[10]-42*c[8]*c[9]*c[10]+18*c[10]*c[8]+18*c[10]*c[9]+18*c[ 8]*c[9]-42*c[11]*c[8]*c[9]-42*c[11]*c[10]*c[8]-42*c[11]*c[10]*c[9])*(- 15-28*c[12]*c[8]+20*c[8]+20*c[9]+20*c[10]+20*c[11]+20*c[12]-70*c[8]*c[ 9]*c[10]*c[11]-70*c[12]*c[8]*c[9]*c[10]+140*c[12]*c[8]*c[9]*c[10]*c[11 ]-28*c[10]*c[9]-28*c[11]*c[8]-28*c[11]*c[9]-28*c[11]*c[10]-28*c[10]*c[ 8]+42*c[8]*c[9]*c[10]+42*c[12]*c[8]*c[9]+42*c[10]*c[9]*c[12]+42*c[10]* c[12]*c[8]+42*c[11]*c[10]*c[9]+42*c[11]*c[10]*c[8]+42*c[11]*c[8]*c[9]+ 42*c[11]*c[10]*c[12]+42*c[11]*c[9]*c[12]+42*c[11]*c[12]*c[8]-28*c[9]*c [12]-28*c[10]*c[12]-70*c[11]*c[12]*c[8]*c[9]-70*c[11]*c[10]*c[12]*c[8] -70*c[11]*c[10]*c[9]*c[12]-28*c[11]*c[12]-28*c[8]*c[9]) = -1/8467200*( -8*c[8]-8*c[9]-8*c[10]-8*c[11]+5+70*c[8]*c[9]*c[10]*c[11]+14*c[11]*c[8 ]+14*c[11]*c[9]+14*c[11]*c[10]-28*c[8]*c[9]*c[10]+14*c[10]*c[8]+14*c[1 0]*c[9]+14*c[8]*c[9]-28*c[11]*c[8]*c[9]-28*c[11]*c[10]*c[8]-28*c[11]*c [10]*c[9])*(-10-24*c[12]*c[8]+15*c[8]+15*c[9]+15*c[10]+15*c[11]+15*c[1 2]-84*c[8]*c[9]*c[10]*c[11]-84*c[12]*c[8]*c[9]*c[10]+210*c[12]*c[8]*c[ 9]*c[10]*c[11]-24*c[10]*c[9]-24*c[11]*c[8]-24*c[11]*c[9]-24*c[11]*c[10 ]-24*c[10]*c[8]+42*c[8]*c[9]*c[10]+42*c[12]*c[8]*c[9]+42*c[10]*c[9]*c[ 12]+42*c[10]*c[12]*c[8]+42*c[11]*c[10]*c[9]+42*c[11]*c[10]*c[8]+42*c[1 1]*c[8]*c[9]+42*c[11]*c[10]*c[12]+42*c[11]*c[9]*c[12]+42*c[11]*c[12]*c [8]-24*c[9]*c[12]-24*c[10]*c[12]-84*c[11]*c[12]*c[8]*c[9]-84*c[11]*c[1 0]*c[12]*c[8]-84*c[11]*c[10]*c[9]*c[12]-24*c[11]*c[12]-24*c[8]*c[9]): \n\nSOeqs2 := [seq(add(a[i,j],j= 1..i-1)=c[i],i=12..15),op(StageOrderC onditions(2,12..15,'expanded')),\n op(StageOrderConditions(3,1 2..15,'expanded')),op(StageOrderConditions(4,12..15,'expanded')),\n \+ op(StageOrderConditions(5,12..15,'expanded'))]:\nord_cdns \+ := [seq(SO9_15[i],i=[102,106,125,212,223,239,245,251,253])]:\nsimp_eqs := [add(b[i]*a[i,1],i=1+1..15)=b[1],seq(add(b[i]*a[i,j],i=j+1..15)=b[ j]*(1-c[j]),j=[$7..13])]:\ncdns := [op(SOeqs2),op(ord_cdns),op(simp_eq s)]:\n\nSO_eqs2 := [add(a[14,j],j=1..13)=c[14],add(a[14,j]*c[j],j=2..1 3)=1/2*c[14]^2,\n add(a[14,j]*c[j]^2,j=2..13)=1/3*c[14]^3,add(a [14,j]*c[j]^3,j=2..13)=1/4*c[14]^4,\n add(a[14,j]*c[j]^4,j=2..1 3)=1/5*c[14]^5]:\n`simp_eqs*` := [add(`b*`[i]*a[i,1],i=2..14)=`b*`[1], seq(add(`b*`[i]*a[i,j],i=j+1..14)=`b*`[j]*(1-c[j]),j=[9,11,13])]:\n`cd ns*` := [op(`simp_eqs*`),op(SO_eqs2),op(`Qeqs*`)]:\n\nerrterms9_16 := \+ PrincipalErrorTerms(9,16,'expanded'):\n`errterms9_16*` :=subs(b=`b*`,e rrterms9_16):\n`errterms8_16*` := subs(b=`b*`,PrincipalErrorTerms(8,16 ,'expanded')):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 13 "calc_RKcoeffs" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3476 "calc_RKcoeffs := proc()\n \+ local eqns,quadeqns,eqns2,eqns3,sm,ct,Rz,stb9,stb8,nrm,nmB,snmB,\n \+ dnB,sdnB,nmC,snmC,B_10,C_10;\n global e1,e2,e3,e4,e5,e6,e7,e8 ,e9,e10,e11,e12,e13,e14,e15,e16,e17,e18;\n\n e1 := \{c[2]=c_2,c[5]=c _5,c[8]=c_8,c[10]=c_10,c[11]=c_11,\n seq(a[i,2]=0,i=4..11), seq(a[i,3]=0,i=6..11),\n seq(a[i,4]=0,i=8..11),seq(a[i,5]=0 ,i=8..11),a[11,a_11[1]]=a_11[2]\};\n if nargs>0 and member('switch_n odes',[args]) then\n eqns := expand(rationalize(subs(e1,[op(RSeqs ),op(SOeqs),op(node_eqsB)])));\n else\n eqns := expand(rational ize(subs(e1,[op(RSeqs),op(SOeqs),op(node_eqsA)])));\n end if;\n e2 := expand(rationalize(solve(\{op(eqns)\})));\n e3 := `union`(e1,e2) ;\n e4 := solve(subs(e3,value(Ceq)),\{c[12]\});\n e5 := `union`(e3 ,e4);\n e6 := \{seq(b[i]=0,i=2..7),c[13]=c_13,c[14]=c_14,c[15]=1\}; \n e7 := `union`(e5,e6);\n quadeqns := subs(e7,Qeqs);\n e8 := so lve(\{op(quadeqns)\});\n e9 := `union`(e7,e8);\n e10 :=\{seq(seq(a [i,j]=0,i=12..15),j=2..5)\};\n e11 := `union`(e9,e10);\n eqns2 := \+ subs(e11,cdns);\n e12 := solve(\{op(eqns2)\});\n e13 := `union`(e1 1,e12);\n e14 := `union`(remove(u_->member(op(1,lhs(u_)),[14,15]) or op(0,lhs(u_))=b,e13),\n \{c[14]=1,seq(a[14,i]=0,i=2..5),seq(` b*`[i]=0,i=2..7)\});\n eqns3 := subs(e14,`cdns*`);\n e15 := solve( \{op(eqns3)\});\n e16 := `union`(e14,e15);\n e17 := \{c[16]=1,seq( a[16,i]=subs(e16,a[14,i]),i=1..13),a[16,14]=0,a[16,15]=0,b[16]=0,\n \+ seq(`b*`[i]=subs(e16,`b*`[i]),i=1..13),`b*`[14]=0,`b*`[15]=0,`b*`[16]= subs(e16,`b*`[14])\};\n e18 := `union`(e13,e17);\n Digits := 14;\n sm := 0:\n for ct to nops(errterms9_16) do\n sm := sm+(evalf (expand(subs(e18,errterms9_16[ct]))))^2;\n end do:\n Rz := subs(e1 8,StabilityFunction(9,16,'expanded'));\n stb9 := max(fsolve(Rz=1,z=- 8..-1e-7),fsolve(Rz=-1,z=-8..-1e-7));\n stb9 := evalf[8](stb9);\n \+ Rz := subs(e18,subs(b=`b*`,StabilityFunction(8,16,'expanded')));\n s tb8 := max(fsolve(Rz=1,z=-8..-1e-7),fsolve(Rz=-1,z=-8..-1e-7));\n st b8 := evalf[8](stb8);\n print(`nodes:`,c[2]=c_2,c[3]=subs(e18,c[3]), c[4]=subs(e18,c[4]),c[5]=c_5,c[6]=subs(e18,c[6]),\n c[ 7]=subs(e18,c[7]),c[8]=c_8,c[9]=subs(e18,c[9]),c[10]=c_10,c[11]=c_11, \n c[12]=subs(e18,c[12]),c[13]=c_13,c[14]=c_14);\n p rint(`order 9 weights:`,seq(b[i]=evalf[6](subs(e18,b[i])),i=[1,$8..15] ));\n print(`order 8 weights:`,seq(`b*`[i]=evalf[6](subs(e18,`b*`[i] )),i=[1,$8..13,16]));\n nrm := max(seq(seq(subs(e18,abs(a[i,j])),j=1 ..i-1),i=2..16));\n print(infinity*`-norm of linking coeffs`=evalf[1 0](nrm));\n print(`2-norm of principal error of order 9 scheme` = ev alf[10](sqrt(sm)));\n print(`order 9 stability interval` = [stb9,0]) ;\n print(`order 8 stability interval` = [stb8,0]);\n if nargs>0 a nd member('short_version',[args]) then return end if;\n nmB := 0; \+ \n for ct to nops(`errterms9_16*`) do\n nmB := nmB+evalf(expand (subs(e18,`errterms9_16*`[ct])))^2;\n end do:\n snmB := sqrt(nmB); \n dnB := 0;\n for ct to nops(`errterms8_16*`) do\n dnB := dn B+evalf(expand(subs(e18,`errterms8_16*`[ct])))^2;\n end do;\n sdnB := sqrt(dnB);\n print(`2-norm of principal error of order 8 scheme` = evalf[10](sdnB));\n nmC := 0;\n for ct to nops(errterms9_16) do \n nmC := nmC+(evalf(expand(subs(e18,`errterms9_16*`[ct])))-evalf (expand(subs(e18,errterms9_16[ct]))))^2;\n end do;\n snmC := sqrt( nmC);\n B_10 := evalf[8](snmB/sdnB):\n C_10 := evalf[8](snmC/sdnB) :\n print('B[10]'=B_10,'C[10]'=C_10)\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 23 "(1) When the argument " }{TEXT 262 15 "'short_version'" } {TEXT -1 72 " is supplied, the procedure returns without giving the c haracterstics " }{XPPEDIT 18 0 "B[10]" "6#&%\"BG6#\"#5" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "C[10]" "6#&%\"CG6#\"#5" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 23 "(2) When the argument " }{TEXT 262 14 "' switch_nodes'" }{TEXT -1 71 " is supplied, the procedure uses the nod e equations that ensure that " }{XPPEDIT 18 0 "c[7] " 0 "" {MPLTEXT 1 0 145 "c_2 := 1/25: \+ c_5 := 72/125: c_8 := 16/25: c_10 := 3377/50000: c_11 := 1/4: c_13 := \+ 1623/2000:\nc_14 := 453/500: a_11 := [6,1/30]:\ncalc_RKcoeffs();" }} {PARA 12 "" 1 "" {XPPMATH 20 "60%'nodes:G/&%\"cG6#\"\"##\"\"\"\"#D/&F& 6#\"\"$,&#\"#[\"$N$F**(\"#KF*\"%v;!\"\"\"\"'#F*F(F7/&F&6#\"\"%,&#\"#sF 3F**(F2F*F6F7F8F9F7/&F&6#\"\"&#F@\"$D\"/&F&6#F8,&#F2FGF**(\"\")F*FGF7F 8F9F7/&F&6#\"\"(,&FLF**(FNF*FGF7F8F9F*/&F&6#FN#\"#;F+/&F&6#\"\"*#\"#7F +/&F&6#\"#5#\"%xL\"&++&/&F&6#\"#6#F*F=/&F&6#Fin#\"/y)*\\Gqhd\"/*pry#[4 &)/&F&6#\"#8#\"%B;\"%+?/&F&6#\"#9#\"$`%\"$+&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6,%1order~9~weights:G/&%\"bG6#\"\"\"$\"'*)e9!\"(/&F&6#\" \")$\"'?C?!\")/&F&6#\"\"*$\"'0y@!\"'/&F&6#\"#5$\"'!\\F\"F9/&F&6#\"#6$ \"'iWAF9/&F&6#\"#7$\"'D(y\"F9/&F&6#\"#8$\"'M%f(F+/&F&6#\"#9$\"'&[H\"F9 /&F&6#\"#:$\"'uZHF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6+%1order~8~weigh ts:G/&%#b*G6#\"\"\"$\"'nM?!\"(/&F&6#\"\")$\"'ip5!\"&/&F&6#\"\"*$\"'$3o (F+/&F&6#\"#5$\"'yI6!\"'/&F&6#\"#6$\"'f_DF?/&F&6#\"#7$!'!f#)*F?/&F&6# \"#8$\"'b\")RF?/&F&6#\"#;$\"'gK\\F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/*&%)infinityG\"\"\"%8-norm~of~linking~coeffsGF&$\"+>Ar!Q#!\")" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~ 9~schemeG$\"+/q\"=^$!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~9~ stability~intervalG7$$!)yW@X!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~8~stability~intervalG7$$!)IS4R!\"(\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~8~schemeG$\"+qF 4^T!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6#\"#5$\")j^CM!\"(/&% \"CGF&$\")A!fT$F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "#-------------------------------------------------------- " }}{PARA 0 "" 0 "" {TEXT -1 32 "Verner's \"most efficient\" scheme" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 159 "c_2 := 1731/50000: c_5 := \+ 561/1000: c_8 := 129/200: c_10 := 6757/100000: c_11 := 1/4: \nc_13 := \+ 4103/5000: c_14 := 2253/2500: a_11 := [6,1/30]:\ncalc_RKcoeffs();" }} {PARA 12 "" 1 "" {XPPMATH 20 "60%'nodes:G/&%\"cG6#\"\"##\"%J<\"&++&/&F &6#\"\"$,&#\"(\\+j(\")++\"Q&\"\"\"*(\"'RN)*F4F3!\"\"\"\"'#F4F(F7/&F&6# \"\"%,&#\")Z,*G#\"*++i2\"F4*(\"(<1&HF4FAF7F8F9F7/&F&6#\"\"&#\"$h&\"%+5 /&F&6#F8,&#\"$(QFJF4*(\"$H\"F4\"%+?F7F8F9F7/&F&6#\"\"(,&FOF4*(FRF4FSF7 F8F9F4/&F&6#\"\")#FR\"$+#/&F&6#\"\"*#FP\"$+)/&F&6#\"#5#\"%dn\"'++5/&F& 6#\"#6#F4F=/&F&6#\"#7#\"4s$e7&4lrzU\"\"4,Zbihkim;#/&F&6#\"#8#\"%.T\"%+ ]/&F&6#\"#9#\"%`A\"%+D" }}{PARA 11 "" 1 "" {XPPMATH 20 "6,%1order~9~we ights:G/&%\"bG6#\"\"\"$\"'?h9!\"(/&F&6#\"\")$!'@:R!\"'/&F&6#\"\"*$\"'$ 4J#F2/&F&6#\"#5$\"'xu7F2/&F&6#\"#6$\"'VYAF2/&F&6#\"#7$\"'N%o&F2/&F&6# \"#8$\"'(e#eF+/&F&6#\"#9$\"'Kk8F2/&F&6#\"#:$\"',dIF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6+%1order~8~weights:G/&%#b*G6#\"\"\"$\"'+(*>!\"(/&F&6# \"\")$\"']\">#!\"&/&F&6#\"\"*$\"'2d))F+/&F&6#\"#5$\"'cS6!\"'/&F&6#\"#6 $\"';LDF?/&F&6#\"#7$!'cc?F2/&F&6#\"#8$\"'53MF?/&F&6#\"#;$\"'BM[F+" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-norm~of~linking ~coeffsGF&$\"+)45\"*e$!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-nor m~of~principal~error~of~order~9~schemeG$\"+#[L0\\$!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~9~stability~intervalG7$$!)F " 0 "" {MPLTEXT 1 0 159 "c_2 := 1/50: c _5 := 28/90: c_8 := 52/125: c_10 := 21/200: c_11 := 280/477: c_13 := 2 47/281:\nc_14 := 229/250: a_11 := [10,-3/20]:\ncalc_RKcoeffs('switch_n odes');" }}{PARA 12 "" 1 "" {XPPMATH 20 "60%'nodes:G/&%\"cG6#\"\"##\" \"\"\"#]/&F&6#\"\"$,&#\"(Os$Q\")v$H%[F**(\"(o8.\"F*\"*D\")GX\"!\"\"\" \"'#F*F(F*/&F&6#\"\"%,&#\"(='=>\")DJ9;F**(\"'%o:&F*F3F7F8F9F*/&F&6#\" \"&#\"#9\"#X/&F&6#F8,&#\"$c\"\"$D'F**(\"#EF*FQF7F8F9F*/&F&6#\"\"(,&FOF **(FSF*FQF7F8F9F7/&F&6#\"\")#\"#_\"$D\"/&F&6#\"\"*#\"#RFjn/&F&6#\"#5# \"#@\"$+#/&F&6#\"#6#\"$!G\"$x%/&F&6#\"#7#\"1:P0NqAeO\"1K!*H>Zq\\`/&F&6 #\"#8#\"$Z#\"$\"G/&F&6#FI#\"$H#\"$]#" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6,%1order~9~weights:G/&%\"bG6#\"\"\"$\"'viH!\"(/&F&6#\"\")$\"'Q3yF+/&F &6#\"\"*$\"'?w?!\"'/&F&6#\"#5$\"'*op\"F8/&F&6#\"#6$\"'1<:F8/&F&6#\"#7$ \"'YU9F8/&F&6#\"#8$\"'in;F8/&F&6#\"#9$\"'MD>F+/&F&6#\"#:$\"'E,LF+" }} {PARA 11 "" 1 "" {XPPMATH 20 "6+%1order~8~weights:G/&%#b*G6#\"\"\"$\"' .')H!\"(/&F&6#\"\")$\"'cxlF+/&F&6#\"\"*$\"'dR@!\"'/&F&6#\"#5$\"'A(o\"F 8/&F&6#\"#6$\"'o<&!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~8~stability~ intervalG7$$!)^C9W!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-no rm~of~principal~error~of~order~8~schemeG$\"+'ea:A\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6#\"#5$\")FFN " 0 "" {MPLTEXT 1 0 147 "c_2 := 1/49: c_5 := 3/7: c_8 := 40/63: c_10 := 19/18: c_11 := 7/9: c_13 := 15/16: c_14 := 39/40:\na_11 := [6,76/100]:\ncalc_RKcoeff s('switch_nodes');" }}{PARA 11 "" 1 "" {XPPMATH 20 "60%'nodes:G/&%\"cG 6#\"\"##\"\"\"\"#\\/&F&6#\"\"$,&#\"#k\"$0(F**(\"#;F*\"&0[\"!\"\"\"\"'# F*F(F7/&F&6#\"\"%,&#\"#K\"$N#F**(\"\")F*\"%N\\F7F8F9F7/&F&6#\"\"&#F/\" \"(/&F&6#F8,&#FC\"#@F**(F=F*\"#jF7F8F9F*/&F&6#FJ,&FOF**(F=F*FRF7F8F9F7 /&F&6#FC#\"#SFR/&F&6#\"\"*#\"#5FP/&F&6#F\\o#\"#>\"#=/&F&6#\"#6#FJFjn/& F&6#\"#7#\"*'y***>$\"+8@rq@/&F&6#\"#8#\"#:F5/&F&6#\"#9#\"#RFfn" }} {PARA 11 "" 1 "" {XPPMATH 20 "6,%1order~9~weights:G/&%\"bG6#\"\"\"$\"' c`T!\"(/&F&6#\"\")$!'G_U!\"'/&F&6#\"\"*$\"'F6\\F2/&F&6#\"#5$\"'yTXF2/& F&6#\"#6$\"'.15!\"&/&F&6#\"#7$\"')pR#F2/&F&6#\"#8$!'\\bWFE/&F&6#\"#9$ \"')*)G*FE/&F&6#\"#:$!'$3k&FE" }}{PARA 11 "" 1 "" {XPPMATH 20 "6+%1ord er~8~weights:G/&%#b*G6#\"\"\"$\"'%p4%!\"(/&F&6#\"\")$!'#[a#!\"'/&F&6# \"\"*$\"'V\\XF2/&F&6#\"#5$!'S=)*F+/&F&6#\"#6$\"'u\\aF2/&F&6#\"#7$\"'5A CF2/&F&6#\"#8$!'))3JF2/&F&6#\"#;$\"'e/QF2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-norm~of~linking~coeffsGF&$\"+rX5 GE!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error ~of~order~9~schemeG$\"+Z^'[k$!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/% ;order~9~stability~intervalG7$$!)RzRR!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~8~stability~intervalG7$$!),MDR!\"(\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~order~ 8~schemeG$\"+N)pWg\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6#\" #5$\")YW@K!\"(/&%\"CGF&$\")\\^@KF*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 57 "#--------------------------------------- -----------------" }}{PARA 0 "" 0 "" {TEXT -1 19 "Tsitouras' scheme B " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "c_2 := 1/46: c_5 := 71/ 136: c_8 := 92/143: c_10 := 3/44: c_11 := 103/411: c_13 := 59/69: c_14 := 44/49:\na_11 := [6,1/30]:\ncalc_RKcoeffs();" }}{PARA 12 "" 1 "" {XPPMATH 20 "60%'nodes:G/&%\"cG6#\"\"##\"\"\"\"#Y/&F&6#\"\"$,&#\",WHDb n*\"-&p$*\\W=(F**(\",WfAc7\"F*F3!\"\"\"\"'#F*F(F6/&F&6#\"\"%,&#\",skix $[\"-lXm\"[R#F**(\"+sH6GcF*F@F6F7F8F6/&F&6#\"\"&#\"#r\"$O\"/&F&6#F7,&# \"$w#\"$:(F**(F+F*FPF6F7F8F6/&F&6#\"\"(,&FNF**(F+F*FPF6F7F8F*/&F&6#\" \")#\"##*\"$V\"/&F&6#\"\"*#\"#pFhn/&F&6#\"#5#F/\"#W/&F&6#\"#6#\"$.\"\" $6%/&F&6#\"#7#\"/,(>)[#e-$\"/8>)HtR`%/&F&6#\"#8#\"#fF^o/&F&6#\"#9#Fdo \"#\\" }}{PARA 11 "" 1 "" {XPPMATH 20 "6,%1order~9~weights:G/&%\"bG6# \"\"\"$\"'!4\\\"!\"(/&F&6#\"\")$!'!3/#!\"'/&F&6#\"\"*$\"'9!H#F2/&F&6# \"#5$\"'1!G\"F2/&F&6#\"#6$\"'1QAF2/&F&6#\"#7$\"'KbRF2/&F&6#\"#8$\"'l;a F+/&F&6#\"#9$\"'9p7F2/&F&6#\"#:$\"'FtJF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6+%1order~8~weights:G/&%#b*G6#\"\"\"$\"'IJ7F2/&F&6#\"#6$\"'YmBF2/&F&6# \"#7$!'T!=#F2/&F&6#\"#8$\"'\\(G#F2/&F&6#\"#;$\"'%R?%F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-norm~of~linking~coeffsGF&$ \"+Gs&4!>!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principa l~error~of~order~9~schemeG$\"+PYS;O!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~9~stability~intervalG7$$!)8M,X!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;order~8~stability~intervalG7$$!)Bo+R!\"(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%L2-norm~of~principal~error~of~orde r~8~schemeG$\"+\")[5*f$!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/&%\"BG6 #\"#5$\")w+,I!\"(/&%\"CGF&$\")At4IF*" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 23 "#======================" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "#============================" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "Test-bed procedures for the examples" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 37 "#------------------------------------" }}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 10 "RK9_15step" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5936 "rk9_15step := proc(x_rk9step::rea lcons)\n local c2,c3,c4,c5,c6,c7,c8,c9,cA,cB,cC,cD,cE,cF,a21,a31,a32 ,a41,a42,\n a43,a51,a52,a53,a54,a61,a62,a63,a64,a65,a71,a72,a73,a74, a75,a76,a81,a82,\n a83,a84,a85,a86,a87,a91,a92,a93,a94,a95,a96,a97,a 98,aA1,aA2,aA3,aA4,aA5,\n aA6,aA7,aA8,aA9,aB1,aB2,aB3,aB4,aB5,aB6,aB 7,aB8,aB9,aBA,aC1,aC2,aC3,aC4,\n aC5,aC6,aC7,aC8,aC9,aCA,aCB,aD1,aD2 ,aD3,aD4,aD5,aD6,aD7,aD8,aD9,aDA,aDB,\n aDC,aE1,aE2,aE3,aE4,aE5,aE6, aE7,aE8,aE9,aEA,aEB,aEC,aED,aF1,aF2,aF3,aF4,\n aF5,aF6,aF7,aF8,aF9,a FA,aFB,aFC,aFD,aFE,f1,f2,f3,f4,f5,f6,f7,f8,f9,fA,\n fB,fC,fD,fE,fF,b 1,b2,b3,b4,b5,b6,b7,b8,b9,bA,bB,bC,bD,bE,bF,xk,yk,t,jF,\n jM,jS,n,h, data,fn,xx,ys,saveDigits;\n options `Copyright 2009 by Peter Stone`; \n \n data := SOLN_;\n\n saveDigits := Digits;\n Digits := max (trunc(evalhf(Digits)),Digits+5);\n\n # procedure to evaluate the sl ope field\n fn := proc(X_,Y_)\n local val; \n val := trape rror(evalf(FXY_));\n if val=lasterror or not type(val,numeric) th en\n error \"evaluation of slope field failed at %1\",evalf([X _,Y_],saveDigits);\n end if;\n val;\n end proc;\n\n x x := evalf(x_rk9step);\n n := nops(data);\n\n if (data[1,1]data[n,1] or xxdata[1,1])) then\n error \"independent v ariable 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_; c 5 := c5_; c6 := c6_; c7 := c7_; c8 := c8_;\n c9 := c9_; cA := cA_; c B := cB_; cC := cC_; cD := cD_; cE := cE_; cF := cF_;\n a21 := c2; a 31 := a31_; a32 := a32_; a41 := a41_; a42 := a42_; a43 := a43_;\n a5 1 := 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 := \+ a94_; a95 := a95_; a96 := a96_; \n a97 := a97_; a98 := a98_; \n \+ aA1 := aA1_; aA2 := aA2_; aA3 := aA3_; aA4 := aA4_; aA5 := aA5_; aA6 : = aA6_; \n aA7 := aA7_; aA8 := aA8_; aA9 := aA9_;\n aB1 := aB1_; a B2 := aB2_; aB3 := aB3_; aB4 := aB4_; aB5 := aB5_; aB6 := aB6_; \n a B7 := aB7_; aB8 := aB8_; aB9 := aB9_; aBA := aBA_;\n aC1 := aC1_; aC 2 := aC2_; aC3 := aC3_; aC4 := aC4_; aC5 := aC5_; aC6 := aC6_; \n aC 7 := aC7_; aC8 := aC8_; aC9 := aC9_; aCA := aCA_; aCB := aCB_;\n aD1 := aD1_; aD2 := aD2_; aD3 := aD3_; aD4 := aD4_; aD5 := aD5_; aD6 := a D6_; \n aD7 := aD7_; aD8 := aD8_; aD9 := aD9_; aDA := aDA_; aDB := a DB_; aDC := aDC_;\n aE1 := aE1_; aE2 := aE2_; aE3 := aE3_; aE4 := aE 4_; aE5 := aE5_; aE6 := aE6_; \n aE7 := aE7_; aE8 := aE8_; aE9 := aE 9_; aEA := aEA_; aEB := aEB_; aEC := aEC_;\n aED := aED_;\n aF1 := aF1_; aF2 := aF2_; aF3 := aF3_; aF4 := aF4_; aF5 := aF5_; aF6 := aF6_ ; \n aF7 := aF7_; aF8 := aF8_; aF9 := aF9_; aFA := aFA_; aFB := aFB_ ; aFC := aFC_;\n aFD := aFD_; aFE := aFE_;\n b1 := b1_; b2 := b2_; b3 := b3_; b4 := b4_; b5 := b5_; b6 := b6_; b7 := b7_; \n b8 := b8_ ; b9 := b9_; bA := bA_; bB := bB_; bC := bC_; bD := bD_; bE := bE_;\n \+ bF := bF_;\n # Perform a binary search for the interval containing x.\n n := nops(data);\n jF := 0;\n jS := n+1;\n\n if data[1,1 ]1 do\n jM := trunc((jF+jS)/ 2);\n if xx>=data[jM,1] then jF := jM else jS := jM end if;\n \+ end do;\n if jM = n then jF := n-1; jS := n end if;\n else \n while jS-jF> 1 do\n jM := trunc((jF+jS)/2);\n if xx<=data[jM,1] then jF := jM else jS := jM end if;\n end do;\n \+ if jM = n then jF := n-1; jS := n end if;\n end if;\n \n # Ge t 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 := a 61*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 + a 84*f4 + a85*f5 + a86*f6 + a87*f7;\n f8 := fn(xk + c8*h,yk + t*h);\n \+ t := a91*f1 + a92*f2 + a93*f3 + a94*f4 + a95*f5 + a96*f6 + a97*f7 + \+ a98*f8;\n f9 := fn(xk + c9*h,yk + t*h);\n t := aA1*f1 + aA2*f2 + a A3*f3 + aA4*f4 + aA5*f5 + aA6*f6 + aA7*f7 + aA8*f8 + aA9*f9;\n fA := fn(xk + cA*h,yk + t*h);\n t := aB1*f1 + aB2*f2 + aB3*f3 + aB4*f4 + \+ aB5*f5 + aB6*f6 + aB7*f7 + aB8*f8 + aB9*f9\n \+ + aBA*fA;\n fB : = fn(xk + cB*h,yk + t*h);\n t := aC1*f1 + aC2*f2 + aC3*f3 + aC4*f4 + aC5*f5 + aC6*f6 + aC7*f7 + aC8*f8 + aC9*f9\n \+ + aCA*fA + aCB*fB;\n fC \+ := fn(xk + cC*h,yk + t*h); \n t := aD1*f1 + aD2*f2 + aD3*f3 + aD4* f4 + aD5*f5 + aD6*f6 + aD7*f7 + aD8*f8 + aD9*f9\n \+ + aDA*fA + aDB*fB + aDC*fC;\n \+ fD := fn(xk + cD*h,yk + t*h);\n\n t := aE1*f1 + aE2*f2 + aE3*f3 + a E4*f4 + aE5*f5 + aE6*f6 + aE7*f7 + aE8*f8 + aE9*f9\n \+ + aEA*fA + aEB*fB + aEC*fC + aED*fD; \n fE := fn(xk + cE*h,yk + t*h);\n t := aF1*f1 + aF2*f2 + aF3*f3 + aF4*f4 + aF5*f5 + aF6*f6 + aF7*f7 + aF8*f8 + aF9*f9\n \+ + aFA*fA + aFB*fB + aFC*fC + aFD*fD + aFE*fE ;\n fF := fn(xk + cF*h,yk + t*h);\n \n t := b1*f1 + b2*f2 + b3*f3 + b4*f4 + b5*f5 + b6*f6 + b7*f7 + b8*f8 + b9*f9 + bA*fA\n \+ + bB*fB + bC*fC + bD*fD + bE*fE + bF *fF;\n \n ys := yk + t*h;\n\n evalf[saveDigits](ys);\nend pro c: # of rk9_15step" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 37 " RK9_1 Verner's \"most robust\" scheme " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17954 "RK9_1 := proc(fxy,x, y,xx,yy,h,stps,bb)\n local c2,c3,c4,c5,c6,c7,c8,c9,cA,cB,cC,cD,cE,cF ,a21,a31,a32,a41,a42,a43,\n a51,a52,a53,a54,a61,a62,a63,a64,a65,a71, a72,a73,a74,a75,a76,a81,a82,a83,\n a84,a85,a86,a87,a91,a92,a93,a94,a 95,a96,a97,a98,aA1,aA2,aA3,aA4,aA5,aA6,\n aA7,aA8,aA9,aB1,aB2,aB3,aB 4,aB5,aB6,aB7,aB8,aB9,aBA,aC1,aC2,aC3,aC4,aC5,\n aC6,aC7,aC8,aC9,aCA ,aCB,aD1,aD2,aD3,aD4,aD5,aD6,aD7,aD8,aD9,aDA,aDB,aDC,\n aE1,aE2,aE3, aE4,aE5,aE6,aE7,aE8,aE9,aEA,aEB,aEC,aED,aF1,aF2,aF3,aF4,aF5,\n aF6,a F7,aF8,aF9,aFA,aFB,aFC,aFD,aFE,f1,f2,f3,f4,f5,f6,f7,f8,f9,fA,fB,fC,\n \+ fD,fE,fF,b1,b2,b3,b4,b5,b6,b7,b8,b9,bA,bB,bC,bD,bE,bF,t,k,fn,xk,yk,s oln,\n eqns,A,saveDigits;\n\n saveDigits := Digits;\n Digits := \+ max(trunc(evalhf(Digits)),Digits+5);\n\n fn := unapply(fxy,x,y);\n\n A := matrix([[1/25,1/25,0,0,0,0,0,0,0,0,0,0,0,0,0,0],\n[48/335-32/1 675*6^(1/2),-15792/112225+5536/112225*6^(1/2),31872/112225-1536/22445* 6^(1/2),0,0,0,0,0,0,0,0,0,0,0,0,0],\n[72/335-48/1675*6^(1/2),18/335-12 /1675*6^(1/2),0,54/335-36/1675*6^(1/2),0,0,0,0,0,0,0,0,0,0,0,0],\n[72/ 125,4014/3125+252/625*6^(1/2),0,-14742/3125-972/625*6^(1/2),12528/3125 +144/125*6^(1/2),0,0,0,0,0,0,0,0,0,0,0],\n[48/125-8/125*6^(1/2),1232/1 6875-152/16875*6^(1/2),0,0,29684/106875-13372/320625*6^(1/2),2132/6412 5-284/21375*6^(1/2),0,0,0,0,0,0,0,0,0,0],\n[48/125+8/125*6^(1/2),2032/ 16875+152/16875*6^(1/2),0,0,-7348/98325-33652/294975*6^(1/2),10132/641 25-716/21375*6^(1/2),2592/14375+2912/14375*6^(1/2),0,0,0,0,0,0,0,0,0], \n[16/25,16/225,0,0,0,0,64/225+4/225*6^(1/2),64/225-4/225*6^(1/2),0,0, 0,0,0,0,0,0],\n[12/25,57/800,0,0,0,0,177/800+69/1600*6^(1/2),177/800-6 9/1600*6^(1/2),-27/800,0,0,0,0,0,0,0],\n[3377/50000,284453082904607402 2657/58982400000000000000000,0,0,0,0,4287156859652598464203/5898240000 0000000000000-1598864762333658025459/117964800000000000000000*6^(1/2), 4287156859652598464203/58982400000000000000000+1598864762333658025459/ 117964800000000000000000*6^(1/2),-141033886218604337343/65536000000000 00000000,-21409264848554971927/204800000000000000000,0,0,0,0,0,0],\n[1 /4,-72189389771/9959178240000-459663572789/59755069440000*6^(1/2),0,0, 0,0,1/30,-14201240926266911/557169364500480000-31790792357660029/55716 9364500480000*6^(1/2),22414436941/1563197440000+459663572789/562751078 40000*6^(1/2),154180604903/2534154240000+459663572789/11403694080000*6 ^(1/2),21871487332435000000/125536952879579583419+18386542911560000000 /1129832575916216250771*6^(1/2),0,0,0,0,0],\n[57617028499878/850948278 71699,-178144571353393080183496267158614821877982611914666395752937745 405391408707734804982447062502773/124771801011299405474614551641042535 3598134947600568397156373491324203879120405304413829778240000+35259119 4575569317115651180991223097026568880384478484650944931412648046608828 531034562538513357/811016706573446135584994585666776479838787715940369 4581516427693607325214282634478689893558560000*6^(1/2),0,0,0,0,-391150 2254564577968858583198897514088250224516183112270320143559103652021094 5751850583137867/38425428798268102367071108589483043166777031255688218 331140907046999467997785653437677908480-490826700396287454540598331961 129757400186839154291202145347161815801053957874498846640625/153701715 1930724094682844343579321726671081250227528733245636281879978719911426 1375071163392*6^(1/2),-62206429668093251619352586547306026465246348663 5595667447855699920023546211607797417635623004817316459/90745170356705 9050763665262359998604536360756710836393258911136378360869155327677312 519401429106240000+643210041535328932923955834959360270277930780334485 030265105750796567439186095378077346484617022694073/181490340713411810 1527330524719997209072721513421672786517822272756721738310655354625038 802858212480000*6^(1/2),1203943546728385294644268186854769106596033156 989737459970836017592684437096940490927391123081521/654673243150492794 0789740125933173479539636700187880198168132038132534360122730704444373 657280000-151110511960958278763850506139095613011386663021919350564690 684891134877118069370443383945077153/327336621575246397039487006296658 6739769818350093940099084066019066267180061365352222186828640000*6^(1/ 2),6368146615670137844952590316092896179010043437749793620142714030331 6098275608362465676059840449/11905763493189349085278056977778076911485 0167606078444507680127107913165342297705956314829715000-35259119457556 9317115651180991223097026568880384478484650944931412648046608828531034 562538513357/154774925411461538108614740711114999849305217887901977859 9841652402871149449870177432092786295000*6^(1/2),102789340487632397056 6863593065627643827980253781207669195240625566060297608283559817962257 9836784640000000/20706607333558650004563853916074751077994608347553087 587301432465002027701979512351053114927376343490460197-247579462796834 0439266926692582942523578559086951126639651892181256353826702448565344 4025561440747520000000/26918589533626245005933010090897176401392990851 8190138634918622045026360125733660563690494055892465375982561*6^(1/2), 1963500009650946638084395645582909493284711388363243988292674544480982 2658280742880787456/15009933124323477487137151792766813737022277834253 210285601916815234167186635020874092933,0,0,0,0],\n[1623/2000,-2548851 1950948766602163761842966272037005387677568247/25343364340644945281003 771622773961523200000000000000+252560824194956338630896462443861744352 7/12416017897756354830807859200000000000000*6^(1/2),0,0,0,0,-261754608 1675469247418718340204655213/431392587109884206933606400000000000-5524 70350996365859393640759989/2400793528263703412500070400000*6^(1/2),-45 969294618407232267578352626581642155421231201/101877311541337815894061 92186163200000000000000+1540493536582818303738906021531546510663696889 /885889665576850572991842798796800000000000000*6^(1/2),203013873418014 401588800777489550153682185921397719/600378351748013431188109373696811 9910400000000000000-3968812951635028178485515838403541696971/183746280 07211630439078297600000000000000*6^(1/2),57340072791914637839492204156 400815228449519011392389/197442388396931261862836350912136101888000000 00000000-677602211254760908521917338264019314117/635714271434389910412 902400000000000000*6^(1/2),2410078871503919222575819785626174078680305 9875758180717/9662717001210818114953908042643919942065136939286292000- 8800028717594297513271653743688562521/20449362746897197076153911415852 248500*6^(1/2),124358916033523439225154730110040589545064737/199359124 49945669485002510178704400000000000,5947161392974866744750823561035920 29886330739685357945223223985387172161751824241784555387316314439/8180 1984711967835963172576910123831338658520874379559669405294358203115600 0597320370585600000000000,0,0,0],\n[453/500,18459267936147075367459423 9208189459317466722387929369596771/12789121709183065623336401420573165 5064571321600000000000000-293880952873230803935166416296767553377/8669 38749696854853908947200000000000000*6^(1/2),0,0,0,0,596699598636738018 1027852718263477/684582962942931480729600000000000+2897291244884828193 281565089/19049265055803310768128000000*6^(1/2),9900924061444672761159 4225452277068906413179/16167053833464253010532287404800000000000000-37 48619435624523040662096435375495632409181/1405830768127326348741938035 200000000000000*6^(1/2),41982993267604400562166630899538221911/1166358 22311401951029305600000000000000*6^(1/2)+46326458231132344862693578442 83603936661991916850927460433/5096189922215990577140952099757378096638 035200000000000000,-12135475677184688070559819501757154968443992507349 800863475059/333753408855726885777511338284367247352697475480000000000 0000+293880952873230803935166416296767553377/1654471223996727647705454 00000000000000*6^(1/2),-8031428603597180964147750321204281053557209400 385686643427859804352/251358644017839130690017816490908811374993398194 9109368563389778875+21495292553013453087829315020563569618432/29973653 446264566614372595657785433238875*6^(1/2),-448666321289612691585995958 25347949153517725172080973/5429607175821308141232283339099249860726338 769531250,-35998853393475254381336822594505813796150444067516777157056 3740579175484411998973149563531328527853237660249760109643/23412813744 5668240736067711662442379892483470931740696615716602345729933343308008 758732590194294958617150200000000000,101113680735918918122257191691468 8/2927578889427871359661497484969729,0,0],\n[1,42203349401687175255637 19481764309232553/5553268895030690235789996003491120640000*6^(1/2)-431 17938494612449223384237106139955697243762787772847383/1403010486229469 0648579566521727488445420965260989440000,0,0,0,0,-49217050443104424835 8505817476691843/26311016381549020957251979880325120-65372649360291914 372644744384375/457582893592156886213077910962176*6^(1/2),-60242025722 206083647640080390153021797730167/467187034609144869550624035596951557 6320000+312292306982904450161582808948983992497619279/5403119617653588 4913246084116864832317440000*6^(1/2),-42203349401687175255637194817643 09232553/5229862652007483519857107084128791040000*6^(1/2)-101284860197 54425336362855829794634260766582925227304979/2051440458733049774272501 566471676463052668060718080000,-42203349401687175255637194817643092325 53/1059789239915401287502749181781142480000*6^(1/2)+901643967752248253 17874747969895939573106537263617688583/9932194582195572820411492715910 530662743409752296080000,822220421784353922416106346105112190310210366 765138782108892460117760000000/112539961436930436130682455392076582934 435837857314380718907401253008522853-432162297873276674617724874932665 26541342720000000/26879922067906624005297106621487120800647611685231*6 ^(1/2),209814213871916216679569640811090009729840605120/11651974660960 738096987230622554824075229159217,394155089286395228118412271548581315 0214519858148244578543766661999079625453432169420046111603168071904019 820197053706993310801913/576348806945747835561749398160966945390916610 7652101824938575939967327573639769764715055303370220747306306384755484 89174498085584,-475944389207705069529277276691227549889712128000000000 0/4614026158486489304426198097242323507384603506591241137,387336141443 15448443538220691259283298308750000000/9380629116032422006554046539278 1147766752569954001,0],\n[0,100976787617015984669475787/69215029524032 62310437464576,0,0,0,0,0,0,8877148253451235588984375/43855140382087824 82187638272,961916572949681511747758515625/441641771550458751503680976 2176,1967337516701564001434375000000000000000000000000/154313648631198 54943071851131903908575429289017877,2323713252076974806855457536/10352 378514220126928031114081,283086002932414569543119391171389373911412640 5432515812108885979809005699950463699380263412267180656606126698463122 6363619/15839154654067008096806575700717389088587763113250985823298798 2213169380898029758612356596464561682203552186100842929847264,29613635 2341197653422080000000000000/3899432561650270968394778037550931439,117 048651891177050452812500000000/903958175807874008864483503817601,14577 8296653275182685983/4945417885871057962703934]]);\n\n c2 := evalf(A[ 1,1]);\n c3 := evalf(A[2,1]);\n c4 := evalf(A[3,1]);\n c5 := eva lf(A[4,1]);\n c6 := evalf(A[5,1]);\n c7 := evalf(A[6,1]);\n c8 : = evalf(A[7,1]);\n c9 := evalf(A[8,1]);\n cA := evalf(A[9,1]);\n \+ cB := evalf(A[10,1]);\n cC := evalf(A[11,1]);\n cD := evalf(A[12, 1]);\n cE := evalf(A[13,1]);\n cF := evalf(A[14,1]);\n a21 := c2 ;\n a31 := evalf(A[2,2]);\n a32 := evalf(A[2,3]);\n a41 := evalf (A[3,2]);\n a42 := evalf(A[3,3]);\n a43 := evalf(A[3,4]);\n a51 \+ := evalf(A[4,2]);\n a52 := evalf(A[4,3]);\n a53 := evalf(A[4,4]); \n a54 := evalf(A[4,5]);\n a61 := evalf(A[5,2]);\n a62 := evalf( A[5,3]);\n a63 := evalf(A[5,4]);\n a64 := evalf(A[5,5]);\n a65 : = evalf(A[5,6]);\n a71 := evalf(A[6,2]);\n a72 := evalf(A[6,3]);\n a73 := evalf(A[6,4]);\n a74 := evalf(A[6,5]);\n a75 := evalf(A[ 6,6]);\n a76 := evalf(A[6,7]);\n a81 := evalf(A[7,2]);\n a82 := \+ evalf(A[7,3]);\n a83 := evalf(A[7,4]);\n a84 := evalf(A[7,5]);\n \+ a85 := evalf(A[7,6]);\n a86 := evalf(A[7,7]);\n a87 := evalf(A[7, 8]);\n a91 := evalf(A[8,2]);\n a92 := evalf(A[8,3]);\n a93 := ev alf(A[8,4]);\n a94 := evalf(A[8,5]);\n a95 := evalf(A[8,6]);\n a 96 := 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 := eval f(A[9,4]);\n aA4 := evalf(A[9,5]);\n aA5 := evalf(A[9,6]);\n aA6 := evalf(A[9,7]);\n aA7 := evalf(A[9,8]);\n aA8 := evalf(A[9,9]); \n aA9 := evalf(A[9,10]);\n aB1 := evalf(A[10,2]);\n aB2 := eval f(A[10,3]);\n aB3 := evalf(A[10,4]);\n aB4 := evalf(A[10,5]);\n \+ aB5 := evalf(A[10,6]);\n aB6 := evalf(A[10,7]);\n aB7 := evalf(A[1 0,8]);\n aB8 := evalf(A[10,9]);\n aB9 := evalf(A[10,10]);\n aBA \+ := evalf(A[10,11]);\n aC1 := evalf(A[11,2]);\n aC2 := evalf(A[11,3 ]);\n aC3 := evalf(A[11,4]);\n aC4 := evalf(A[11,5]);\n aC5 := e valf(A[11,6]);\n aC6 := evalf(A[11,7]);\n aC7 := evalf(A[11,8]);\n aC8 := evalf(A[11,9]);\n aC9 := evalf(A[11,10]);\n aCA := evalf (A[11,11]);\n aCB := evalf(A[11,12]);\n aD1 := evalf(A[12,2]);\n \+ aD2 := evalf(A[12,3]);\n aD3 := evalf(A[12,4]);\n aD4 := evalf(A[ 12,5]);\n aD5 := evalf(A[12,6]);\n aD6 := evalf(A[12,7]);\n aD7 \+ := evalf(A[12,8]);\n aD8 := evalf(A[12,9]);\n aD9 := evalf(A[12,10 ]);\n aDA := evalf(A[12,11]);\n aDB := evalf(A[12,12]);\n aDC := evalf(A[12,13]);\n aE1 := evalf(A[13,2]);\n aE2 := evalf(A[13,3]) ;\n aE3 := evalf(A[13,4]);\n aE4 := evalf(A[13,5]);\n aE5 := eva lf(A[13,6]);\n aE6 := evalf(A[13,7]);\n aE7 := evalf(A[13,8]);\n \+ aE8 := evalf(A[13,9]);\n aE9 := evalf(A[13,10]);\n aEA := evalf(A [13,11]);\n aEB := evalf(A[13,12]);\n aEC := evalf(A[13,13]);\n \+ aED := evalf(A[13,14]);\n aF1 := evalf(A[14,2]);\n aF2 := evalf(A[ 14,3]);\n aF3 := evalf(A[14,4]);\n aF4 := evalf(A[14,5]);\n aF5 \+ := evalf(A[14,6]);\n aF6 := evalf(A[14,7]);\n aF7 := evalf(A[14,8] );\n aF8 := evalf(A[14,9]);\n aF9 := evalf(A[14,10]);\n aFA := e valf(A[14,11]);\n aFB := evalf(A[14,12]);\n aFC := evalf(A[14,13]) ;\n aFD := evalf(A[14,14]);\n aFE := evalf(A[14,15]);\n b1 := ev alf(A[15,2]);\n b2 := evalf(A[15,3]);\n b3 := evalf(A[15,4]);\n \+ b4 := evalf(A[15,5]);\n b5 := evalf(A[15,6]);\n b6 := evalf(A[15,7 ]);\n b7 := evalf(A[15,8]);\n b8 := evalf(A[15,9]);\n b9 := eval f(A[15,10]);\n bA := evalf(A[15,11]);\n bB := evalf(A[15,12]);\n \+ bC := evalf(A[15,13]);\n bD := evalf(A[15,14]);\n bE := evalf(A[1 5,15]);\n bF := evalf(A[15,16]);\n xk := evalf(xx);\n yk := eval f(yy);\n soln := [xk,yk]; \n for k from 1 to stps do\n f1 := \+ fn(xk,yk);\n t := a21*f1;\n f2 := fn(xk + c2*h,yk + t*h);\n \+ t := a31*f1 + a32*f2;\n f3 := fn(xk + c3*h,yk + t*h);\n \+ t := a41*f1 + a42*f2 + a43*f3;\n f4 := fn(xk + c4*h,yk + t*h);\n t := a51*f1 + a52*f2 + a53*f3 + a54*f4;\n f5 := fn(xk + c5* h,yk + t*h);\n t := a61*f1 + a62*f2 + a63*f3 + a64*f4 + a65*f5;\n f6 := fn(xk + c6*h,yk + t*h);\n t := a71*f1 + a72*f2 + a73* f3 + a74*f4 + a75*f5 + a76*f6;\n f7 := fn(xk + c7*h,yk + t*h);\n \+ t := a81*f1 + a82*f2 + a83*f3 + a84*f4 + a85*f5 + a86*f6 + a87*f7 ;\n f8 := fn(xk + c8*h,yk + t*h);\n t := a91*f1 + a92*f2 + a 93*f3 + a94*f4 + a95*f5 + a96*f6 + a97*f7 + a98*f8;\n f9 := fn(xk + c9*h,yk + t*h);\n t := aA1*f1 + aA2*f2 + aA3*f3 + aA4*f4 + aA5 *f5 + aA6*f6 + aA7*f7 + aA8*f8 + aA9*f9;\n fA := fn(xk + cA*h,yk \+ + t*h);\n t := aB1*f1 + aB2*f2 + aB3*f3 + aB4*f4 + aB5*f5 + aB6*f 6 + aB7*f7 + aB8*f8 + aB9*f9\n \+ + aBA*fA;\n fB := fn(xk \+ + cB*h,yk + t*h);\n t := aC1*f1 + aC2*f2 + aC3*f3 + aC4*f4 + aC5* f5 + aC6*f6 + aC7*f7 + aC8*f8 + aC9*f9\n \+ + aCA*fA + aCB*fB;\n fC := fn(xk + cC*h,yk + t*h);\n \n t := aD1*f1 + aD2*f2 + aD3*f3 \+ + aD4*f4 + aD5*f5 + aD6*f6 + aD7*f7 + aD8*f8 + aD9*f9\n \+ + aDA*fA + aDB*fB + aD C*fC;\n fD := fn(xk + cD*h,yk + t*h);\n\n t := aE1*f1 + aE2* f2 + aE3*f3 + aE4*f4 + aE5*f5 + aE6*f6 + aE7*f7 + aE8*f8 + aE9*f9\n \+ + aEA*fA + aEB*fB + aEC*fC + aED*fD;\n fE := fn(xk + cE*h,yk + t*h);\n t := aF1 *f1 + aF2*f2 + aF3*f3 + aF4*f4 + aF5*f5 + aF6*f6 + aF7*f7 + aF8*f8 + a F9*f9\n + aFA*fA + aFB*fB \+ + aFC*fC + aFD*fD + aFE*fE;\n fF := fn(xk + cF*h,yk + t*h);\n \+ \n t := b1*f1 + b2*f2 + b3*f3 + b4*f4 + b5*f5 + b6*f6 + b7*f7 + \+ b8*f8 + b9*f9 + bA*fA\n \+ + bB*fB + bC*fC + bD*fD + bE*fE + bF*fF;\n\n yk := yk + t*h;\n \+ xk := xk + h:\n soln := soln,[xk,yk];\n end do;\n if bb= true then\n eqns := \{SOLN_=[soln],FXY_=fxy,X_=x,Y_=y,c2_=c2,c3_= c3,\n c4_=c4,c5_=c5,c6_=c6,c7_=c7,c8_=c8,c9_=c9,cA_=cA,cB_=cB, cC_=cC,\n cD_=cD,cE_=cE,cF_=cF,a31_=a31,a32_=a32,a41_=a41,a42_ =a42,a43_=a43,\n a51_=a51,a52_=a52,a53_=a53,a54_=a54,a61_=a61, a62_=a62,a63_=a63,\n a64_=a64,a65_=a65,a71_=a71,a72_=a72,a73_= a73,a74_=a74,a75_=a75,\n a76_=a76,a81_=a81,a82_=a82,a83_=a83,a 84_=a84,a85_=a85,a86_=a86,\n a87_=a87,a91_=a91,a92_=a92,a93_=a 93,a94_=a94,a95_=a95,a96_=a96,\n a97_=a97,a98_=a98,aA1_=aA1,aA 2_=aA2,aA3_=aA3,aA4_=aA4,aA5_=aA5,\n aA6_=aA6,aA7_=aA7,aA8_=aA 8,aA9_=aA9,aB1_=aB1,aB2_=aB2,aB3_=aB3,\n aB4_=aB4,aB5_=aB5,aB6 _=aB6,aB7_=aB7,aB8_=aB8,aB9_=aB9,aBA_=aBA,\n aC1_=aC1,aC2_=aC2 ,aC3_=aC3,aC4_=aC4,aC5_=aC5,aC6_=aC6,aC7_=aC7,\n aC8_=aC8,aC9_ =aC9,aCA_=aCA,aCB_=aCB,aD1_=aD1,aD2_=aD2,aD3_=aD3,\n aD4_=aD4, aD5_=aD5,aD6_=aD6,aD7_=aD7,aD8_=aD8,aD9_=aD9,aDA_=aDA,\n aDB_= aDB,aDC_=aDC,aE1_=aE1,aE2_=aE2,aE3_=aE3,aE4_=aE4,aE5_=aE5,\n a E6_=aE6,aE7_=aE7,aE8_=aE8,aE9_=aE9,aEA_=aEA,aEB_=aEB,aEC_=aEC,\n \+ aED_=aED,aF1_=aF1,aF2_=aF2,aF3_=aF3,aF4_=aF4,aF5_=aF5,aF6_=aF6,\n \+ aF7_=aF7,aF8_=aF8,aF9_=aF9,aFA_=aFA,aFB_=aFB,aFC_=aFC,aFD_=aFD, \n aFE_=aFE,b1_=b1,b2_=b2,b3_=b3,b4_=b4,b5_=b5,b6_=b6,b7_=b7,b 8_=b8,\n b9_=b9,bA_=bA,bB_=bB,bC_=bC,bD_=bD,bE_=bE,bF_=bF\};\n return subs(eqns,eval(rk9_15step)); \n else\n return eval f[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 39 "RK9_2 Verner's \"most efficient\" sche me" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21408 "RK9_2 := proc(fxy,x,y,xx,yy,h,stps,bb)\n local c 2,c3,c4,c5,c6,c7,c8,c9,cA,cB,cC,cD,cE,cF,a21,a31,a32,a41,a42,a43,\n \+ a51,a52,a53,a54,a61,a62,a63,a64,a65,a71,a72,a73,a74,a75,a76,a81,a82,a8 3,\n a84,a85,a86,a87,a91,a92,a93,a94,a95,a96,a97,a98,aA1,aA2,aA3,aA4 ,aA5,aA6,\n aA7,aA8,aA9,aB1,aB2,aB3,aB4,aB5,aB6,aB7,aB8,aB9,aBA,aC1, aC2,aC3,aC4,aC5,\n aC6,aC7,aC8,aC9,aCA,aCB,aD1,aD2,aD3,aD4,aD5,aD6,a D7,aD8,aD9,aDA,aDB,aDC,\n aE1,aE2,aE3,aE4,aE5,aE6,aE7,aE8,aE9,aEA,aE B,aEC,aED,aF1,aF2,aF3,aF4,aF5,\n aF6,aF7,aF8,aF9,aFA,aFB,aFC,aFD,aFE ,f1,f2,f3,f4,f5,f6,f7,f8,f9,fA,fB,fC,\n fD,fE,fF,b1,b2,b3,b4,b5,b6,b 7,b8,b9,bA,bB,bC,bD,bE,bF,t,k,fn,xk,yk,soln,\n 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([[1731/50000,1 731/50000,0,0,0,0,0,0,0,0,0,0,0,0,0,0],\n[7630049/53810000-983539/5381 0000*6^(1/2),-177968356965557/1002427673820000+14180534491313/25060691 8455000*6^(1/2),64021741529527/200485534764000-7504450763411/100242767 382000*6^(1/2),0,0,0,0,0,0,0,0,0,0,0,0,0],\n[22890147/107620000-295061 7/107620000*6^(1/2),22890147/430480000-2950617/430480000*6^(1/2),0,686 70441/430480000-8851851/430480000*6^(1/2),0,0,0,0,0,0,0,0,0,0,0,0],\n[ 561/1000,592203994261020339/513126355505556250+730386990293623641/2052 505422022225000*6^(1/2),0,-8712153884182794903/2052505422022225000-284 3421359195851533/2052505422022225000*6^(1/2),1873698362223295443/51312 6355505556250+528258592225556973/513126355505556250*6^(1/2),0,0,0,0,0, 0,0,0,0,0,0],\n[387/1000-129/2000*6^(1/2),11380823631/157617812000-339 148869/39404453000*6^(1/2),0,0,16193232887091831/58864341808507450-235 5345717024309/58864341808507450*6^(1/2),165912282616977/41790752303080 00-33181894472511/2089537615154000*6^(1/2),0,0,0,0,0,0,0,0,0,0],\n[387 /1000+129/2000*6^(1/2),26523528363/231790900000+863255358/123138915625 *6^(1/2),0,0,-38208748178016484817787/842517966262441068418750-8611878 8556282369822807/842517966262441068418750*6^(1/2),92362336407446913/29 0322814529044000-232039320950012997/2467743923496874000*6^(1/2),-36292 5891/1690350537500+857800423623/3380701075000*6^(1/2),0,0,0,0,0,0,0,0, 0],\n[129/200,43/600,0,0,0,0,43/150+43/2400*6^(1/2),43/150-43/2400*6^( 1/2),0,0,0,0,0,0,0,0],\n[387/800,7353/102400,0,0,0,0,22833/102400+8901 /204800*6^(1/2),22833/102400-8901/204800*6^(1/2),-3483/102400,0,0,0,0, 0,0,0],\n[6757/100000,376708742472214988700853/77884560281250000000000 00,0,0,0,0,187914666753956840195279/2596152009375000000000000-21044084 6556290693268911/15576912056250000000000000*6^(1/2),187914666753956840 195279/2596152009375000000000000+210440846556290693268911/155769120562 50000000000000*6^(1/2),-18552667221896744226647/8653840031250000000000 00,-3167799860072183913409/30423656359863281250000,0,0,0,0,0,0],\n[1/4 ,-426968570497/54394415898750-92754382349/12087647977500*6^(1/2),0,0,0 ,0,1/30,-2865012129681958/114898584332330625-12962517687655099/2297971 68664661250*6^(1/2),4389715333607/309890657317500+92754382349/11477431 752500*6^(1/2),4990058173976/83757096376875+371017529396/9306344041875 *6^(1/2),1099523524595993125000/6257667909869756018891+100957348037989 687500/6257667909869756018891*6^(1/2),0,0,0,0,0],\n[142797165095125837 2/2166662646162554701,183820311047984038699385390091546565875214985735 9559506316407788280031537278728468323843947895514151799719800710862376 1931447163756/13974256944499724344918960993890933614161025322970450047 9326889980950085286208212396047346081112917694447061874978078691795508 41329375+4078857781851586092107938925175825953058964707564676126367962 59611491408260896413446883450891351622914818800693274034252252905536/2 8084926388601226073624096169175002956970191576455110633226765141161372 294098693275117181239385312198137508846535933127837167926875*6^(1/2),0 ,0,0,0,-33388131178984941197157347286812828143820221072172312325174214 5367734582887577395547778228760174068758086134389952015563403904/22708 7200460810303712768984860403962308663903544137293405018059381649379612 9405349914148981460714202232988727738778494557727635+48192728924777681 7137330866672068912142109195362579297027804407154995064019505647295552 3769829034800621890424847009130000000/23162894447002650978702436455761 2041554837181615020039273118420569282367205199345691243196108992848627 76485022935540644488821877*6^(1/2),-1366666074964636222701356088637720 7644362546879813948039042674099302480394698176320934836471610872131282 2619845726151693667598437699964416/37192864653424042747885853272541808 2819528242734205765019485563491782111356343287068137204351252040188714 1437067106105683944802332422369375+16984508556536133680555600929639437 4527636952379388961026066628725155521832762086875632366996477567928657 535912191396155566765457826139904/159397991371817326062367942596607749 7797978183146596135797795272107637620098614087434873732933937315094489 187314474045293119200999609586875*6^(1/2),-558747641349532341384649167 8323049250765705078855720721052003556321800113162964567765526724539063 327600257543743479921263738432/365303089362201664516413596925286161494 4735753371152962505117528597281088686969296140248032551227854032323598 17965288739565550625*6^(1/2)+56109878992732785254119605280814429021985 6759480976437975619567367326570055107681288392558337025376570255323559 4764427173637673766208/92881598198144033018278804740626334135423356791 6395981093588677703616092328460126267323324508442642938404565749560363 49633197336361875,-652617245096253774737270228028132152489434353210348 1802188740153783862532174342615150135214261625966637100811092384548036 046488576/864909328430372818360283879213205026685796531766248922845664 8746817034128576286937426571324705771222895418404433420637223081654437 5*6^(1/2)+545985398180836152335661486022032448966969589107343397540652 7098543350794516270773775946921467448080727221064814847749923878327625 9328/30124791909229885263488687512995931079466293201418449982714507585 1637298698312074030567479239502011693447423026416040794479934024058125 ,939166734840458401095542221032870712500612066161106190888975080561941 8785820948002455890360939221912190524731087070645107486913457760000000 /581572669687730206124190285037387083035152858549707256623268015312953 8726578484984317222364519327722935843448874220309127298193173915258478 3-81088251450850881043447210481663252251737294956893646964267201611120 12414227752328969720658987315654179873760357725235734000399440000000/2 6555829666106402106127410275679775481057208152954669252204018964061820 6693081506133206500662983001047298787619827411375675716583283801757*6^ (1/2),1234617126598879151772713393966068608104790287778693480148704506 06260914019560285661288212498128400476015695960341952/2816291066703206 7475424520935884070370423514730783889674107551122082605682904720561432 4978253226176275078922716132461,0,0,0,0],\n[4103/5000,2814045797346992 32141455524604487724159024972527/1478009944832743180452316204077188415 527343750000*6^(1/2)-5604277267532204213922762997804258633063362270605 3363946766144416933631/58808540772323190525590122613223430507352118534 557342666015625000000000,0,0,0,0,-102716390022975035656123823794722533 2675621517/179261894431132664078747698292867431640625000-2745292391641 202525373103979336813513372321/117022164684643403110606497445583859375 00000*6^(1/2),-1572299998537482273051657733644269252823780722383329301 21/36699907367985458573273204094330716033963413238525390625+5757606442 802795095318986067317837904184278650664590252101/352319110732660402303 4227593055748739260487670898437500000*6^(1/2),-93114481685939341460159 65019904013602133802943325818346622781285907057/4255970849010124217193 135449668739985401313363005576159362792968750000-844213739204097696424 366573813463172477074917581/421018835994657833697686816496616302490234 3750000*6^(1/2),885774233856672590222951867695327816457340130391639153 070521335485617578/301098541380295011015469248465465290112505656143757 799934635162353515625-281404579734699232141455524604487724159024972527 /284481916364737983221402322504830303192138671875*6^(1/2),315479116729 780153956412124052199685097744239386639023787359107959254802182/134481 850506505848012587842215515574380212543200894932329128471154748828125- 2940396453647872276646068776592292229737651937934623/73454650587819837 10795837429530784777245286520703125*6^(1/2),22509961634065453786165320 39018846586217631599453822541/3824913037970959935633041482042756364335 04028320312500,2689340957307691853294902388334454003959378146957529866 2335292519863593923360441517089497209588097479705143662934584242721740 24493/9595163860195788085005691147808717084668947522804828351054080278 15194895319055443842782227102120493960805649575561796875000000000,0,0, 0],\n[2253/2500,473420038480243914987079768476888930130830744411597794 65719863625051668939887702630319/4480254687392605073040122263665685576 0802419993852060264615320801485392456054687500000-86636953098707799112 5562402829092187100493209601/33255223758736721560177114591736739349365 23437500*6^(1/2),0,0,0,0,871779321807802447463310035318238762878527157 /134446420823349498059060773719650573730468750+10764126848099939608184 8975271849857994818/1097082793918531904161935913552348681640625*6^(1/2 ),496103786351862292800034805114190705484800743513354117014/1100997221 03956375719819612282992148101890239715576171875-1329938412606197485769 312599390307351191540891599374831099/660598332623738254318917673697952 888611341438293457031250*6^(1/2),1237670758552968558750803432612988838 71499029943/451091609994276250390378731960660324096679687500*6^(1/2)+4 0774077277747636354598451708891165494123131383777235229538611989392175 193285994266471/152642905462481621010589859415880795182567412553770317 36357946125713524703979492187500,-105220386085005564598286490383020684 73735749030796372764961618751973793724796364606986664/3899417425005422 2540345740003973828622358928296533758351973409182715560555076599121093 75+3465478123948311964502249611316368748401972838404/25603372472826418 48992620902543472728729248046875*6^(1/2),-2784376447126269318936520113 5620670490328475323282820219474851621693895769527094334687108984/12257 0410662851642220025943006055939294341391930221663178021214129993570247 04596261133984375+574774300271998598683873114105472016699241495055292/ 1049352151254569101542262489932969253892183788671875*6^(1/2),-34241134 351848245624232809437676889009431930503529853032576417589898516/561334 7824358651981100985009024281007603230062439942682713165283203125,-3432 0443758939323781023685680522865010338509105169992020885327052116334327 9392054770280096153243800840188373734185468897263960533460016393861026 8855705742764072609/11431741063416822609716476904105672921439261986509 2777892082326746111137127590759980171487016581339414751906821093176684 4494994616580258435518181434575195312500000,47469308760239193350794516 12726717649218264199984/1859206553840704975520014438813408934643275559 4877,0,0],\n[1,1234273058981860170179592598535508631343082535549881956 /2105633771469628744518390642968552144069898845895808125*6^(1/2)-25188 3292492588254437485270381424098799230121337389853132654309322802508557 08601/1137064132557446931205696187407729855082764230877464731699571703 6347558064286250,0,0,0,0,-54821142119685055562477216205428613949905430 396088/3959439837009461289085587746748097947393101278095-1511276753825 982856072891469504471256664975925000/403862863374965051486729950168305 99063409633036569*6^(1/2),-6092242427406159991860352404939065730543126 2635197540405697952/64848617474890321697745846247599531485315640324174 61909516875+8455857575163597873310996189398423878692955046261537569934 1616/19454585242467096509323753874279859445594692097252385728550625*6^ (1/2),-176324722711694310025656085505072661620440362221411708/28561940 6719829107485771207042040133465420149964555625*6^(1/2)-116118147575045 169733222875835719955334334798191459879782123534889390467935109772/881 0626901954835245672275131295870892503713957512170681453300814988417642 493125,-19748368943709762722873481576568138101489320568798111296/64845 54262322259071286545935997129135111813687175650625*6^(1/2)+17769448722 5138983422768374906650972869276072470733356185669871434672949001830332 16/2551217008137889615056342146084561867122485163596619283719957742418 751029506356875,976592661391240748181932648019295477816599265437863815 10190954184218570746215033823993530000000/1856007665446970620596348290 8787056850812308205603127326855360961727608242796551101182080033599-85 297084611782122474911131363078900058888025224607913745000000/692106594 50201393843166746722954036326338355649915383851733911*6^(1/2),47338974 9049752963256114649231353822492912259509649519870869750525/35412440882 360341799798842428365422941216508121322622479260846291,333514392451584 3824807349405678414409787291277341590453640072838769033456396839411470 2414108807505158106385116468732853458202899966748488718531545706559142 895903144848764637/231661102532728742771480201132225288609079390498990 0621592365627649097578102163572190502232425490606773312310665593424982 745744299371285598588298606088543376742054644818966,-38714992656958413 389743252726016897599283911682945255636643554687500000/485404949269715 87499294589382572212036169135429877901702347521300421767,1480025020094 0323717124616175641261235119295795768814717803955078125/33565577125141 877760287380588632421223433194078156948298488471160489,0],\n[0,8198160 366203173411119943711500331/561057579384085860167277847128765528,0,0,0 ,0,0,0,-455655493073428838813281446213740000000/1163808011150910561240 464225837312497869,19965163648706008081135075746915614720000000/863944 04190537086868394686205782432516544599,8923110791998141870556697080434 3750000000000000000000000/69997987098833567444559467985644506056259769 3583175985391,47104273954945906713184913871143492/20968463912233960193 4631113492763467,20845004421404500464010584740796750650832176798370383 0842263512947307311966736473110623309727407347372795031193876271463816 78677156136042524139311907482802844083/3667084989113637302023822532826 5100250605144718501926305140966586758054847604681466336103169284755987 753542321202462371554120593858149755539878561976786592389608,605303728 2142306509795911286909179687500000000/10389925735051806345529007757377 5162739725126989,917401104920993498360358406096725463867187500/6724249 815911346653315790737453607382989551463,258544955766526895137169959649 3957/84574345160764140163208606048427531]]);\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 := eva lf(A[7,1]);\n c9 := evalf(A[8,1]);\n cA := evalf(A[9,1]);\n cB : = evalf(A[10,1]);\n cC := evalf(A[11,1]);\n cD := evalf(A[12,1]); \n cE := evalf(A[13,1]);\n cF := evalf(A[14,1]);\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 := e valf(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 := eva lf(A[5,6]);\n a71 := evalf(A[6,2]);\n a72 := evalf(A[6,3]);\n a7 3 := evalf(A[6,4]);\n a74 := evalf(A[6,5]);\n a75 := evalf(A[6,6]) ;\n a76 := evalf(A[6,7]);\n a81 := evalf(A[7,2]);\n a82 := evalf (A[7,3]);\n a83 := evalf(A[7,4]);\n a84 := evalf(A[7,5]);\n a85 \+ := evalf(A[7,6]);\n a86 := evalf(A[7,7]);\n a87 := evalf(A[7,8]); \n a91 := evalf(A[8,2]);\n a92 := evalf(A[8,3]);\n a93 := evalf( A[8,4]);\n a94 := evalf(A[8,5]);\n a95 := evalf(A[8,6]);\n a96 : = evalf(A[8,7]);\n a97 := evalf(A[8,8]);\n a98 := evalf(A[8,9]);\n aA1 := evalf(A[9,2]);\n aA2 := evalf(A[9,3]);\n aA3 := evalf(A[ 9,4]);\n aA4 := evalf(A[9,5]);\n aA5 := evalf(A[9,6]);\n aA6 := \+ evalf(A[9,7]);\n aA7 := evalf(A[9,8]);\n aA8 := evalf(A[9,9]);\n \+ aA9 := evalf(A[9,10]);\n aB1 := evalf(A[10,2]);\n aB2 := evalf(A[ 10,3]);\n aB3 := evalf(A[10,4]);\n aB4 := evalf(A[10,5]);\n aB5 \+ := evalf(A[10,6]);\n aB6 := evalf(A[10,7]);\n aB7 := evalf(A[10,8] );\n aB8 := evalf(A[10,9]);\n aB9 := evalf(A[10,10]);\n aBA := e valf(A[10,11]);\n aC1 := evalf(A[11,2]);\n aC2 := evalf(A[11,3]); \n aC3 := evalf(A[11,4]);\n aC4 := evalf(A[11,5]);\n aC5 := eval f(A[11,6]);\n aC6 := evalf(A[11,7]);\n aC7 := evalf(A[11,8]);\n \+ aC8 := evalf(A[11,9]);\n aC9 := evalf(A[11,10]);\n aCA := evalf(A[ 11,11]);\n aCB := evalf(A[11,12]);\n aD1 := evalf(A[12,2]);\n aD 2 := evalf(A[12,3]);\n aD3 := evalf(A[12,4]);\n aD4 := evalf(A[12, 5]);\n aD5 := evalf(A[12,6]);\n aD6 := evalf(A[12,7]);\n aD7 := \+ evalf(A[12,8]);\n aD8 := evalf(A[12,9]);\n aD9 := evalf(A[12,10]); \n aDA := evalf(A[12,11]);\n aDB := evalf(A[12,12]);\n aDC := ev alf(A[12,13]);\n aE1 := evalf(A[13,2]);\n aE2 := evalf(A[13,3]);\n aE3 := evalf(A[13,4]);\n aE4 := evalf(A[13,5]);\n aE5 := evalf( A[13,6]);\n aE6 := evalf(A[13,7]);\n aE7 := evalf(A[13,8]);\n aE 8 := evalf(A[13,9]);\n aE9 := evalf(A[13,10]);\n aEA := evalf(A[13 ,11]);\n aEB := evalf(A[13,12]);\n aEC := evalf(A[13,13]);\n aED := evalf(A[13,14]);\n aF1 := evalf(A[14,2]);\n aF2 := evalf(A[14, 3]);\n aF3 := evalf(A[14,4]);\n aF4 := evalf(A[14,5]);\n aF5 := \+ evalf(A[14,6]);\n aF6 := evalf(A[14,7]);\n aF7 := evalf(A[14,8]); \n aF8 := evalf(A[14,9]);\n aF9 := evalf(A[14,10]);\n aFA := eva lf(A[14,11]);\n aFB := evalf(A[14,12]);\n aFC := evalf(A[14,13]); \n aFD := evalf(A[14,14]);\n aFE := evalf(A[14,15]);\n b1 := eva lf(A[15,2]);\n b2 := evalf(A[15,3]);\n b3 := evalf(A[15,4]);\n b 4 := evalf(A[15,5]);\n b5 := evalf(A[15,6]);\n b6 := evalf(A[15,7] );\n b7 := evalf(A[15,8]);\n b8 := evalf(A[15,9]);\n b9 := evalf (A[15,10]);\n bA := evalf(A[15,11]);\n bB := evalf(A[15,12]);\n \+ bC := evalf(A[15,13]);\n bD := evalf(A[15,14]);\n bE := evalf(A[15 ,15]);\n bF := evalf(A[15,16]);\n xk := evalf(xx);\n yk := evalf (yy);\n soln := [xk,yk]; \n for k from 1 to stps do\n f1 := f n(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*f 3 + 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 + a9 3*f3 + a94*f4 + a95*f5 + a96*f6 + a97*f7 + a98*f8;\n f9 := fn(xk \+ + c9*h,yk + t*h);\n t := aA1*f1 + aA2*f2 + aA3*f3 + aA4*f4 + aA5* f5 + aA6*f6 + aA7*f7 + aA8*f8 + aA9*f9;\n fA := fn(xk + cA*h,yk + t*h);\n t := aB1*f1 + aB2*f2 + aB3*f3 + aB4*f4 + aB5*f5 + aB6*f6 + aB7*f7 + aB8*f8 + aB9*f9\n \+ + aBA*fA;\n fB := fn(xk + cB*h,yk + t*h);\n t := aC1*f1 + aC2*f2 + aC3*f3 + aC4*f4 + aC5*f 5 + aC6*f6 + aC7*f7 + aC8*f8 + aC9*f9\n \+ + aCA*fA + aCB*fB;\n fC \+ := fn(xk + cC*h,yk + t*h);\n \n t := aD1*f1 + aD2*f2 + aD3*f3 + aD4*f4 + aD5*f5 + aD6*f6 + aD7*f7 + aD8*f8 + aD9*f9\n \+ + aDA*fA + aDB*fB + aDC *fC;\n fD := fn(xk + cD*h,yk + t*h);\n\n t := aE1*f1 + aE2*f 2 + aE3*f3 + aE4*f4 + aE5*f5 + aE6*f6 + aE7*f7 + aE8*f8 + aE9*f9\n \+ + aEA*fA + aEB*fB + \+ aEC*fC + aED*fD;\n fE := fn(xk + cE*h,yk + t*h);\n t := aF1* f1 + aF2*f2 + aF3*f3 + aF4*f4 + aF5*f5 + aF6*f6 + aF7*f7 + aF8*f8 + aF 9*f9\n + aFA*fA + aFB*fB + aFC*fC + aFD*fD + aFE*fE;\n fF := fn(xk + cF*h,yk + t*h);\n \+ \n t := b1*f1 + b2*f2 + b3*f3 + b4*f4 + b5*f5 + b6*f6 + b7*f7 + b 8*f8 + b9*f9 + bA*fA\n \+ + bB*fB + bC*fC + bD*fD + bE*fE + bF*fF;\n\n yk := yk + t*h;\n \+ xk := xk + h:\n soln := soln,[xk,yk];\n end do;\n if bb=t rue then\n eqns := \{SOLN_=[soln],FXY_=fxy,X_=x,Y_=y,c2_=c2,c3_=c 3,\n c4_=c4,c5_=c5,c6_=c6,c7_=c7,c8_=c8,c9_=c9,cA_=cA,cB_=cB,c C_=cC,\n cD_=cD,cE_=cE,cF_=cF,a31_=a31,a32_=a32,a41_=a41,a42_= a42,a43_=a43,\n a51_=a51,a52_=a52,a53_=a53,a54_=a54,a61_=a61,a 62_=a62,a63_=a63,\n a64_=a64,a65_=a65,a71_=a71,a72_=a72,a73_=a 73,a74_=a74,a75_=a75,\n a76_=a76,a81_=a81,a82_=a82,a83_=a83,a8 4_=a84,a85_=a85,a86_=a86,\n a87_=a87,a91_=a91,a92_=a92,a93_=a9 3,a94_=a94,a95_=a95,a96_=a96,\n a97_=a97,a98_=a98,aA1_=aA1,aA2 _=aA2,aA3_=aA3,aA4_=aA4,aA5_=aA5,\n aA6_=aA6,aA7_=aA7,aA8_=aA8 ,aA9_=aA9,aB1_=aB1,aB2_=aB2,aB3_=aB3,\n aB4_=aB4,aB5_=aB5,aB6_ =aB6,aB7_=aB7,aB8_=aB8,aB9_=aB9,aBA_=aBA,\n aC1_=aC1,aC2_=aC2, aC3_=aC3,aC4_=aC4,aC5_=aC5,aC6_=aC6,aC7_=aC7,\n aC8_=aC8,aC9_= aC9,aCA_=aCA,aCB_=aCB,aD1_=aD1,aD2_=aD2,aD3_=aD3,\n aD4_=aD4,a D5_=aD5,aD6_=aD6,aD7_=aD7,aD8_=aD8,aD9_=aD9,aDA_=aDA,\n aDB_=a DB,aDC_=aDC,aE1_=aE1,aE2_=aE2,aE3_=aE3,aE4_=aE4,aE5_=aE5,\n aE 6_=aE6,aE7_=aE7,aE8_=aE8,aE9_=aE9,aEA_=aEA,aEB_=aEB,aEC_=aEC,\n \+ aED_=aED,aF1_=aF1,aF2_=aF2,aF3_=aF3,aF4_=aF4,aF5_=aF5,aF6_=aF6,\n \+ aF7_=aF7,aF8_=aF8,aF9_=aF9,aFA_=aFA,aFB_=aFB,aFC_=aFC,aFD_=aFD, \n aFE_=aFE,b1_=b1,b2_=b2,b3_=b3,b4_=b4,b5_=b5,b6_=b6,b7_=b7,b 8_=b8,\n b9_=b9,bA_=bA,bB_=bB,bC_=bC,bD_=bD,bE_=bE,bF_=bF\};\n return subs(eqns,eval(rk9_15step)); \n else\n return eval f[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 21 "RK9_3 Sharp's scheme" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17777 "RK9_3 : = proc(fxy,x,y,xx,yy,h,stps,bb)\n local c2,c3,c4,c5,c6,c7,c8,c9,cA,c B,cC,cD,cE,cF,a21,a31,a32,a41,a42,a43,\n a51,a52,a53,a54,a61,a62,a63 ,a64,a65,a71,a72,a73,a74,a75,a76,a81,a82,a83,\n a84,a85,a86,a87,a91, a92,a93,a94,a95,a96,a97,a98,aA1,aA2,aA3,aA4,aA5,aA6,\n aA7,aA8,aA9,a B1,aB2,aB3,aB4,aB5,aB6,aB7,aB8,aB9,aBA,aC1,aC2,aC3,aC4,aC5,\n aC6,aC 7,aC8,aC9,aCA,aCB,aD1,aD2,aD3,aD4,aD5,aD6,aD7,aD8,aD9,aDA,aDB,aDC,\n \+ aE1,aE2,aE3,aE4,aE5,aE6,aE7,aE8,aE9,aEA,aEB,aEC,aED,aF1,aF2,aF3,aF4,a F5,\n aF6,aF7,aF8,aF9,aFA,aFB,aFC,aFD,aFE,f1,f2,f3,f4,f5,f6,f7,f8,f9 ,fA,fB,fC,\n fD,fE,fF,b1,b2,b3,b4,b5,b6,b7,b8,b9,bA,bB,bC,bD,bE,bF,t ,k,fn,xk,yk,soln,\n eqns,A,saveDigits;\n\n saveDigits := Digits;\n Digits := max(trunc(evalhf(Digits)),Digits+5);\n\n fn := unapply( fxy,x,y);\n\n A := matrix([[1/50,1/50,0,0,0,0,0,0,0,0,0,0,0,0,0,0], \n[3837236/48429375+1031368/145288125*6^(1/2),-24000387317036/28144852 3546875-5917264532296/281448523546875*6^(1/2),46300580261936/281448523 546875+7915204837696/281448523546875*6^(1/2),0,0,0,0,0,0,0,0,0,0,0,0,0 ],\n[1918618/16143125+515684/48429375*6^(1/2),959309/32286250+128921/4 8429375*6^(1/2),0,2877927/32286250+128921/16143125*6^(1/2),0,0,0,0,0,0 ,0,0,0,0,0,0],\n[14/45,2826523628723851/5953434698904030-6845949231747 5/595343469890403*6^(1/2),0,-704240024458145/396895646593602+912775308 07085/198447823296801*6^(1/2),958925642225180/595343469890403-20537310 0103780/595343469890403*6^(1/2),0,0,0,0,0,0,0,0,0,0,0],\n[156/625+26/6 25*6^(1/2),376341108/9406484375+207933466/65845390625*6^(1/2),0,0,4343 545768844529/27892881885795625+469265141246109/27892881885795625*6^(1/ 2),1559927818449/28957835234375+4382126882523/202704846640625*6^(1/2), 0,0,0,0,0,0,0,0,0,0],\n[156/625-26/625*6^(1/2),11781705468/23516210937 5+2328587014/1646134765625*6^(1/2),0,0,23459106068523828440829/3542988 72323611753203125+7870375504052283205581/354298872323611753203125*6^(1 /2),146263465360621089/7558718942052734375-1881455818308499953/5291103 2594369140625*6^(1/2),9444124356888/82889304453125-2459298027368/82889 304453125*6^(1/2),0,0,0,0,0,0,0,0,0],\n[52/125,52/1125,0,0,0,0,208/112 5-13/1125*6^(1/2),208/1125+13/1125*6^(1/2),0,0,0,0,0,0,0,0],\n[39/125, 741/16000,0,0,0,0,2301/16000-897/32000*6^(1/2),2301/16000+897/32000*6^ (1/2),-351/16000,0,0,0,0,0,0,0],\n[21/200,35291978967/748709478400,0,0 ,0,0,23154511989/149741895680+39398793/1772093440*6^(1/2),23154511989/ 149741895680-39398793/1772093440*6^(1/2),-6251205429/149741895680,-981 041103/4679434240,0,0,0,0,0,0],\n[280/477,1601589807329134144752443/16 639785968494158002257920,0,0,0,0,-1736562342312744743536201/1109319064 566277200150528-360257484908262597335743/511993414415204861607936*6^(1 /2),-1736562342312744743536201/1109319064566277200150528+3602574849082 62597335743/511993414415204861607936*6^(1/2),512032742176678555764127/ 369773021522092400050176,248233526294563631278471/10399866230308848751 4112,-3/20,0,0,0,0,0],\n[3658227035053715/5349704719299032,-1319870176 0878669635722542338759463561271938920612860688067043417832133196962788 9057541436355642743061150672386594396559/31875392608799555501514792620 1612010240447228295789462486798116221476093939683123897279961564118214 685494052121351290880,0,0,0,0,-581038619225160876203856834629458675128 9267051434651924507164484661690757973591786160210452910809721214291885 43592047/1011917225676176365127453733973371461080784851732664960275549 575306273314094232139356444322425772110112679530543972352+713482798078 9881496508823372973790665635109665954131159198195323855274480577648002 8282602780942939988708855996996031/51893191060316736673202755588378023 6451684539350084595013102346310909391843195968900740678167062620570604 88745844736*6^(1/2),-5810386192251608762038568346294586751289267051434 65192450716448466169075797359178616021045291080972121429188543592047/1 0119172256761763651274537339733714610807848517326649602755495753062733 14094232139356444322425772110112679530543972352-7134827980789881496508 8233729737906656351096659541311591981953238552744805776480028282602780 942939988708855996996031/518931910603167366732027555883780236451684539 3500845950131023463109093918431959689007406781670626205706048874584473 6*6^(1/2),-18935700826260772432168308633651734522837925089710329104904 4350530935228180690663776657891613652665009511679250229667441/10490208 5728430283184879370421906174798708029629619600881898639306750333561102 065113284728091471708748347777999725133824,-16183509927928156539922841 5225411182739942653401484724580110184517256730426918980054437210005086 9595166981551925667441/19637518660778297585754649024920739916598981028 9370293853473839457873665016411924543859976320751400118741871396189634 56,6883437842714982754414155283530543027800010156600147069119889350771 791431366439329656536871565378282089012991331513/182718148955179478466 9860898707808352423218653885642080180242252960011545073999200946066370 836641132319880849653760,115590271440716912566235566233889746097162479 804636463234298604185457969653794053637008425503953091180886565/315361 3332490718363304114648792457541639644686569637672840787260046309750159 21697525870414526347596936773632,0,0,0,0],\n[247/281,52151747835589184 07997583468635543407988332719241764605769949554629/2028313261321481206 4685094275151111714651171227532533713038580121600,0,0,0,0,-18227070890 226867447840942666790512323422585544257/121857700488461867579360666965 049340968208464609280*6^(1/2)+2843598186227456480865065344408178581293 412110128603/792075053175002139265844335272820716293355019960320,18227 070890226867447840942666790512323422585544257/121857700488461867579360 666965049340968208464609280*6^(1/2)+2843598186227456480865065344408178 581293412110128603/792075053175002139265844335272820716293355019960320 ,932682946442206211824845748135153950427533947675946704732660559568563 3/49019017918582288638576910410293096780103555478957212859191772630220 80,-741604155090542466856213236072374206251235617068304762316465738169 791/141551673163321136844445993892555326037025917405403892742525852712 960,-60585048664412196555955486187624853999747736853070460011793555360 03/2252275720815396172726400694965157641073696835574259179818290290400 ,-72917047186465183128180555150230405657138451692847535142343993/44661 747288016218276854771442831738093234145203222656783563600,273615392054 0927643774133147635296486946660915558253285983742020488887296849241173 151960647763453239551016003889152/248567211069834101529026447046393920 3955869249618375406787169018009688457749866177826801192710345262847046 284166825,0,0,0],\n[229/250,196143162589031568706314157581823240552254 5898155499982338718373117379429883/48005664716707742999059356805540609 3586176318669944422673481728000000000000,0,0,0,0,-86885256061463155300 22414580346392155721271039/2238673811875443318181460760248117624832000 0000-10256190098435854298655077997613296122112139953/11480378522438170 86246902953973393653760000000*6^(1/2),-8688525606146315530022414580346 392155721271039/22386738118754433181814607602481176248320000000+102561 90098435854298655077997613296122112139953/1148037852243817086246902953 973393653760000000*6^(1/2),-108151392092290424953498836380059772609736 403739434481043071361807712075869481/860073549519456344831626147833135 3993230137106756553600705167485829120000000,68321055425793546260025797 5958139742203919396113084127371502375524416129719/26895337200565243662 247103690698994332502640106760065066162305761280000000,-12597103405120 3704183074450363446847441594334546885083244594242327104115033/50660499 34698363488698655054901069679084758735799331062593807151200000000,4322 338495495152743252505005837177994220267688026960252214552638944423/236 867625787508422152958167179676757535142999000924357630500000000000,-88 6824143941836194254416478662433881129172892391614639409444929304921125 47171652363240146123589908870567811533658125375935101390832/9405104776 2301760672023836894444996842386398230976500899543333833779587704919990 76242219980453370248022825420814384818872314453125,2623547564198662518 7247554297838197168935151270802587/31781620957198174033817415268740604 591106877500000000,0,0],\n[1,-2933688768685553737193922190442902414638 569907165819426999847151894747/142396785481313780235051679506520125893 0107696470226170813903745843200,0,0,0,0,590178041984076152291792832462 29064921710388893173/1798714516716377337062141465912811283284194492416 0*6^(1/2)-279050827135618188106138704976571118076242172562777/26980717 750745660055932121988692169249262917386240,-59017804198407615229179283 246229064921710388893173/179871451671637733706214146591281128328419449 24160*6^(1/2)-279050827135618188106138704976571118076242172562777/2698 0717750745660055932121988692169249262917386240,68240477823918559060550 996013166770535743446467404475965020846786328901/696286259008227753162 47857716283393138506964199995817446989013471723520,4853160486533574344 0838806675493568975092395234916265724406574203650554879/75290755690494 50724715447951105730327391304314424323559158913160761835520,7315898198 049114373691779027237206235234893868747090317226910860963581499/432400 219379131684183655517400956805867024783959757582533074834771793600,-87 035912584683752124645187592152267644073875904388006117245587111831/414 68532532723053663401983927439573730970639521941633396843682248800,7930 0605432804165106136013125641247440025308990955400537833272821480610899 5212138291759017448087224471716436232175864384424753159293287828190208 /600635305507048430170531323174007915813446499853348186301444535722932 5524440717897755775080839552802580708447805203719870411490533416177042 19325,1901323869288778426716498142786763035626208187060094642270136445 8/146516308633144198110735805762400905606463733191840985648075179899,3 68176545506575596342007241113258886329861009608750000000/7515329389098 801941975451526298754679007062667248055263091,0],\n[0,3070384338936194 6002220520407/1036329015084155723633962896000,0,0,0,0,0,0,151668188891 3470906364013671875/19423768214582439936604117641536,19299227379984705 73359614532470703125/9295447834009061726737853188569292704,27072397368 129209968072433152000000000/159540891067276798629433718421290211669,34 16676287738448149119878197304164096817920457/2252175244121156627053678 6917243920830369456000,90903490074941164563143999126052497791688659150 2548355130330148829066896764151555292038222333366816993556860935646735 988456500531298304/630197874918897931765938035588221137118814650606654 3226107226217493443986031316306450151922600620534579104501042337690306 078523205079625,9160897746149204383653282352747804858423571/5493411900 2888850773584011583391921191449440,3769686146953412690297035156250000/ 195792979665408643382362918863397227,50782110772148063247179059/153826 6148871578545201811280]]);\n\n c2 := evalf(A[1,1]);\n c3 := evalf( A[2,1]);\n c4 := evalf(A[3,1]);\n c5 := evalf(A[4,1]);\n c6 := e valf(A[5,1]);\n c7 := evalf(A[6,1]);\n c8 := evalf(A[7,1]);\n c9 := evalf(A[8,1]);\n cA := evalf(A[9,1]);\n cB := evalf(A[10,1]); \n cC := evalf(A[11,1]);\n cD := evalf(A[12,1]);\n cE := evalf(A [13,1]);\n cF := evalf(A[14,1]);\n a21 := c2;\n a31 := evalf(A[2 ,2]);\n a32 := evalf(A[2,3]);\n a41 := evalf(A[3,2]);\n a42 := e valf(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 := eva lf(A[5,4]);\n a64 := evalf(A[5,5]);\n a65 := evalf(A[5,6]);\n a7 1 := evalf(A[6,2]);\n a72 := evalf(A[6,3]);\n a73 := evalf(A[6,4]) ;\n a74 := evalf(A[6,5]);\n a75 := evalf(A[6,6]);\n a76 := evalf (A[6,7]);\n a81 := evalf(A[7,2]);\n a82 := evalf(A[7,3]);\n a83 \+ := evalf(A[7,4]);\n a84 := evalf(A[7,5]);\n a85 := evalf(A[7,6]); \n a86 := evalf(A[7,7]);\n a87 := evalf(A[7,8]);\n a91 := evalf( A[8,2]);\n a92 := evalf(A[8,3]);\n a93 := evalf(A[8,4]);\n a94 : = evalf(A[8,5]);\n a95 := evalf(A[8,6]);\n a96 := evalf(A[8,7]);\n a97 := evalf(A[8,8]);\n a98 := evalf(A[8,9]);\n aA1 := evalf(A[ 9,2]);\n aA2 := evalf(A[9,3]);\n aA3 := evalf(A[9,4]);\n aA4 := \+ evalf(A[9,5]);\n aA5 := evalf(A[9,6]);\n aA6 := evalf(A[9,7]);\n \+ aA7 := evalf(A[9,8]);\n aA8 := evalf(A[9,9]);\n aA9 := evalf(A[9, 10]);\n aB1 := evalf(A[10,2]);\n aB2 := evalf(A[10,3]);\n aB3 := evalf(A[10,4]);\n aB4 := evalf(A[10,5]);\n aB5 := evalf(A[10,6]); \n aB6 := evalf(A[10,7]);\n aB7 := evalf(A[10,8]);\n aB8 := eval f(A[10,9]);\n aB9 := evalf(A[10,10]);\n aBA := evalf(A[10,11]);\n \+ aC1 := evalf(A[11,2]);\n aC2 := evalf(A[11,3]);\n aC3 := evalf(A [11,4]);\n aC4 := evalf(A[11,5]);\n aC5 := evalf(A[11,6]);\n aC6 := evalf(A[11,7]);\n aC7 := evalf(A[11,8]);\n aC8 := evalf(A[11,9 ]);\n aC9 := evalf(A[11,10]);\n aCA := evalf(A[11,11]);\n aCB := evalf(A[11,12]);\n aD1 := evalf(A[12,2]);\n aD2 := evalf(A[12,3]) ;\n aD3 := evalf(A[12,4]);\n aD4 := evalf(A[12,5]);\n aD5 := eva lf(A[12,6]);\n aD6 := evalf(A[12,7]);\n aD7 := evalf(A[12,8]);\n \+ aD8 := evalf(A[12,9]);\n aD9 := evalf(A[12,10]);\n aDA := evalf(A [12,11]);\n aDB := evalf(A[12,12]);\n aDC := evalf(A[12,13]);\n \+ aE1 := evalf(A[13,2]);\n aE2 := evalf(A[13,3]);\n aE3 := evalf(A[1 3,4]);\n aE4 := evalf(A[13,5]);\n aE5 := evalf(A[13,6]);\n aE6 : = evalf(A[13,7]);\n aE7 := evalf(A[13,8]);\n aE8 := evalf(A[13,9]) ;\n aE9 := evalf(A[13,10]);\n aEA := evalf(A[13,11]);\n aEB := e valf(A[13,12]);\n aEC := evalf(A[13,13]);\n aED := evalf(A[13,14]) ;\n aF1 := evalf(A[14,2]);\n aF2 := evalf(A[14,3]);\n aF3 := eva lf(A[14,4]);\n aF4 := evalf(A[14,5]);\n aF5 := evalf(A[14,6]);\n \+ aF6 := evalf(A[14,7]);\n aF7 := evalf(A[14,8]);\n aF8 := evalf(A[ 14,9]);\n aF9 := evalf(A[14,10]);\n aFA := evalf(A[14,11]);\n aF B := evalf(A[14,12]);\n aFC := evalf(A[14,13]);\n aFD := evalf(A[1 4,14]);\n aFE := evalf(A[14,15]);\n b1 := evalf(A[15,2]);\n b2 : = evalf(A[15,3]);\n b3 := evalf(A[15,4]);\n b4 := evalf(A[15,5]); \n b5 := evalf(A[15,6]);\n b6 := evalf(A[15,7]);\n b7 := evalf(A [15,8]);\n b8 := evalf(A[15,9]);\n b9 := evalf(A[15,10]);\n bA : = evalf(A[15,11]);\n bB := evalf(A[15,12]);\n bC := evalf(A[15,13] );\n bD := evalf(A[15,14]);\n bE := evalf(A[15,15]);\n bF := eva lf(A[15,16]);\n xk := evalf(xx);\n yk := evalf(yy);\n soln := [x k,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 + a3 2*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 + a 52*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 + c 6*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 + a8 2*f2 + a83*f3 + a84*f4 + a85*f5 + a86*f6 + a87*f7;\n f8 := fn(xk \+ + c8*h,yk + t*h);\n t := a91*f1 + a92*f2 + a93*f3 + a94*f4 + a95* f5 + a96*f6 + a97*f7 + a98*f8;\n f9 := fn(xk + c9*h,yk + t*h);\n \+ t := aA1*f1 + aA2*f2 + aA3*f3 + aA4*f4 + aA5*f5 + aA6*f6 + aA7*f7 + aA8*f8 + aA9*f9;\n fA := fn(xk + cA*h,yk + t*h);\n t := a B1*f1 + aB2*f2 + aB3*f3 + aB4*f4 + aB5*f5 + aB6*f6 + aB7*f7 + aB8*f8 + aB9*f9\n \+ + aBA*fA;\n fB := fn(xk + cB*h,yk + t*h);\n \+ t := aC1*f1 + aC2*f2 + aC3*f3 + aC4*f4 + aC5*f5 + aC6*f6 + aC7*f7 \+ + aC8*f8 + aC9*f9\n \+ + aCA*fA + aCB*fB;\n fC := fn(xk + cC*h,yk + t*h);\n \n t := aD1*f1 + aD2*f2 + aD3*f3 + aD4*f4 + aD5*f5 + a D6*f6 + aD7*f7 + aD8*f8 + aD9*f9\n \+ + aDA*fA + aDB*fB + aDC*fC;\n fD := fn (xk + cD*h,yk + t*h);\n\n t := aE1*f1 + aE2*f2 + aE3*f3 + aE4*f4 \+ + aE5*f5 + aE6*f6 + aE7*f7 + aE8*f8 + aE9*f9\n \+ + aEA*fA + aEB*fB + aEC*fC + aED*fD;\n \+ fE := fn(xk + cE*h,yk + t*h);\n t := aF1*f1 + aF2*f2 + aF3*f3 + aF4*f4 + aF5*f5 + aF6*f6 + aF7*f7 + aF8*f8 + aF9*f9\n \+ + aFA*fA + aFB*fB + aFC*fC + aFD*fD + a FE*fE;\n fF := fn(xk + cF*h,yk + t*h);\n \n t := b1*f1 + b2*f2 + b3*f3 + b4*f4 + b5*f5 + b6*f6 + b7*f7 + b8*f8 + b9*f9 + bA*fA \n + bB*fB + bC*fC + b D*fD + bE*fE + bF*fF;\n\n yk := yk + t*h;\n xk := xk + h:\n \+ soln := soln,[xk,yk];\n end do;\n if bb=true then\n eqns := \{SOLN_=[soln],FXY_=fxy,X_=x,Y_=y,c2_=c2,c3_=c3,\n c4_=c4, c5_=c5,c6_=c6,c7_=c7,c8_=c8,c9_=c9,cA_=cA,cB_=cB,cC_=cC,\n cD_ =cD,cE_=cE,cF_=cF,a31_=a31,a32_=a32,a41_=a41,a42_=a42,a43_=a43,\n \+ a51_=a51,a52_=a52,a53_=a53,a54_=a54,a61_=a61,a62_=a62,a63_=a63,\n \+ a64_=a64,a65_=a65,a71_=a71,a72_=a72,a73_=a73,a74_=a74,a75_=a75 ,\n a76_=a76,a81_=a81,a82_=a82,a83_=a83,a84_=a84,a85_=a85,a86_ =a86,\n a87_=a87,a91_=a91,a92_=a92,a93_=a93,a94_=a94,a95_=a95, a96_=a96,\n a97_=a97,a98_=a98,aA1_=aA1,aA2_=aA2,aA3_=aA3,aA4_= aA4,aA5_=aA5,\n aA6_=aA6,aA7_=aA7,aA8_=aA8,aA9_=aA9,aB1_=aB1,a B2_=aB2,aB3_=aB3,\n aB4_=aB4,aB5_=aB5,aB6_=aB6,aB7_=aB7,aB8_=a B8,aB9_=aB9,aBA_=aBA,\n aC1_=aC1,aC2_=aC2,aC3_=aC3,aC4_=aC4,aC 5_=aC5,aC6_=aC6,aC7_=aC7,\n aC8_=aC8,aC9_=aC9,aCA_=aCA,aCB_=aC B,aD1_=aD1,aD2_=aD2,aD3_=aD3,\n aD4_=aD4,aD5_=aD5,aD6_=aD6,aD7 _=aD7,aD8_=aD8,aD9_=aD9,aDA_=aDA,\n aDB_=aDB,aDC_=aDC,aE1_=aE1 ,aE2_=aE2,aE3_=aE3,aE4_=aE4,aE5_=aE5,\n aE6_=aE6,aE7_=aE7,aE8_ =aE8,aE9_=aE9,aEA_=aEA,aEB_=aEB,aEC_=aEC,\n aED_=aED,aF1_=aF1, aF2_=aF2,aF3_=aF3,aF4_=aF4,aF5_=aF5,aF6_=aF6,\n aF7_=aF7,aF8_= aF8,aF9_=aF9,aFA_=aFA,aFB_=aFB,aFC_=aFC,aFD_=aFD,\n aFE_=aFE,b 1_=b1,b2_=b2,b3_=b3,b4_=b4,b5_=b5,b6_=b6,b7_=b7,b8_=b8,\n b9_= b9,bA_=bA,bB_=bB,bC_=bC,bD_=bD,bE_=bE,bF_=bF\};\n return subs(eqn s,eval(rk9_15step)); \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 26 "RK9_4 Tsitouras' scheme A" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14943 "RK9_4 := proc(fxy,x, y,xx,yy,h,stps,bb)\n local c2,c3,c4,c5,c6,c7,c8,c9,cA,cB,cC,cD,cE,cF ,a21,a31,a32,a41,a42,a43,\n a51,a52,a53,a54,a61,a62,a63,a64,a65,a71, a72,a73,a74,a75,a76,a81,a82,a83,\n a84,a85,a86,a87,a91,a92,a93,a94,a 95,a96,a97,a98,aA1,aA2,aA3,aA4,aA5,aA6,\n aA7,aA8,aA9,aB1,aB2,aB3,aB 4,aB5,aB6,aB7,aB8,aB9,aBA,aC1,aC2,aC3,aC4,aC5,\n aC6,aC7,aC8,aC9,aCA ,aCB,aD1,aD2,aD3,aD4,aD5,aD6,aD7,aD8,aD9,aDA,aDB,aDC,\n aE1,aE2,aE3, aE4,aE5,aE6,aE7,aE8,aE9,aEA,aEB,aEC,aED,aF1,aF2,aF3,aF4,aF5,\n aF6,a F7,aF8,aF9,aFA,aFB,aFC,aFD,aFE,f1,f2,f3,f4,f5,f6,f7,f8,f9,fA,fB,fC,\n \+ fD,fE,fF,b1,b2,b3,b4,b5,b6,b7,b8,b9,bA,bB,bC,bD,bE,bF,t,k,fn,xk,yk,s oln,\n eqns,A,saveDigits;\n\n saveDigits := Digits;\n Digits := \+ max(trunc(evalhf(Digits)),Digits+5);\n\n fn := unapply(fxy,x,y);\n\n A := matrix([[1/49,1/49,0,0,0,0,0,0,0,0,0,0,0,0,0,0],\n[64/705-16/1 4805*6^(1/2),-165952/1491075+38896/10437525*6^(1/2),301312/1491075-716 8/1491075*6^(1/2),0,0,0,0,0,0,0,0,0,0,0,0,0],\n[32/235-8/4935*6^(1/2), 8/235-2/4935*6^(1/2),0,24/235-2/1645*6^(1/2),0,0,0,0,0,0,0,0,0,0,0,0], \n[3/7,38937/44800+171/5600*6^(1/2),0,-149931/44800-81/700*6^(1/2),650 97/22400+477/5600*6^(1/2),0,0,0,0,0,0,0,0,0,0,0],\n[8/21+4/63*6^(1/2), 176/5103-29/5103*6^(1/2),0,0,364520/1674351+87715/5023053*6^(1/2),1940 224/15069159+779264/15069159*6^(1/2),0,0,0,0,0,0,0,0,0,0],\n[8/21-4/63 *6^(1/2),4336/127575+479/127575*6^(1/2),0,0,90731944/400648275-1701427 39/8413613775*6^(1/2),8245504/62429373-22187008/437005611*6^(1/2),-393 6/340025+11464/3060225*6^(1/2),0,0,0,0,0,0,0,0,0],\n[40/63,40/567,0,0, 0,0,160/567-10/567*6^(1/2),160/567+10/567*6^(1/2),0,0,0,0,0,0,0,0],\n[ 10/21,95/1344,0,0,0,0,295/1344-115/2688*6^(1/2),295/1344+115/2688*6^(1 /2),-15/448,0,0,0,0,0,0,0],\n[19/18,52918819/138240000,0,0,0,0,-145304 7743/103680000-4153586941/829440000*6^(1/2),-1453047743/103680000+4153 586941/829440000*6^(1/2),44599023/5120000,518179039/25920000,0,0,0,0,0 ,0],\n[7/9,258780283/8618400000+585428803/51710400000*6^(1/2),0,0,0,0, 19/25,1180508473123/443296800000-136404911099/147765600000*6^(1/2),-10 6856621/190800000+585428803/2289600000*6^(1/2),-1260561943/591300000+5 85428803/886950000*6^(1/2),13167297224/792049782825-9366860848/2376149 348475*6^(1/2),0,0,0,0,0],\n[319999786/2170712113,11910753332681922251 0639750832411974467191643469020133053/29137664905764716334007503363213 354847664112851105748578125*6^(1/2)+3072133953285828679644307658474730 84972824867512957518186088963/5126364212860621132939944111710304798478 633358572140981841000000,0,0,0,0,-326846378808790716888590231940475422 3684763360609854606186699/41847871125392825575019951932329018763090884 559772579443600000*6^(1/2)+1035482118210023049302666700037918495550562 2050895245676387169/14646754893887488951256983176315156567081809595920 4028052600000,54237115726095689129801119753077769917434333557040118823 5055068891/71570321592203079939677872792265704339576185318351053993416 9000000-25571039199675674205714455677983806004177793498080540925688106 87111/1001984502290843119155490219091719860754066594456914755907836600 0000*6^(1/2),203673881988860870493193973923424476338897710332024427520 63/220613748572218566528913954035758258132313997301229239234375*6^(1/2 )-892225578009519154676238995901578841509244882985862325637048827/3881 3900468801845720831005417235164902766795429189067433939000000,72417380 262706087286468968506106480476052519229164240896224/303864219731546327 483221106502082129125640034018674235171875*6^(1/2)-5506886052357703380 34863642917595195825050633798073448796494914/8353227400420208542513748 21774223772966384453517335472487484375,-878156021711972763726444754937 2070053517105489683916369731584/61696858574806530092961398698756477329 94358307038155314262445125*6^(1/2)+12339689511549573843454922971558704 0998178289648593721222723693824/16960466422214315122555088502288155618 001490986047888958907461648625,-24189012942629864713848561037755140616 5672225318019246672/36616887234718722378142457940787891417704523989801 00701315,0,0,0,0],\n[15/16,1493491403898138129099/13100021190238236835 840*6^(1/2)+45077846760256141387004276823/1103158941439921335917399244 80,0,0,0,0,-15062887306567756845/5628365710091616256*6^(1/2)+285307321 23103900185/9849639992660328448,-8920823473649531766699837/13476277437 95786138255360*6^(1/2)+530875502237315716994493/2406478113921046675456 0,13441422635083243161891/5220309196109974077440*6^(1/2)-1558502517539 28802974915857362119/174015086605340016477040216637440,149349140389813 8129099/224694912332563742720*6^(1/2)-493074073683718697930133408597/2 7602712116408194083051274240,200609996314078300148532240828075/1019933 691979646265167106381709312-336035565877081079047275/84650664247949736 77551616*6^(1/2),-1259978731825102407292471875/94764207560034314320294 7072,-193916214235317468987992391599053188049367133486207120889311375/ 42748092349455088111344007455417233641020280098816132254793728,0,0,0], \n[39/40,1221461237263884679751555607/9994523002806272000000000000*6^( 1/2)+36716621212098036093935018687105425505961/72248275402215258274603 114496000000000000,0,0,0,0,-15668946773152185221466849/429410225684480 0000000000*6^(1/2)+8283471074731862302286097/7514678949478400000000000 ,-6493922587539771225254133441201/1028158373867634688000000000000*6^(1 /2)+397408075485926915758262202639/18359970961922048000000000000,10993 151135374962117764000463/3982779843223552000000000000*6^(1/2)+24352784 93903047909370803905780425755549361787/6951941874960961660657094977773 568000000000000,1221461237263884679751555607/1714286135349760000000000 00*6^(1/2)-558123239069103416347126929975086912148938889/3418477140693 2232290260692794368000000000000,-10993151135374962117764000463/2583333 25951995046312000000000*6^(1/2)+30673458616172717370414682337838233088 9519/1406266800626214225846914661737584000000000,-16384577883526466025 5510638493965671483/114196173990354810149157741209600000000,-379727098 6915804513041293376628177197844518996782506306850218942331696541095234 05323/7498094372633797605306278132471605294704710333028576185307953569 1020475760640000000,-235412270220829707518634576/100049213779824637253 22265625,0,0],\n[1,5933645037523445166666379/5232675730499930792800000 0*6^(1/2)+1564746779443331677794753119798867/2864199537049451295689544 004000000,0,0,0,0,-6047516944575480929793/1405122376611152200000*6^(1/ 2)-57628625604267458078211/39343426545112261600000,-665090476518894340 8799448943/1345741904975564908028000000*6^(1/2)+1691425739887134646566 682083/96124421783968922002000000,53402805337711006499997411/208520160 68909498648000000*6^(1/2)+4336304776662958741869045159571925984937/227 1064142891395647108842129832843500000,5933645037523445166666379/897521 918060373467750000*6^(1/2)-253787748130868165717697094592697839427/218 39824320042045467560266343639250000,-854444885403376103999958576/21640 244757394101989947629875*6^(1/2)+2479701613016438411235353770858147265 76/1172015708277609400708896218214309893875,-5296663573769760545511761 5341/39091452325689672618552125990,-4438603995312738130229112719866504 3342539469696141788200181019629996219153064322715843/92609628361826304 67435788682851407364301742047595974505212083507724183850275361816608,2 5469705993361596208461643776/2398429987672964863327432743875,-10513898 964163619809241937920000000/255387066048362821838491552709590753,0],\n [0,173734691637390647/4182794002754640000,0,0,0,0,0,0,-722631631417150 44860361/169939769455665013040000,14586697891849999254003/297004623905 76849520000,102209317997264953344/225042304099487188475,18835705376932 11021/1872275755054959100,17109990417889849939560223376925306674323804 078983341334325755071278367152457480027/713814271258088281460765347031 73195056090466069876919551824450642795947233961810560,-106782640999939 89152768/2396652442219114419375,1212545712242913280000000/130535954653 501897388343,-369769046476619/65552230622340]]);\n\n c2 := evalf(A[1 ,1]);\n c3 := evalf(A[2,1]);\n c4 := evalf(A[3,1]);\n c5 := eval f(A[4,1]);\n c6 := evalf(A[5,1]);\n c7 := evalf(A[6,1]);\n c8 := evalf(A[7,1]);\n c9 := evalf(A[8,1]);\n cA := evalf(A[9,1]);\n \+ cB := evalf(A[10,1]);\n cC := evalf(A[11,1]);\n cD := evalf(A[12,1 ]);\n cE := evalf(A[13,1]);\n cF := evalf(A[14,1]);\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 aB1 := evalf(A[10,2]);\n aB2 := evalf( A[10,3]);\n aB3 := evalf(A[10,4]);\n aB4 := evalf(A[10,5]);\n aB 5 := evalf(A[10,6]);\n aB6 := evalf(A[10,7]);\n aB7 := evalf(A[10, 8]);\n aB8 := evalf(A[10,9]);\n aB9 := evalf(A[10,10]);\n aBA := evalf(A[10,11]);\n aC1 := evalf(A[11,2]);\n aC2 := evalf(A[11,3]) ;\n aC3 := evalf(A[11,4]);\n aC4 := evalf(A[11,5]);\n aC5 := eva lf(A[11,6]);\n aC6 := evalf(A[11,7]);\n aC7 := evalf(A[11,8]);\n \+ aC8 := evalf(A[11,9]);\n aC9 := evalf(A[11,10]);\n aCA := evalf(A [11,11]);\n aCB := evalf(A[11,12]);\n aD1 := evalf(A[12,2]);\n a D2 := evalf(A[12,3]);\n aD3 := evalf(A[12,4]);\n aD4 := evalf(A[12 ,5]);\n aD5 := evalf(A[12,6]);\n aD6 := evalf(A[12,7]);\n aD7 := evalf(A[12,8]);\n aD8 := evalf(A[12,9]);\n aD9 := evalf(A[12,10]) ;\n aDA := evalf(A[12,11]);\n aDB := evalf(A[12,12]);\n aDC := e valf(A[12,13]);\n aE1 := evalf(A[13,2]);\n aE2 := evalf(A[13,3]); \n aE3 := evalf(A[13,4]);\n aE4 := evalf(A[13,5]);\n aE5 := eval f(A[13,6]);\n aE6 := evalf(A[13,7]);\n aE7 := evalf(A[13,8]);\n \+ aE8 := evalf(A[13,9]);\n aE9 := evalf(A[13,10]);\n aEA := evalf(A[ 13,11]);\n aEB := evalf(A[13,12]);\n aEC := evalf(A[13,13]);\n a ED := evalf(A[13,14]);\n aF1 := evalf(A[14,2]);\n aF2 := evalf(A[1 4,3]);\n aF3 := evalf(A[14,4]);\n aF4 := evalf(A[14,5]);\n aF5 : = evalf(A[14,6]);\n aF6 := evalf(A[14,7]);\n aF7 := evalf(A[14,8]) ;\n aF8 := evalf(A[14,9]);\n aF9 := evalf(A[14,10]);\n aFA := ev alf(A[14,11]);\n aFB := evalf(A[14,12]);\n aFC := evalf(A[14,13]); \n aFD := evalf(A[14,14]);\n aFE := evalf(A[14,15]);\n b1 := eva lf(A[15,2]);\n b2 := evalf(A[15,3]);\n b3 := evalf(A[15,4]);\n b 4 := evalf(A[15,5]);\n b5 := evalf(A[15,6]);\n b6 := evalf(A[15,7] );\n b7 := evalf(A[15,8]);\n b8 := evalf(A[15,9]);\n b9 := evalf (A[15,10]);\n bA := evalf(A[15,11]);\n bB := evalf(A[15,12]);\n \+ bC := evalf(A[15,13]);\n bD := evalf(A[15,14]);\n bE := evalf(A[15 ,15]);\n bF := evalf(A[15,16]);\n xk := evalf(xx);\n yk := evalf (yy);\n soln := [xk,yk]; \n for k from 1 to stps do\n f1 := f n(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*f 3 + 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 + a9 3*f3 + a94*f4 + a95*f5 + a96*f6 + a97*f7 + a98*f8;\n f9 := fn(xk \+ + c9*h,yk + t*h);\n t := aA1*f1 + aA2*f2 + aA3*f3 + aA4*f4 + aA5* f5 + aA6*f6 + aA7*f7 + aA8*f8 + aA9*f9;\n fA := fn(xk + cA*h,yk + t*h);\n t := aB1*f1 + aB2*f2 + aB3*f3 + aB4*f4 + aB5*f5 + aB6*f6 + aB7*f7 + aB8*f8 + aB9*f9\n \+ + aBA*fA;\n fB := fn(xk + cB*h,yk + t*h);\n t := aC1*f1 + aC2*f2 + aC3*f3 + aC4*f4 + aC5*f 5 + aC6*f6 + aC7*f7 + aC8*f8 + aC9*f9\n \+ + aCA*fA + aCB*fB;\n fC \+ := fn(xk + cC*h,yk + t*h);\n \n t := aD1*f1 + aD2*f2 + aD3*f3 + aD4*f4 + aD5*f5 + aD6*f6 + aD7*f7 + aD8*f8 + aD9*f9\n \+ + aDA*fA + aDB*fB + aDC *fC;\n fD := fn(xk + cD*h,yk + t*h);\n\n t := aE1*f1 + aE2*f 2 + aE3*f3 + aE4*f4 + aE5*f5 + aE6*f6 + aE7*f7 + aE8*f8 + aE9*f9\n \+ + aEA*fA + aEB*fB + \+ aEC*fC + aED*fD;\n fE := fn(xk + cE*h,yk + t*h);\n t := aF1* f1 + aF2*f2 + aF3*f3 + aF4*f4 + aF5*f5 + aF6*f6 + aF7*f7 + aF8*f8 + aF 9*f9\n + aFA*fA + aFB*fB + aFC*fC + aFD*fD + aFE*fE;\n fF := fn(xk + cF*h,yk + t*h);\n \+ \n t := b1*f1 + b2*f2 + b3*f3 + b4*f4 + b5*f5 + b6*f6 + b7*f7 + b 8*f8 + b9*f9 + bA*fA\n \+ + bB*fB + bC*fC + bD*fD + bE*fE + bF*fF;\n\n yk := yk + t*h;\n \+ xk := xk + h:\n soln := soln,[xk,yk];\n end do;\n if bb=t rue then\n eqns := \{SOLN_=[soln],FXY_=fxy,X_=x,Y_=y,c2_=c2,c3_=c 3,\n c4_=c4,c5_=c5,c6_=c6,c7_=c7,c8_=c8,c9_=c9,cA_=cA,cB_=cB,c C_=cC,\n cD_=cD,cE_=cE,cF_=cF,a31_=a31,a32_=a32,a41_=a41,a42_= a42,a43_=a43,\n a51_=a51,a52_=a52,a53_=a53,a54_=a54,a61_=a61,a 62_=a62,a63_=a63,\n a64_=a64,a65_=a65,a71_=a71,a72_=a72,a73_=a 73,a74_=a74,a75_=a75,\n a76_=a76,a81_=a81,a82_=a82,a83_=a83,a8 4_=a84,a85_=a85,a86_=a86,\n a87_=a87,a91_=a91,a92_=a92,a93_=a9 3,a94_=a94,a95_=a95,a96_=a96,\n a97_=a97,a98_=a98,aA1_=aA1,aA2 _=aA2,aA3_=aA3,aA4_=aA4,aA5_=aA5,\n aA6_=aA6,aA7_=aA7,aA8_=aA8 ,aA9_=aA9,aB1_=aB1,aB2_=aB2,aB3_=aB3,\n aB4_=aB4,aB5_=aB5,aB6_ =aB6,aB7_=aB7,aB8_=aB8,aB9_=aB9,aBA_=aBA,\n aC1_=aC1,aC2_=aC2, aC3_=aC3,aC4_=aC4,aC5_=aC5,aC6_=aC6,aC7_=aC7,\n aC8_=aC8,aC9_= aC9,aCA_=aCA,aCB_=aCB,aD1_=aD1,aD2_=aD2,aD3_=aD3,\n aD4_=aD4,a D5_=aD5,aD6_=aD6,aD7_=aD7,aD8_=aD8,aD9_=aD9,aDA_=aDA,\n aDB_=a DB,aDC_=aDC,aE1_=aE1,aE2_=aE2,aE3_=aE3,aE4_=aE4,aE5_=aE5,\n aE 6_=aE6,aE7_=aE7,aE8_=aE8,aE9_=aE9,aEA_=aEA,aEB_=aEB,aEC_=aEC,\n \+ aED_=aED,aF1_=aF1,aF2_=aF2,aF3_=aF3,aF4_=aF4,aF5_=aF5,aF6_=aF6,\n \+ aF7_=aF7,aF8_=aF8,aF9_=aF9,aFA_=aFA,aFB_=aFB,aFC_=aFC,aFD_=aFD, \n aFE_=aFE,b1_=b1,b2_=b2,b3_=b3,b4_=b4,b5_=b5,b6_=b6,b7_=b7,b 8_=b8,\n b9_=b9,bA_=bA,bB_=bB,bC_=bC,bD_=bD,bE_=bE,bF_=bF\};\n return subs(eqns,eval(rk9_15step)); \n else\n return eval f[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 26 "RK9_5 Tsitouras' scheme B" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20116 "RK 9_5 := proc(fxy,x,y,xx,yy,h,stps,bb)\n local c2,c3,c4,c5,c6,c7,c8,c9 ,cA,cB,cC,cD,cE,cF,a21,a31,a32,a41,a42,a43,\n a51,a52,a53,a54,a61,a6 2,a63,a64,a65,a71,a72,a73,a74,a75,a76,a81,a82,a83,\n a84,a85,a86,a87 ,a91,a92,a93,a94,a95,a96,a97,a98,aA1,aA2,aA3,aA4,aA5,aA6,\n aA7,aA8, aA9,aB1,aB2,aB3,aB4,aB5,aB6,aB7,aB8,aB9,aBA,aC1,aC2,aC3,aC4,aC5,\n a C6,aC7,aC8,aC9,aCA,aCB,aD1,aD2,aD3,aD4,aD5,aD6,aD7,aD8,aD9,aDA,aDB,aDC ,\n aE1,aE2,aE3,aE4,aE5,aE6,aE7,aE8,aE9,aEA,aEB,aEC,aED,aF1,aF2,aF3, aF4,aF5,\n aF6,aF7,aF8,aF9,aFA,aFB,aFC,aFD,aFE,f1,f2,f3,f4,f5,f6,f7, f8,f9,fA,fB,fC,\n fD,fE,fF,b1,b2,b3,b4,b5,b6,b7,b8,b9,bA,bB,bC,bD,bE ,bF,t,k,fn,xk,yk,soln,\n eqns,A,saveDigits;\n\n saveDigits := Digi ts;\n Digits := max(trunc(evalhf(Digits)),Digits+5);\n\n fn := una pply(fxy,x,y);\n\n A := matrix([[1/46,1/46,0,0,0,0,0,0,0,0,0,0,0,0,0 ,0],\n[96755252944/718444993695-11256225944/718444993695*6^(1/2),-1632 87951175938724532816/516163208965408589753025+42011574289334042817176/ 516163208965408589753025*6^(1/2),232801278267248934720896/516163208965 408589753025-50098553466700618240256/516163208965408589753025*6^(1/2), 0,0,0,0,0,0,0,0,0,0,0,0,0],\n[48377626472/239481664565-5628112972/2394 81664565*6^(1/2),12094406618/239481664565-1407028243/239481664565*6^(1 /2),0,36283219854/239481664565-4221084729/239481664565*6^(1/2),0,0,0,0 ,0,0,0,0,0,0,0,0],\n[71/136,450479172821804238979159483/48998547173293 5255816699904+65404175703680378526395577/244992735866467627908349952*6 ^(1/2),0,-1663285823745576633021875313/489985471732935255816699904-258 991054585998425691922779/244992735866467627908349952*6^(1/2),734303944 921586208649981787/244992735866467627908349952+96793439441159023582763 601/122496367933233813954174976*6^(1/2),0,0,0,0,0,0,0,0,0,0,0],\n[276/ 715-46/715*6^(1/2),188634486760257/2753187875656075-40451003556679/550 6375751312150*6^(1/2),0,0,890541395040155939974909749/3404930508779360 011084250045-235414842445143790083329443/6809861017558720022168500090* 6^(1/2),127509164130554343284736/2278805333809176804299525-51090254569 210884816896/2278805333809176804299525*6^(1/2),0,0,0,0,0,0,0,0,0,0],\n [276/715+46/715*6^(1/2),523150756520001/5294592068569375+3722056750028 61/137659393782803750*6^(1/2),0,0,121832502441158811994748302664452173 /6319431229672072722127362725145820625-1205400814135515666268035792222 4203047/164305211971473890775311430853791336250*6^(1/2),-7345188891123 909155979140554752/52428978281511938535235507146875+713821951824578894 88943971467264/681576717659655200958061592909375*6^(1/2),8421175214349 8940768/206389046233053165625+567839841668979868/18762640566641196875* 6^(1/2),0,0,0,0,0,0,0,0,0],\n[92/143,92/1287,0,0,0,0,368/1287+23/1287* 6^(1/2),368/1287-23/1287*6^(1/2),0,0,0,0,0,0,0,0],\n[69/143,1311/18304 ,0,0,0,0,4071/18304+1587/36608*6^(1/2),4071/18304-1587/36608*6^(1/2),- 621/18304,0,0,0,0,0,0,0],\n[3/44,2451872601/50434064384,0,0,0,0,843293 49/1146228736-1383050643/100868128768*6^(1/2),84329349/1146228736+1383 050643/100868128768*6^(1/2),-1098320769/50434064384,-333490521/3152129 024,0,0,0,0,0,0],\n[103/411,-11290810941252792923651/16694694614145777 48900000-76218489460616423924209/10016816768487466493400000*6^(1/2),0, 0,0,0,1/30,-44608220078798131601386867/1778327431680661626219300000-30 2663621648107819403033939/5334982295041984878657900000*6^(1/2),4768550 623191902657077/335320789258483564950000+76218489460616423924209/93889 82099237539818600000*6^(1/2),76371166597983496297729/12681546870360734 82337500+76218489460616423924209/1902232030554110223506250*6^(1/2),128 37092726068800321242176/73489499260117750229428125+2243872329720547520 32871296/13889515360162254793361915625*6^(1/2),0,0,0,0,0],\n[302582488 19701/45339732981913,-355843792738780589211336013211266011894384892859 307606673616682840321037292934264483298005378753134077/611498050133846 6999761271314291724602877525435623381536957589416556451501095664913528 943447961301150000+163425300686746737060984103973048211168317072573898 8407797492220074165441589604495769296514008733014761/62508689569237662 6642263289905376292738591488974834557111220251470215042334223524494069 77468048856200000*6^(1/2),0,0,0,0,-17594918026040863488775323023688367 263938903884097169692619657825340943285779242173824532021369934883/346 2019729988547470634073605629776390552199015860622162462142931219652542 1587764433517710597688597280-17188002444689712939111901065805768959625 84281439946985382458420029691073118839588510586218092229/1344473681548 9504740326499439338937439037666081012124902765603616387000163723403663 5020235330829504*6^(1/2),-13820730750259787241446624871190711710123828 9950946956575872497287994333538943652935288845283125425999567221/44944 5892634106961164222874370868126852218761612180707797390011848931329871 568905581758203549930436500950000+373422609934822782117113512174693455 2518699213931065868445711261717561735324793853258746229836835407333258 99/1797783570536427844656891497483472507408875046448722831189560047395 725319486275622327032814199721746003800000*6^(1/2),1263244583627961911 3294210780168171565726892406599945732333835965116776912265868042131556 830549327654393/112996477298237313277639902405971868302745384545450862 247028276227308103806571175581620305423011393900000-630354731220308842 9495101153246145287920801370707526715790327134571780988988474483681572 268319398771221/225992954596474626555279804811943736605490769090901724 494056552454616207613142351163240610846022787800000*6^(1/2),1115379785 2776277138749876364320845224044108260104646285465373849752391924503161 38834225701046102632948257/3347520639960280405850082108776916598468832 017158981794068212683234002575269671076605501548156712544287500-163425 3006867467370609841039730482111683170725738988407797492220074165441589 604495769296514008733014761/118706405672350369001775961304145978669107 51833897098560525576890900718352020110200728728894172739518750*6^(1/2) ,111953957936197928184195726875956182710949222537008671066128508919127 59357989068567623193453186950145251776/3714676274546537921638426673756 7863919875050348378042013698628944987922182903823702648363618722076517 534375-687320121745403419867910288709505619542179230939368838936545299 414049008577113662220686991034529999350912/123822542484884597387947555 7918928797329168344945934733789954298166264072763460790088278787290735 8839178125*6^(1/2),738251041874747688759674210057303751565868055282720 61538441201406502469378399602291332635087412/9287127253902959914328914 4740668914525146202470165909630170359355065425993729905138177287271317 ,0,0,0,0],\n[59/69,-64719070744144335733962214412431202035561004419937 320357188840883/101209800434111984325751891003355158288493635359529771 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3400+2893908603034923903437694020377944093049/446537803730292850953036 80366320956891120*6^(1/2),82096401403548497101641865315099485067606311 06345347/2089832016538282470163966831380108687881472614625000-73264258 89801057568096943993446608268878122947239059/4179664033076564940327933 662760217375762945229250000*6^(1/2),9656355719858106181793772673915704 646989403181/39976896846716122011518307352108347913593312500*6^(1/2)+3 5838730146139312307525393375183357494472205437196407446046250796739244 1417/60038122073498036770225353795012178236450255253187133063989752661 1625875000,-6428514041253116051017238172002206137209817585683623769030 35228606467992291/2746471448654456711714239942033442928622109850609548 52952360191727428234375+7725084575886484945435018139132563717591522544 8/64795380679365872530827165335902056306758765625*6^(1/2),-36996958930 1763518639761285997824520507173970793320519753635225871563412913152/18 0850870268987312974476662632186804296491491034759756207691410652763681 078125+454852979828196233587213868032125351691788847437824/94623202717 4365750372191179393662736769436409578125*6^(1/2),-77153034582811990417 57411869319392732268755939408305903151657039/1483062857858019956599354 292891821472831958379628421680653323310,-94543290861814042087512801900 3679821034774098061354410087876969999388668430388640158852256997513450 369741874915618358283/118333625607368137927111484394931198556993205054 7818490887959987766109714104308279282406274069868573023638087923030436 570,1056911827593717127690972016166243945915857152155/7359790814129537 930306952068888958882079118176267,0,0],\n[1,18418541509225838359706930 946702924369912480521/39508190287667470693939925841123267455183000000* 6^(1/2)-35844288448275172110290039888693420028897572948383502697746736 49/2085716341227133258652824258948052402773350356190879990897000000,0, 0,0,0,207046340886115163444903302653766033657147/390892087327123713034 5900952525773317746560*6^(1/2)-109506544804218207328507858426712184332 96327603/1006547124867343561064069495275386629319739200,17891127852837 9700355580290396510513302568946148710869/52268734010455066946605798802 781733427287406982000000*6^(1/2)-1907876364659215409988833049875997409 85291747006984827/2613436700522753347330289940139086671364370349100000 0,-497300620749097635712087135560978957987636974067/999861123434045989 100479661671504230211939000000*6^(1/2)-1157338419312435372868206758992 74760337634770259165918831723224267751/7941600225359840800631673172720 3547297279769620940498280027513000000,-1841854150922583835970693094670 2924369912480521/7502757290206082175051091686174851271537156250*6^(1/2 )+20283630417023233099057095727795854554028063632221885527696504052311 1/40483452265866865051188583915057088814854119415174645941812575437500 ,-54224186203160868130977204707093409345022342653824/54782834560687628 399797684955475647938658789640625*6^(1/2)+9985470767253984503216770090 1378988710484675887497768961753983967005376/23989269043634452040903285 829718875782291016518318453275155443387390625,474324561414678755109830 553528867775624956239150831585597139/451579116356903666403376148747297 21281201275171059286117565,5838559500251120707358738694419063212491522 3533408383221102363939423894748980604430394490848526064113167169126734 415639048389/224741650740382631378562225386536931859861163472113576869 92324141917918392966746421148746670582600045180015758041428626487315,- 2061784272050868275289038565787612564079661263925986910/63058180411532 25448392022471538861648368501130406546697,2144782169029430053686260375 092251514835023987332030/525539085247001422007381644286290748516695808 9934203,0],\n[0,385924436255198461459913/25885297292164750617319296,0, 0,0,0,0,0,-81508791888782942071778080019859673/39939540077799955278767 2219983509760,935936315524449978576662571821361001/4086801409502444670 525951816974611200,2172547024243858864854526674870272/1697228340841402 7681310867548214525,1193746724713997342094811077921918673563219/533383 9543124606397779553737025343484595200,78998885900843720607956191287550 9208028937668327456911860695435130268907660397023313561421879843370942 61434894685526783/1997283537594056933833443573570503668060657340439012 44628659238274456345277244226082448918708806038102409307770530790400,1 1409994679937666036993318622713183/21064682981931652352331725782966400 0,10048608923923592706010638721995991/79176273134521955696793568162336 000,957935979810312917705761/30187687238921485361088000]]);\n\n c2 : = evalf(A[1,1]);\n c3 := evalf(A[2,1]);\n c4 := evalf(A[3,1]);\n \+ c5 := evalf(A[4,1]);\n c6 := evalf(A[5,1]);\n c7 := evalf(A[6,1]) ;\n c8 := evalf(A[7,1]);\n c9 := evalf(A[8,1]);\n cA := evalf(A[ 9,1]);\n cB := evalf(A[10,1]);\n cC := evalf(A[11,1]);\n cD := e valf(A[12,1]);\n cE := evalf(A[13,1]);\n cF := evalf(A[14,1]);\n \+ a21 := c2;\n a31 := evalf(A[2,2]);\n a32 := evalf(A[2,3]);\n a4 1 := evalf(A[3,2]);\n a42 := evalf(A[3,3]);\n a43 := evalf(A[3,4]) ;\n a51 := evalf(A[4,2]);\n a52 := evalf(A[4,3]);\n a53 := evalf (A[4,4]);\n a54 := evalf(A[4,5]);\n a61 := evalf(A[5,2]);\n a62 \+ := evalf(A[5,3]);\n a63 := evalf(A[5,4]);\n a64 := evalf(A[5,5]); \n a65 := evalf(A[5,6]);\n a71 := evalf(A[6,2]);\n a72 := evalf( A[6,3]);\n a73 := evalf(A[6,4]);\n a74 := evalf(A[6,5]);\n a75 : = evalf(A[6,6]);\n a76 := evalf(A[6,7]);\n a81 := evalf(A[7,2]);\n a82 := evalf(A[7,3]);\n a83 := evalf(A[7,4]);\n a84 := evalf(A[ 7,5]);\n a85 := evalf(A[7,6]);\n a86 := evalf(A[7,7]);\n a87 := \+ evalf(A[7,8]);\n a91 := evalf(A[8,2]);\n a92 := evalf(A[8,3]);\n \+ a93 := evalf(A[8,4]);\n a94 := evalf(A[8,5]);\n a95 := evalf(A[8, 6]);\n a96 := evalf(A[8,7]);\n a97 := evalf(A[8,8]);\n a98 := ev alf(A[8,9]);\n aA1 := evalf(A[9,2]);\n aA2 := evalf(A[9,3]);\n a A3 := 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 := eval f(A[9,9]);\n aA9 := evalf(A[9,10]);\n aB1 := evalf(A[10,2]);\n a B2 := evalf(A[10,3]);\n aB3 := evalf(A[10,4]);\n aB4 := evalf(A[10 ,5]);\n aB5 := evalf(A[10,6]);\n aB6 := evalf(A[10,7]);\n aB7 := evalf(A[10,8]);\n aB8 := evalf(A[10,9]);\n aB9 := evalf(A[10,10]) ;\n aBA := evalf(A[10,11]);\n aC1 := evalf(A[11,2]);\n aC2 := ev alf(A[11,3]);\n aC3 := evalf(A[11,4]);\n aC4 := evalf(A[11,5]);\n \+ aC5 := evalf(A[11,6]);\n aC6 := evalf(A[11,7]);\n aC7 := evalf(A [11,8]);\n aC8 := evalf(A[11,9]);\n aC9 := evalf(A[11,10]);\n aC A := evalf(A[11,11]);\n aCB := evalf(A[11,12]);\n aD1 := evalf(A[1 2,2]);\n aD2 := evalf(A[12,3]);\n aD3 := evalf(A[12,4]);\n aD4 : = evalf(A[12,5]);\n aD5 := evalf(A[12,6]);\n aD6 := evalf(A[12,7]) ;\n aD7 := evalf(A[12,8]);\n aD8 := evalf(A[12,9]);\n aD9 := eva lf(A[12,10]);\n aDA := evalf(A[12,11]);\n aDB := evalf(A[12,12]); \n aDC := evalf(A[12,13]);\n aE1 := evalf(A[13,2]);\n aE2 := eva lf(A[13,3]);\n aE3 := evalf(A[13,4]);\n aE4 := evalf(A[13,5]);\n \+ aE5 := evalf(A[13,6]);\n aE6 := evalf(A[13,7]);\n aE7 := evalf(A[ 13,8]);\n aE8 := evalf(A[13,9]);\n aE9 := evalf(A[13,10]);\n aEA := evalf(A[13,11]);\n aEB := evalf(A[13,12]);\n aEC := evalf(A[13 ,13]);\n aED := evalf(A[13,14]);\n aF1 := evalf(A[14,2]);\n aF2 \+ := evalf(A[14,3]);\n aF3 := evalf(A[14,4]);\n aF4 := evalf(A[14,5] );\n aF5 := evalf(A[14,6]);\n aF6 := evalf(A[14,7]);\n aF7 := ev alf(A[14,8]);\n aF8 := evalf(A[14,9]);\n aF9 := evalf(A[14,10]);\n aFA := evalf(A[14,11]);\n aFB := evalf(A[14,12]);\n aFC := eval f(A[14,13]);\n aFD := evalf(A[14,14]);\n aFE := evalf(A[14,15]);\n b1 := evalf(A[15,2]);\n b2 := evalf(A[15,3]);\n b3 := evalf(A[1 5,4]);\n b4 := evalf(A[15,5]);\n b5 := evalf(A[15,6]);\n b6 := e valf(A[15,7]);\n b7 := evalf(A[15,8]);\n b8 := evalf(A[15,9]);\n \+ b9 := evalf(A[15,10]);\n bA := evalf(A[15,11]);\n bB := evalf(A[1 5,12]);\n bC := evalf(A[15,13]);\n bD := evalf(A[15,14]);\n bE : = evalf(A[15,15]);\n bF := evalf(A[15,16]);\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,y k + 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 + a7 2*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 + a A4*f4 + aA5*f5 + aA6*f6 + aA7*f7 + aA8*f8 + aA9*f9;\n fA := fn(xk + cA*h,yk + t*h);\n t := aB1*f1 + aB2*f2 + aB3*f3 + aB4*f4 + aB5 *f5 + aB6*f6 + aB7*f7 + aB8*f8 + aB9*f9\n \+ + aBA*fA;\n f B := fn(xk + cB*h,yk + t*h);\n t := aC1*f1 + aC2*f2 + aC3*f3 + aC 4*f4 + aC5*f5 + aC6*f6 + aC7*f7 + aC8*f8 + aC9*f9\n \+ + aCA*fA + aCB*fB ;\n fC := fn(xk + cC*h,yk + t*h);\n \n t := aD1*f1 + aD2*f 2 + aD3*f3 + aD4*f4 + aD5*f5 + aD6*f6 + aD7*f7 + aD8*f8 + aD9*f9\n \+ + aDA*fA + \+ aDB*fB + aDC*fC;\n fD := fn(xk + cD*h,yk + t*h);\n\n t := aE 1*f1 + aE2*f2 + aE3*f3 + aE4*f4 + aE5*f5 + aE6*f6 + aE7*f7 + aE8*f8 + \+ aE9*f9\n + aEA*fA + aEB*fB + aEC*fC + aED*fD;\n fE := fn(xk + cE*h,yk + t*h);\n \+ t := aF1*f1 + aF2*f2 + aF3*f3 + aF4*f4 + aF5*f5 + aF6*f6 + aF7*f7 + aF8*f8 + aF9*f9\n + aFA*f A + aFB*fB + aFC*fC + aFD*fD + aFE*fE;\n fF := fn(xk + cF*h,yk + \+ t*h);\n \n t := b1*f1 + b2*f2 + b3*f3 + b4*f4 + b5*f5 + b6*f6 + b7*f7 + b8*f8 + b9*f9 + bA*fA\n \+ + bB*fB + bC*fC + bD*fD + bE*fE + bF*fF;\n\n yk := y k + t*h;\n xk := xk + h:\n soln := soln,[xk,yk];\n end do; \n if bb=true then\n eqns := \{SOLN_=[soln],FXY_=fxy,X_=x,Y_=y, c2_=c2,c3_=c3,\n c4_=c4,c5_=c5,c6_=c6,c7_=c7,c8_=c8,c9_=c9,cA_ =cA,cB_=cB,cC_=cC,\n cD_=cD,cE_=cE,cF_=cF,a31_=a31,a32_=a32,a4 1_=a41,a42_=a42,a43_=a43,\n a51_=a51,a52_=a52,a53_=a53,a54_=a5 4,a61_=a61,a62_=a62,a63_=a63,\n a64_=a64,a65_=a65,a71_=a71,a72 _=a72,a73_=a73,a74_=a74,a75_=a75,\n a76_=a76,a81_=a81,a82_=a82 ,a83_=a83,a84_=a84,a85_=a85,a86_=a86,\n a87_=a87,a91_=a91,a92_ =a92,a93_=a93,a94_=a94,a95_=a95,a96_=a96,\n a97_=a97,a98_=a98, aA1_=aA1,aA2_=aA2,aA3_=aA3,aA4_=aA4,aA5_=aA5,\n aA6_=aA6,aA7_= aA7,aA8_=aA8,aA9_=aA9,aB1_=aB1,aB2_=aB2,aB3_=aB3,\n aB4_=aB4,a B5_=aB5,aB6_=aB6,aB7_=aB7,aB8_=aB8,aB9_=aB9,aBA_=aBA,\n aC1_=a C1,aC2_=aC2,aC3_=aC3,aC4_=aC4,aC5_=aC5,aC6_=aC6,aC7_=aC7,\n aC 8_=aC8,aC9_=aC9,aCA_=aCA,aCB_=aCB,aD1_=aD1,aD2_=aD2,aD3_=aD3,\n \+ aD4_=aD4,aD5_=aD5,aD6_=aD6,aD7_=aD7,aD8_=aD8,aD9_=aD9,aDA_=aDA,\n \+ aDB_=aDB,aDC_=aDC,aE1_=aE1,aE2_=aE2,aE3_=aE3,aE4_=aE4,aE5_=aE5, \n aE6_=aE6,aE7_=aE7,aE8_=aE8,aE9_=aE9,aEA_=aEA,aEB_=aEB,aEC_= aEC,\n aED_=aED,aF1_=aF1,aF2_=aF2,aF3_=aF3,aF4_=aF4,aF5_=aF5,a F6_=aF6,\n aF7_=aF7,aF8_=aF8,aF9_=aF9,aFA_=aFA,aFB_=aFB,aFC_=a FC,aFD_=aFD,\n aFE_=aFE,b1_=b1,b2_=b2,b3_=b3,b4_=b4,b5_=b5,b6_ =b6,b7_=b7,b8_=b8,\n b9_=b9,bA_=bA,bB_=bB,bC_=bC,bD_=bD,bE_=bE ,bF_=bF\};\n return subs(eqns,eval(rk9_15step)); \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 37 "#------------------------------------" }}{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 8 method for error correction." }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "Test 1 of 15 stage, order 9 Runge -Kutta methods " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=12*x*cos(4*x)*exp(-x)*y" "6#/*&%# dyG\"\"\"%#dxG!\"\"*,\"#7F&%\"xGF&-%$cosG6#*&\"\"%F&F+F&F&-%$expG6#,$F +F(F&%\"yGF&" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "y(0)=1" "6#/-%\"yG6 #\"\"!\"\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y=exp(-12/17*x* cos(4*x)*exp(-x)+180/289*exp(-x)*cos(4*x)+48/17*exp(-x)*sin(4*x)*x+96/ 289*exp(-x)*sin(4*x)-180/289)" "6#/%\"yG-%$expG6#,,*,\"#7\"\"\"\"# " 0 "" {MPLTEXT 1 0 229 "de := diff(y(x),x)=12*x*cos(4*x)*exp(-x)*y(x); \nic := y(0)=1;\ndsolve(\{de,ic\},y(x)):\ny(x)=simplify(numer(rhs(%))/ convert(denom(rhs(%)),exp));\nf := unapply(rhs(%),x):\nplot(f(x),x=0.. 5,0..1.45,font=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,$*,\"#7\"\"\"F,F0-%$ cosG6#,$*&\"\"%F0F,F0F0F0-%$expG6#,$F,!\"\"F0F)F0F0" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%$expG6#,,*&#\"#7\"#<\"\"\"*(F'F0-%$cosG 6#,$*&\"\"%F0F'F0F0F0-F)6#,$F'!\"\"F0F0F;*&#\"$!=\"$*GF0*&F8F0F2F0F0F0 *&#\"#[F/F0*(F8F0-%$sinGF4F0F'F0F0F0*&#\"#'*F?F0*&F8F0FEF0F0F0#F>F?F; " }}{PARA 13 "" 1 "" {GLPLOT2D 577 286 286 {PLOTDATA 2 "6&-%'CURVESG6$ 7er7$$\"\"!F)$\"\"\"F)7$$\"3gmmTN@Ki8!#>$\"3Fk>e\"G.6+\"!#<7$$\"3ALL$3 FWYs#F/$\"3!H*fm:2P/5F27$$\"3%)***\\iSmp3%F/$\"3Qn()\\Dat45F27$$\"3Wmm mT&)G\\aF/$\"34$Q7t`Dr,\"F27$$\"3m****\\7G$R<)F/$\"3S2-*\\9jw.\"F27$$ \"3GLLL3x&)*3\"!#=$\"3U([#>C\\El5F27$$\"3))**\\i!R(*Rc\"FJ$\"3>&=^@[0u 7\"F27$$\"3umm\"H2P\"Q?FJ$\"3k\\#o#G?)=?\"F27$$\"3!***\\PMnNrDFJ$\"3s_ j<)f!R*G\"F27$$\"3MLL$eRwX5$FJ$\"37'\\4u:c`O\"F27$$\"3_LLe*[`HP$FJ$\"3 [!\\'y0#yNR\"F27$$\"3rLLL$eI8k$FJ$\"3N\"Ha_9o@T\"F27$$\"3_L$3-8>bx$FJ$ \"3@))>@pAD<9F27$$\"3*QL$3xwq4RFJ$\"3a@g!fsi#>9F27$$\"3EM$eRA'*Q/%FJ$ \"3^DvP/8/=9F27$$\"33ML$3x%3yTFJ$\"3bF0p:\"oMT\"F27$$\"3h+]PfyG7ZFJ$\" 3e=U+Y19h8F27$$\"3emm\"z%4\\Y_FJ$\"3Yii#4W6uD\"F27$$\"3'QLL3FGT\\&FJ$ \"3c!QStI8]>\"F27$$\"32++v$flWv*FJ7$$\"3I++vVVX$\\'FJ$ \"3w/21T*\\F&*)FJ7$$\"31nm\"zWo)\\nFJ$\"3E>3;k'H:;)FJ7$$\"3%QL$3_DG1qF J$\"31le1yn9(R(FJ7$$\"3]***\\il'pisFJ$\"3E!)4GzFfsmFJ7$$\"3+MLe*[!)y_( FJ$\"3CJpN=**=vfFJ7$$\"3Qnm\"HKkIz(FJ$\"3'oU:>LtrL&FJ7$$\"3!3+]i:[#e!) 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PMfE5F27$$\"3?+](=xpe=#F2$\"3ES3-I\\16#*FJ7$$\"3mLeRA9WRAF2$\"3IMhv&[? ^3)FJ7$$\"37nm\"H28IH#F2$\"3H\\m$Q)R4@qFJ7$$\"3$p;a8d3AM#F2$\"39j2HRJ+ ZhFJ7$$\"3um;zpSS\"R#F2$\"3#>07(=j$QR&FJ7$$\"3-+v$41oWW#F2$\"3QVRl9U0B ZFJ7$$\"3GLL3_?`(\\#F2$\"3/\\cKWs=$>%FJ7$$\"3AL3_D1l_DF2$\"3o6E$fFc$yP FJ7$$\"3fL$e*)>pxg#F2$\"3ym)*p(*f`&[$FJ7$$\"3%omm\"z+vbEF2$\"3jG&[,$f< =LFJ7$$\"33+]Pf4t.FF2$\"3%R>3YHT'HFF2$\"3/C%)f*f*e+K FJ7$$\"3om\"zWi^bv#F2$\"3-Gu$[oUh>$FJ7$$\"3)*\\7.d>Y\"y#F2$\"3#p*R$)o? n4KFJ7$$\"3uLLe*Gst!GF2$\"3>.X!=mk1C$FJ7$$\"3)om\"H2\"34'GF2$\"3'[>IF2$\"3a-&\\&*p%H, TFJ7$$\"3F+]i!RU07$F2$\"3'fkDHe#=P[FJ7$$\"3+++v=S2LKF2$\"3K%\\5FaXpw&F J7$$\"3Jmmm\"p)=MLF2$\"3))zmB`6`OlFJ7$$\"3GLLeR%p\")Q$F2$\"3#o,C;(=8fo FJ7$$\"3B++](=]@W$F2$\"3#G%=QV$\\;4(FJ7$$\"3C$ekyZ2mY$F2$\"3u,muc\"4C(FJ7$$\"3hTgx.2vFNF2$\"3/^M\"Q[;l C(FJ7$$\"35L$e*[$z*RNF2$\"3=wJ%fi2nC(FJ7$$\"3)*\\PMFwrmNF2$\"3R[i&\\xl (GsFJ7$$\"3%o;Hd!fX$f$F2$\"3IEKi0hy'=(FJ7$$\"3r$e9T=%>?OF2$\"3(>gS`&3d ArFJ7$$\"3e++]iC$pk$F2$\"3ma\\oRiHQqFJ7$$\"3ILe*[t\\sp$F2$\"3'e9/wG(3M oFJ7$$\"3[m;H2qcZPF2$\"3CYQ8S*3be'FJ7$$\"3O+]7.\"fF&QF2$\"3**Q8E[N&3+' FJ7$$\"3Ymm;/OgbRF2$\"3kN#z0%oN^aFJ7$$\"3w**\\ilAFjSF2$\"3[i8#)*p//*\\ FJ7$$\"3ym\"zW7@^6%F2$\"3>C%QCunR#[FJ7$$\"3yLLL$)*pp;%F2$\"3g*yCm#3E'p %FJ7$$\"3)QL3-$H**>UF2$\"3$*o:W?mr0YFJ7$$\"3)RL$3xe,tUF2$\"3!\\Bp&*))o Xb%FJ7$$\"3h+v=n(*fDVF2$\"3kIpK$)H$3a%FJ7$$\"3Cn;HdO=yVF2$\"3u&G6!oNOh XFJ7$$\"3MMe9\"z-lU%F2$\"3kC\">#=Lu2YFJ7$$\"3a+++D>#[Z%F2$\"3w_(eqj7vn %FJ7$$\"3SnmT&G!e&e%F2$\"3W>T$>g**p!\\FJ7$$\"3#RLLL)Qk%o%F2$\"3'yDBP_q :;&FJ7$$\"37+]iSjE!z%F2$\"3J;fP@m(pV&FJ7$$\"3a+]P40O\"*[F2$\"3!>+$=fU- gcFJ7$$\"\"&F)$\"3h(Q0fOqh\"eFJ-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%%FONT G6$%*HELVETICAG\"\"*-%+AXESLABELSG6$%\"xG%%y(x)G-%%VIEWG6$;F(F]am;F($ \"$X\"!\"#" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "C urve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The following code constructs a " }{TEXT 260 17 "discrete solution " }{TEXT -1 44 " based on each of the methods and gives the " }{TEXT 260 22 "root mean square error" }{TEXT -1 18 " of each solution." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 688 "F := (x,y) -> 12*x*cos(4*x) *exp(-x)*y: hh := 0.01: numsteps := 500: x0 := 0: y0 := 1:\nmatrix([[` slope field: `,F(x,y)],[`initial point: `,``(x0,y0)],[`step width: \+ `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Verner's \"mo st robust\" scheme`,`Verner \"most efficient\" scheme`,`Sharp's scheme `,\n`Tsitouras' scheme A`,`Tsitouras' scheme B`]: errs := []:\nDigits \+ := 25:\nfor ct to 5 do\n Fn_RK9_||ct := RK9_||ct(F(x,y),x,y,x0,y0,hh ,numsteps,false);\n sm := 0: numpts := nops(Fn_RK9_||ct):\n for ii to numpts do\n sm := sm+(Fn_RK9_||ct[ii,2]-f(Fn_RK9_||ct[ii,1])) ^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\nend do:\nDigi ts := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,$*,\"#7\"\"\"%\"xG F,-%$cosG6#,$*&\"\"%F,F-F,F,F,-%$expG6#,$F-!\"\"F,%\"yGF,F,7$%0initial ~point:~G-%!G6$\"\"!F,7$%/step~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G \"$+&Q(pprint26\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%>Verner's~\"most~robust\"~schem eG$\"+-G2YF!#H7$%?Verner~\"most~efficient\"~schemeG$\"+`?%=i#F+7$%/Sha rp's~schemeG$\"+!RiV_\"!#G7$%4Tsitouras'~scheme~AG$\"+#)3v:nF+7$%4Tsit ouras'~scheme~BG$\"+I)p0O\"F+Q(pprint36\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code constructs \+ " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for solutions ba sed on each of the Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each of the methods at the poin t where " }{XPPEDIT 18 0 "x = 4.999;" "6#/%\"xG-%&FloatG6$\"%**\\!\"$ " }{TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 619 "F := (x,y) -> 12*x*cos(4*x)*exp(-x)*y: hh := 0.01: numsteps : = 500: x0 := 0: y0 := 1:\nmatrix([[`slope field: `,F(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numst eps]]);``;\nmthds := [`Verner's \"most robust\" scheme`,`Verner \"most efficient\" scheme`,`Sharp's scheme`,\n`Tsitouras' scheme A`,`Tsitour as' scheme B`]: errs := []:\nDigits := 30:\nfor ct to 5 do\n fn_RK9_ ||ct := RK9_||ct(F(x,y),x,y,x0,y0,hh,numsteps,true);\nend do:\nxx := 4 .999: fxx := evalf(f(xx)):\nfor ct to 5 do\n errs := [op(errs),abs(f n_RK9_||ct(xx)-fxx)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthd s,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0 slope~field:~~~G,$*,\"#7\"\"\"%\"xGF,-%$cosG6#,$*&\"\"%F,F-F,F,F,-%$ex pG6#,$F-!\"\"F,%\"yGF,F,7$%0initial~point:~G-%!G6$\"\"!F,7$%/step~widt h:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$+&Q(pprint46\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7' 7$%>Verner's~\"most~robust\"~schemeG$\"+)4;>/$!#H7$%?Verner~\"most~eff icient\"~schemeG$\"+>\\*)**GF+7$%/Sharp's~schemeG$\"+<^)*p;!#G7$%4Tsit ouras'~scheme~AG$\"+iP\">E(F+7$%4Tsitouras'~scheme~BG$\"+o=Ui9F+Q(ppri nt56\"" }}}{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 p rocedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perform numerical in tegration by the 7 point Newton-Cotes method over 200 equal subinterva ls." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 363 "mthds := [`Verner's \+ \"most robust\" scheme`,`Verner \"most efficient\" scheme`,`Sharp's sc heme`,\n`Tsitouras' scheme A`,`Tsitouras' scheme B`]: errs := []:\nDig its := 20:\nfor ct to 5 do\n sm := NCint((f(x)-'fn_RK9_||ct'(x))^2,x =0..5,adaptive=false,numpoints=7,factor=200);\n errs := [op(errs),sq rt(sm/5)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(err s)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%>Verner's~\" most~robust\"~schemeG$\"+`]&Hu#!#H7$%?Verner~\"most~efficient\"~scheme G$\"+#\\;)=EF+7$%/Sharp's~schemeG$\"+dR(Q_\"!#G7$%4Tsitouras'~scheme~A G$\"+/jh:nF+7$%4Tsitouras'~scheme~BG$\"+/acd8F+Q(pprint66\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following e rror graphs are constructed using the numerical procedures for the sol utions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 450 "evalf[30](plot([ 'fn_RK9_1'(x)-f(x),'fn_RK9_2'(x)-f(x),'fn_RK9_3'(x)-f(x),\n'fn_RK9_4'( x)-f(x),'fn_RK9_5'(x)-f(x)],x=0..5,-1e-20..3.5e-19,\ncolor=[COLOR(RGB, .45,0,1),COLOR(RGB,.85,.45,0),COLOR(RGB,0,.65,0),COLOR(RGB,0,.55,.95), red],\nlegend=[`Verner's \"most robust\" scheme`,`Verner \"most effici ent\" scheme`,`Sharp's scheme`,\n`Tsitouras' scheme A`,`Tsitouras' sch eme B`],font=[HELVETICA,9],\ntitle=`error curves for 15 stage order 9 \+ Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 858 641 641 {PLOTDATA 2 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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 0 691 "G := (x,y ) -> x/y: hh := 0.05: numsteps := 200: x0 := 0: y0 := 1:\nmatrix([[`sl ope field: `,G(x,y)],[`initial point: `,``(x0,y0)],[`step width: ` ,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Verner's \"most robust\" scheme`,`Verner \"most efficient\" scheme`,`Sharp's scheme`, \n`Tsitouras' scheme A`,`Tsitouras' scheme B`]: errs := []:\nDigits := 20:\ng := x -> sqrt(1+x^2):\nfor ct to 5 do\n Gn_RK9_||ct := RK9_|| ct(G(x,y),x,y,x0,y0,hh,numsteps,false);\n sm := 0: numpts := nops(Gn _RK9_||ct):\n for ii to numpts do\n sm := sm+(Gn_RK9_||ct[ii,2] -g(Gn_RK9_||ct[ii,1]))^2;\n end do:\n errs := [op(errs),sqrt(sm/nu mpts)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)] );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~ ~G*&%\"xG\"\"\"%\"yG!\"\"7$%0initial~point:~G-%!G6$\"\"!F+7$%/step~wid th:~~~G$\"\"&!\"#7$%1no.~of~steps:~~~G\"$+#Q(pprint76\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrix G6#7'7$%>Verner's~\"most~robust\"~schemeG$\"+K7p59!#H7$%?Verner~\"most ~efficient\"~schemeGF)7$%/Sharp's~schemeG$\"+\"oYv)\\F+7$%4Tsitouras'~ scheme~AG$\"+$f$*\\8&F+7$%4Tsitouras'~scheme~BG$\"+X2?x:F+Q(pprint86\" " }}}{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 "9.99;" "6#-%&F loatG6$\"$***!\"#" }{TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 621 "G := (x,y) -> x/y: hh := 0.05: numsteps := 2 00: x0 := 0: y0 := 1:\nmatrix([[`slope field: `,G(x,y)],[`initial po int: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps ]]);``;\nmthds := [`Verner's \"most robust\" scheme`,`Verner \"most ef ficient\" scheme`,`Sharp's scheme`,\n`Tsitouras' scheme A`,`Tsitouras' scheme B`]: errs := []:\nDigits := 30:\nfor ct to 5 do\n gn_RK9_||c t := RK9_||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 err s := [op(errs),abs(gn_RK9_||ct(xx)-gxx)];\nend do:\nDigits := 10:\nlin alg[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(pprint96\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%>Verner's~\"most~robust\"~s chemeG$\"(k0j#!#G7$%?Verner~\"most~efficient\"~schemeG$\"(e62(F+7$%/Sh arp's~schemeG$\")k1QyF+7$%4Tsitouras'~scheme~AG$\")4M%R*F+7$%4Tsitoura s'~scheme~BG$\"'qf'*F+Q)pprint106\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square er ror" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 10]" "6#7 $\"\"!\"#5" }{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 0 388 "mthds := [`Verner's \"most robust\" scheme`,`Verner \"most ef ficient\" scheme`,`Sharp's scheme`,\n`Tsitouras' scheme A`,`Tsitouras' scheme B`]: errs := []:\nDigits := 20:\ng := x -> sqrt(1+x^2):\nfor c t to 5 do\n sm := NCint((g(x)-'gn_RK9_||ct'(x))^2,x=0..10,adaptive=f alse,numpoints=7,factor=100);\n errs := [op(errs),sqrt(sm/10)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%>Verner's~\"most~robust\"~s chemeG$\"+y#R)[5!#H7$%?Verner~\"most~efficient\"~schemeG$\"+=QA,5F+7$% /Sharp's~schemeG$\"+\\mg!*GF+7$%4Tsitouras'~scheme~AG$\"+b?b2JF+7$%4Ts itouras'~scheme~BG$\"+%))=^r(!#IQ)pprint116\"" }}}{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 0 453 "evalf[25](plot(['gn_RK9_1' (x)-g(x),'gn_RK9_2'(x)-g(x),'gn_RK9_3'(x)-g(x),\n'gn_RK9_4'(x)-g(x),'g n_RK9_5'(x)-g(x)],x=0..10,-2.3e-20..9.5e-20,\ncolor=[COLOR(RGB,.45,0,1 ),COLOR(RGB,.85,.45,0),COLOR(RGB,0,.65,0),COLOR(RGB,0,.55,.95),red],\n legend=[`Verner's \"most robust\" scheme`,`Verner \"most efficient\" s cheme`,`Sharp's scheme`,\n`Tsitouras' scheme A`,`Tsitouras' scheme B`] ,font=[HELVETICA,9],\ntitle=`error curves for 15 stage order 9 Runge-K utta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 915 664 664 {PLOTDATA 2 "6+-%'CURVESG6%7^s7$$\"\"!F)F(7$$\":mmmmmm;a8ABO\"!#EF(7$$\":KLLLLLL 3FWYs#F-$\"#p!#C7$$\":lmmmmmT&Q`!eS$F-$\"$X'F37$$\":)***********\\iSmp 3%F-$\"%pRF37$$\":KLLLLLeRZF\"oZF-$\"&G%=F37$$\":lmmmmmm;a)G\\aF-$\"&b &HF37$$\":ILLLLL$3x1h6oF-$\"&J&HF37$$\":+++++++D\"G$R<)F-$\"&q(HF37$$ \":ILLLLL3-)Q4b))F-$\"&I8$F37$$\":lmmmmm;z%\\DO&*F-$\"&r'QF37$$\":++++ +]i:gT<-\"!#D$\"&\\M&F37$$\":LLLLLLL$3x&)*3\"Fin$\"&6M&F37$$\":mmmmm;/ ^\"Q(z:\"Fin$\"&rL&F37$$\":++++++v=#**3E7Fin$\"&ML&F37$$\":LLLLL$ekGg? %H\"Fin$\"&lL&F37$$F,Fin$\"&qQ&F37$$\":+++++](=U#Q/V\"Fin$\"&\"ecF37$$ \":LLLLLLe*[Vb)\\\"Fin$\"&5s'F37$$\":mmmmm;HdXqmc\"Fin$\"&Sv'F37$$\":+ ++++++Dc'yM;Fin$\"&ou'F37$$\":LLLLL$3FpE!Hq\"Fin$\"&%RnF37$$\":mmmmmmT 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yg&F37$$\":++++]PMx#\\%HU)Fin$\"&7h&F37$$\":ILLLL3-QJ?(*[)Fin$\"&*zcF3 7$$\":lmmm;zp)*p&\\c&)Fin$\"&ro&F37$$\":+++++]Pf3rKi)Fin$\"&%ocF37$$\" :lmmmm\"H2e=#ov)Fin$\"&7j&F37$$\":ILLLLL3-js.*))Fin$\"&Og&F37$$\":lmmm ;/wi,[r&*)Fin$\"&8i&F37$$\":+++++vVBSBR-*Fin$\"&\"ocF37$$\":ILLL$e9T)y )p!4*Fin$\"&$\\cF37$$\":lmmmm;zWF37$Fijl$\"&m$=F37$F^[m$\"&Zv\"F37$Fc[m$\"&m n\"F37$Fh[m$\"&0h\"F37$F]\\m$\"&[a\"F37$Fb\\m$\"&>[\"F37$Fg\\m$\"&5V\" F37$F\\]m$\"&*z8F37$Fa]m$\"&1L\"F37$Ff]m$\"&dG\"F37$F[^m$\"&]C\"F37$F` ^m$\"&F?\"F37$Fe^m$\"&q;\"F37$Fj^m$\"&78\"F37$F__m$\"&15\"F37$Fd_m$\"& *o5F37$Fi_m$\"&2/\"F37$F^`m$\"&G,\"F37$Fc`m$\"%p)*F37$Fh`m$\"%6'*F37$F ]am$\"%v$*F37$Fbam$\"%X\"*F37$Fgam$\"%H*)F37$F\\bm$\"%Q()F37$Fabm$\"%I &)F37$Ffbm$\"%^$)F37$F[cm$\"%q\")F37$F`cm$\"%.!)F37$Fecm$\"$%yFicm-F[d m6&F]dmF($F\\emF`dmF(-Fddm6#%/Sharp's~schemeG-F$6%7_rF'F*7$F/$F3F37$F: $!%'>\"F37$FD$!%ByF37$FI$!%'=a>4gFin$\"&B#yF37$ $\":NLLLL3-j(y%e2'Fin$\"&$*z(F37$$\":++++]P%)RL+D9'Fin$\"&ix(F37$F^al$ 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p$\"&%)3%F37$Fap$\"&Z7%F37$Fep$\"&$>VF37$Fjp$\"&o2&F37$F_q$\"&**4&F37$ Fdq$\"&W4&F37$F^r$\"&P3&F37$Fhr$\"&$*4&F37$F]s$\"&[;&F37$Fbs$\"&R?&F37 $Fgs$\"&o>&F37$F\\t$\"&&*=&F37$Fat$\"&32&F37$Fft$\"&r^%F37$F[u$\"&-^%F 37$F`u$\"&8]%F37$Feu$\"&#zWF37$Fju$\"&vR%F37$F_v$\"&u4%F37$Fdv$\"&0O$F 37$Fiv$\"&]N$F37$F^w$\"&&\\LF37$Fcw$\"&OM$F37$Fhw$\"&oL$F37$F]x$\"&EK$ F37$Fbx$\"&.F$F37$Fgx$\"&&oIF37$F\\y$\"&IQ#F37$Fay$\"&*4?F37$Ffy$\"&g+ #F37$F[z$\"&B+#F37$F`z$\"&\")*>F37$Fez$\"&2*>F37$Fjz$\"&Z'>F37$F_[l$\" &%e=F37$Fd[l$\"&WZ\"F37$Fi[l$\"%(e(F37$F^\\l$\"%rvF37$Fh\\l$\"%MvF37$F b]l$\"%ksF37$Fg]l$\"%Z9F37$F\\^l$!%iAF37$Fa^l$!%dAF37$Ff^l$!%bAF37$F[_ l$!%/BF37$F`_l$!%WEF37$Fe_l$!%QUF37$Fj_l$!%1))F37$F_`l$!%#y)F37$Fd`l$! %f()F37$Fi`l$!&/@\"F37$F^al$!&_H\"F37$Ffcl$!&G.\"F37$F`dl$!%DgF37$Fedl $!%bCF37$Fjdl$!#$*F37$F_el$\"$=$F37$Fdel$\"$#fF37$Fiel$\"$h(F37$F^fl$ \"$a)F37$Fcfl$\"$,*F37$Fhfl$\"$J)F37$F]gl$\"$e'F37$Fbgl$\"$M&F37$Fggl$ \"$`%F37$F\\hl$\"$1%F37$Fahl$\"$n$F37$Ffhl$\"$Q$F37$F[il$\"$;$F37$F`il $\"$)HF37$Feil$\"$y#F37$Fjil$\"$k#F37$F_jl$\"$]#F37$Fdjl$\"$Q#F37$Fijl $\"$E#F37$F^[m$\"$;#F37$Fc[m$\"$2#F37$Fh[m$\"$)>F37$F]\\m$\"$\">F37$Fb \\m$\"$$=F37$Fg\\m$\"$w\"F37$F\\]m$\"$r\"F37$Fa]m$\"$k\"F37$Ff]m$\"$f \"F37$F[^m$\"$`\"F37$F`^m$\"$[\"F37$Fe^m$\"$W\"F37$Fj^m$\"$R\"F37$F__m $\"$O\"F37$Fd_m$\"$J\"F37$Fi_m$\"$G\"F37$F^`m$\"$E\"F37$Fc`m$\"$A\"F37 $Fh`m$\"$=\"F37$F]am$\"$:\"F37$Fbam$\"$7\"F37$Fgam$\"$5\"F37$F\\bm$\"$ 2\"F37$Fabm$\"$0\"F37$Ffbm$\"$.\"F37$F[cm$\"$,\"F37$F`cm$\"#**F37$Fecm $FfcmFicm-%'COLOURG6&F]dm$\"*++++\"!\")F(F(-Fddm6#%4Tsitouras'~scheme~ BG-%&TITLEG6#%Verror~curves~for~15~stage~order~9~Runge-Kutta~methodsG- %%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6$Q\"x6\"Q!Fg`s-%%VIEWG6$;F(Fe cm;$Ficm!#@$F_iqF_as" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Verner's \"most robust\" scheme" "Verner \"most efficie nt\" scheme" "Sharp's scheme" "Tsitouras' scheme A" "Tsitouras' scheme B" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "Test 3 of \+ 15 stage, order 9 Runge-Kutta methods " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = -x*y;" "6#/*&%#dyG\"\"\"%#dxG!\"\",$*&% \"xGF&%\"yGF&F(" }{TEXT -1 11 ", " }{XPPEDIT 18 0 "y(0)=1" "6 #/-%\"yG6#\"\"!\"\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 " Solution: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = exp (-x^2/2);" "6#/%\"yG-%$expG6#,$*&%\"xG\"\"#F+!\"\"F," }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The f ollowing code constructs a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " based on each of the methods and gives the " }{TEXT 260 22 "ro ot mean square error" }{TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 691 "H := (x,y) -> -x*y: hh := 0.1: num steps := 100: x0 := 0: y0 := 1:\nmatrix([[`slope field: `,H(x,y)],[` initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: \+ `,numsteps]]);``;\nmthds := [`Verner's \"most robust\" scheme`,`Verner \"most efficient\" scheme`,`Sharp's scheme`,\n`Tsitouras' scheme A`,` Tsitouras' scheme B`]: errs := []:\nDigits := 20:\nh := x -> exp(-x^2/ 2):\nfor ct to 5 do\n Hn_RK9_||ct := RK9_||ct(H(x,y),x,y,x0,y0,hh,nu msteps,false);\n sm := 0: numpts := nops(Hn_RK9_||ct):\n for ii to numpts do\n sm := sm+(Hn_RK9_||ct[ii,2]-h(Hn_RK9_||ct[ii,1]))^2; \n end do:\n errs := [op(errs),sqrt(sm/numpts)];\nend do:\nDigits \+ := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,$*&%\"xG\"\"\"%\"yGF ,!\"\"7$%0initial~point:~G-%!G6$\"\"!F,7$%/step~width:~~~G$F,F.7$%1no. ~of~steps:~~~G\"$+\"Q)pprint126\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% !G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%>Verner's~\"mos t~robust\"~schemeG$\"+\"eP;L%!#D7$%?Verner~\"most~efficient\"~schemeG$ \"+vw*)4QF+7$%/Sharp's~schemeG$\"++!RR#=!#C7$%4Tsitouras'~scheme~AG$\" +o0&)3bF47$%4Tsitouras'~scheme~BG$\"+x[a#\\#F+Q)pprint136\"" }}}{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 "9.99;" "6#-%&FloatG6$\"$***! \"#" }{TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 621 "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 := [`Verner's \"most robust\" scheme`,`Verner \"most efficient \" scheme`,`Sharp's scheme`,\n`Tsitouras' scheme A`,`Tsitouras' scheme B`]: errs := []:\nDigits := 20:\nfor ct to 5 do\n hn_RK9_||ct := RK 9_||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 := [o p(errs),abs(hn_RK9_||ct(xx)-hxx)];\nend do:\nDigits := 10:\nlinalg[tra nspose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'ma trixG6#7&7$%0slope~field:~~~G,$*&%\"xG\"\"\"%\"yGF,!\"\"7$%0initial~po int:~G-%!G6$\"\"!F,7$%/step~width:~~~G$F,F.7$%1no.~of~steps:~~~G\"$+\" Q)pprint246\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%>Verner's~\"most~robust\"~schemeG$ \"+CIL@]!#S7$%?Verner~\"most~efficient\"~schemeG$\"+-ca!4$!#T7$%/Sharp 's~schemeG$\"+W!pD7#!#Q7$%4Tsitouras'~scheme~AG$\"+89)*)H'!#P7$%4Tsito uras'~scheme~BG$\"+H*3eX\"!#RQ)pprint256\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean s quare error" }{TEXT -1 110 " over the interval [0, 0.5] of each Rung e-Kutta method is estimated as follows using the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 97 " to perform numerical integration by the 7 point Newton-Cotes method over 50 equal subintervals." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 387 "mthds := [`Verner's \"most \+ robust\" scheme`,`Verner \"most efficient\" scheme`,`Sharp's scheme`, \n`Tsitouras' scheme A`,`Tsitouras' scheme B`]: errs := []:\nDigits := 20:\nh := x -> exp(-x^2/2):\nfor ct to 5 do\n sm := NCint((h(x)-'hn _RK9_||ct'(x))^2,x=0..10,adaptive=false,numpoints=7,factor=50);\n er rs := [op(errs),sqrt(sm/10)];\nend do:\nDigits := 10:\nlinalg[transpos e]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG 6#7'7$%>Verner's~\"most~robust\"~schemeG$\"+VGbmS!#D7$%?Verner~\"most~ efficient\"~schemeG$\"+J\"*4xNF+7$%/Sharp's~schemeG$\"+7EB- " 0 "" {MPLTEXT 1 0 448 "evalf[25](plot(['hn_RK9_1'(x)-h(x),'hn_RK9_2'(x)-h(x),'hn_RK9_3'( x)-h(x),\n'hn_RK9_4'(x)-h(x),'hn_RK9_5'(x)-h(x)],x=0..6,numpoints=150, \ncolor=[COLOR(RGB,.45,0,1),COLOR(RGB,.85,.45,0),COLOR(RGB,0,.65,0),CO LOR(RGB,0,.55,.95),red],\nlegend=[`Verner's \"most robust\" scheme`,`V erner \"most efficient\" scheme`,`Sharp's scheme`,\n`Tsitouras' scheme A`,`Tsitouras' scheme B`],font=[HELVETICA,9],\ntitle=`error curves fo r 15 stage order 9 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 604 604 604 {PLOTDATA 2 "6+-%'CURVESG6%7`el7$$\"\"!F)F(7$$\" :&yn^F\"p`!3798U!#E$\"%%\\'!#D7$$\":\\uh#*QePc%G(*yyF-$\"(&y:MF07$$\": +)p/dN.b#Qy-%**F-$\")@M0NF07$$\"::A$[s3j%>Re,?\"F0$\")v39PF07$$\":%**[ B&yn^FJbwS\"F0$\")Cb/PF07$$\":sd')zp/dNB_^h\"F0$\"),VBPF07$$\":G\"p`!3 7=x&Hm@=F0$\")Q,YUF07$$\":$[s3j%>z=ot\"G?F0$\")(=xu(F07$$\":Uk'\\uh#*Q ebj>AF0$\")!yir(F07$$\":-/1f)G$**[V(46CF0$\")0y#o(F07$$\":;v7p`!37=^M4 EF0$\")n.xwF07$$\":JY>z=GU8!Gf2GF0$\")I0l\")F07$$\":&HWm\\uh#*Q3i7IF0$ \"*4(RS7F07$$\":fR49r15l()[w@$F0$\"*=+DB\"F07$$\":!p`!37=xlQ>?U$F0$\"* RmUA\"F07$$\":AM,-`HWm*)*QEOF0$\"*/#p?7F07$$\":!p`!37=xl35m$QF0$\"*kgR I\"F07$$\":fR49r15lFIo/%F0$\"*$*=#*y\"F07$$\":'3j%>z=GUL#*>B%F0$\"*'zb vRarT%F0$\"*<0:w\"F07$$\":3j%>z=GU8IgDYF0$\"*(zC^zB^wu&F0$\"*f\"*[Q#F07$$\":?K[s3j%>zV BReF0$\"*W^8Z#F07$$\":?\"=xl)zp/_<3$fF0$\"*tuJs#F07$$\":>I&HWm\\uh1SAg 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?RkF-7$Fjfl$!2k&[i%eo(GkF-7$F_gl$!2&RQ)*QNRAkF-7$Fdgl$!2kE8 [gK^V'F-7$Figl$!2V!e&>(opClF-7$F^hl$!2%Rx$ed++$oF--%'COLOURG6&Ffhl$\"* ++++\"!\")F(F(-F\\il6#%4Tsitouras'~scheme~BG-%&TITLEG6#%inrelative~err or~curves~for~15~stage~order~9~Runge-Kutta~methodsG-%%FONTG6$%*HELVETI CAG\"\"*-%+AXESLABELSG6$Q\"x6\"Q!Fgeq-%%VIEWG6$;F(F^hl;$!\"(Ffdq$\"\"& !\"*" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Verner' s \"most robust\" scheme" "Verner \"most efficient\" scheme" "Sharp's \+ scheme" "Tsitouras' scheme A" "Tsitouras' scheme B" }}}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "Test 4 of 15 stage, order 9 Runge -Kutta methods " }}{PARA 0 "" 0 "" {TEXT -1 81 "F. G. Lether: Mathemat ics of Computation, Vol. 20, no. 95, (July 1966) page 381. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = -32*x*y*ln(2);" "6#/*&%#dyG\"\"\"%#dxG!\"\",$**\"#KF&%\" xGF&%\"yGF&-%#lnG6#\"\"#F&F(" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "y(- 1) = 1/8;" "6#/-%\"yG6#,$\"\"\"!\"\"*&F(F(\"\")F)" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y = 2^(13-6*x^2);" "6#/%\"yG)\"\"#,&\"#8\"\"\"* &\"\"'F)*$%\"xGF&F)!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 191 "de := diff(y(x),x)=- 32*x*y(x)*ln(2);\nic := y(-1)=1/8;\ndsolve(\{de,ic\},y(x)):\ny(x)=2^si mplify(log[2](rhs(%)));\nk := unapply(rhs(%),x):\nplot(k(x),x=-1..1,fo nt=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,$**\"#K\"\"\"F,F0F)F0-%#lnG6#\" \"#F0!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#!\"\"#\" \"\"\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG)\"\"#,&\"# 8\"\"\"*&\"#;F,)F'F)F,!\"\"" }}{PARA 13 "" 1 "" {GLPLOT2D 577 286 286 {PLOTDATA 2 "6&-%'CURVESG6$7eo7$$!\"\"\"\"!$\"3+++++++]7!#=7$$!3ommm;p 0k&*F-$\"3!Hg[[\"[$)=KF-7$$!3wKL$3$3(F-7$$!3mmmmT%p \"e()F-$\"3!=E-TWD`l\"!#<7$$!3:mmm\"4m(G$)F-$\"3M\"fONp()[t$F=7$$!3\"Q LL3i.9!zF-$\"3A!e'4A%*Rg!)F=7$$!3\"ommT!R=0vF-$\"3%z2Mbncie\"!#;7$$!3u ****\\P8#\\4(F-$\"3C>dT>$)H#3$FM7$$!3+nm;/siqmF-$\"3gp%*z`g)4*eFM7$$!3 [++](y$pZiF-$\"3%R6L-Y$zz5!#:7$$!33LLL$yaE\"eF-$\"3xvp\"p)==K>Fgn7$$!3 hmmm\">s%HaF-$\"3dBW_P%Gb6$Fgn7$$!3Q+++]$*4)*\\F-$\"3e;N4:OFap7$$!3]++]PYx\"\\#F-$\"37]-4,Tp9TFap7$$ !3QnmTNz>&H#F-$\"3y(*QMk^JnXFap7$$!3EMLLL7i)4#F-$\"3yCPsPtXE]Fap7$$!3# pm;aVXH)=F-$\"3_cYryDpGbFap7$$!3c****\\P'psm\"F-$\"38i9x*[!p=gFap7$$!3 s*****\\F&*=Y\"F-$\"3K`b3X@JjkFap7$$!3')****\\74_c7F-$\"3co2Qfx9woFap7 $$!3ZmmT5VBU5F-$\"3E!>K?nVAE(Fap7$$!3)3LLL3x%z#)!#>$\"3C'Q/NU&H#f(Fap7 $$!3gKL$e9d;J'Fft$\"3@erx_1&z$yFap7$$!3KMLL3s$QM%Fft$\"3<%pUH&HNA!)Fap 7$$!3'ym;aQdDG$Fft$\"3%eWuwq(o%4)Fap7$$!3T,+]ivF@AFft$\"3[vW[$G&HZ\")F ap7$$!3=o;/^wj!p\"Fft$\"3]3j^OK2m\")Fap7$$!3'\\L$eRx**f6Fft$\"355#oX4% yz\")Fap7$$!3S<+D\"GyNH'!#?$\"3#QE)R9AS)=)Fap7$$!3]^omm;zr)*!#@$\"3;#) *eHY6>>)Fap7$$\"3o'H$3x\"yY_%Fjv$\"3Q>*>/AS,>)Fap7$$\"3&yK$3_Nl.5Fft$ \"3eqLS$Q`G=)Fap7$$\"3/E$ekGR[b\"Fft$\"3-l$f@nl+<)Fap7$$\"3@CL$3-Dg5#F ft$\"3kJX?)*G!=:)Fap7$$\"3e?Le*['R3KFft$\"3E'yGoI5!*4)Fap7$$\"3%pJL$ez w5VFft$\"3U_-I6(**[-)Fap7$$\"3L`mmmJ+IiFft$\"3%pB(\\hv&o%yFap7$$\"3s*) ***\\PQ#\\\")Fft$\"3!QM&=wHL5wFap7$$\"3ilm\"z\\1A-\"F-$\"3#*[#H(\\2i&H (Fap7$$\"3GKLLe\"*[H7F-$\"3))\\\\;@heFpFap7$$\"3ylm;HCjV9F-$\"3)e+$\\9 -Y,lFap7$$\"3I*******pvxl\"F-$\"3S%z:5s)zRgFap7$$\"3g)***\\7JFn=F-$\"3 1))p(30[[c&Fap7$$\"3#z****\\_qn2#F-$\"3ae5F\"zuv2&Fap7$$\"3=)**\\P/q%z AF-$\"3ZUhzOe!Rg%Fap7$$\"3U)***\\i&p@[#F-$\"3r&f%4uLbOTFap7$$\"3L)**\\ (=GB2FF-$\"3WV]5@%**Rj$Fap7$$\"3B)****\\2'HKHF-$\"3ul]=$GLo:$Fap7$$\"3 uJL$3UDX8$F-$\"3sKZjodBbFFap7$$\"3ElmmmZvOLF-$\"3!>\\-t_7IQ#Fap7$$\"3i ******\\2goPF-$\"3Q>G9F7l&p\"Fap7$$\"3UKL$eR<*fTF-$\"3?\"Fap 7$$\"3m******\\)Hxe%F-$\"3V-?C_;$p$zFgn7$$\"3ckm;H!o-*\\F-$\"31MiF2c]v ^Fgn7$$\"3y)***\\7k.6aF-$\"3#pB[/J``=$Fgn7$$\"3#emmmT9C#eF-$\"3&*=.D]9 +3>Fgn7$$\"33****\\i!*3`iF-$\"3%HX+j$our5Fgn7$$\"3%QLLL$*zym'F-$\"3!o4 *yfd(\\\"fFM7$$\"3wKLL3N1#4(F-$\"3!\\\\K5**)='4$FM7$$\"3Nmm;HYt7vF-$\" 3%o[)olFVm:FM7$$\"3Y*******p(G**yF-$\"3)3H-pcT.4)F=7$$\"3]mmmT6KU$)F-$ \"35omE\\#[Ck$F=7$$\"3fKLLLbdQ()F-$\"3TxwT%Qu%>ei< " 0 "" {MPLTEXT 1 0 698 "K := (x,y) -> -32*x*y(x)*ln(2): hh := 0.01: numsteps := 200: \+ x0 := -1: y0 := 1/8:\nmatrix([[`slope field: `,K(x,y)],[`initial poi nt: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps] ]);``;\nmthds := [`Verner's \"most robust\" scheme`,`Verner \"most eff icient\" scheme`,`Sharp's scheme`,\n`Tsitouras' scheme A`,`Tsitouras' \+ scheme B`]: errs := []:\nDigits := 20:\nfor ct to 5 do\n Kn_RK9_||ct := RK9_||ct(evalf(K(x,y)),x,y,x0,evalf(y0),hh,numsteps,false);\n sm := 0: numpts := nops(Kn_RK9_||ct):\n for ii to numpts do\n sm \+ := sm+(Kn_RK9_||ct[ii,2]-k(Kn_RK9_||ct[ii,1]))^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transp ose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matri xG6#7&7$%0slope~field:~~~G,$**\"#K\"\"\"%\"xGF,-%\"yG6#F-F,-%#lnG6#\" \"#F,!\"\"7$%0initial~point:~G-%!G6$F5#F,\"\")7$%/step~width:~~~G$F,! \"#7$%1no.~of~steps:~~~G\"$+#Q)pprint176\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$ %>Verner's~\"most~robust\"~schemeG$\"+r6;$R\"!#?7$%?Verner~\"most~effi cient\"~schemeG$\"+2z22AF+7$%/Sharp's~schemeG$\"+[o9v))F+7$%4Tsitouras '~scheme~AG$\"+e^\"G2#!#@7$%4Tsitouras'~scheme~BG$\"+;uNk>F+Q)pprint18 6\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "T he following code constructs " }{TEXT 260 20 "numerical procedures" } {TEXT -1 56 " for solutions based on each of the Runge-Kutta schemes. " }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by ea ch of the methods at the point where " }{XPPEDIT 18 0 "x = 0;" "6#/% \"xG\"\"!" }{TEXT -1 20 ".995 is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 629 "K := (x,y) -> -32*x*y(x)*ln(2): hh := 0.01: num steps := 200: x0 := -1: y0 := 1/8:\nmatrix([[`slope field: `,K(x,y)] ,[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Verner's \"most robust\" scheme`,`Ver ner \"most efficient\" scheme`,`Sharp's scheme`,\n`Tsitouras' scheme A `,`Tsitouras' scheme B`]: errs := []:\nDigits := 25:\nfor ct to 5 do\n kn_RK9_||ct := RK9_||ct(evalf(K(x,y)),x,y,x0,evalf(y0),hh,numsteps, true);\nend do:\nxx := 0.995: kxx := evalf(k(xx)):\nfor ct to 5 do\n \+ errs := [op(errs),abs(kn_RK9_||ct(xx)-kxx)];\nend do:\nDigits := 10: \nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,$**\"#K\"\"\"%\"xGF, -%\"yG6#F-F,-%#lnG6#\"\"#F,!\"\"7$%0initial~point:~G-%!G6$F5#F,\"\")7$ %/step~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$+#Q)pprint196\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #K%'matrixG6#7'7$%>Verner's~\"most~robust\"~schemeG$\"+'RF7F\"!#C7$%?V erner~\"most~efficient\"~schemeG$\"+*oFK)=F+7$%/Sharp's~schemeG$\"+f^? +aF+7$%4Tsitouras'~scheme~AG$\"+=a*[T\"!#B7$%4Tsitouras'~scheme~BG$\"+ #pGdr\"F+Q)pprint206\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[-1,1]" "6#7$,$\"\"\"!\" \"F%" }{TEXT -1 82 " of each Runge-Kutta method is estimated as follo ws using the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " t o perform numerical integration by the 7 point Newton-Cotes method ove r 100 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 364 "mthds := [`Verner's \"most robust\" scheme`,`Verner \"most effici ent\" scheme`,`Sharp's scheme`,\n`Tsitouras' scheme A`,`Tsitouras' sch eme B`]: errs := []:\nDigits := 20:\nfor ct to 5 do\n sm := NCint((k (x)-'kn_RK9_||ct'(x))^2,x=-1..1,adaptive=false,numpoints=7,factor=100) ;\n errs := [op(errs),sqrt(sm/2)];\nend do:\nDigits := 10:\nlinalg[t ranspose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%' matrixG6#7'7$%>Verner's~\"most~robust\"~schemeG$\"+5@q'R\"!#?7$%?Verne r~\"most~efficient\"~schemeG$\"+/6l7AF+7$%/Sharp's~schemeG$\"+/UD(*))F +7$%4Tsitouras'~scheme~AG$\"+(Rk&y?!#@7$%4Tsitouras'~scheme~BG$\"+QPKp >F+Q)pprint216\"" }}}{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 0 435 "evalf[20](plot(['kn_RK9_1'(x)-k(x),'kn_RK9_2'(x)-k(x ),'kn_RK9_3'(x)-k(x),\n'kn_RK9_4'(x)-k(x),'kn_RK9_5'(x)-k(x)],x=-1..1, \ncolor=[COLOR(RGB,.45,0,1),COLOR(RGB,.85,.45,0),COLOR(RGB,0,.65,0),CO LOR(RGB,0,.55,.95),red],\nlegend=[`Verner's \"most robust\" scheme`,`V erner \"most efficient\" scheme`,`Sharp's scheme`,\n`Tsitouras' scheme A`,`Tsitouras' scheme B`],font=[HELVETICA,9],\ntitle=`error curves fo r 15 stage order 9 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 748 569 569 {PLOTDATA 2 "6+-%'CURVESG6%7eo7$$!\"\"\"\"!$F*F* 7$$!5nmmmm;p0k&*!#?$\"&sw&F/7$$!5LLLL$3d!#>7$$!5nmmmm\"4m(G$)F/$\"'(*49F<7$$!5LLLL$3i.9!zF/$\"'')y JF<7$$!5mmmm;/R=0vF/$\"&vK'!#=7$$!5++++]P8#\\4(F/$\"';K7FL7$$!5mmmm;/s iqmF/$\"'))[BFL7$$!5++++](y$pZiF/$\"&&*G%!#<7$$!5LLLLL$yaE\"eF/$\"&uk( Ffn7$$!5mmmmm\">s%HaF/$\"')*H7Ffn7$$!5+++++]$*4)*\\F/$\"'5@?Ffn7$$!5++ +++]_&\\c%F/$\"';'>$Ffn7$$!5+++++]1aZTF/$\"&5y%!#;7$$!5mmmm;/#)[oPF/$ \"&lm'F`p7$$!5LLLLL$=exJ$F/$\"&$)\\*F`p7$$!5LLLLLeW%o7$F/$\"'x)3\"F`p7 $$!5LLLLLL2$f$HF/$\"'\"yB\"F`p7$$!5mmmmT&o_Qr#F/$\"'pA9F`p7$$!5******* *\\PYx\"\\#F/$\"'A<;F`p7$$!5mmmmTNz>&H#F/$\"'+&z\"F`p7$$!5LLLLLL7i)4#F 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "de := diff(y(x),x)=16/((16*x+1)*y( x));\nic := y(0)=1;\ndsolve(\{de,ic\},y(x));\ns := unapply(rhs(%),x): \nplot(s(x),x=0..0.5,0..2.6,font=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,$*( \"#;\"\"\",&*&F/F0F,F0F0F0F0!\"\"F)F3F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/- %\"yG6#%\"xG*$,&*&\"\"#\"\"\"-%#lnG6#,&*&\"#;F,F'F,F,F,F,F,F,F,F,#F,F+ " }}{PARA 13 "" 1 "" {GLPLOT2D 503 318 318 {PLOTDATA 2 "6&-%'CURVESG6$ 7U7$$\"\"!F)$\"\"\"F)7$$\"3WmmmT&)G\\a!#?$\"3EP,(Qyl.3\"!#<7$$\"3ILLL3 x&)*3\"!#>$\"3?25A!pa&\\6F27$$\"3-+]i!R(*Rc\"F6$\"3oz*p77wF?\"F27$$\"3 umm\"H2P\"Q?F6$\"3]_vibZz]7F27$$\"3MLL$eRwX5$F6$\"3!Qb+fY'3W8F27$$\"3C LL$3x%3yTF6$\"31#\\\\E7=EU\"F27$$\"3=mm\"z%4\\Y_F6$\"3s(e$4OUg*[\"F27$ $\"3)HL$eR-/PiF6$\"3.fPtw=4W:F27$$\"3A***\\il'pisF6$\"3/07@a`R%f\"F27$ $\"3`KLe*)>VB$)F6$\"3K!\\`od36k\"F27$$\"3!))**\\7`l2Q*F6$\"3#HUv\"fmC$ o\"F27$$\"3smm;/j$o/\"!#=$\"3:'H!f>cuAjU6Fco$\"3K$o8QC! za=F27$$\"3)*****\\P[6j9Fco$\"39iuo+OIZ=F27$$\"3KL$e*[z(yb\" Fco$\"3Q:]fA\\>F27$$\"3))**\\iSj0x=Fco$\"3-5Hbh&QF%>F27$$\"3Wmmm\"pW` (>Fco$\"3So#znsrC'>F27$$\"35+]i!f#=$3#Fco$\"3w)>Y)R!pI)>F27$$\"3/+](=x pe=#Fco$\"3?*eB@.[<+#F27$$\"3smm\"H28IH#Fco$\"3/Fyh^(\\.-#F27$$\"3km;z pSS\"R#Fco$\"3)4US+%ypO?F27$$\"3GLL3_?`(\\#Fco$\"3#4Cj+a0O0#F27$$\"3#H Le*)>pxg#Fco$\"3ab\\mG7Vq?F27$$\"3u**\\Pf4t.FFco$\"3Cx7m@=^%3#F27$$\"3 2LLe*Gst!GFco$\"3Q>IFco$\"3&ocGC'[]F@F27$$\"3h**\\i!RU07$Fco$\"3HCH$Q\")f .9#F27$$\"3b***\\(=S2LKFco$\"3C`wrWc9a@F27$$\"3Kmmm\"p)=MLFco$\"3;=S,I A7m@F27$$\"3!*****\\(=]@W$Fco$\"3w4%eC\"p]y@F27$$\"35L$e*[$z*RNFco$\"3 UyOr,.R*=#F27$$\"3#*****\\iC$pk$Fco$\"3wIdFs1%4?#F27$$\"39m;H2qcZPFco$ \"3Qbx\"QY%\\6AF27$$\"3q**\\7.\"fF&QFco$\"3f+!e(oz@AAF27$$\"3Ymm;/OgbR Fco$\"36qG(yA8CB#F27$$\"3y**\\ilAFjSFco$\"3v.zLgjzUAF27$$\"3YLLL$)*pp; %Fco$\"3IImU*yHDD#F27$$\"3?LL3xe,tUFco$\"3I%R!fhiAiAF27$$\"3em;HdO=yVF co$\"3?ogo1xfrAF27$$\"3))*****\\#>#[Z%Fco$\"3EO%fx0/+G#F27$$\"3immT&G! e&e%Fco$\"3)zsS%e\"3%*G#F27$$\"3;LLL$)Qk%o%Fco$\"35e8d5)>wH#F27$$\"37+ ]iSjE!z%Fco$\"3e%4h.zwhI#F27$$\"35+]P40O\"*[Fco$\"3Nwd2,K=9BF27$$\"3++ ++++++]Fco$\"3m'>())[`fABF2-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%%FONTG6$% *HELVETICAG\"\"*-%+AXESLABELSG6$%\"xG%%y(x)G-%%VIEWG6$;F($\"\"&Fj[l;F( $\"#EFj[l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Cu rve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The following code constructs a " }{TEXT 260 17 "discrete solution " }{TEXT -1 44 " based on each of the methods and gives the " }{TEXT 260 22 "root mean square error" }{TEXT -1 18 " of each solution." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 681 "S := (x,y) -> 16/((16*x+1)* y): hh := 0.005: numsteps := 100: x0 := 0: y0 := 1:\nmatrix([[`slope f ield: `,S(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh], \n[`no. of steps: `,numsteps]]);``;\nmthds := [`Verner's \"most robu st\" scheme`,`Verner \"most efficient\" scheme`,`Sharp's scheme`,\n`Ts itouras' scheme A`,`Tsitouras' scheme B`]: errs := []:\nDigits := 20: \nfor ct to 5 do\n Sn_RK9_||ct := RK9_||ct(S(x,y),x,y,x0,y0,hh,numst eps,false);\n sm := 0: numpts := nops(Sn_RK9_||ct):\n for ii to nu mpts do\n sm := sm+(Sn_RK9_||ct[ii,2]-s(Sn_RK9_||ct[ii,1]))^2;\n \+ end do:\n errs := [op(errs),sqrt(sm/numpts)];\nend do:\nDigits := \+ 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,$*(\"#;\"\"\",&*&F+F ,%\"xGF,F,F,F,!\"\"%\"yGF0F,7$%0initial~point:~G-%!G6$\"\"!F,7$%/step~ width:~~~G$\"\"&!\"$7$%1no.~of~steps:~~~G\"$+\"Q)pprint226\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'ma trixG6#7'7$%>Verner's~\"most~robust\"~schemeG$\"+d?MgC!#C7$%?Verner~\" most~efficient\"~schemeG$\"+[0)*GDF+7$%/Sharp's~schemeG$\"+2a`%=#F+7$% 4Tsitouras'~scheme~AG$\"+2nFy?F+7$%4Tsitouras'~scheme~BG$\"+i7UuBF+Q)p print236\"" }}}{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 21 ".4995 is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 613 "S := (x,y) -> 16/((16*x+1)*y): hh := 0.005 : numsteps := 100: x0 := 0: y0 := 1:\nmatrix([[`slope field: `,S(x,y )],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of step s: `,numsteps]]);``;\nmthds := [`Verner's \"most robust\" scheme`,`V erner \"most efficient\" scheme`,`Sharp's scheme`,\n`Tsitouras' scheme A`,`Tsitouras' scheme B`]: errs := []:\nDigits := 25:\nfor ct to 5 do \n sn_RK9_||ct := RK9_||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 := [o p(errs),abs(sn_RK9_||ct(xx)-sxx)];\nend do:\nDigits := 10:\nlinalg[tra nspose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'ma trixG6#7&7$%0slope~field:~~~G,$*(\"#;\"\"\",&*&F+F,%\"xGF,F,F,F,!\"\"% \"yGF0F,7$%0initial~point:~G-%!G6$\"\"!F,7$%/step~width:~~~G$\"\"&!\"$ 7$%1no.~of~steps:~~~G\"$+\"Q)pprint246\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%>Verner' s~\"most~robust\"~schemeG$\"+1*o0,#!#C7$%?Verner~\"most~efficient\"~sc hemeG$\"+3elm?F+7$%/Sharp's~schemeG$\"+JVX&y\"F+7$%4Tsitouras'~scheme~ AG$\"+k([Qp\"F+7$%4Tsitouras'~scheme~BG$\"+4rSS>F+Q)pprint256\"" }}} {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 0 366 "mthds := [`Verner's \"most robust\" scheme`,`Verner \"most efficient\" scheme` ,`Sharp's scheme`,\n`Tsitouras' scheme A`,`Tsitouras' scheme B`]: errs := []:\nDigits := 20:\nfor ct to 5 do\n sm := NCint((s(x)-'sn_RK9_| |ct'(x))^2,x=0..0.5,adaptive=false,numpoints=7,factor=50);\n errs := [op(errs),sqrt(sm/0.5)];\nend do:\nDigits := 10:\nlinalg[transpose]([ mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7' 7$%>Verner's~\"most~robust\"~schemeG$\"+#Rk'fC!#C7$%?Verner~\"most~eff icient\"~schemeG$\"+#G#HGDF+7$%/Sharp's~schemeG$\"+gn#R=#F+7$%4Tsitour as'~scheme~AG$\"+EW!z2#F+7$%4Tsitouras'~scheme~BG$\"+GYxtBF+Q)pprint26 6\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "T he following error graphs are constructed using the numerical procedur es for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 445 "ev alf[20](plot([s(x)-'sn_RK9_1'(x),s(x)-'sn_RK9_2'(x),s(x)-'sn_RK9_3'(x) ,\ns(x)-'sn_RK9_4'(x),s(x)-'sn_RK9_5'(x)],x=0..0.5,0..4e-15,\ncolor=[C OLOR(RGB,.45,0,1),COLOR(RGB,.85,.45,0),COLOR(RGB,0,.65,0),COLOR(RGB,0, .55,.95),red],\nlegend=[`Verner's \"most robust\" scheme`,`Verner \"mo st efficient\" scheme`,`Sharp's scheme`,\n`Tsitouras' scheme A`,`Tsito uras' scheme B`],font=[HELVETICA,9],\ntitle=`error curves for 15 stage order 9 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 894 466 466 {PLOTDATA 2 "6+-%'CURVESG6%7jo7$$\"\"!F)F(7$$\"5ommmTN@Ki8!#A$ \"\"\"!#>7$$\"5NLLL$3FWYs#F-$\"$8\"F07$$\"5-++vo/[AlIF-$\"$Y$F07$$\"5o mm;aQ`!eS$F-$\"$N*F07$$\"5NLLeRseQYPF-$\"%'G#F07$$\"5-+++D1k'p3%F-$\"% Z^F07$$\"5pmmT5SpaFWF-$\"&63\"F07$$\"5OLL$eRZF\"oZF-$\"&89#F07$$\"50++ D\"y+3(3^F-$\"&lI$F07$$\"5qmmmmT&)G\\aF-$\"&CH$F07$$\"5SLLL3x1h6oF-$\" &xB$F07$$\"50+++]7G$R<)F-$\"&i>$F07$$\"5qmmm\"z%\\DO&*F-$\"&hT$F07$$\" 5MLLLL3x&)*3\"!#@$\"&Vx$F07$$\"5++]i!*GER37Fjo$\"&%HPF07$$\"5nmm\"z%\\ 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&_c$F07$Fit$\"&qc$F07$F^u$\"&?X$F07$Fcu$\"&cM$F07$Fhu$\"&d;$F07$F]v$\" &W-$F07$Fbv$\"&\"=HF07$Fgv$\"&h#GF07$F\\w$\"&du#F07$Faw$\"&qn#F07$Ffw$ \"&ch#F07$F\\x$\"&yc#F07$Fax$\"&)>DF07$Ffx$\"&nZ#F07$F[y$\"&#RCF07$F`y $\"&\"3CF07$Fey$\"&VP#F07$Fjy$\"&zM#F07$F_z$\"&%>BF07$Fdz$\"&hH#F07$Fi z$\"&AF#F07$F^[l$\"&5D#F07$Fc[l$\"&.B#F07$Fh[l$\"&C@#F07$F]\\l$\"&V>#F 07$Fb\\l$\"&k<#F07$Fg\\l$\"&<;#F07$F\\]l$\"&m9#F07$Fa]l$\"&<8#F07$Ff]l $\"&!=@F07$F[^l$\"&`5#F07$F`^l$\"&=4#F07$Fe^l$\"&.3#F07$Fj^l$\"&%o?F07 $F__l$\"&#e?F07$Fd_l$\"&t/#F07$Fi_l$\"&w.#F07$F^`l$\"&x-#F07$Fc`l$\"&% =?F07$Fh`l$\"&\"4?F07$F]al$\"&/+#F07$Fbal$\"&>*>F07$Fgal$\"&O)>F07$F\\ bl$\"&k(>F07$Fabl$\"&$o>F07$Ffbl$\"&7'>F07$F[cl$\"&R&>F07$F`cl$\"&r%>F 07$Fecl$\"&,%>F0-%'COLOURG6&F]dl$\"*++++\"!\")F(F(-Fcdl6#%4Tsitouras'~ scheme~BG-%&TITLEG6#%Verror~curves~for~15~stage~order~9~Runge-Kutta~me thodsG-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6$Q\"x6\"Q!Fc_p-%%VIEWG 6$;F(Fecl;F($\"\"%!#:" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Verner's \"most robust\" scheme" "Verner \"most efficie nt\" scheme" "Sharp's scheme" "Tsitouras' scheme A" "Tsitouras' scheme B" }}}}{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 15 stage, order 9 Runge-Kutta methods" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "dy/dx = (1+2*(x+1)*sin(3*x))*exp(-y);" "6#/*&%# dyG\"\"\"%#dxG!\"\"*&,&F&F&*(\"\"#F&,&%\"xGF&F&F&F&-%$sinG6#*&\"\"$F&F .F&F&F&F&-%$expG6#,$%\"yGF(F&" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "y( 0) = 0;" "6#/-%\"yG6#\"\"!F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y=ln(x+2/9*sin(3*x)-2/3*x*cos(3*x)-2/3*cos(3*x)+5/3)" "6#/%\"yG-%#lnG 6#,,%\"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- 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^6%FI$\"3w4XE;/Rz()F-7$$\"3w3F>RL3GTFI$\"3JeP:9JjA()F-7$$\"3t]i!RbX59% FI$\"3mH1#H$\\k'o)F-7$$\"3#=z>'ox+aTFI$\"3oLr_-o*=n)F-7$$\"3yLLL$)*pp; %FI$\"3A7j1wipy')F-7$$\"3!Q3_+sD-=%FI$\"32pcM,k23()F-7$$\"3#Q$3xc9[$>% FI$\"3Gri,**=4g()F-7$$\"3'Qe*[$>Pn?%FI$\"3se,X+?^M))F-7$$\"3)QL3-$H**> UFI$\"3Z**e,OD#4$*)F-7$$\"3#R$ek.W]YUFI$\"3i#fiyx0s=*F-7$$\"3)RL$3xe,t UFI$\"3[2R[)*eVA&*F-7$$\"3Cn;HdO=yVFI$\"3#)>Y<=$f\\9\"FI7$$\"3MMe9\"z- lU%FI$\"3)4DVDmlMD\"FI7$$\"3a+++D>#[Z%FI$\"3qZKS'GmoO\"FI7$$\"3TM$3_5, -`%FI$\"3CFB-Gn\\(\\\"FI7$$\"3SnmT&G!e&e%FI$\"3t\\(p9r/Xi\"FI7$$\"3m+] P%37^j%FI$\"3_eaMDR_K " 0 " " {MPLTEXT 1 0 693 "U := (x,y) -> (1+2*(x+1)*sin(3*x))*exp(-y): hh := \+ 0.01: numsteps := 500: x0 := 0: y0 := 0:\nmatrix([[`slope field: `,U (x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of \+ steps: `,numsteps]]);``;\nmthds := [`Verner's \"most robust\" scheme `,`Verner \"most efficient\" scheme`,`Sharp's scheme`,\n`Tsitouras' sc heme A`,`Tsitouras' scheme B`]: errs := []:\nDigits := 25:\nfor ct to \+ 5 do\n Un_RK9_||ct := RK9_||ct(U(x,y),x,y,x0,y0,hh,numsteps,false); \n sm := 0: numpts := nops(Un_RK9_||ct):\n for ii to numpts do\n \+ sm := sm+(Un_RK9_||ct[ii,2]-u(Un_RK9_||ct[ii,1]))^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg [transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K %'matrixG6#7&7$%0slope~field:~~~G*&,&\"\"\"F+*(\"\"#F+,&%\"xGF+F+F+F+- %$sinG6#,$*&\"\"$F+F/F+F+F+F+F+-%$expG6#,$%\"yG!\"\"F+7$%0initial~poin t:~G-%!G6$\"\"!FA7$%/step~width:~~~G$F+!\"#7$%1no.~of~steps:~~~G\"$+&Q )pprint276\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#K%'matrixG6#7'7$%>Verner's~\"most~robust\"~schemeG$\" +BJFO7!#I7$%?Verner~\"most~efficient\"~schemeG$\"+G*QnL\"F+7$%/Sharp's ~schemeG$\"+.L(R5\"F+7$%4Tsitouras'~scheme~AG$\"+#fV$Rn!#J7$%4Tsitoura s'~scheme~BG$\"+pe_-()F8Q)pprint286\"" }}}{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 = 4.999;" "6#/%\"xG-%&FloatG6$\"%**\\!\"$" } {TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 624 "U := (x,y) -> (1+2*(x+1)*sin(3*x))*exp(-y): hh := 0.01: numstep s := 500: x0 := 0: y0 := 0:\nmatrix([[`slope field: `,U(x,y)],[`init ial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,nu msteps]]);``;\nmthds := [`Verner's \"most robust\" scheme`,`Verner \"m ost efficient\" scheme`,`Sharp's scheme`,\n`Tsitouras' scheme A`,`Tsit ouras' scheme B`]: errs := []:\nDigits := 30:\nfor ct to 5 do\n un_R K9_||ct := RK9_||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 errs := [op(errs),ab s(un_RK9_||ct(xx)-uxx)];\nend do:\nDigits := 10:\nlinalg[transpose]([m thds,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)pprint296\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%>Verner's~\"most~robust\"~schemeG$\"*T%[67!#H7$%?V erner~\"most~efficient\"~schemeG$\"*EC!)H\"F+7$%/Sharp's~schemeG$\"*aB ME\"F+7$%4Tsitouras'~scheme~AG$\")g=upF+7$%4Tsitouras'~scheme~BG$\")0T p#)F+Q)pprint306\"" }}}{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$\"\"!\"\"&" } {TEXT -1 82 " of each Runge-Kutta method is estimated as follows usin g the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perfo rm numerical integration by the 7 point Newton-Cotes method over 200 e qual subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 363 "mthds := [`Verner's \"most robust\" scheme`,`Verner \"most efficient\" sche me`,`Sharp's scheme`,\n`Tsitouras' scheme A`,`Tsitouras' scheme B`]: e rrs := []:\nDigits := 20:\nfor ct to 5 do\n sm := NCint((u(x)-'un_RK 9_||ct'(x))^2,x=0..5,adaptive=false,numpoints=7,factor=200);\n errs \+ := [op(errs),sqrt(sm/5)];\nend do:\nDigits := 10:\nlinalg[transpose]([ mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7' 7$%>Verner's~\"most~robust\"~schemeG$\"+#fm)H7!#I7$%?Verner~\"most~eff icient\"~schemeG$\"+)**Q&H8F+7$%/Sharp's~schemeG$\"+Qn@*4\"F+7$%4Tsito uras'~scheme~AG$\"+r832n!#J7$%4Tsitouras'~scheme~BG$\"+0cJ_')F8Q)pprin t316\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following error graphs are constructed using the numerical pro cedures for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 452 "evalf[25](plot([u(x)-'un_RK9_1'(x),u(x)-'un_RK9_2'(x),u(x)-'un_RK 9_3'(x),\nu(x)-'un_RK9_4'(x),u(x)-'un_RK9_5'(x)],x=0..5,-2.1e-21..4.3e -21,\ncolor=[COLOR(RGB,.45,0,1),COLOR(RGB,.85,.45,0),COLOR(RGB,0,.65,0 ),COLOR(RGB,0,.55,.95),red],\nlegend=[`Verner's \"most robust\" scheme `,`Verner \"most efficient\" scheme`,`Sharp's scheme`,\n`Tsitouras' sc heme A`,`Tsitouras' scheme B`],font=[HELVETICA,9],\ntitle=`error curve s for 15 stage order 9 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 918 700 700 {PLOTDATA 2 "6+-%'CURVESG6%7ds7$$\"\"!F)F(7$$\": NLLLLLL3FWYs#!#E$!%:8F-7$$\":qmmmmmm;a)G\\aF-$!%iAF-7$$\":SLLLLL$3x1h6 oF-$!%!H#F-7$$\":0++++++D\"G$R<)F-$!%/?F-7$$\":qmmmmm;z%\\DO&*F-$!$k\" !#D7$$\":MLLLLLL$3x&)*3\"FD$!##*FD7$$\":++++++D1R(*Rc\"FD$\"$@#FD7$$\" :nmmmmm;H2P\"Q?FD$\"$j&FD7$$\":++++++vVtc8d#FD$\"$j(FD7$$\":MLLLLLLeRw X5$FD$\"$_)FD7$$\":MLLLLLLLeI8k$FD$\"$A)FD7$$\":NLLLLLL3x%3yTFD$\"$i(F D7$$\":ommmmm;z%4\\Y_FD$\"$N'FD7$$\":NLLLLL$eR-/PiFD$\"#b!#C7$$\":-+++ ++]il'pisFD$\"#ZF\\p7$$\":OLLLLL$e*)>VB$)FD$\"#TF\\p7$$\":.+++++]7`l2Q 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" }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " y = 1/3" "6#/%\"yG*&\"\"\"F&\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-4/3*sin(3*x)+8/3*sin(3/2*x)*cos(3/2*x))+2/3" "6#,&-%$expG6#,&* (\"\"%\"\"\"\"\"$!\"\"-%$sinG6#*&F+F*%\"xGF*F*F,**\"\")F*F+F,-F.6#*(F+ F*\"\"#F,F1F*F*-%$cosG6#*(F+F*F7F,F1F*F*F*F**&F7F*F+F,F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-4/3*sin(3*x)-x)" "6#-%$expG6#,&*(\"\"%\"\"\" \"\"$!\"\"-%$sinG6#*&F*F)%\"xGF)F)F+F0F+" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 181 "de : = diff(y(x),x)=-(1+4*cos(3*x))*(y(x)-1/3);\nic := y(0)=1;\nsimplify(ds olve(\{de,ic\},y(x)));\nv := unapply(rhs(%),x):\nplot(v(x),x=0..5,0..1 .1,font=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,$*&,&\"\"\"F0*&\"\"%F0 -%$cosG6#,$*&\"\"$F0F,F0F0F0F0F0,&F)F0#F0F8!\"\"F0F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*&#\"\"\"\"\"$F+-%$expG6#,&*&#\"\"%F,F+ -%$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$$\"3ALL$3FWYs#F/$\"3K tP[t*Q;:*F27$$\"3%)***\\iSmp3%F/$\"3g.\"H>f!3q()F27$$\"3WmmmT&)G\\aF/$ \"36p*p.:G\\T)F27$$\"3m****\\7G$R<)F/$\"3a?glh]$zx(F27$$\"3GLLL3x&)*3 \"F2$\"3IM[S(o-#HsF27$$\"3em\"z%\\v#pK\"F2$\"3i=)H'*Q$=:oF27$$\"3))** \\i!R(*Rc\"F2$\"3w,'pRB0LX'F27$$\"3&edVF27$$\"3%QL$3_DG1qF2$\"3'fN^hMLe*)>VB$)F2$\"3DB(Rfp)*\\j%F27$$\"3Y++DJbw!Q*F2$\"3%GsCu$*)zK]F27$$ \"3+N$ekGkX#**F2$\"3u>+\\,YW?`F27$$\"3%ommTIOo/\"!#<$\"3q]2x8ZEqcF27$$ \"3E+]7GTt%4\"Fgt$\"39b$=$pWlHgF27$$\"3YLL3_>jU6Fgt$\"3nwYdkc=KkF27$$ \"3ym;HdNb'>\"Fgt$\"3l[hQOW]BpF27$$\"37++]i^Z]7Fgt$\"3IVnF)*yXIuF27$$ \"35+++v\"=YI\"Fgt$\"3ahS!3L%e=zF27$$\"33++](=h(e8Fgt$\"3l&QV-<82M)F27 $$\"3&*****\\7!Q4T\"Fgt$\"3^]H\"3wS2k)F27$$\"3/++]P[6j9Fgt$\"3ur)[IAj$ )z)F27$$\"3'=HKkAg\\Z\"Fgt$\"33z^;ogY6))F27$$\"3W$ek`h0o[\"Fgt$\"3h=q? g>u:))F27$$\"3/voH/5l)\\\"Fgt$\"3p\\\\U!)G36))F27$$\"3%o;HKR'\\5:Fgt$ \"3%G4&GMdV(z)F27$$\"3-]P4rr=M:Fgt$\"3Erd.MaCV()F27$$\"3UL$e*[z(yb\"Fg t$\"3m)))[\\1qQl)F27$$\"34+Dc,#>Uh\"Fgt$\"3(fTb\\\\y3J)F27$$\"3wmm;a/c q;Fgt$\"3-!y\"yF27$$\"3\"pm;a)))G=BtF27$$\"3%om mmJFgt$\"3%RlX>.MR=&F27$$\"3gmmm \"pW`(>Fgt$\"3+6YS9:C2[F27$$\"3dLe9TOEH?Fgt$\"3!eWte3T%oWF27$$\"3K+]i! f#=$3#Fgt$\"3:XZ<;2j,UF27$$\"3?+](=xpe=#Fgt$\"3E#Q(H44MbQF27$$\"37nm\" H28IH#Fgt$\"3MH4)f2==l$F27$$\"3um;zpSS\"R#Fgt$\"3wpxg#Fgt$\"37l*=e[EHY$F27$$ \"33+]Pf4t.FFgt$\"35!4Ne]qiX$F27$$\"3uLLe*Gst!GFgt$\"3U+pq))z7kMF27$$ \"30+++DRW9HFgt$\"37'z:1TS%*[$F27$$\"3:++DJE>>IFgt$\"3N!o4Joz]`$F27$$ \"3F+]i!RU07$Fgt$\"3=,?;D0\"Qg$F27$$\"3+++v=S2LKFgt$\"3wRH=fZn5PF27$$ \"3Jmmm\"p)=MLFgt$\"3RsXuk([b#QF27$$\"3B++](=]@W$Fgt$\"3%4[=*QOMSRF27$ $\"3mm\"H#oZ1\"\\$Fgt$\"3QK??D+QyRF27$$\"35L$e*[$z*RNFgt$\"3UAxt;S)>+% F27$$\"3%o;Hd!fX$f$Fgt$\"3+h91z&\\y+%F27$$\"3e++]iC$pk$Fgt$\"3eIRs#H!Q \"*RF27$$\"3ILe*[t\\sp$Fgt$\"3m\"Rx)H&*[cRF27$$\"3[m;H2qcZPFgt$\"3w))) [$RF!f!RF27$$\"3O+]7.\"fF&QFgt$\"3+Efp,iIqPF27$$\"3Ymm;/OgbRFgt$\"3W-T ml[`MOF27$$\"3w**\\ilAFjSFgt$\"3&zNMj#[Z%Fgt$\"3ADU\\K%G5O$F27$$\"3SnmT&G!e&e%Fgt$\"3 5gRzc#\\LF27$$\"\"&F)$\"3Ii# 4)y!3AN$F2-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%%FONTG6$%*HELVETICAG\"\"*- %+AXESLABELSG6$%\"xG%%y(x)G-%%VIEWG6$;F(Fiel;F($\"#6Fcfl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The following code c onstructs a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " based o n each of the methods and gives the " }{TEXT 260 22 "root mean square \+ error" }{TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 688 "V := (x,y) -> -(1+4*cos(3*x))*(y-1/3): hh := 0.02: n umsteps := 250: x0 := 0: y0 := 1:\nmatrix([[`slope field: `,V(x,y)], [`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: \+ `,numsteps]]);``;\nmthds := [`Verner's \"most robust\" scheme`,`Vern er \"most efficient\" scheme`,`Sharp's scheme`,\n`Tsitouras' scheme A` ,`Tsitouras' scheme B`]: errs := []:\nDigits := 30:\nfor ct to 5 do\n \+ Vn_RK9_||ct := RK9_||ct(V(x,y),x,y,x0,y0,hh,numsteps,false);\n sm \+ := 0: numpts := nops(Vn_RK9_||ct):\n for ii to numpts do\n sm : = sm+(Vn_RK9_||ct[ii,2]-v(Vn_RK9_||ct[ii,1]))^2;\n end do:\n errs \+ := [op(errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpo se]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrix G6#7&7$%0slope~field:~~~G,$*&,&\"\"\"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\"$]#Q)pprint326\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%>Verner's~\"most~robust\"~schemeG$\"+8>+1@!#F7$%?V erner~\"most~efficient\"~schemeG$\"+_D,DBF+7$%/Sharp's~schemeG$\"+$zM* >pF+7$%4Tsitouras'~scheme~AG$\"+T!*R?vF+7$%4Tsitouras'~scheme~BG$\"+oZ $H@#F+Q)pprint336\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 30 "The following code constructs " }{TEXT 260 20 "numerica l procedures" }{TEXT -1 56 " for solutions based on each of the Runge- Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value \+ obtained by each of the methods at the point where " }{XPPEDIT 18 0 " x = 4.999;" "6#/%\"xG-%&FloatG6$\"%**\\!\"$" }{TEXT -1 16 " is also g iven." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 619 "V := (x,y) -> -(1+ 4*cos(3*x))*(y-1/3): hh := 0.02: numsteps := 250: x0 := 0: y0 := 1:\nm atrix([[`slope field: `,V(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Vern er's \"most robust\" scheme`,`Verner \"most efficient\" scheme`,`Sharp 's scheme`,\n`Tsitouras' scheme A`,`Tsitouras' scheme B`]: errs := []: \nDigits := 30:\nfor ct to 5 do\n vn_RK9_||ct := RK9_||ct(V(x,y),x,y ,x0,y0,hh,numsteps,true);\nend do:\nxx := 4.999: vxx := evalf(v(xx)): \nfor ct to 5 do\n errs := [op(errs),abs(vn_RK9_||ct(xx)-vxx)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,$*&,&\"\" \"F,*&\"\"%F,-%$cosG6#,$*&\"\"$F,%\"xGF,F,F,F,F,,&%\"yGF,#F,F4!\"\"F,F 97$%0initial~point:~G-%!G6$\"\"!F,7$%/step~width:~~~G$\"\"#!\"#7$%1no. ~of~steps:~~~G\"$]#Q)pprint346\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%! G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%>Verner's~\"most ~robust\"~schemeG$\"+ " 0 "" {MPLTEXT 1 0 363 "mthds := [`Verner's \"most robust\" sche me`,`Verner \"most efficient\" scheme`,`Sharp's scheme`,\n`Tsitouras' \+ scheme A`,`Tsitouras' scheme B`]: errs := []:\nDigits := 20:\nfor ct t o 5 do\n sm := NCint((v(x)-'vn_RK9_||ct'(x))^2,x=0..5,adaptive=false ,numpoints=7,factor=100);\n errs := [op(errs),sqrt(sm/5)];\nend do: \nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%>Verner's~\"most~robust\"~schem eG$\"+H;\"y4#!#F7$%?Verner~\"most~efficient\"~schemeG$\"+=Y)oJ#F+7$%/S harp's~schemeG$\"+2EPdoF+7$%4Tsitouras'~scheme~AG$\"+2J9puF+7$%4Tsitou ras'~scheme~BG$\"+%eyj?#F+Q)pprint366\"" }}}{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 0 446 "evalf[30](plot(['vn_RK9_1'(x)-v(x) ,'vn_RK9_2'(x)-v(x),'vn_RK9_3'(x)-v(x),\n'vn_RK9_4'(x)-v(x),'vn_RK9_5' (x)-v(x)],x=0..5,0..1.65e-17,\ncolor=[COLOR(RGB,.45,0,1),COLOR(RGB,.85 ,.45,0),COLOR(RGB,0,.65,0),COLOR(RGB,0,.55,.95),red],\nlegend=[`Verner 's \"most robust\" scheme`,`Verner \"most efficient\" scheme`,`Sharp's scheme`,\n`Tsitouras' scheme A`,`Tsitouras' scheme B`],font=[HELVETIC A,9],\ntitle=`error curves for 15 stage order 9 Runge-Kutta methods`)) ;" }}{PARA 13 "" 1 "" {GLPLOT2D 904 466 466 {PLOTDATA 2 "6+-%'CURVESG6 %7dt7$$\"\"!F)F(7$$\"?SLLLLLLLL3x1h6o!#K$\")oLI7!#I7$$\"?ommmmmmmmTN@K i8!#J$\",]aR?\\\"F07$$\"?NLLLLLLL3FpE!Hq\"F4$\"-0OMn)\\\"F07$$\"?-++++ +++]7.K[V?F4$\"-#GT(oVzF07$$\"?ommmmmmm\"zptjSQ#F4$\"-h`[z4yF07$$\"?NL LLLLLLL$3FWYs#F4$\"-uP'*[ywF07$$\"?-+++++++vo/[AlIF4$\"-')pBphvF07$$\" ?ommmmmmm;aQ`!eS$F4$\"-\"\\I\"*pj(F07$$\"?NLLLLLLLeRseQYPF4$\"-(*zF1x# *F07$$\"?-++++++++D1k'p3%F4$\".Z[WY0^\"F07$$\"?OLLLLLLL$eRZF\"oZF4$\". #GFOOg9F07$$\"?qmmmmmmmmmT&)G\\aF4$\".G\"Hz0S9F07$$\"?SLLLLLLL3_v!p)*y &F4$\".!z*zH$G;F07$$\"?0+++++++]P4'\\/8'F4$\".by3DC5#F07$$\"?qmmmmmmm \"HK9I5Z'F4$\".&)G*zUn?F07$$F,F4$\".!H;B9L?F07$$\"?0+++++++v$4@\">_rF4 $\".Jmui=+#F07$$\"?qmmmmmmm;zW'RHEF07$$\"?++++++++Dc,;u@5F0$\".=#H#*4QGF 07$$\"?nmmmmmm;z%\\l*zb5F0$\".cAB6?z#F07$$\"?MLLLLLLLLL3x&)*3\"F0$\".M 5oQpu#F07$$\"?++++++]ilZQ9\\>6F0$\".9wz\"p5FF07$$\"?nmmmmmm\"z>'o^7\\6 F0$\".n`z?Jp#F07$$\"?MLLLLL$3-j()*)e(y6F0$\".n2(>,rFF07$$\"?+++++++]i! *GER37F0$\".9C65N,$F07$$\"?MLLLLLL3F>*3gwE\"F0$\".\\iMf-$HF07$$\"?nmmm mmmm\"z%\\v#pK\"F0$\".[yy.L&GF07$$\"?+++++++Dcw4]>'Q\"F0$\".\\*oGeCHF0 7$$\"?MLLLLLL$3_+ZiaW\"F0$\".CH'*3p*HF07$$\"?nmmmmmmT&Q.$*HZ]\"F0$\".* Rh_W;HF07$$\"?++++++++]i!R(*Rc\"F0$\".z1r5(oGF07$$\"?nmmmmmm;z>6B`#o\" F0$\".m@:/'*)GF07$$\"?MLLLLLLL3xJs1,=F0$\".\\Z.8v'GF07$$\"?+++++++]PM_ @g>>F0$\".J:F(fAFF07$$\"?nmmmmmmmm\"H2P\"Q?F0$\".YXXw2l#F07$$\"?++++++ ++]PMnNrDF0$\".*)=qh^9#F07$$\"?MLLLLLLLL$eRwX5$F0$\".s+]H6r\"F07$$\"?M LLLLLLLLL$eI8k$F0$\".(z')\\:89F07$$\"?NLLLLLLLL$3x%3yTF0$\"._$e3RK7F07 $$\"?-+++++++]PfyG7ZF0$\"._:NV,7\"F07$$\"?ommmmmmmm\"z%4\\Y_F0$\".='*f [!\\5F07$$\"?NLLLLLLLL$3FGT\\&F0$\".W:KSn-\"F07$$\"?+++++++++v$flVB$)F0$\".,]-uYD\"F07$$\"?qmmmmmmmmTg()4_))F0$\". 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47$F``q$\"/(>,))\\:.#F47$F\\`m$\"/&*)*GA^&3#F47$Fh`q$\"/(4$o-k\\AF47$F a`m$\"/HQ@xg!\\#F47$Ff`m$\"/xCV9]GKF47$Fcaq$\"/DMqD0QPF47$F[am$\"/h1M0 %GP%F47$F[bq$\"/o1Fa*z?&F47$F`am$\"/dfhPN)='F47$Fcbq$\"/&\\N,\"pbrF47$ Feam$\"/PX``rS\")F47$F[cq$\"/+#)G?@h\"*F47$Fjam$\"0<[?v+^+\"F47$F_bm$ \"0CcPAKo1\"F47$Fdbm$\"0xy$Hx$G5\"F47$Ficq$\"0.YK*Gh36F47$Fibm$\"0V!RN N466F47$Fadq$\"0$fV>oO66F47$F^cm$\"0`\\Hc[.6\"F47$Fhcm$\"0()z\"3c$**4 \"F47$Fbdm$\"06JjVC:3\"F47$Fgdm$\"0\")\\&H=rC5F47$F\\em$\"/:qL>6T%*F47 $Faem$\"/jHiR`H%)F47$Ffem$\"/$f()e+zK(F47$Fafq$\"/E%pH=TR'F47$F[fm$\"/ x0]5V@bF47$F`fm$\"/K[3q@')[F47$Fefm$\"/sr'4a^K%F47$Figm$\"/'30S#p**HF4 7$Fchm$\"0r#f1&*fI=F-7$Fhhm$\"0x8TQIH:\"F-7$F]im$\"/\"=I?W;5)F-7$Fbim$ \"/\"y'>T![$eF-7$Fgim$\"/f$[!>(f\"[F-7$F\\jm$\"/&))R2=#oWF-7$Fajm$\"/) \\yR#4%p%F-7$Ffjm$\"/&3t\\adc&F--F[[n6&F][nF($\"#bF`[n$\"#&*F`[n-Fd[n6 #%4Tsitouras'~scheme~AG-F$6%7fsF'7$F+$\"(\">\"f)F07$F2$\",qwx'Q6F07$F8 $\"-Q4^0#=\"F07$F=$\"-QAlC?kF07$FB$\"-he=.7jF07$FG$\"-s,y*e?'F07$FL$\" -`I[F6hF07$FQ$\"-9Bv!G<'F07$FV$\"-eCccQvF07$Fen$\".CpADrC\"F07$Fjn$\". dGQ$p07F07$F_o$\".`R0j'*=\"F07$Fcp$\".^gX4Pr\"F07$Ffq$\"./$Q\"Q5>#F07$ F[r$\".)\\cuk>@F07$F`r$\".Wiy-q3#F07$Fer$\".cj\"*[))G#F07$Fjr$\".j5BCO \\#F07$F_s$\".=$z-8`CF07$Fds$\".X]F0$\".iqCH#)e#F 07$Fjw$\".@Y@&)=g#F07$F_x$\".dw]2w?#F07$Fdx$\".JFig`%=F07$Fix$\".X&4x4 j:F07$F^y$\".qXQP,P\"F07$Fcy$\".3/!)3^C\"F07$Fhy$\".k)Q52m6F07$F]z$\". qX9\"F07$Fbz$\".L\")f]d7\"F07$Fgz$\"/+Rb'*R;6F47$F\\[l$\"/I:K8v96F4 7$Fa[l$\".\"F07$Fe\\l$\".*)*)Q2CG\"F07$Fj\\l$\".Q9/UVT\"F07$F_]l$\".l n]H**e\"F07$Fd]l$\".hDk3h#=F07$Fi]l$\".:\"G!)eO@F07$F^^l$\".6>/aJ^#F07 $Fd^l$\".#pPEk))GF07$Fi^l$\".,&)=G)yKF07$F^_l$\".c^C7Yv$F07$Fc_l$\"./D Lu,B%F07$Fh_l$\".#\\b5<t=&F07$Fb`l$\".s$H7%fd&F07$F g`l$\".X&)ptx)eF07$F\\al$\".9+5-!ofF07$Faal$\".DqZpu,'F07$Ffal$\".QEOs 'QgF07$F[bl$\".*RA\")4dgF07$F`bl$\".:L(z]nfF07$Febl$\".v%>!*QudF07$Fjb l$\".HVI5#HaF07$F_cl$\".OXlS^'[F07$Ficl$\".Enfw%>WF07$Fcdl$\".HMq,Q$QF 07$F]el$\".H$pHW*H$F07$Fgel$\".RP>/Ut#F07$F\\fl$\".w0wEyV#F07$Fafl$\". 8j)*H:5#F07$Fffl$\".-Pt.d)>F07$F[gl$\".)3)p\\]o\"F07$F`gl$\".y\"ek#)G8 F07$Fegl$\".vy__-.\"F07$Fjgl$\"-[pcF07$Fihl$\".IG&oHE`F47$F]jl$\".$eG8+m_F47$Fa[m$\".dE!*p\"yaF47$ Fd[q$\".Bppc@^&F47$Ff[m$\".FCoH?j&F47$$\"?,++++++]i!*y?OZ@F`^l$\".#yVy #*\\gF47$F\\\\q$\".%*Gq?9Y'F47$$\"?,++++++D1RDfhm@F`^l$\".j9#3\"3E'F47 $$\"?,++++++](=U(Q.t@F`^l$\".vNFcj2'F47$$\"?,+++++]7Gj[GCw@F`^l$\".xx< 5'egF47$$\"?,++++++vo/B=Xz@F`^l$\".$[50K7kF47$$\"?,+++++]P4Y(zgE=#F`^l $\".a))\\i^['F47$F[\\m$\".p!4B!\\Q'F47$F`\\m$\".fTHZ%*R'F47$Fe\\m$\".R oy'F47$Fd]m$\".(fRUDUeF47$ Fi]m$\".#))Gk'[Q&F47$F^^m$\".t+Xx'yQF47$Fc^m$\".8)GO_mKF47$Fh^m$\"..3P (*f$GF47$F]_m$\".\"o5$R'QDF47$Fb_m$\".3@/8=N#F47$Fg_m$\".XjT\"fDAF47$F \\`m$\".D=hYMN#F47$Fa`m$\".zmc%Q$y#F47$Ff`m$\".d!)Ru!oNF47$F[am$\".TmS Jny%F47$F`am$\".2w\"F47$F^cm$\"/j&ep VZ?\"F47$Fccm$\"/87#*e!)47F47$Fhcm$\"/x&)zun87F47$F]dm$\"/R6&Qd^?\"F47 $Fbdm$\"/J^*4I]?\"F47$Fgdm$\"/J*3V77<\"F47$F\\em$\"/Nr^(o87\"F47$Faem$ \"/`js&G?/\"F47$Ffem$\".$efW,Z\"*F47$F[fm$\".Pml=m)eF47$F`fm$\".#f#\\i +.%F47$Fefm$\".-jZcMmSz$F-7$Fajm$\".)pDh&G(RF-7$Ffjm$\". &=I" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Verner's \"most robust\" scheme" "Verner \"most efficient\" schem e" "Sharp's scheme" "Tsitouras' scheme A" "Tsitouras' scheme B" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "Test 8 of 15 stage, ord er 9 Runge-Kutta methods " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx=x*(9-x^2)/(1+y^2)" "6#/*&%#dyG\"\"\"%#dxG!\"\"*(% \"xGF&,&\"\"*F&*$F*\"\"#F(F&,&F&F&*$%\"yGF.F&F(" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "y(0)=0" "6#/-%\"yG6#\"\"!F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "y = rho(x)/2-2/rho(x);" "6#/%\"yG,&*&-%$rhoG6#%\"xG\"\" \"\"\"#!\"\"F+*&F,F+-F(6#F*F-F-" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 7 "where " }{XPPEDIT 18 0 "rho(x) = (54*x^2-3*x^4+sqrt(64+9* x^8-324*x^6+2916*x^4))^(1/3);" "6#/-%$rhoG6#%\"xG),(*&\"#a\"\"\"*$F'\" \"#F,F,*&\"\"$F,*$F'\"\"%F,!\"\"-%%sqrtG6#,*\"#kF,*&\"\"*F,*$F'\"\")F, F,*&\"$C$F,*$F'\"\"'F,F3*&\"%;HF,*$F'F2F,F,F,*&F,F,F0F3" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 "de := 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/F 0F/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$$\"3G(H(H(p](oPF> $\"3XzNO%Rmzr&F>7$$\"3/I(H(Hj=>VF>$\"3n)f#*4g%))4rF>7$$\"373\"3\"3n&y' [F>$\"3[oK(4I9f[)F>7$$\"3oKCVK;BKaF>$\"3y!44&)e3J')*F>7$$\"3etH(HPL$Hf F>$\"3xCnB_*)['F>$\"3g[#>`Yr+B\"Fho7$$\"3Ul['[') o30(F>$\"3?yA^q$e=N\"Fho7$$\"3)yH(H(pzBf(F>$\"3)GF.C0:VY\"Fho7$$\"3m53 \"3TBT3)F>$\"3ipKusL^i:Fho7$$\"3u\\'['[U&)o')F>$\"3_Iow_)\\Zn\"Fho7$$ \"3ynvcnz>k\"*F>$\"31*=-!RxBm$\"3+Ep.yT!)o=Fho7$ $\"3\">*=*=tV]-\"Fho$\"3eT(\\)>LNc>Fho7$$\"3!z$y$y&G+\"3\"Fho$\"3!4zI& *RE\"\\?Fho7$$\"37>*=*y\"*GM6Fho$\"3tVizMbXM@Fho7$$\"3'\\'['['y))*=\"F ho$\"3fa-R]1_?AFho7$$\"3s%f%f9[%4C\"Fho$\"3#y2[?x6qH#Fho7$$\"3[KCVKm,' H\"Fho$\"3)*e,ywZ!pP#Fho7$$\"3c8N^t2A`8Fho$\"3WUCG,38dCFho7$$\"3Y'['[Y r,.9Fho$\"3@#)=I[fvCDFho7$$\"3q8N^$f)zc9Fho$\"3%3DX;Fho$\"3iG:8:1'ez#Fho7$$\"3c%f%fuKqx;Fho$\"3[Z4\"[C.I'GFh o7$$\"3=^8N\"ft,t\"Fho$\"3!\\dJru-7#HFho7$$\"3EaS0ao>'y\"Fho$\"3!*)Rs, )f8\")HFho7$$\"3tcnvcA'p$=Fho$\"3`s'\\F\\'[LIFho7$$\"3363\"3DiC*=Fho$ \"3wC-.o]f)3$Fho7$$\"3$)******>MoW>Fho$\"3Mca,!\\@%QJFho7$$\"3!*['[')e p#**>Fho$\"3Or<<>%)R)=$Fho7$$\"31Yf%faPE0#Fho$\"3!)Qt2wy;NKFho7$$\"3C^ 8N^)3&3@Fho$\"3Ik#>&=s*=G$Fho7$$\"3E>*=*e&>B;#Fho$\"3HYm&4R?ZK$Fho7$$ \"3_f%f%z([t@#Fho$\"3-!>y6]piO$Fho7$$\"3oq-FIB#>F#Fho$\"3%>b4%)\\*>0MF ho7$$\"3vvcnb(p?K#Fho$\"35nW;n-%*QMFho7$$\"3eCVKkVazBFho$\"3))))QO[-#QDFho$\"3]n/[Ud8hNFho7$$\"3AdnvO#pkf#Fho$\"3[Y' [5$oF(e$Fho7$$\"3'>;i@A4pk#Fho$\"3e\"*)HR3Wug$Fho7$$\"3$ovcn0fTq#Fho$ \"3E1Ex&>gui$Fho7$$\"3;Yf%f1Ojv#Fho$\"3)*3G:eW%Hk$Fho7$$\"3[aS09&4M\"G Fho$\"3#G5xO+Znl$Fho7$$\"3OnvcZUliGFho$\"3?xe_?\\\"fm$Fho7$$\"3163\"3D Q(=HFho$\"3I;`e!RAJn$Fho7$$\"3k*=*=HE\"H(HFho$\"3IP\"\\xLemn$Fho7$$\"3 %pvcnh^q-$Fho$\"3e^O4S8lwOFho7$$\"3-dnvO9*43$Fho$\"3y,I^j&GHn$Fho7$$\" 3m(H(H<2\"G8$Fho$\"3]\\w?2VllOFho7$$\"3'>*=*=tG))=$Fho$\"3#R\\Gi]bMl$F ho7$$\"3-A;ihz@UKFho$\"3q^o+4DNPOFho7$$\"3!>*=*=h2%)H$Fho$\"3%>QOHhJ`h $Fho7$$\"3A;i@wFF\\LFho$\"3YxQ[8&f0f$Fho7$$\"3)\\'['[I)[0MFho$\"3q>yn% )HPdNFho7$$\"3c0aS&)HLfMFho$\"3a59(yGT$>NFho7$$\"3['['[1m/8NFho$\"3uI! *3pTpuMFho7$$\"3E#*=*=p]\"pNFho$\"3s<,a='3,U$Fho7$$\"37.Fq-Y#3i$Fho$\" 3w<#3Q&zxhLFho7$$\"31wcnbdutOFho$\"3?^'pWBqHH$Fho7$$\"32dnv'*p'o+i*\\/FFFho7$$\"\"%F)$\"3CxC=rRoRDFho-%'COLOURG6&%$RGB G$\"#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 the methods and gives the " }{TEXT 260 22 "root mean square error" }{TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 682 "W := \+ (x,y) -> x*(9-x^2)/(1+y^2): hh := 0.01: numsteps := 400: x0 := 0: y0 : = 0:\nmatrix([[`slope field: `,W(x,y)],[`initial point: `,``(x0,y0)] ,[`step width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Verner's \"most robust\" scheme`,`Verner \"most efficient\" scheme` ,`Sharp's scheme`,\n`Tsitouras' scheme A`,`Tsitouras' scheme B`]: errs := []:\nDigits := 30:\nfor ct to 5 do\n Wn_RK9_||ct := RK9_||ct(W(x ,y),x,y,x0,y0,hh,numsteps,false);\n sm := 0: numpts := nops(Wn_RK9_| |ct):\n for ii to numpts do\n sm := sm+(Wn_RK9_||ct[ii,2]-w(Wn_ RK9_||ct[ii,1]))^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)] ;\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G*(% \"xG\"\"\",&\"\"*F+*$)F*\"\"#F+!\"\"F+,&F+F+*$)%\"yGF0F+F+F17$%0initia l~point:~G-%!G6$\"\"!F;7$%/step~width:~~~G$F+!\"#7$%1no.~of~steps:~~~G \"$+%Q)pprint426\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%>Verner's~\"most~robust\"~sche meG$\"+=tiNw!#J7$%?Verner~\"most~efficient\"~schemeG$\"+p9W'[(F+7$%/Sh arp's~schemeG$\"+F+(Q*)*F+7$%4Tsitouras'~scheme~AG$\"+\"Qi6a\"!#I7$%4T sitouras'~scheme~BG$\"+o\\*>\\&F+Q)pprint436\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code constr ucts " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for solutio ns based on each of the Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each of the methods at the p oint where " }{XPPEDIT 18 0 "x = 3.499;" "6#/%\"xG-%&FloatG6$\"%*\\$! \"$" }{TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 613 "W := (x,y) -> x*(9-x^2)/(1+y^2): hh := 0.01: numstep s := 400: x0 := 0: y0 := 0:\nmatrix([[`slope field: `,W(x,y)],[`init ial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,nu msteps]]);``;\nmthds := [`Verner's \"most robust\" scheme`,`Verner \"m ost efficient\" scheme`,`Sharp's scheme`,\n`Tsitouras' scheme A`,`Tsit ouras' scheme B`]: errs := []:\nDigits := 35:\nfor ct to 5 do\n wn_R K9_||ct := RK9_||ct(W(x,y),x,y,x0,y0,hh,numsteps,true);\nend do:\nxx : = 3.499: wxx := evalf(w(xx)):\nfor ct to 5 do\n errs := [op(errs),ab s(wn_RK9_||ct(xx)-wxx)];\nend do:\nDigits := 10:\nlinalg[transpose]([m thds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7 $%0slope~field:~~~G*(%\"xG\"\"\",&\"\"*F+*$)F*\"\"#F+!\"\"F+,&F+F+*$)% \"yGF0F+F+F17$%0initial~point:~G-%!G6$\"\"!F;7$%/step~width:~~~G$F+!\" #7$%1no.~of~steps:~~~G\"$+%Q)pprint466\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%>Verner' s~\"most~robust\"~schemeG$\"+B#4T]&!#K7$%?Verner~\"most~efficient\"~sc hemeG$\"+`#*f2qF+7$%/Sharp's~schemeG$\"+a\"H'e=!#L7$%4Tsitouras'~schem e~AG$\"+OLX+[!#J7$%4Tsitouras'~scheme~BG$\"+s4+gGF+Q)pprint476\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " } {TEXT 260 22 "root mean square error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0,4]" "6#7$\"\"!\"\"%" }{TEXT -1 82 " of each Ru nge-Kutta method is estimated as follows using the special procedure \+ " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perform numerical integration \+ by the 7 point Newton-Cotes method over 200 equal subintervals." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 363 "mthds := [`Verner's \"most \+ robust\" scheme`,`Verner \"most efficient\" scheme`,`Sharp's scheme`, \n`Tsitouras' scheme A`,`Tsitouras' scheme B`]: errs := []:\nDigits := 20:\nfor ct to 5 do\n sm := NCint((w(x)-'wn_RK9_||ct'(x))^2,x=0..4, adaptive=false,numpoints=7,factor=200);\n errs := [op(errs),sqrt(sm/ 4)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%>Verner's~\"most~ robust\"~schemeG$\"+\\OL@r!#J7$%?Verner~\"most~efficient\"~schemeG$\"+ Wh/OpF+7$%/Sharp's~schemeG$\"+'Q/Qq*F+7$%4Tsitouras'~scheme~AG$\"+\"=; 9]\"!#I7$%4Tsitouras'~scheme~BG$\"+RLI-_F+Q)pprint486\"" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following erro r graphs are constructed using the numerical procedures for the soluti ons." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 448 "evalf[30](plot(['wn _RK9_1'(x)-w(x),'wn_RK9_2'(x)-w(x),'wn_RK9_3'(x)-w(x),\n'wn_RK9_4'(x)- w(x),'wn_RK9_5'(x)-w(x)],x=0..4,-2e-21..6e-21,\ncolor=[COLOR(RGB,.45,0 ,1),COLOR(RGB,.85,.45,0),COLOR(RGB,0,.65,0),COLOR(RGB,0,.55,.95),red], \nlegend=[`Verner's \"most robust\" scheme`,`Verner \"most efficient\" scheme`,`Sharp's scheme`,\n`Tsitouras' scheme A`,`Tsitouras' scheme B `],font=[HELVETICA,9],\ntitle=`error curves for 15 stage order 9 Runge -Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 855 532 532 {PLOTDATA 2 "6+-%'CURVESG6%7`r7$$\"\"!F)F(7$$\"?LLLLLLLLLLL3VfV!#J$\"* nUz1&!#I7$$\"?mmmmmmmmmmm;')=()F-$\"*+#[:*)F07$$\"?0++++++++DJ59n'*F-$ \"*3LIA*F07$$\"?MLLLLLLLLeR?ah5F0$\"*'eA&4*F07$$\"?nmmmmmmm;/w*pj:\"F0 $\"*+8oU)F07$$\"?+++++++++]7z>^7F0$\"*k:/9(F07$$\"?mmmmmmmmmT&y`3W\"F0 $\"*c4jo#F07$$\"?LLLLLLLLLLe'40j\"F0$!*3&=*R$F07$$\"?++++++++]i!f`rt\" F0$!*UdpG'F07$$\"?mmmmmmmmm\"H_(zV=F0$!*\"*3hW)F07$$\"?LLLLLLL$e*)f]e/ (=F0$!*he-Z)F07$$\"?++++++++D1*[>r*=F0$!*xy%[\"*F07$$\"?mmmmmmm;a8s/yB >F0$!*O#ze$*F07$$\"?LLLLLLLL$3_XT/&>F0$!*v$zW$*F07$$\"?+++++++]7GQC5x> 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F0$\"*U'\\.)*F07$F<$\"*TSYz*F07$FA$\"*0w42*F07$FF$\"*Ov^m(F07$Fh^p$\"* G8?`&F07$FK$\"*Po9q#F07$F`_p$!)a))[pF07$FP$!*'\\%[S%F07$FU$!*gE'[!)F07 $FZ$!+RAi96F07$Fho$!+6#elJ\"F07$F\\q$!+`K\"*f8F07$Faq$!+3k707F07$Ffq$! *mQIJ)F07$$\"?+++++++]i:NU.(H#F0$!*KFqQ%F07$F[r$!*>&[DHF07$$\"?MLLLLLL $3-8?c.N#F0$!*nA9\"HF07$F`r$!*W=SS#F07$$\"?nmmmmmm;zWn\"yOS#F0$\"*ePW8 %F07$Fer$\"*CYP7%F07$$\"?+++++++]PfL,+dCF0$\"*KN29%F07$Fjr$\"*\"G*4T&F 07$$\"?nmmmmmmm;/E))\\5DF0$\"+FC+!>\"F07$F_s$\"+,Qf'=\"F07$$\"?nmmmmmm m;zWU\"F07$Fds$\"+[l_z9F07$$\"?nmmmmmmm;aj'\\yh#F0$\"+!* ffS>F07$Fis$\"+R\"HY$>F07$$\"?nmmmmmmm;H#3D:n#F0$\"+zK\"=&>F07$F^t$\"+ ')RexCF07$Fct$\"+cJ(*)*HF07$Fht$\"+B[OiJF07$F]u$\"+-%yB0$F07$Fbu$\"+== E%p#F07$Fgu$\"+@^*pg#F07$F\\v$\"+V`8X@F07$Fav$\"+#>9Z$>F07$Ffv$\"+*zO7 [\"F07$F[w$\"+1naB5F07$F`w$\"*fXJ$yF07$Few$\"*MvRP(F07$Fjw$\"*s$Qi7F07 $F_x$\"*5:cD\"F07$Fdx$\"*Av.B\"F07$$\"?nmmmmmm;/^1#fGe$F0$\")rMvNF07$$ \"?++++++++v$4Op&4OF0$!*H*odWF07$$\"?MLLLLLL$ek`^zij$F0$!*@5KV%F07$Fix $!*+?bX%F07$$\"?+++++++](=U#)*p*o$F0$!*D'zWfF07$$\"?MLLLLLLLeky*4kr$F0 $!*D=X!*)F07$$\"?nmmmmmm;H2L,7VPF0$!*g]W&))F07$F^y$!*.\"e)*))F07$Fcy$! +=)zp>\"F07$Fhy$!+5()*yO\"F07$F]z$!+m*>RV\"F07$Fbz$!+Of;D9F07$Fgz$!+Fy f;9F07$F\\[l$!+d^y49F07$Ff[l$!+;E!RF07$Fbjl$!)go2YF07$Fgjl$!)ot\"Q'F07$F\\[m$ !)yR7)*F07$Fa[m$!*e0>H\"F07$Ff[m$!*u#GxLQF07$Fe\\m$!*#G-8gF07$Fj\\m$!*+)pd)*F07$F_]m$!+I8T\"H\"F07$Fd]m$!+! >l3u\"F07$Fi]m$!+WA@\"=#F07$F^^m$!+s66wCF07$Fc^m$!+-3R$*GF07$Fh^m$!+I \"fK\"HF07$F]_m$!+SN)*pHF07$Fb_m$!+[-tsQF0-%'COLOURG6&Fi_m$\"*++++\"! \")F(F(-F``m6#%4Tsitouras'~scheme~BG-%&TITLEG6#%Verror~curves~for~15~s tage~order~9~Runge-Kutta~methodsG-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLAB ELSG6$Q\"x6\"Q!F_^s-%%VIEWG6$;F(Fb_m;$F\\`m!#@$\"\"'Fg^s" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Verner's \"most robust\" scheme" "Verner \"most efficient\" scheme" "Sharp's scheme" "Tsitoura s' scheme A" "Tsitouras' scheme B" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "Test 9 of 15 stage, order 9 Runge-Kutta methods " }} {PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "dy/dx=-(1+cos(2*x)) *y^3" "6#/*&%#dyG\"\"\"%#dxG!\"\",$*&,&F&F&-%$cosG6#*&\"\"#F&%\"xGF&F& F&*$%\"yG\"\"$F&F(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(0) = sqrt(2); " "6#/-%\"yG6#\"\"!-%%sqrtG6#\"\"#" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = 1/sqrt(sin(2*x)+2*x+1/2)" "6#/%\"yG*&\"\"\"F&-%%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#%\"xGF,,$*&,&\"\"\"F0-%$cosG6# ,$*&\"\"#F0F,F0F0F0F0)F)\"\"$F0!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#icG/-%\"yG6#\"\"!*$\"\"##\"\"\"F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*&\"\"\"F)*$,(*(\"\"#F)-%$cosGF&F)-%$sinGF&F)F)*&F -F)F'F)F)#F)F-F)#F)F-!\"\"" }}{PARA 13 "" 1 "" {GLPLOT2D 503 318 318 {PLOTDATA 2 "6&-%'CURVESG6$7Y7$$\"\"!F)$\"3:&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%=M 2W@)FJ7$$\"3%)***\\(oX*y9$FJ$\"31u2v$Q9&GwFJ7$$\"3z***\\P9CAu$FJ$\"3=X IMTf7+sFJ7$$\"3!)***\\P*zhdVFJ$\"3P$G(zQ8#4%oFJ7$$\"31++v$>fS*\\FJ$\"3 X'3%RcqqPlFJ7$$\"3$)***\\(=$f%GcFJ$\"3mYY%G?7\"*G'FJ7$$\"3Q+++Dy,\"G'F J$\"3?u@-35zxgFJ7$$\"33++]7[X$ocbFJ7$$\"3))***\\PpnsM*FJ$\"3!\\;$Q)fJR[&FJ7$$\"3,++] siL-5F,$\"3&3j/q(Qq8aFJ7$$\"3-+++!R5'f5F,$\"3q`:6QhHm`FJ7$$\"3)***\\P/ QBE6F,$\"3@Igj*yDKK&FJ7$$\"3!******\\\"o?&=\"F,$\"3i/K.-M\\%H&FJ7$$\"3 1+]Pa&4*\\7F,$\"3OjcS#)ygr_FJ7$$\"33+]7j=_68F,$\"3'e4m\")R`oD&FJ7$$\"3 3++vVy!eP\"F,$\"3a@U-1/NZ_FJ7$$\"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\"3g SMou)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@OtBF,$\"3m09))4iC_[FJ7$$\" 3')**\\PfL'zV#F,$\"3%Gjf])o8tZFJ7$$\"3>+++!*>=+DF,$\"3[G/4+_V#p%FJ7$$ \"3-++DE&4Qc#F,$\"3!**R*=7x[1YFJ7$$\"3=+]P%>5pi#F,$\"3f7E:iH**=XFJ7$$ \"39+++bJ*[o#F,$\"3cgVvc$ovV%FJ7$$\"33++Dr\"[8v#F,$\"3Ln\\jDQ5WVFJ7$$ \"3++++Ijy5GF,$\"3OZ!)Q%zK7E%FJ7$$\"31+]P/)fT(GF,$\"3)*4_&egIW<%FJ7$$ \"31+]i0j\"[$HF,$\"3qns]&)H\\$4%FJ7$$\"\"$F)$\"3ntdq;jW4SFJ-%'COLOURG6 &%$RGBG$\"#5!\"\"F(F(-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6$%\"xG% %y(x)G-%%VIEWG6$;F(Fc\\l;F($\"$U\"!\"#" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The following code constructs a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " based on each of t he methods and gives the " }{TEXT 260 22 "root mean square error" } {TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 695 "M := (x,y) -> -(1+cos(2*x))*y^3: hh := 0.01: numsteps := 300: x0 := 0: y0 := sqrt(2):\nmatrix([[`slope field: `,M(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numst eps]]);``;\nmthds := [`Verner's \"most robust\" scheme`,`Verner \"most efficient\" scheme`,`Sharp's scheme`,\n`Tsitouras' scheme A`,`Tsitour as' scheme B`]: errs := []:\nDigits := 20:\nfor ct to 5 do\n Mn_RK9_ ||ct := RK9_||ct(M(x,y),x,y,x0,evalf(y0),hh,numsteps,false);\n sm := 0: numpts := nops(Mn_RK9_||ct):\n for ii to numpts do\n sm := \+ sm+(Mn_RK9_||ct[ii,2]-m(Mn_RK9_||ct[ii,1]))^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpose ]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6 #7&7$%0slope~field:~~~G,$*&,&\"\"\"F,-%$cosG6#,$*&\"\"#F,%\"xGF,F,F,F, )%\"yG\"\"$F,!\"\"7$%0initial~point:~G-%!G6$\"\"!*$F2#F,F27$%/step~wid th:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$+$Q)pprint496\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6 #7'7$%>Verner's~\"most~robust\"~schemeG$\"+&*e7`W!#F7$%?Verner~\"most~ efficient\"~schemeG$\"+$o)p1YF+7$%/Sharp's~schemeG$\"+p9*>v%F+7$%4Tsit ouras'~scheme~AG$\"+l-x65F+7$%4Tsitouras'~scheme~BG$\"+%3$[#e$F+Q)ppri nt506\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code constructs " }{TEXT 260 20 "numerical procedure s" }{TEXT -1 56 " for solutions based on each of the Runge-Kutta schem es." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each of the methods at the point where " }{XPPEDIT 18 0 "x = 2.999; " "6#/%\"xG-%&FloatG6$\"%**H!\"$" }{TEXT -1 16 " is also given." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 626 "M := (x,y) -> -(1+cos(2*x)) *y^3: hh := 0.01: numsteps := 300: x0 := 0: y0 := sqrt(2):\nmatrix([[` slope field: `,M(x,y)],[`initial point: `,``(x0,y0)],[`step width: \+ `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Verner's \"mo st robust\" scheme`,`Verner \"most efficient\" scheme`,`Sharp's scheme `,\n`Tsitouras' scheme A`,`Tsitouras' scheme B`]: errs := []:\nDigits \+ := 30:\nfor ct to 5 do\n mn_RK9_||ct := RK9_||ct(M(x,y),x,y,x0,evalf (y0),hh,numsteps,true);\nend do:\nxx := 2.999: mxx := evalf(m(xx)):\nf or ct to 5 do\n errs := [op(errs),abs(mn_RK9_||ct(xx)-mxx)];\nend do :\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,$*&,&\"\"\"F ,-%$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)pprint516\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%>Verner's~\"most~robust\"~sche meG$\"+g/c;z!#G7$%?Verner~\"most~efficient\"~schemeG$\"+!)*oJ=)F+7$%/S harp's~schemeG$\"+7$fRX)F+7$%4Tsitouras'~scheme~AG$\"+\\T?g=F+7$%4Tsit ouras'~scheme~BG$\"+^eskjF+Q)pprint526\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean squa re error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 3]; " "6#7$\"\"!\"\"$" }{TEXT -1 82 " of each Runge-Kutta method is estim ated as follows using the special procedure " }{TEXT 0 5 "NCint" } {TEXT -1 98 " to perform numerical integration by the 7 point Newton- Cotes method over 150 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 363 "mthds := [`Verner's \"most robust\" scheme`,`Verner \+ \"most efficient\" scheme`,`Sharp's scheme`,\n`Tsitouras' scheme A`,`T sitouras' scheme B`]: errs := []:\nDigits := 20:\nfor ct to 5 do\n s m := NCint((m(x)-'mn_RK9_||ct'(x))^2,x=0..3,adaptive=false,numpoints=7 ,factor=150);\n errs := [op(errs),sqrt(sm/3)];\nend do:\nDigits := 1 0:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%>Verner's~\"most~robust\"~schemeG$\"+; Ox*Q%!#F7$%?Verner~\"most~efficient\"~schemeG$\"+')e`QXF+7$%/Sharp's~s chemeG$\"+,69$o%F+7$%4Tsitouras'~scheme~AG$\"+7!es'**!#G7$%4Tsitouras' ~scheme~BG$\"+?_DINF+Q)pprint536\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 92 "The following error graphs are construct ed using the numerical procedures for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 450 "evalf[25](plot([m(x)-'mn_RK9_1'(x),m(x)- 'mn_RK9_2'(x),m(x)-'mn_RK9_3'(x),\nm(x)-'mn_RK9_4'(x),m(x)-'mn_RK9_5'( x)],x=0..3,-5e-18..2.2e-17,\ncolor=[COLOR(RGB,.45,0,1),COLOR(RGB,.85,. 45,0),COLOR(RGB,0,.65,0),COLOR(RGB,0,.55,.95),red],\nlegend=[`Verner's \"most robust\" scheme`,`Verner \"most efficient\" scheme`,`Sharp's s cheme`,\n`Tsitouras' scheme A`,`Tsitouras' scheme B`],font=[HELVETICA, 9],\ntitle=`error curves for 15 stage order 9 Runge-Kutta methods`)); " }}{PARA 13 "" 1 "" {GLPLOT2D 795 594 594 {PLOTDATA 2 "6+-%'CURVESG6% 7fo7$$\"\"!F)F(7$$\":++++++DJ?$[V?!#F$\"\"#!#C7$$\":++++++]iSmp3%F-$\" %KAF07$$\":++++++v$4'\\/8'F-$\"'087F07$$\":+++++++D\"G$R<)F-$\"(x6.#F0 7$$\":+++++]i:gT<-\"!#E$\")]uW9F07$$\":++++++v=#**3E7FC$\")kg79F07$$\" :+++++](=U#Q/V\"FC$\"),\"=Q\"F07$$\":+++++++Dc'yM;FC$\")4\\f8F07$$\":+ ++++]7G)[8R=FC$\")aqT9F07$$F,FC$\")/C\\>F07$$\":+++++]PM_JyC#FC$\")a)) 3>F07$$\":++++++]P%)z@X#FC$\")d.q=F07$$\":+++++]iS;Gll#FC$\")(Qt$=F07$ $\":++++++vV[w3'GFC$\")[fo=F07$$\":+++++](o/[AlIFC$\")u+t?F07$$\":++++ +++]7t&pKFC$\"):)G.#F07$$\":+++++++vofV!\\FC$\")w#*H>F07$$\":++++++++D 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nf&4\"F07$FV$\")U(3;\"F07$Fen$\")3^p:F07$Fin$\")u,P:F07$F^o$\")Dt0:F07 $Fco$\")kJz9F07$Fho$\")A!R]\"F07$F]p$\")M5o;F07$Fbp$\")\\\"ej\"F07$Fgp $\")`4_:F07$F\\q$\").zF9F07$Faq$\")>Q38F07$Ffq$\")\"F07$F[r$\")%e5 4\"F07$Far$\")v)G+\"F07$Ffr$\")Ek^\"*F]r7$F[s$\")26*R)F]r7$F`s$\")eRWx F]r7$Fes$\")%z3<(F]r7$Fjs$\")4)y@'F]r7$F_t$\")0wnaF]r7$Fdt$\")C[o[F]r7 $Fit$\")\"H&zVF]r7$F^u$\")ZU#o$F]r7$Fcu$\")1MeJF]r7$Fhu$\")grcFF]r7$F] v$\")B0aCF]r7$Fbv$\")Z*[@#F]r7$Fgv$\")#[!\\?F]r7$F\\w$\")AS+>F]r7$Faw$ \")&REy\"F]r7$Ffw$\")Hj#p\"F]r7$F[x$\")l,F;F]r7$F`x$\")jIl:F]r7$Fex$\" )KaC:F]r7$Fjx$\")!H\")[\"F]r7$F_y$\")?;k9F]r7$Fdy$\")tDX9F]r7$Fiy$\")f :L9F]r7$F^z$\")xRD9F]r7$Fcz$\")kf@9F]r7$Fhz$\")h$*>9F]r7$F][l$\")Ak>9F ]r7$Fb[l$\")h`>9F]r7$Fg[l$\")?]=9F]r7$F\\\\l$\")KH:9F]r7$Fa\\l$\")\"f* 39F]r7$Ff\\l$\")#3*)R\"F]r7$F[]l$\")GM#Q\"F]r7$F`]l$\")*R?O\"F]r7$Fe]l $\")%zVL\"F]r7$Fj]l$\")v'RI\"F]r7$F_^l$\").=l7F]r7$Fd^l$\")%4RA\"F]r7$ Fi^l$\")ydw6F]r7$F^_l$\")J0F6F]r7$Fc_l$\");#G2\"F]r7$Fh_l$\")0K>5F]r7$ F]`l$\"()HV'*F]r7$Fb`l$\"(\"=/\"*F]r7$Fg`l$\"(q3i)F]r7$F\\al$\"(Sv3)F] r7$Faal$\"(kMj(F]r7$Ffal$\"(;k<(F]r7$F[bl$\"(Hqw'F]r7$F`bl$\"(6(ejF]r- %'COLOURG6&Fgbl$\"*++++\"!\")F(F(-F^cl6#%4Tsitouras'~scheme~BG-%&TITLE G6#%Verror~curves~for~15~stage~order~9~Runge-Kutta~methodsG-%%FONTG6$% *HELVETICAG\"\"*-%+AXESLABELSG6$Q\"x6\"Q!F_]p-%%VIEWG6$;F(F`bl;$!\"&!# =$\"#AFh]p" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "V erner's \"most robust\" scheme" "Verner \"most efficient\" scheme" "Sh arp's scheme" "Tsitouras' scheme A" "Tsitouras' scheme B" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 49 "Test 10 of 15 stage, order 9 Ru nge-Kutta methods " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = -(2*sin(5*x)+3*cos(7*x))*sinh(y);" "6#/*&%#dyG\"\"\"%#dxG!\" \",$*&,&*&\"\"#F&-%$sinG6#*&\"\"&F&%\"xGF&F&F&*&\"\"$F&-%$cosG6#*&\"\" (F&F3F&F&F&F&-%%sinhG6#%\"yGF&F(" }{TEXT -1 5 " , " }{XPPEDIT 18 0 " y(0)=sqrt(5)/2" "6#/-%\"yG6#\"\"!*&-%%sqrtG6#\"\"&\"\"\"\"\"#!\"\"" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 207 "de : = diff(y(x),x)=-(2*sin(5*x)+3*cos(7*x))*sinh(y(x));\nic := y(0)=sqrt(5 )/2;\ndsolve(\{de,ic\},y(x));\nsimplify(convert(%,exp));\np := unapply (rhs(%),x):\nplot(p(x),x=0..5,font=[HELVETICA,9],labels=[`x`,`y(x)`]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"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 "6#/-%\"yG6#%\"xG-%#lnG6#-%%tanhG6#,**&#\"\"\"\"\"& F1-%$cosG6#,$*&F2F1F'F1F1F1!\"\"*&#\"\"$\"#9F1-%$sinG6#,$*&\"\"(F1F'F1 F1F1F1#F1F2F1*&#F1\"\"#F1-F)6#,$*&,&-%$expG6#,$*&FFF8F2FEF1F1F1F1F1,&F LF1F1F8F8F8F1F1" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%#ln 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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\"$+&Q(pprint06\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #K%'matrixG6#7'7$%>Verner's~\"most~robust\"~schemeG$\"+L!ofP\"!#E7$%?V erner~\"most~efficient\"~schemeG$\"+'yrEX\"F+7$%/Sharp's~schemeG$\"+:L x\\FF+7$%4Tsitouras'~scheme~AG$\"+pjc];!#F7$%4Tsitouras'~scheme~BG$\"+ 2wm-))F8Q(pprint16\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code constructs " }{TEXT 260 20 "numeric al procedures" }{TEXT -1 56 " for solutions based on each of the Runge -Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each of the methods at the point where " }{XPPEDIT 18 0 "x = 4.999;" "6#/%\"xG-%&FloatG6$\"%**\\!\"$" }{TEXT -1 16 " is also \+ given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 643 "P := (x,y) -> -(2 *sin(5*x)+3*cos(7*x))*sinh(y): hh := 0.01: numsteps := 500: x0 := 0: y 0 := sqrt(5)/2:\nmatrix([[`slope field: `,P(x,y)],[`initial point: ` ,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);`` ;\nmthds := [`Verner's \"most robust\" scheme`,`Verner \"most efficien t\" scheme`,`Sharp's scheme`,\n`Tsitouras' scheme A`,`Tsitouras' schem e B`]: errs := []:\nDigits := 30:\nfor ct to 5 do\n pn_RK9_||ct := R K9_||ct(P(x,y),x,y,x0,evalf(y0),hh,numsteps,true);\nend do:\nxx := 4.9 99: pxx := evalf(p(xx)):\nfor ct to 5 do\n errs := [op(errs),abs(pn_ RK9_||ct(xx)-pxx)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds, evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0sl ope~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\"$+&Q)pprint566\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%! G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%>Verner's~\"most ~robust\"~schemeG$\"+\"ff-S$!#E7$%?Verner~\"most~efficient\"~schemeG$ \"+FM(od$F+7$%/Sharp's~schemeG$\"+\"3xVv(F+7$%4Tsitouras'~scheme~AG$\" +q8qg6!#F7$%4Tsitouras'~scheme~BG$\"+wY[U?F+Q)pprint576\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square error" }{TEXT -1 20 " over the interval " } {XPPEDIT 18 0 "[0, 5];" "6#7$\"\"!\"\"&" }{TEXT -1 82 " of each Runge -Kutta method is estimated as follows 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 0 363 "mthds := [`Verner's \"most \+ robust\" scheme`,`Verner \"most efficient\" scheme`,`Sharp's scheme`, \n`Tsitouras' scheme A`,`Tsitouras' scheme B`]: errs := []:\nDigits := 20:\nfor ct to 5 do\n sm := NCint((p(x)-'pn_RK9_||ct'(x))^2,x=0..5, adaptive=false,numpoints=7,factor=200);\n errs := [op(errs),sqrt(sm/ 5)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%>Verner's~\"most~ robust\"~schemeG$\"+XnTf8!#E7$%?Verner~\"most~efficient\"~schemeG$\"+c O'\\V\"F+7$%/Sharp's~schemeG$\"+?_f@FF+7$%4Tsitouras'~scheme~AG$\"+&ph /o\"!#F7$%4Tsitouras'~scheme~BG$\"+-29#p)F8Q)pprint586\"" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following erro r graphs are constructed using the numerical procedures for the soluti ons." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 455 "evalf[30](plot(['pn _RK9_1'(x)-p(x),'pn_RK9_2'(x)-p(x),'pn_RK9_3'(x)-p(x),\n'pn_RK9_4'(x)- p(x),'pn_RK9_5'(x)-p(x)],x=0..2.2,-3.5e-19..1.05e-18,\ncolor=[COLOR(RG B,.45,0,1),COLOR(RGB,.85,.45,0),COLOR(RGB,0,.65,0),COLOR(RGB,0,.55,.95 ),red],\nlegend=[`Verner's \"most robust\" scheme`,`Verner \"most effi cient\" scheme`,`Sharp's scheme`,\n`Tsitouras' scheme A`,`Tsitouras' s cheme B`],font=[HELVETICA,9],\ntitle=`error curves for 15 stage order \+ 9 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 746 518 518 {PLOTDATA 2 "6+-%'CURVESG6%7^t7$$\"\"!F)F(7$$\"?mmmmmmmmm;zM%))>\"!#J$ !+ccW\"3$!#H7$$\"?LLLLLLLLLLepo(R#F-$!++*e\"*y%F07$$\"?mmmmmmmm\"HKRUv a#F-$!+%zrlv%F07$$\"?++++++++]7GyR(p#F-$!+oH$F-$!,iidHg&FJ7$$\"?+++++++++]P/`'f$F-$!,R$e?CbFJ7$$\"?LLLLLLLLL3x @&f>%F-$!,h3P7z&FJ7$$\"?mmmmmmmmmm;RP&z%F-$!,P'pcRcFJ7$$\"?+++++++++v= &)e\")oF-$!,(e[2n\\FJ7$$\"?LLLLLLLLL$37.y'*)F-$!,C@8$GSFJ7$$\"?mmmmmmm mTgP')395FJ$!,c%QC]OFJ7$$\"?++++++++]7jpRJ6FJ$!,*)*==3MFJ7$$\"?MLLLLLL Lek)G0([7FJ$!,g(yBRKFJ7$$\"?nmmmmmmmm;9O,m8FJ$!,\")p`/8$FJ7$$\"?nmmmmm mmmmca=-;FJ$!,,gPd7$FJ7$$\"?nmmmmmmmm;*Hd$Q=FJ$!,\"[]U!4$FJ7$$\"?LLLLL LLLL3$FJ$!,H^S2M#FJ7$$\"?mmmmmmmmmTv+JiOFJ$!,tE;0C#FJ7$$\"?++++++++ +vLo`FTFJ$!,]]Z4E#FJ7$$\"?LLLLLLLLLLQ(zgg%FJ$!,%\\wG\"=#FJ7$$\"?mmmmmm mmm;*e!eF]FJ$!,@&)3)))>FJ7$$\"?++++++++++:24-bFJ$!,O!eN\"p\"FJ7$$\"?++ 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G. Lether: Mathematics of Computation, Vol. 20, no. 9 5, (July 1966) page 382. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=exp(-x)/(x-1)^2" "6#/ *&%#dyG\"\"\"%#dxG!\"\"*&-%$expG6#,$%\"xGF(F&*$,&F.F&F&F(\"\"#F(" } {TEXT -1 2 " " }{XPPEDIT 18 0 "cos(1/(x-1))-y" "6#,&-%$cosG6#*&\"\"\" F(,&%\"xGF(F(!\"\"F+F(%\"yGF+" }{TEXT -1 5 ", " }{XPPEDIT 18 0 "y(0 )=sin*1" "6#/-%\"yG6#\"\"!*&%$sinG\"\"\"F*F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 2 " \+ " }{XPPEDIT 18 0 "y = -exp(-x)*sin(1/(x-1))" "6#/%\"yG,$*&-%$expG6#,$% \"xG!\"\"\"\"\"-%$sinG6#*&F-F-,&F+F-F-F,F,F-F," }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 185 "de := diff(y(x),x)=exp(-x)/(x-1)^2*cos(1/(x-1))-y(x);\nic := y(0) =sin(1);\ndsolve(\{de,ic\},y(x));\nq := unapply(rhs(%),x):\nplot(q(x), x=0..1-1/(6*Pi),font=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,&*(-%$expG6#,$F ,!\"\"\"\"\",&F,F4F4F3!\"#-%$cosG6#*&F4F4F5F3F4F4F)F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!-%$sinG6#\"\"\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,$*&-%$expG6#,$F'!\"\"\"\"\"-%$sin G6#*&F/F/,&F'F/F/F.F.F/F." }}{PARA 13 "" 1 "" {GLPLOT2D 603 396 396 {PLOTDATA 2 "6&-%'CURVESG6$7[r7$$\"\"!F)$\"30l*y![)4ZT)!#=7$$\"3#>=\"* )>z2k?!#>$\"3m%p=Qu6DN)F,7$$\"3kMQU\"3@+'QF0$\"3m=O<,5$yO#)F,7$$\"38UQ!)p4'G\"zF0$\"3/Z^$zTlU<)F,7$$\" 3BY$*R0>JO**F0$\"36ty1)z*36\")F,7$$\"3wbXC%*4B\"=\"F,$\"3A;o(=P!Q^!)F, 7$$\"3M!)fxC(zaP\"F,$\"3E8^!ydw!))zF,7$$\"3kgswR?Pw:F,$\"3T8>lD8j?zF,7 $$\"3Qnlb!4?mx\"F,$\"3%p4\\s^P4&yF,7$$\"3OsvSC)*f#)>F,$\"3/$H(=wa6wxF, 7$$\"3)Rk+h)o-k@F,$\"3LR8d$>`qq(F,7$$\"3Q^Vo'yq#oBF,$\"3YB)Qc;#3DwF,7$ $\"3?0sMKLNtDF,$\"3,;%fG`C(F,7$$ \"3S+dSsVlWLF,$\"3&36sy[X09(F,7$$\"3EOur83&\\b$F,$\"37)QgTzpp+(F,7$$\" 3c#**Qwz)4TPF,$\"3M]xfz#>l(oF,7$$\"3wx#p)QELXRF,$\"3UR-VbS%zr'F,7$$\"3 \"yy;Gb6)RTF,$\"3%40quzD%\\lF,7$$\"3p2KM(*)HFM%F,$\"3W'4!o9@F_jF,7$$\" 3`G+(=Gs!HXF,$\"3S.Rv)o&))[hF,7$$\"3&\\%Rcec>>`F,7$$\"39 pTj@J(oJ&F,$\"3-j_%RM!Rk\\F,7$$\"3QD(p)Qdl>bF,$\"34_#)R=svWXF,7$$\"3#) Qm@o*Q!=dF,$\"3Iba)Q0\")Q2%F,7$$\"3#oP&GV\\)*4fF,$\"3;HPYk\"[Jb$F,7$$ \"3qUjqA#3J7'F,$\"3iI9Us9n*)GF,7$$\"3-koIo*3YJ'F,$\"3G0-$>\\f\"3AF,7$$ \"3fQ6D*)o2>lF,$\"3CG]Vmkt$Q\"F,7$$\"35Y]`:_N/nF,$\"33Dv;1s&pZ&F07$$\" 3.\"Q#ekL\"p!pF,$!3OST&zF\\gd%F07$$\"3%yB5rz/v4(F,$!3cn!**p^J2Z\"F,7$$ \"33-p_gxs'H(F,$!3A9e\"Q#e(4b#F,7$$\"3%324>i/:\\(F,$!3S!=W0GJd`$F,7$$ \"39FEN$GhMf(F,$!3U'\\`Wd^A(RF,7$$\"3c%='zWzT&p(F,$!3f\\'**z(4Q8VF,7$$ \"3,%e*>Oh^WxF,$!3G(>*pycjJWF,7$$\"3K#)HgFVh$z(F,$!3)Q%ReLd!G^%F,7$$\" 3h#o/LUj\"=yF,$!3p>V&oBmw`%F,7$$\"3y\"Q1!>DrUyF,$!3MH+5_!>5b%F,7$$\"3% 433Zhhs'yF,$!3KtI_'\\k?b%F,7$$\"3?\"y4/r5=*yF,$!3)*fMuc7)*RXF,7$$\"3@o 'GDq??%zF,$!3)R70$\\E%3Z%F,7$$\"3=bvk%pIA*zF,$!3(yFHocWfL%F,7$$\"3IVkw 'oSC/)F,$!3&[P'\\m/)z7%F,7$$\"3II`))y1l#4)F,$!3E`XP\"[()*RQF,7$$\"3?gE &\\8RA>)F,$!3'*pxl2UD4IF,7$$\"37!**>5fF=H)F,$!3Fu3OFE=:=F,7$$\"3$)fW:: LeP$)F,$!374'4ii(p\\6F,7$$\"3kI*)GR!RLQ)F,$!3q\\t;%4rO@%F07$$\"3Z,MUjZ 4H%)F,$\"3c]$R4W&G]NF07$$\"3=ryb([][Z)F,$\"37Ccp$Rpv:\"F,7$$\"3%QzY4r \"HF&)F,$\"3CcG)))y\"yp?F,7$$\"3g()F,$\"3y_cgRCTxTF,7$$\"3Wx*okV>:t)F,$\"37VUul#=X<%F,7$$\"3q% oYTYr\\v)F,$\"3Ri.A'QV25%F,7$$\"3'>RC=\\B%y()F,$\"3x'=W7Fx&HRF,7$$\"3] 1)zraF`#))F,$\"3QFG%plB4F$F,7$$\"39A_`-;Bs))F,$\"3700_RQTz@F,7$$\"31^1 [bjB(*))F,$\"3OuNH?R:M9F,7$$\"34\"3E%36CA*)F,$\"3E<)GsFg<(fF07$$\"376: PheCZ*)F,$!3ocHHl.]EIF07$$\"39TpJ91Ds*)F,$!38U'y=D^^A\"F,7$$\"3=rBEn`D (**)F,$!3j&4Nt%[@=@F,7$$\"3@,y??,EA!*F,$!3!e@9jLE#=HF,7$$\"3EJK:t[EZ!* F,$!3I(>(y]J<_NF,7$$\"3Gh')4E'pA2*F,$!3U`$=&457URF,7$$\"3'3]u3\"GDy!*F ,$!3U'ykJ>F2*RF,7$$\"3cT.l&*fB%3*F,$!3AH^N$RP+-%F,7$$\"3E#=E/=>-4*F,$! 3&Rn/8H]\"HSF,7$$\"3'=--_O-i4*F,$!39=u$p*4B>*F,$!39yV6L@mxvF07$$\"3oYr;\"p**Q?*F,$!3_9 %**Gult/#!#?7$$\"3(p#)=21me@*F,$\"3=-0hZ^rztF07$$\"3E20FIC$yA*F,$\"3ML bSaq^*[\"F,7$$\"3b(=A)*z)zR#*F,$\"3Qccz;X$>?#F,7$$\"3%*oQPp^w^#*F,$\"3 \\RTgHcmRGF,7$$\"3C\\b#*Q:tj#*F,$\"3M$o6)eV:lLF,7$$\"3oE3CP5fw#*F,$\"3 Ma)HpV]]I!H!G$*F,$\"3a%4t 07BP*GF,7$$\"3J=s\")G&))3M*F,$\"3f(3)H;[%o+#F,7$$\"3w&\\Kr-[PN*F,$\"33 n1kl[]%*F,$!3m (=[SoWqQ#F,7$$\"3%>saO,CmX*F,$!3GDS5eP#en\"F,7$$\"3AhB\"Gw`IY*F,$!3U$3 !Gg0_(o)F07$$\"3]++(>^$[p%*F,$!3V'=8$[D+C:!#C-%'COLOURG6&%$RGBG$\"#5! \"\"F(F(-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6$%\"xG%%y(x)G-%%VIEW G6$;F($\"+BN[p%*!#5%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 93 "The following code constructs a discrete \+ solution based on each of the methods and gives the " }{TEXT 260 22 "r oot mean square error" }{TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 735 "Q := (x,y) -> exp(-x)/(x-1)^2*cos( 1/(x-1))-y: hh := 1/500-1/(3000*Pi): numsteps := 500: x0 := 0: y0 := s in(1):\nmatrix([[`slope field: `,Q(x,y)],[`initial point: `,``(x0,y0 )],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds \+ := [`Verner's \"most robust\" scheme`,`Verner \"most efficient\" schem e`,`Sharp's scheme`,\n`Tsitouras' scheme A`,`Tsitouras' scheme B`]: er rs := []:\nDigits := 30:\nfor ct to 5 do\n Qn_RK9_||ct := RK9_||ct(Q (x,y),x,y,x0,evalf[33](y0),evalf[33](hh),numsteps,false);\n sm := 0: numpts := nops(Qn_RK9_||ct):\n for ii to numpts do\n sm := sm+ (Qn_RK9_||ct[ii,2]-q(Qn_RK9_||ct[ii,1]))^2;\n end do:\n errs := [o p(errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpose]([ mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7& 7$%0slope~field:~~~G,&*(-%$expG6#,$%\"xG!\"\"\"\"\",&F/F1F1F0!\"#-%$co sG6#*&F1F1F2F0F1F1%\"yGF07$%0initial~point:~G-%!G6$\"\"!-%$sinG6#F17$% /step~width:~~~G,&#F1\"$+&F1*&F1F1*&\"%+IF1%#PiGF1F0F07$%1no.~of~steps :~~~GFFQ(pprint06\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%>Verner's~\"most~robust\"~s chemeG$\"+f=s;x!#C7$%?Verner~\"most~efficient\"~schemeG$\"+xzFozF+7$%/ Sharp's~schemeG$\"+=K.5ZF+7$%4Tsitouras'~scheme~AG$\"+!>'QP?F+7$%4Tsit ouras'~scheme~BG$\"+@t\"yY)F+Q(pprint16\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code constructs \+ " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for solutions ba sed on each of the Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each of the methods at the poin t where " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" }{TEXT -1 21 ".9469 is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 659 "Q := (x ,y) -> exp(-x)/(x-1)^2*cos(1/(x-1))-y: hh := 1/500-1/(3000*Pi): numste ps := 500: x0 := 0: y0 := sin(1):\nmatrix([[`slope field: `,Q(x,y)], [`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: \+ `,numsteps]]);``;\nmthds := [`Verner's \"most robust\" scheme`,`Vern er \"most efficient\" scheme`,`Sharp's scheme`,\n`Tsitouras' scheme A` ,`Tsitouras' scheme B`]: errs := []:\nDigits := 30:\nfor ct to 5 do\n \+ qn_RK9_||ct := RK9_||ct(Q(x,y),x,y,x0,evalf(y0),evalf(hh),numsteps,t rue);\nend do:\nxx := 0.9469: qxx := evalf(q(xx)):\nfor ct to 5 do\n \+ errs := [op(errs),abs(qn_RK9_||ct(xx)-qxx)];\nend do:\nDigits := 10: \nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,&*(-%$expG6#,$%\"xG! \"\"\"\"\",&F/F1F1F0!\"#-%$cosG6#*&F1F1F2F0F1F1%\"yGF07$%0initial~poin t:~G-%!G6$\"\"!-%$sinG6#F17$%/step~width:~~~G,&#F1\"$+&F1*&F1F1*&\"%+I F1%#PiGF1F0F07$%1no.~of~steps:~~~GFFQ(pprint26\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$ %>Verner's~\"most~robust\"~schemeG$\"+eg!eO\"!#A7$%?Verner~\"most~effi cient\"~schemeG$\"+gnF59F+7$%/Sharp's~schemeG$\"+q(3,K)!#B7$%4Tsitoura s'~scheme~AG$\"+/&=Qv\"F47$%4Tsitouras'~scheme~BG$\"+,dh)\\\"F+Q(pprin t36\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 " The " }{TEXT 260 22 "root mean square error" }{TEXT -1 19 " over the i nterval " }{XPPEDIT 18 0 " [0, 1-1/(6*Pi)] " "6#7$\"\"!,&\"\"\"F&*&F&F &*&\"\"'F&%#PiGF&!\"\"F+" }{TEXT -1 82 " of each Runge-Kutta method i s estimated as follows using the special procedure " }{TEXT 0 5 "NCin t" }{TEXT -1 98 " to perform numerical integration by the 7 point New ton-Cotes method over 200 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 383 "mthds := [`Verner's \"most robust\" scheme`,`Ve rner \"most efficient\" scheme`,`Sharp's scheme`,\n`Tsitouras' scheme \+ A`,`Tsitouras' scheme B`]: errs := []:\nDigits := 20:\nfor ct to 5 do \n sm := NCint((q(x)-'qn_RK9_||ct'(x))^2,x=0..1-1/(6*Pi),adaptive=fa lse,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$%>Verner's~\"most~ robust\"~schemeG$\"+[>8rO!#C7$%?Verner~\"most~efficient\"~schemeG$\"+Q 5>$z$F+7$%/Sharp's~schemeG$\"+H'y'*G#F+7$%4Tsitouras'~scheme~AG$\"+@!G /k\"F+7$%4Tsitouras'~scheme~BG$\"+&pDS.%F+Q(pprint46\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following error \+ graphs are constructed using the numerical procedures for the solution s." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 452 "evalf[30](plot(['qn_R K9_1'(x)-q(x),'qn_RK9_2'(x)-q(x),'qn_RK9_3'(x)-q(x),\n'qn_RK9_4'(x)-q( x),'qn_RK9_5'(x)-q(x)],x=0..0.5,-9e-28..2.3e-27,\ncolor=[COLOR(RGB,.45 ,0,1),COLOR(RGB,.85,.45,0),COLOR(RGB,0,.65,0),COLOR(RGB,0,.55,.95),red ],\nlegend=[`Verner's \"most robust\" scheme`,`Verner \"most efficient \" scheme`,`Sharp's scheme`,\n`Tsitouras' scheme A`,`Tsitouras' scheme B`],font=[HELVETICA,9],\ntitle=`error curves for 15 stage order 9 Run ge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 783 536 536 {PLOTDATA 2 "6+-%'CURVESG6%7en7$$\"\"!F)F(7$$\"?MLLLLLLLLL3x&)*3\"!#J$ !\"#!#I7$$\"?nmmmmmmmm\"H2P\"Q?F-$!\"$F07$$\"?MLLLLLLLL$eRwX5$F-F(7$$ \"?NLLLLLLLL$3x%3yTF-F47$$\"?ommmmmmmm\"z%4\\Y_F-$!\"\"F07$$\"?NLLLLLL LLeR-/PiF-$!\"%F07$$\"?-++++++++DcmpisF-F?7$$\"?OLLLLLLLLe*)>VB$)F-$! 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" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 679 "R := (x,y) -> 5*y*sin(7 *x)^7: hh := 0.01: numsteps := 500: x0 := 0: y0 := 1:\nmatrix([[`slope field: `,R(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh ],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Verner's \"most ro bust\" scheme`,`Verner \"most efficient\" scheme`,`Sharp's scheme`,\n` Tsitouras' scheme A`,`Tsitouras' scheme B`]: errs := []:\nDigits := 30 :\nfor ct to 5 do\n Rn_RK9_||ct := RK9_||ct(R(x,y),x,y,x0,y0,hh,nums teps,false);\n sm := 0: numpts := nops(Rn_RK9_||ct):\n for ii to n umpts do\n sm := sm+(Rn_RK9_||ct[ii,2]-r(Rn_RK9_||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,$*(\"\"&\"\"\"%\"yGF ,)-%$sinG6#,$*&\"\"(F,%\"xGF,F,F4F,F,7$%0initial~point:~G-%!G6$\"\"!F, 7$%/step~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$+&Q(pprint56\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #K%'matrixG6#7'7$%>Verner's~\"most~robust\"~schemeG$\"+f^Ax:!#D7$%?Ver ner~\"most~efficient\"~schemeG$\"+*zbf%e!#E7$%/Sharp's~schemeG$\"+2p$y i\"!#C7$%4Tsitouras'~scheme~AG$\"+0f&HU%F+7$%4Tsitouras'~scheme~BG$\"+ K-E98F+Q(pprint66\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 30 "The following code constructs " }{TEXT 260 20 "numerica l procedures" }{TEXT -1 56 " for solutions based on each of the Runge- Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value \+ obtained by each of the methods at the point where " }{XPPEDIT 18 0 " x = 4.999;" "6#/%\"xG-%&FloatG6$\"%**\\!\"$" }{TEXT -1 16 " is also g iven." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 610 "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 := [`Verner's \"mo st robust\" scheme`,`Verner \"most efficient\" scheme`,`Sharp's scheme `,\n`Tsitouras' scheme A`,`Tsitouras' scheme B`]: errs := []:\nDigits \+ := 25:\nfor ct to 5 do\n rn_RK9_||ct := RK9_||ct(R(x,y),x,y,x0,y0,hh ,numsteps,true);\nend do:\nxx := 4.999: rxx := evalf(r(xx)):\nfor ct t o 5 do\n errs := [op(errs),abs(rn_RK9_||ct(xx)-rxx)];\nend do:\nDigi ts := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,$*(\"\"&\"\"\"%\"y GF,)-%$sinG6#,$*&\"\"(F,%\"xGF,F,F4F,F,7$%0initial~point:~G-%!G6$\"\"! F,7$%/step~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$+&Q(pprint76\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #K%'matrixG6#7'7$%>Verner's~\"most~robust\"~schemeG$\"*gOnC$!#C7$%?Ver ner~\"most~efficient\"~schemeG$\")6 " 0 "" {MPLTEXT 1 0 363 "mthds := [`Verner's \"most robust\" scheme`,`Verner \"most efficient\" sche me`,`Sharp's scheme`,\n`Tsitouras' scheme A`,`Tsitouras' scheme B`]: e rrs := []:\nDigits := 20:\nfor ct to 5 do\n sm := NCint((r(x)-'rn_RK 9_||ct'(x))^2,x=0..5,adaptive=false,numpoints=7,factor=200);\n errs \+ := [op(errs),sqrt(sm/5)];\nend do:\nDigits := 10:\nlinalg[transpose]([ mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7' 7$%>Verner's~\"most~robust\"~schemeG$\"+kH=k:!#D7$%?Verner~\"most~effi cient\"~schemeG$\"+rVx@d!#E7$%/Sharp's~schemeG$\"+VP`A;!#C7$%4Tsitoura s'~scheme~AG$\"+j&=VV%F+7$%4Tsitouras'~scheme~BG$\"+?ZW.8F+Q(pprint96 \"" }}}{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 0 454 "eva lf[20](plot([r(x)-'rn_RK9_1'(x),r(x)-'rn_RK9_2'(x),r(x)-'rn_RK9_3'(x), \nr(x)-'rn_RK9_4'(x),r(x)-'rn_RK9_5'(x)],x=0..5,-1.05e-15..4.05e-15,\n color=[COLOR(RGB,.45,0,1),COLOR(RGB,.85,.45,0),COLOR(RGB,0,.65,0),COLO R(RGB,0,.55,.95),red],\nlegend=[`Verner's \"most robust\" scheme`,`Ver ner \"most efficient\" scheme`,`Sharp's scheme`,\n`Tsitouras' scheme A `,`Tsitouras' scheme B`],font=[HELVETICA,9],\ntitle=`error curves for \+ 15 stage order 9 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 909 533 533 {PLOTDATA 2 "6+-%'CURVESG6%7^dm7$$\"\"!F)F(7$$\" 5ommmTN@Ki8!#@$\"#R!#>7$$\"5NLLL$3FWYs#F-$\"#nF07$$\"5omm;aQ`!eS$F-$\" #uF07$$\"5-+++D1k'p3%F-$\"#cF07$$\"5pmmT5SpaFWF-F>7$$\"5OLL$eRZF\"oZF- $\"#aF07$$\"50++D\"y+3(3^F-$\"#'o^7\\6Fco$\"#@F07$$\"5++]i!*GER37Fco$\"#rF07 $$\"5ML3F>*3gwE\"Fco$\"#sF07$$\"5nmm\"z%\\v#pK\"Fco$\"#eF07$$\"5ML$3_+ ZiaW\"Fco$!#AF07$$\"5+++]i!R(*Rc\"Fco$!$E\"F07$$\"5MLL3xJs1,=Fco$\"#>F 07$$\"5nmmm\"H2P\"Q?Fco$\"$L\"F07$$\"5MLe*)fI&*y/@Fco$!$:#F07$$\"5++]7 G))>Wr@Fco$!$U#F07$$\"5M$eRAr@oZ?#Fco$!$l)F07$$\"5nmTN'fW%4QAFco$!$!)) 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" }}{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 d ifferential equation " }{XPPEDIT 18 0 "dy/dx = cos*x+2*y;" "6#/*&%# dyG\"\"\"%#dxG!\"\",&*&%$cosGF&%\"xGF&F&*&\"\"#F&%\"yGF&F&" }{TEXT -1 64 " contains an exponential term, but with the initial condition " }{XPPEDIT 18 0 "y(0) = -2/5" "6#/-%\"yG6#\"\"!,$*&\"\"#\"\"\"\"\"&!\" \"F-" }{TEXT -1 23 " this term disappears." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "de := diff(y(x),x) =cos(x)+2*y(x);\ndsolve(de,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#deG/-%%diffG6$-%\"yG6#%\"xGF,,&-%$cosGF+\"\"\"*&\"\"#F0F)F0F0" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,(*&#\"\"#\"\"&\"\"\"-%$ cosGF&F-!\"\"*&#F-F,F--%$sinGF&F-F-*&-%$expG6#,$*&F+F-F'F-F-F-%$_C1GF- F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 105 "A ny slight deviation of a numerical solution from the correct solution \+ tends to become rapidly magnified." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 154 "de := diff(y(x),x)=cos(x)+2 *y(x);\nic := y(0)=-2/5;\ndsolve(\{de,ic\},y(x));\ne := unapply(rhs(%) ,x):\nplot(e(x),x=0..8,font=[HELVETICA,9],labels=[`x`,`y(x)`]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,&-%$c osGF+\"\"\"*&\"\"#F0F)F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/- %\"yG6#\"\"!#!\"#\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\" xG,&*&#\"\"#\"\"&\"\"\"-%$cosGF&F-!\"\"*&#F-F,F--%$sinGF&F-F-" }} {PARA 13 "" 1 "" {GLPLOT2D 703 312 312 {PLOTDATA 2 "6&-%'CURVESG6$7gn7 $$\"\"!F)$!3A+++++++S!#=7$$\"3ELLLLBxV5E$F,$!35;%fC]([[JF,7$$\"3MLLLLAKn\\F,$!3C&4%=OwYjDF,7$$\"3=LLLLc$\\ o'F,$!31c1[)*fT**=F,7$$\"3)emmm^&Q%R)F,$!39J7$$\"3))*****\\YJ?;\"!#<$\"3m!=?Y3*>`CFK7$$\"3?LL L=\"\\g**FK7$$\"3\")*****\\[A4]\"FO$\"3Xgu?U;&er\"F ,7$$\"3wmmm'3Q\\n\"FO$\"3S\\4.g(y\\S#F,7$$\"3OLLLB6@G=FO$\"3e*[f2BGC&H F,7$$\"3&)******f-w+?FO$\"375@EVOJ&[$F,7$$\"3%*********y,u@FO$\"3VG2]n #=i\"RF,7$$\"3)*******RP)4M#FO$\"3ym!)\\t%R1A%F,7$$\"3Umm;HUz;CFO$\"3: @(\\YT,0K%F,7$$\"3ILLL=Zg#\\#FO$\"3++xVHVa&R%F,7$$\"3;++]A2v#e#FO$\"3+ <'Hh4))=X%F,7$$\"3cmmmEn*Gn#FO$\"3a5#zx'*y?Z%F,7$$\"3qmmm;AE\\FFO$\"35 ^%H>#ywgWF,7$$\"3Tmmm1xiDGFO$\"3(3\\(>4bXBWF,7$$\"3LLL$e#*eW\"HFO$\"3! 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$!3;B6I^7jsMF,7$$\"3aLLLt>:nmFO$!37+2hu:afHF,7$$\"35LLL.a#o$oFO$!3;\"e /Z#4*3N#F,7$$\"3ammm^Q40qFO$!3!4`1I$pa!o\"F,7$$\"3y******z]rfrFO$!3pfL '*)RTA-\"F,7$$\"3gmmmc%GpL(FO$!3?j;%3XMsQ#FK7$$\"3/LLL8-V&\\(FO$\"3qi( R>/(R\"p%FK7$$\"3=+++XhUkwFO$\"3ZX^U-))=F,7$$\"\")F)$\"3s<7[GmrgDF,-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%%FON TG6$%*HELVETICAG\"\"*-%+AXESLABELSG6$%\"xG%%y(x)G-%%VIEWG6$;F(Fg]l%(DE FAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "T he following code constructs a " }{TEXT 260 17 "discrete solution" } {TEXT -1 44 " based on each of the methods and gives the " }{TEXT 260 22 "root mean square error" }{TEXT -1 18 " of each solution." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 685 "E := (x,y) -> cos(x)+2*y: h h := 0.02: numsteps := 400: x0 := 0: y0 := -2/5:\nmatrix([[`slope fiel d: `,E(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[ `no. of steps: `,numsteps]]);``;\nmthds := [`Verner's \"most robust \" scheme`,`Verner \"most efficient\" scheme`,`Sharp's scheme`,\n`Tsit ouras' scheme A`,`Tsitouras' scheme B`]: errs := []:\nDigits := 20:\nf or ct to 5 do\n En_RK9_||ct := RK9_||ct(E(x,y),x,y,x0,evalf(y0),hh,n umsteps,false);\n sm := 0: numpts := nops(En_RK9_||ct):\n for ii t o numpts do\n sm := sm+(En_RK9_||ct[ii,2]-e(En_RK9_||ct[ii,1]))^2 ;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,&-%$cosG6#%\"xG\"\" \"*&\"\"#F.%\"yGF.F.7$%0initial~point:~G-%!G6$\"\"!#!\"#\"\"&7$%/step~ width:~~~G$F0F97$%1no.~of~steps:~~~G\"$+%Q)pprint106\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6 #7'7$%>Verner's~\"most~robust\"~schemeG$\"+%ogZ+\"!#D7$%?Verner~\"most ~efficient\"~schemeG$\"+0YK=5F+7$%/Sharp's~schemeG$\"+Mv^TSF+7$%4Tsito uras'~scheme~AG$\"+#=8[=\"F+7$%4Tsitouras'~scheme~BG$\"++@6ik!#EQ)ppri nt116\"" }}}{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 = 7.999; " "6#/%\"xG-%&FloatG6$\"%**z!\"$" }{TEXT -1 16 " is also given." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 616 "E := (x,y) -> cos(x)+2*y: h h := 0.02: numsteps := 400: x0 := 0: y0 := -2/5:\nmatrix([[`slope fiel d: `,E(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[ `no. of steps: `,numsteps]]);``;\nmthds := [`Verner's \"most robust \" scheme`,`Verner \"most efficient\" scheme`,`Sharp's scheme`,\n`Tsit ouras' scheme A`,`Tsitouras' scheme B`]: errs := []:\nDigits := 25:\nf or ct to 5 do\n en_RK9_||ct := RK9_||ct(E(x,y),x,y,x0,evalf(y0),hh,n umsteps,true);\nend do:\nxx := 7.999: exx := evalf(e(xx)):\nfor ct to \+ 5 do\n errs := [op(errs),abs(en_RK9_||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)pprint126\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6 #7'7$%>Verner's~\"most~robust\"~schemeG$\"+zO\")ob!#D7$%?Verner~\"most ~efficient\"~schemeG$\"+H')[[cF+7$%/Sharp's~schemeG$\"+5zDRA!#C7$%4Tsi touras'~scheme~AG$\"+FGZ6mF+7$%4Tsitouras'~scheme~BG$\"+\")pz(e$F+Q)pp rint136\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 8];" "6#7$\"\"!\"\")" }{TEXT -1 82 " of each Runge-Kutta method is estimated as follows using the spe cial procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perform numeri cal integration by the 7 point Newton-Cotes method over 200 equal subi ntervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 363 "mthds := [`Ver ner's \"most robust\" scheme`,`Verner \"most efficient\" scheme`,`Shar p's scheme`,\n`Tsitouras' scheme A`,`Tsitouras' scheme B`]: errs := [] :\nDigits := 20:\nfor ct to 5 do\n sm := NCint((e(x)-'en_RK9_||ct'(x ))^2,x=0..8,adaptive=false,numpoints=7,factor=200);\n errs := [op(er rs),sqrt(sm/8)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,eva lf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%>Verne r's~\"most~robust\"~schemeG$\"+mN2k)*!#E7$%?Verner~\"most~efficient\"~ schemeG$\"+O-_+5!#D7$%/Sharp's~schemeG$\"+8BTmRF07$%4Tsitouras'~scheme ~AG$\"+`T4r6F07$%4Tsitouras'~scheme~BG$\"+Ep3bjF+Q)pprint146\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The follo wing error graphs are constructed using the numerical procedures for t he solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 453 "evalf[25]( plot(['en_RK9_1'(x)-e(x),'en_RK9_2'(x)-e(x),'en_RK9_3'(x)-e(x),\n'en_R K9_4'(x)-e(x),'en_RK9_5'(x)-e(x)],x=0..2,-5.5e-21..1.55e-20,\ncolor=[C OLOR(RGB,.45,0,1),COLOR(RGB,.85,.45,0),COLOR(RGB,0,.65,0),COLOR(RGB,0, .55,.95),red],\nlegend=[`Verner's \"most robust\" scheme`,`Verner \"mo st efficient\" scheme`,`Sharp's scheme`,\n`Tsitouras' scheme A`,`Tsito uras' scheme B`],font=[HELVETICA,9],\ntitle=`error curves for 15 stage order 9 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 979 587 587 {PLOTDATA 2 "6+-%'CURVESG6%7S7$$\"\"!F)F(7$$\":MLLLLLLL$3VfV!# E$!%=*)!#F7$$\":nmmmmmm;H[D:)F-$!%M=F-7$$\":LLLLLLL$e0$=C\"!#D$!%kGF-7 $$\":LLLLLLL$3RBr;F9$!%_RF-7$$\":nmmmmmm\"zjf)4#F9$!%?^F-7$$\":MLLLLLL e4;[\\#F9$!%EjF-7$$\":+++++++Dmy]!HF9$!%CwF-7$$\":MLLLLLLezs$HLF9$!%K! *F-7$$\":+++++++D@1Bv$F9$!&g0\"F-7$$\":nmmmmmmm@Xt=%F9$!&KB\"F-7$$\":M LLLLLL$3y_qXF9$!&WQ\"F-7$$\":++++++++l+>+&F9$!&=f\"F-7$$\":++++++++vW] V&F9$!%*y\"F97$$\":++++++++NfC&eF9$!%'*>F97$$\":MLLLLLLez6:B'F9$!%*>#F 97$$\":nmmmmmmm\"=C#o'F9$!%^CF97$$\":nmmmmmmmEpS1(F9$!%%o#F97$$\":,+++ +++DOD#3vF9$!%qHF97$$\":nmmmmmmmwy8!zF9$!%XKF97$$\":,++++++DOIFL)F9$!% mNF97$$\":,++++++v3zMu)F9$!%&*QF97$$\":nmmmmmm;H_?<*F9$!%lUF97$$\":nmm mmmm\"zihl&*F9$!%GYF97$$\":MLLLLLL$3#G,***F9$!%`]F97$$\":LLLLLL$ezw5V5 !#C$!%CbF97$$\":++++++]PQ#\\\"3\"Fbs$!%mfF97$$\":LLLLLLLe\"*[H7\"Fbs$! %!['F97$$\":++++++++dxd;\"Fbs$!%^qF97$$\":+++++++D0xw?\"Fbs$!%bwF97$$ \":++++++]i&p@[7Fbs$!%)G)F97$$\":+++++++vgHKH\"Fbs$!%[!*F97$$\":nmmmmm mmZvOL\"Fbs$!%'y*F97$$\":+++++++]2goP\"Fbs$!&Q1\"F97$$\":LLLLLL$eR<*fT \"Fbs$!&t9\"F97$$\":+++++++])Hxe9Fbs$!&`C\"F97$$\":nmmmmm;H!o-*\\\"Fbs $!&aM\"F97$$\":++++++]7k.6a\"Fbs$!&#e9F97$$\":nmmmmmm;WTAe\"Fbs$!&&y:F 97$$\":++++++]i!*3`i\"Fbs$!&_r\"F97$$\":LLLLLLLL*zym;Fbs$!&y&=F97$$\": LLLLLLL3N1#4Fbs$!&\"3IF97$$\":++++++]P?Wl&>Fbs$!&MD$F97 $$\"\"#F)$!&r`$F9-%&COLORG6&%$RGBG$\"#X!\"#F($\"\"\"F)-%'LEGENDG6#%>Ve rner's~\"most~robust\"~schemeG-F$6%7SF'7$F+$!%=$*F07$F2$!%/>F-7$F7$!%k HF-7$F=$!%#4%F-7$FB$!%+`F-7$FG$!%clF-7$FL$!%%*yF-7$FQ$!%U$*F-7$FV$!&+4 \"F-7$Fen$!&KF\"F-7$Fjn$!&%G9F-7$F_o$!&3k\"F-7$Fdo$!%V=F97$Fio$!%b?F97 $F^p$!%iAF97$Fcp$!%>DF97$Fhp$!%eFF97$F]q$!%]IF97$Fbq$!%ILF97$Fgq$!%dOF 97$F\\r$!%$*RF97$Far$!%qVF97$Ffr$!%SZF97$F[s$!%r^F97$F`s$!%]cF97$Ffs$! %+hF97$F[t$!%AmF97$F`t$!%-sF97$Fet$!%;yF97$Fjt$!%e%)F97$F_u$!%J#*F97$F du$!%!)**F97$Fiu$!&X3\"F97$F^v$!&#p6F97$Fcv$!&'o7F97$Fhv$!&-P\"F97$F]w $!&X[\"F97$Fbw$!&lg\"F97$Fgw$!&_u\"F97$F\\x$!&**)=F97$Fax$!&20#F97$Ffx $!&QA#F97$F[y$!&`R#F97$F`y$!&%4EF97$Fey$!&m\"GF97$Fjy$!&l0$F97$F_z$!&^ I$F97$Fdz$!&Ef$F9-Fiz6&F[[l$\"#&)F^[lF\\[lF(-Fb[l6#%?Verner~\"most~eff icient\"~schemeG-F$6%7SF'7$F+$\"&#)H(F07$F2$\"&YZ\"F-7$F7$\"&YD#F-7$F= $\"&y1$F-7$FB$\"&I\"RF-7$FG$\"&9w%F-7$FL$\"&1l&F-7$FQ$\"&3f'F-7$FV$\"& Sd(F-7$Fen$\"&Gn)F-7$Fjn$\"&Og*F-7$F_o$\"'7z5F-7$Fdo$\"&Q>\"F97$Fio$\" &(48F97$F^p$\"&,U\"F97$Fcp$\"&jb\"F97$Fhp$\"&sn\"F97$F]q$\"&`#=F97$Fbq $\"&@'>F97$Fgq$\"&67#F97$F\\r$\"&*zAF97$Far$\"&NX#F97$Ffr$\"&Ii#F97$F[ 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Rb#F97$F[y$\"&^v#F97$F`y$\"&d+$F97$Fey$\"&!\\KF97$Fjy$\"&)HNF97$F_z$\" &<#QF97$Fdz$\"&4;%F9-Fiz6&F[[lF($\"#bF^[l$\"#&*F^[l-Fb[l6#%4Tsitouras' ~scheme~AG-F$6%7SF'7$F+$!%=nF07$F2$!%k8F-7$F7$!%C@F-7$F=$!%AHF-7$FB$!% !y$F-7$FG$!%YYF-7$FL$!%%e&F-7$FQ$!%#e'F-7$FV$!%gwF-7$Fen$!%-*)F-7$Fjn$ !%a**F-7$F_o$!&)Q6F-7$Fdo$!%u7F97$Fio$!%<9F97$F^p$!%a:F97$Fcp$!%EF97$F_z$!&'=@F97$Fdz$!&-I#F9-%'COLOURG6& F[[l$\"*++++\"!\")F(F(-Fb[l6#%4Tsitouras'~scheme~BG-%&TITLEG6#%Verror~ curves~for~15~stage~order~9~Runge-Kutta~methodsG-%%FONTG6$%*HELVETICAG \"\"*-%+AXESLABELSG6$Q\"x6\"Q!F^cn-%%VIEWG6$;F(Fdz;$!#b!#A$\"$b\"Fgcn " 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Verner's \" most robust\" scheme" "Verner \"most efficient\" scheme" "Sharp's sche me" "Tsitouras' scheme A" "Tsitouras' scheme B" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 452 "evalf[25](p lot(['en_RK9_1'(x)-e(x),'en_RK9_2'(x)-e(x),'en_RK9_3'(x)-e(x),\n'en_RK 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0*x*cos*x-10*y;" "6#/*&%#dyG\"\"\"%#dxG!\"\",&**\"#5F&%\"xGF&%$cosGF&F ,F&F&*&F+F&%\"yGF&F(" }{TEXT -1 8 ", " }{XPPEDIT 18 0 "y(0) = sq rt(5);" "6#/-%\"yG6#\"\"!-%%sqrtG6#\"\"&" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "y=100/101" "6#/%\"yG*&\"$+\"\"\"\"\"$,\"!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x*cos*x-990/10201" "6#,&*(%\"xG\"\"\"%$cosGF& F%F&F&*&\"$!**F&\"&,-\"!\"\"F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "cos*x+ 10/101" "6#,&*&%$cosG\"\"\"%\"xGF&F&*&\"#5F&\"$,\"!\"\"F&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x*sin*x-200/10201" "6#,&*(%\"xG\"\"\"%$sinGF&F%F &F&*&\"$+#F&\"&,-\"!\"\"F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin*x+(990 /10201+sqrt(5))*exp(-10*x)" "6#,&*&%$sinG\"\"\"%\"xGF&F&*&,&*&\"$!**F& \"&,-\"!\"\"F&-%%sqrtG6#\"\"&F&F&-%$expG6#,$*&\"#5F&F'F&F-F&F&" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 163 "de := diff(y(x),x)=10*x*cos(x)-10*y(x);\nic := \+ y(0)=sqrt(5);\ndsolve(\{de,ic\},y(x));\nb := unapply(rhs(%),x):\nplot( b(x),x=0..5,font=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,&*(\"#5\"\"\"F,F0-% $cosGF+F0F0*&F/F0F)F0!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/- %\"yG6#\"\"!*$\"\"&#\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-% \"yG6#%\"xG,,*&#\"$+\"\"$,\"\"\"\"*&F'F--%$cosGF&F-F-F-*&#\"$!**\"&,- \"F-F/F-!\"\"*&#\"#5F,F-*&-%$sinGF&F-F'F-F-F-*&#\"$+#F4F-F:F-F5*&-%$ex pG6#,$*&F8F-F'F-F5F-,&#F3F4F-*$\"\"&#F-\"\"#F-F-F-" }}{PARA 13 "" 1 " " {GLPLOT2D 703 312 312 {PLOTDATA 2 "6&-%'CURVESG6$7hn7$$\"\"!F)$\"3\" )*y*\\xz1OA!#<7$$\"3ALL$3FWYs#!#>$\"3*pc)\\jF;12h\"H\"48F,7$$\"3m****\\7G$R<)F0$\"3<_u(oLbK,\"F,7$$\"3GLLL3x &)*3\"!#=$\"3(**[ro!GyVzF@7$$\"3))**\\i!R(*Rc\"F@$\"3A'ysO]2xW&F@7$$\" 3umm\"H2P\"Q?F@$\"3/$)oqvSKmSF@7$$\"3YLek.pu/BF@$\"3$Qjx*Gs<7OF@7$$\"3 !***\\PMnNrDF@$\"3M:4%*3rt@LF@7$$\"3MmT5ll'z$GF@$\"3/?Np5C\\bJF@7$$\"3 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#*y!45Ut-E$F,7$$\"3e++]iC$pk$F,$!3LO[nw')*)RKF,7$$\"3ILe*[t\\sp$F,$!3D 1>x`HA5KF,7$$\"3[m;H2qcZPF,$!3/[q%\\V.-<$F,7$$\"3O+]7.\"fF&QF,$!3KL?tX &>E0$F,7$$\"3Ymm;/OgbRF,$!3KQEMNc$G*GF,7$$\"3w**\\ilAFjSF,$!3/QR)44g!y EF,7$$\"3yLLL$)*pp;%F,$!30,GW_`#fU#F,7$$\"3)RL$3xe,tUF,$!3*G#*H@1([B@F ,7$$\"3Cn;HdO=yVF,$!35Q!)*4x]5y\"F,7$$\"3a+++D>#[Z%F,$!3(y*pyl_QJ9F,7$ $\"3SnmT&G!e&e%F,$!3]X/0\"RC%G**F@7$$\"3#RLLL)Qk%o%F,$!3u!*)Q\"4WH+dF@ 7$$\"37+]iSjE!z%F,$!3+r[gfMO'=*F07$$\"3a+]P40O\"*[F,$\"3+2*eSHde(QF@7$ $\"\"&F)$\"3&Q8`\">jC3#*F@-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%%FONTG6$%* HELVETICAG\"\"*-%+AXESLABELSG6$%\"xG%%y(x)G-%%VIEWG6$;F(F\\^l%(DEFAULT G" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" } }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The fo llowing code constructs a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " based on each of the methods and gives the " }{TEXT 260 22 "ro ot mean square error" }{TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 701 "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 := [`Verner's \"most robu st\" scheme`,`Verner \"most efficient\" scheme`,`Sharp's scheme`,\n`Ts itouras' scheme A`,`Tsitouras' scheme B`]: errs := []:\nDigits := 20: \nfor ct to 5 do\n Bn_RK9_||ct := RK9_||ct(B(x,y),x,y,x0,evalf(y0),h h,numsteps,false);\n sm := 0: numpts := nops(Bn_RK9_||ct):\n for i i to numpts do\n sm := sm+(Bn_RK9_||ct[ii,2]-evalf(b(Bn_RK9_||ct[ ii,1])))^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\nend d o:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,&*(\"#5\" \"\"%\"xGF,-%$cosG6#F-F,F,*&F+F,%\"yGF,!\"\"7$%0initial~point:~G-%!G6$ \"\"!*$\"\"&#F,\"\"#7$%/step~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$ +&Q(pprint26\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%>Verner's~\"most~robust\"~schemeG $\"+p%**3+(!#G7$%?Verner~\"most~efficient\"~schemeG$\"+n0ON()F+7$%/Sha rp's~schemeG$\"+-)f#oU!#F7$%4Tsitouras'~scheme~AG$\"+\"=jjp)F47$%4Tsit ouras'~scheme~BG$\"+P[QLzF+Q(pprint36\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code constructs " } {TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for solutions based on 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 = 4.999;" "6#/%\"xG-%&FloatG6$\"%**\\!\"$" } {TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 625 "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 po int: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps ]]);``;\nmthds := [`Verner's \"most robust\" scheme`,`Verner \"most ef ficient\" scheme`,`Sharp's scheme`,\n`Tsitouras' scheme A`,`Tsitouras' scheme B`]: errs := []:\nDigits := 30:\nfor ct to 5 do\n bn_RK9_||c t := RK9_||ct(B(x,y),x,y,x0,evalf(y0),hh,numsteps,true);\nend do:\nxx \+ := 4.999: bxx := evalf(b(xx)):\nfor ct to 5 do\n errs := [op(errs),a bs(bn_RK9_||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)pprint266\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$ %>Verner's~\"most~robust\"~schemeG$\"+^u3(>\"!#H7$%?Verner~\"most~effi cient\"~schemeG$\"+q8Emx!#I7$%/Sharp's~schemeG$\"+\"*HExg!#G7$%4Tsitou ras'~scheme~AG$\"+Os(z^%F57$%4Tsitouras'~scheme~BG$\"+Etog\")F0Q)pprin t276\"" }}}{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 p rocedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perform numerical in tegration by the 7 point Newton-Cotes method over 200 equal subinterva ls." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 363 "mthds := [`Verner's \+ \"most robust\" scheme`,`Verner \"most efficient\" scheme`,`Sharp's sc heme`,\n`Tsitouras' scheme A`,`Tsitouras' scheme B`]: errs := []:\nDig its := 20:\nfor ct to 5 do\n sm := NCint((b(x)-'bn_RK9_||ct'(x))^2,x =0..5,adaptive=false,numpoints=7,factor=200);\n errs := [op(errs),sq rt(sm/5)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(err s)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%>Verner's~\" most~robust\"~schemeG$\"+O8_(p'!#G7$%?Verner~\"most~efficient\"~scheme G$\"+^pq\"Q)F+7$%/Sharp's~schemeG$\"+(pW@5%!#F7$%4Tsitouras'~scheme~AG $\"+3qXd$)F47$%4Tsitouras'~scheme~BG$\"+@%z>g(F+Q)pprint196\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The follo wing error graphs are constructed using the numerical procedures for t he solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 470 "evalf[25]( plot([b(x)-'bn_RK9_1'(x),b(x)-'bn_RK9_2'(x),b(x)-'bn_RK9_3'(x),\nb(x)- 'bn_RK9_4'(x),b(x)-'bn_RK9_5'(x)],x=0..0.65,-6.5e-18..4.65e-17,numpoin ts=150,\ncolor=[COLOR(RGB,.45,0,1),COLOR(RGB,.85,.45,0),COLOR(RGB,0,.6 5,0),COLOR(RGB,0,.55,.95),red],\nlegend=[`Verner's \"most robust\" sch eme`,`Verner \"most efficient\" scheme`,`Sharp's scheme`,\n`Tsitouras' scheme A`,`Tsitouras' scheme B`],font=[HELVETICA,9],\ntitle=`error cu rves for 15 stage order 9 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 994 525 525 {PLOTDATA 2 "6+-%'CURVESG6%7\\dl7$$\"\"!F)F(7$$ \":]DQ2hTiV:=@G#!#FF(7$$\":,^w9A$[s3jBkXF-$!$$Q!#C7$$\":?xl)zp/d5d1dbF -$!%)p#F37$$\":N.b#Q2hT7^*)\\lF-$!&QP\"F37$$\":bHWm\\uhU^CFa(F-$!&Oa&F 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\")FdhmFdhm-Fhhm6#%4Tsitouras'~scheme~BG-%&TITLEG6#%Verror~curves~for~ 15~stage~order~9~Runge-Kutta~methodsG-%%FONTG6$%*HELVETICAG\"\"*-%+AXE SLABELSG6$Q\"x6\"Q!Fabo-%%VIEWG6$;F(Figm;$!#8!#?$\"#L!#@" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Verner's \"most robust\" scheme" "Verner \"most efficient\" scheme" "Tsitouras' scheme B" }}}} {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 15 stage, or der 9 Runge-Kutta methods " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 198 "See: Comparing Numerical Methods for Ordinary Di fferential Equations, Hull, Enright, Fellen and Sedgwick,\n Sia m Journal on Numerical Analysis, Vol. 9, No. 4 (Dec. 1972), page 617, \+ Example A5." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "dy/dx = (y-x)/(y+x);" "6#/*&%#dyG\"\" \"%#dxG!\"\"*&,&%\"yGF&%\"xGF(F&,&F+F&F,F&F(" }{TEXT -1 8 ", " } {XPPEDIT 18 0 "y(1) = 1;" "6#/-%\"yG6#\"\"\"F'" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*ln((x^2+y^2)/(x^2))+4*arctan(y/x)+4*ln*x-2*ln* 2-Pi = 0;" "6#/,,*&\"\"#\"\"\"-%#lnG6#*&,&*$%\"xGF&F'*$%\"yGF&F'F'*$F. F&!\"\"F'F'*&\"\"%F'-%'arctanG6#*&F0F'F.F2F'F'*(F4F'F)F'F.F'F'*(F&F'F) F'F&F'F2%#PiGF2\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "de := diff(y(x),x)=(y(x) -x)/(y(x)+x);\nic := y(1)=1;\ndsolve(\{de,ic\},y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,*&,&F)\"\"\"F,!\" \"F/,&F)F/F,F/F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\" \"\"F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%'RootOfG6#,, *&\"\"#\"\"\"-%#lnG6#*&,&*$)F'F-F.F.*$)%#_ZGF-F.F.F.F'!\"#F.!\"\"*&\" \"%F.-%'arctanG6#*&F8F.F'F:F.F:*&F " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "The solution can be given more simply as " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " ln(x^2+y^2)+2*arctan(y/x)=ln*2+Pi/2" " 6#/,&-%#lnG6#,&*$%\"xG\"\"#\"\"\"*$%\"yGF+F,F,*&F+F,-%'arctanG6#*&F.F, F*!\"\"F,F,,&*&F&F,F+F,F,*&%#PiGF,F+F4F," }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 18 "The solution (for " }{TEXT 280 1 "x" }{TEXT -1 47 " increasing) is the section of the polar curve " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "r=sqrt(2)*exp(Pi/4-theta)" "6#/%\"rG*&- %%sqrtG6#\"\"#\"\"\"-%$expG6#,&*&%#PiGF*\"\"%!\"\"F*%&thetaGF2F*" } {TEXT -1 5 ", " }{XPPEDIT 18 0 "-Pi/4<=theta" "6#1,$*&%#PiG\"\"\"\" \"%!\"\"F)%&thetaG" }{XPPEDIT 18 0 "``<=Pi/4" "6#1%!G*&%#PiG\"\"\"\"\" %!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 " ln((x^2+y^2))+2*arctan(y/x)=ln(2)+Pi/2;\nimplicitdiff(%,y,x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%#lnG6#,&*$)%\"xG\"\"#\"\"\"F-*$)% \"yGF,F-F-F-*&F,F--%'arctanG6#*&F0F-F+!\"\"F-F-,&-F&6#F,F-*&F,F6%#PiGF -F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&%\"xG\"\"\"%\"yG!\"\"F',& F(F'F&F'F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 471 "p1 := plot([sqrt(2)*exp(Pi/4-t),t,t=-Pi/4..Pi/4 ],coords=polar,thickness=2,color=red):\np2 := plot([sqrt(2)*exp(Pi/4-t ),t,t=Pi/4..2*Pi],coords=polar,color=black,linestyle=2):\np3 := plot([ sqrt(2)*exp(Pi/4-t),t,t=-Pi/3..-Pi/4],coords=polar,color=black,linesty le=2):\np4 := plot([[[1,1],[uu,-uu]]$4],style=point,symbol=[circle$2,d iamond,cross],\n symbolsize=[12,10$3],color=[black,gr een$3]):\nplots[display]([p1,p2,p3,p4],font=[HELVETICA,9],labels=[`x`, `y(x)`]);" }}{PARA 13 "" 1 "" {GLPLOT2D 567 520 520 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GF+F,F,*&F+F,-%'arctanG6#*&F.F,F*!\"\"F,F,,&*&F&F,F+F,F,*&%#PiGF,F+F4F ," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "for " }{TEXT 278 1 "y " }{TEXT -1 25 " numerically in terms of " }{TEXT 279 1 "x" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 209 "phi := proc(x) local y;\n fsolve(ln(x^2+y^2 )+2*arctan(y/x)=ln(2)+Pi/2,y=-x..7/2-x);\n end proc:\nuu := eval f(exp(Pi/2)):\nplot('phi'(x),x=1..uu,numpoints=100,font=[HELVETICA,9], labels=[`x`,`y(x)`]);" }}{PARA 13 "" 1 "" {GLPLOT2D 515 404 404 {PLOTDATA 2 "6&-%'CURVESG6$7jq7$$\"\"\"\"\"!$\"+++++5!\"*7$$\"+M.FS5F- $\"+!Hsf***!#57$$\"+N$4`2\"F-$\"+]6*f)**F37$$\"+eVr96F-$\"+Zyon**F37$$ \"+#e!Qa6F-$\"+z@\"=%**F37$$\"+!GeQ>\"F-$\"+^qu3**F37$$\"+f\"f/B\"F-$ \"+$)4mr)*F37$$\"+\\sNo7F-$\"+m$=o#)*F37$$\"+4:b28F-$\"+-Eit(*F37$$\"+ s+iY8F-$\"+@hw8(*F37$$\"+\"*o!oQ\"F-$\"+ig9X'*F37$$\"+jM?A9F-$\"+N0$)y &*F37$$\"+>;0i9F-$\"+tvj(\\*F37$$\"+'Rj?]\"F-$\"+2v=4%*F37$$\"+d@iS:F- 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F37$$\"+t#3\\U$F-$!+e$R\\f$F37$$\"+d)[KY$F-$!+\")RRuSF37$$\"+>h\\,NF-$ !+)*o;lXF37$$\"+LbWTNF-$!+6&)e\"4&F37$$\"+,/CyNF-$!+4mR*e&F37$$\"+zL#f h$F-$!+w:g7hF37$$\"+l)Hvl$F-$!+Zam1nF37$$\"+I6?&p$F-$!+q5-gsF37$$\"+aq sLPF-$!+g\\%=%yF37$$\"+Fp!Hx$F-$!+)QD3X)F37$$\"+a3#*3QF-$!+'*3rE!*F37$ $\"+M0JZQF-$!+?;We'*F37$$\"+*)zS&)QF-$!+6MXI5F-7$$\"+C/;ERF-$!+B0!=5\" F-7$$\"+$oA@'RF-$!+qNzm6F-7$$\"+kch.SF-$!+'>,VC\"F-7$$\"+!))f5/%F-$!+- [r;8F-7$$\"+v*3\"ySF-$!+Z4$3R\"F-7$$\"+b'[z6%F-$!+zwYt9F-7$$\"+$\\\\z: %F-$!+'fX(f:F-7$$\"+MVM%>%F-$!+DqOT;F-7$$\"+QS*HB%F-$!+A:fJF-7$$\"+G\"ypM%F-$!+AUgA?F-7$$ \"+M=h(Q%F-$!+:96P@F-7$$\"+`'4eU%F-$!+)[X5D#F-7$$\"+C&QOY%F-$!+d+zqBF- 7$$\"+([(\\,XF-$!+HVk)\\#F-7$$\"+L76SXF-$!+'GZ)QEF-7$$\"+d6/\"e%F-$!+$ pm1!GF-7$$\"+!*)p&=YF-$!+?m5kHF-7$$\"+obhbYF-$!+9%4S9$F-7$$\"+(y;_p%F- $!+/&R[O$F-7$$\"+*zJZt%F-$!+`i>JOF-7$$\"+M`Y_ZF-$!+El5vPF-7$$\"+o))>qZ F-$!+iFfWRF-7$$\"+'\\o-y%F-$!+G<\")eSF-7$$\"+D\"Q.z%F-$!+!p\\]>%F-7$$ \"+SHP&z%F-$!+M@kwUF-7$$\"+axS+[F-$!+ " 0 "" {MPLTEXT 1 0 747 "C := \+ (x,y) -> (y-x)/(y+x): hh := 0.01: numsteps := 375: x0 := 1: y0 := 1:\n matrix([[`slope field: `,C(x,y)],[`initial point: `,``(x0,y0)],[`ste p width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Ver ner's \"most robust\" scheme`,`Verner \"most efficient\" scheme`,`Shar p's scheme`,\n`Tsitouras' scheme A`,`Tsitouras' scheme B`]: \nerrs := \+ []: vals := []:\nDigits := 25:\nfor ct to 5 do\n Cn_RK9_||ct := RK9 _||ct(C(x,y),x,y,x0,y0,hh,numsteps,false);\n sm := 0: numpts := nops (Cn_RK9_||ct):\n for ii to numpts do\n if ct=1 then vals := [op (vals),phi(Cn_RK9_||ct[ii,1])] end if;\n sm := sm+(Cn_RK9_||ct[ii ,2]-vals[ii])^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\n end do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G*&,& %\"yG\"\"\"%\"xG!\"\"F,,&F+F,F-F,F.7$%0initial~point:~G-%!G6$F,F,7$%/s tep~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$v$Q)pprint346\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'ma trixG6#7'7$%>Verner's~\"most~robust\"~schemeG$\"+]!)Gh?!#E7$%?Verner~ \"most~efficient\"~schemeG$\"+,$[N6#F+7$%/Sharp's~schemeG$\"+j&H'*z\"F +7$%4Tsitouras'~scheme~AG$\"+9>DR[F+7$%4Tsitouras'~scheme~BG$\"+[k,z=F +Q)pprint356\"" }}}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code constructs " }{TEXT 260 20 "numerical \+ procedures" }{TEXT -1 56 " for solutions based on each of the Runge-Ku tta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value ob tained by each of the methods at the point where " }{XPPEDIT 18 0 "x \+ = 4.749;" "6#/%\"xG-%&FloatG6$\"%\\Z!\"$" }{TEXT -1 16 " is also give n." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 609 "C := (x,y) -> (y-x)/( y+x): hh := 0.01: numsteps := 375: x0 := 1: y0 := 1:\nmatrix([[`slope \+ field: `,C(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh] ,\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Verner's \"most rob ust\" scheme`,`Verner \"most efficient\" scheme`,`Sharp's scheme`,\n`T sitouras' scheme A`,`Tsitouras' scheme B`]: errs := []:\nDigits := 25: \nfor ct to 5 do\n cn_RK9_||ct := RK9_||ct(C(x,y),x,y,x0,y0,hh,numst eps,true);\nend do:\nxx := 4.749: cxx := evalf(phi(xx)):\nfor ct to 5 \+ do\n errs := [op(errs),abs(cn_RK9_||ct(xx)-cxx)];\nend do:\nDigits : = 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G*&,&%\"yG\"\"\"%\"xG! \"\"F,,&F+F,F-F,F.7$%0initial~point:~G-%!G6$F,F,7$%/step~width:~~~G$F, !\"#7$%1no.~of~steps:~~~G\"$v$Q)pprint366\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$ %>Verner's~\"most~robust\"~schemeG$\"*k%G#*>!#C7$%?Verner~\"most~effic ient\"~schemeG$\"*S!GV?F+7$%/Sharp's~schemeG$\"*!H$pu\"F+7$%4Tsitouras '~scheme~AG$\"*KqLN%F+7$%4Tsitouras'~scheme~BG$\"*ME&>=F+Q)pprint376\" " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The \+ " }{TEXT 260 22 "root mean square error" }{TEXT -1 20 " over the inter val " }{XPPEDIT 18 0 "[1, 4.75];" "6#7$\"\"\"-%&FloatG6$\"$v%!\"#" } {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 0 370 "mthds := [`Verner's \"most robust\" scheme`,`Verner \"most efficient\" sche me`,`Sharp's scheme`,\n`Tsitouras' scheme A`,`Tsitouras' scheme B`]: e rrs := []:\nDigits := 20:\nfor ct to 5 do\n sm := NCint(('phi'(x)-'c n_RK9_||ct'(x))^2,x=1..4.75,adaptive=false,numpoints=7,factor=200);\n \+ errs := [op(errs),sqrt(sm/5)];\nend do:\nDigits := 10:\nlinalg[trans pose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matr ixG6#7'7$%>Verner's~\"most~robust\"~schemeG$\"+j5>5v!#F7$%?Verner~\"mo st~efficient\"~schemeG$\"+dq&=q(F+7$%/Sharp's~schemeG$\"+n5Z*e'F+7$%4T sitouras'~scheme~AG$\"+g@V#o\"!#E7$%4Tsitouras'~scheme~BG$\"+C&Rb&oF+Q )pprint386\"" }}}{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 0 476 "evalf[30](plot(['phi'(x)-'cn_RK9_1'(x),'phi'(x)-'cn_ RK9_2'(x),'phi'(x)-'cn_RK9_3'(x),\n'phi'(x)-'cn_RK9_4'(x),'phi'(x)-'cn _RK9_5'(x)],x=1..3.75,-2.63e-26..3.95e-26,font=[HELVETICA,9],\ncolor=[ COLOR(RGB,.45,0,1),COLOR(RGB,.85,.45,0),COLOR(RGB,0,.65,0),COLOR(RGB,0 ,.55,.95),red],\nlegend=[`Verner's \"most robust\" scheme`,`Verner \"m ost efficient\" scheme`,`Sharp's scheme`,\n`Tsitouras' scheme A`,`Tsit ouras' scheme B`],title=`error curves for 15 stage order 9 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 918 520 520 {PLOTDATA 2 "6+ -%'CURVESG6%7]o7$$\"\"\"\"\"!$F*F*7$$\"?LLLLLLLL$eR**F27$$\"?LLLLLLLLektvWG;F/$\"&!G5F27$$\"?++++++++]PR8w(o \"F/$\"&w1\"F27$$\"?++++++++]7`'=tu\"F/$\"&p5\"F27$$\"?++++++++]igJr/= F/$\"&\\9\"F27$$\"?LLLLLLL$3F>(G$o&=F/$\"&!z6F27$$\"?nmmmmmmm;z\\#3)=> F/$\"&1A\"F27$$\"?nmmmmmmmm;C&48(>F/$\"&mD\"F27$$\"?+++++++]PM()4QK?F/ $\"&#*H\"F27$$\"?nmmmmmmmmT!eRk3#F/$\"&xL\"F27$$\"?+++++++]P%[U]d9#F/$ \"&6Q\"F27$$\"?+++++++]7`u$GA?#F/$\"&RU\"F27$$\"?nmmmmmmmT5!>d6E#F/$\" &,Z\"F27$$\"?nmmmmmm;aQQAF:BF/$\"&P^\"F27$$\"?LLLLLLLLekGEktBF/$\"&Dc \"F27$$\"?LLLLLLL$3F%fIFMCF/$\"&^h\"F27$$\"?+++++++]ilF?0([#F/$\"&Cm\" F27$$\"?LLLLLLLL3FfZ0WDF/$\"&ar\"F27$$\"?+++++++++veT%Hg#F/$\"&Ex\"F27 $$\"?++++++++v=ZfbgEF/$\"&.$=F27$$\"?+++++++]P%[J)H;FF/$\"&)))=F27$$\" ?++++++++DJ52>yFF/$\"&f&>F27$$\"?nmmmmmmmmT!y.Q$GF/$\"&'=?F27$$\"?++++ ++++]7.E=$*GF/$\"&\")3#F27$$\"?LLLLLLL$3F>k))p%HF/$\"&Q:#F27$$\"?+++++ +++]PaG\"e+$F/$\"'vFA!#J7$$\"?nmmmmmm;/,a=;hIF/$\"'7*H#Fgv7$$\"?++++++ +](=n]JF/$\"';xBFgv7$$\"?nmmmmmmm;H#)>evJF/$\"'taCFgv7$$\"?+++++++] P4Y(*zMKF/$\"&p`#F27$$\"?LLLLLLLLL$3\\L=H$F/$\"&eh#F27$$\"?LLLLLLLLeRK 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