{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Dark Red Emphasis" -1 259 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Purple Emphasis" -1 260 "Times " 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 261 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Emphasis" -1 262 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Magenta E mphasis" -1 263 "Times" 1 12 200 0 200 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Orange Emphasis" -1 264 "Times" 1 12 225 100 10 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 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 "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE " " -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Bullet \+ Item" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 3 3 1 0 1 0 2 2 15 2 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 50 "Derivation of 8 stage, order 6 Ru nge-Kutta schemes" }}{PARA 0 "" 0 "" {TEXT -1 45 "by Peter Stone, Gabr iola Island, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 18 "Version: 5.1 2.2011" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " ;" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 58 "load procedures for constru cting Runge-Kutta schemes etc. " }}{PARA 0 "" 0 "" {TEXT -1 18 "The Ma ple 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 ca n be read into a Maple session by commands similar to those that follo w, where each file path gives the location of the m-file." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "read \"C:\\\\Maple/procdrs/butcher .m\";\nread \"C:\\\\Maple/procdrs/roots.m\";\nread \"C:\\\\Maple/procd rs/intg.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 23 "#======================" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 15 "Huta's scheme A" }}{PARA 0 "" 0 "" {TEXT -1 5 "See: " }}{PARA 0 "" 0 "" {TEXT -1 149 "1. Une am\351lioration de la m\351thode de Runge-Kutt a-Nystr\366m pour la r\351solution num\351rique des \351quations diff \351rentielles du premi\350r ordre, by Anton Huta, " }}{PARA 0 "" 0 " " {TEXT -1 70 " Acta Fac. Nat. Univ. Comenian Math., Vol. 1, pages \+ 201-224 (1956)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 "2. Contribution \340 la formule de sixi\350me ordre dans la m\351thode de Runge-Kutta-Nystr\366m, by Anton Huta, " }}{PARA 0 " " 0 "" {TEXT -1 67 " Acta Fac.Nat. Univ. Comenian Math., Vol. 2, pa ges 21-24 (1957)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[1/9, 1/9, ``, ``, ``, ``, `` , ``, ``], [1/6, 1/24, 1/8, ``, ``, ``, ``, ``, ``], [1/3, 1/6, -1/2, \+ 2/3, ``, ``, ``, ``, ``], [1/2, -5/8, 27/8, -3, 3/4, ``, ``, ``, ``], \+ [2/3, 221/9, -109, 289/3, -34/3, 1/9, ``, ``, ``], [5/6, -61/16, 113/8 , -59/6, -11/8, 5/3, 1/16, ``, ``], [1, 358/41, -2079/82, 501/41, 417/ 41, -227/41, -9/82, 36/41, ``], [``, 41/840, 0, 9/35, 9/280, 34/105, 9 /280, 9/35, 41/840]])" "6#-%'matrixG6#7*7+*&\"\"\"F)\"\"*!\"\"*&F)F)F* F+%!GF-F-F-F-F-F-7+*&F)F)\"\"'F+*&F)F)\"#CF+*&F)F)\"\")F+F-F-F-F-F-F-7 +*&F)F)\"\"$F+*&F)F)F0F+,$*&F)F)\"\"#F+F+*&F;F)F7F+F-F-F-F-F-7+*&F)F)F ;F+,$*&\"\"&F)F4F+F+*&\"#FF)F4F+,$F7F+*&F7F)\"\"%F+F-F-F-F-7+*&F;F)F7F +*&\"$@#F)F*F+,$\"$4\"F+*&\"$*GF)F7F+,$*&\"#MF)F7F+F+*&F)F)F*F+F-F-F-7 +*&FAF)F0F+,$*&\"#hF)\"#;F+F+*&\"$8\"F)F4F+,$*&\"#fF)F0F+F+,$*&\"#6F)F 4F+F+*&FAF)F7F+*&F)F)FXF+F-F-7+F)*&\"$e$F)\"#TF+,$*&\"%z?F)\"##)F+F+*& \"$,&F)F`oF+*&\"$<%F)F`oF+,$*&\"$F#F)F`oF+F+,$*&F*F)FdoF+F+*&\"#OF)F`o F+F-7+F-*&F`oF)\"$S)F+\"\"!*&F*F)\"#NF+*&F*F)\"$!GF+*&FQF)\"$0\"F+*&F* F)FgpF+*&F*F)FepF+*&F`oF)FbpF+" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 34 "#------------- --------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "checking \+ the scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "coefficie nts of the scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 550 "ee := \{c[2]=1/9,\nc[3]=1/6,\nc[4]=1/3,\nc[5 ]=1/2,\nc[6]=2/3,\nc[7]=5/6,\nc[8]=1,\n\na[2,1]=1/9,\na[3,1]=1/24,\na[ 3,2]=1/8,\na[4,1]=1/6,\na[4,2]=-1/2,\na[4,3]=2/3,\na[5,1]=-5/8,\na[5,2 ]=27/8,\na[5,3]=-3,\na[5,4]=3/4,\na[6,1]=221/9,\na[6,2]=-981/9,\na[6,3 ]=289/3,\na[6,4]=-34/3,\na[6,5]=1/9,\na[7,1]=-61/16,\na[7,2]=113/8,\na [7,3]=-59/6,\na[7,4]=-11/8,\na[7,5]=5/3,\na[7,6]=1/16,\na[8,1]=358/41, \na[8,2]=-2079/82,\na[8,3]=501/41,\na[8,4]=417/41,\na[8,5]=-227/41,\na [8,6]=-9/82,\na[8,7]=36/41,\n\nb[1]=41/840,\nb[2]=0,\nb[3]=9/35,\nb[4] =9/280,\nb[5]=34/105,\nb[6]=9/280,\nb[7]=9/35,\nb[8]=41/840\}:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "subs(ee,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(9 -i)],i=2..8),\n[``,seq(b[i],i=1..8)]]));\n``;\nevalf[8](%%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7*7+#\"\"\"\"\"*F(%!GF+F+F+F+F+F +7+#F)\"\"'#F)\"#C#F)\"\")F+F+F+F+F+F+7+#F)\"\"$F-#!\"\"\"\"##F8F5F+F+ F+F+F+7+#F)F8#!\"&F2#\"#FF2!\"$#F5\"\"%F+F+F+F+7+F9#\"$@#F*!$4\"#\"$*G F5#!#MF5F(F+F+F+7+#\"\"&F.#!#h\"#;#\"$8\"F2#!#fF.#!#6F2#FMF5#F)FPF+F+7 +F)#\"$e$\"#T#!%z?\"##)#\"$,&Ffn#\"$<%Ffn#!$F#Ffn#!\"*Fin#\"#OFfnF+7+F +#Ffn\"$S)\"\"!#F*\"#N#F*\"$!G#\"#M\"$0\"FjoFhoFeoQ)pprint156\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #K%'matrixG6#7*7+$\")6666!\")F(%!GF+F+F+F+F+F+7+$\")nmm;F*$\")nmmT!\"* $\")++]7F*F+F+F+F+F+F+7+$\")LLLLF*F-$!)+++]F*$\")nmmmF*F+F+F+F+F+7+$\" )+++]F*$!)++]iF*$\")++vL!\"($!\"$\"\"!$\")+++vF*F+F+F+F+7+F9$\")cbbC! \"'$!$4\"FE$\")LLL'*FK$!)LLL6FKF(F+F+F+7+$\")LLL$)F*$!)+]7QFB$\")+]79F K$!)LLL)*FB$!)++v8FB$F.FB$\")++]iF1F+F+7+$\"\"\"FE$\")tqJ()FB$!)fONDFK $\")7&>A\"FK$\")K2<5FK$!)aeObFB$!)5c(4\"F*$\")y[!y)F*F+7+F+$\")C&4)[F1 $FEFE$\")'G9d#F*$\")dG9KF1$\")_4QKF*FapF_pF\\pQ)pprint166\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "RK 6_8eqs := [op(RowSumConditions(8,'expanded')),op(OrderConditions(6,8,' expanded'))]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "simpl ify(subs(ee,RK6_8eqs)):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7N\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#W" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 86 "Next we set-up stage-order condtions to check for stage -orders from 2 to 4 inclusive. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "for ct from 2 to 4 do\n so||ct||_8 := StageOrderConditions(c t,8,'expanded');\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "Stages 3 to 9 have the following respective stage -orders. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "[seq([seq(expa nd(subs(ee,so||i||_8[j])),i=2..4)],j=1..6)]:\nmap(proc(L) local i; for i to nops(L) do if not evalb(L[i]) then break end if end do; i end pr oc,%):\nsimplify(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(\"\"$F$F$F$ F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 " #-------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 27 "The s implifying conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j+1 .. 8) = b[j]*(1-c[j]);" "6#/-%$SumG6$*&& %\"bG6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,\"\")*&&F)6#F0F,,&F,F ,&%\"cG6#F0!\"\"F," }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\" \"\"" }{TEXT -1 7 " . . 7 " }}{PARA 0 "" 0 "" {TEXT -1 8 "(where " } {XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\"\"!" }{TEXT -1 17 " ) are s atisfied." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "[Sum(b[i]*a[i,1],i=2..8)=b[1],seq(Sum(b[i]*a[i,j] ,i=j+1..8)=b[j]*(1-c[j]),j=2..7)];\neval(subs(Sum=add,%)):\nsubs(ee,%) :\nmap(u->`if`(lhs(u)=rhs(u),0,1),%);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7)/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F,F-F-/F,;\"\" #\"\")&F*6#F-/-F&6$*&F)F-&F/6$F,F3F-/F,;\"\"$F4*&&F*6#F3F-,&F-F-&%\"cG FB!\"\"F-/-F&6$*&F)F-&F/6$F,F?F-/F,;\"\"%F4*&&F*6#F?F-,&F-F-&FEFRFFF-/ -F&6$*&F)F-&F/6$F,FOF-/F,;\"\"&F4*&&F*6#FOF-,&F-F-&FEFjnFFF-/-F&6$*&F) F-&F/6$F,FgnF-/F,;\"\"'F4*&&F*6#FgnF-,&F-F-&FEFhoFFF-/-F&6$*&F)F-&F/6$ F,FeoF-/F,;\"\"(F4*&&F*6#FeoF-,&F-F-&FEFfpFFF-/-F&6$*&F)F-&F/6$F,FcpF- /F,;F4F4*&&F*6#FcpF-,&F-F-&FEFcqFFF-" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#7)\"\"!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 simpl ifying condition: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,2],i = 3 .. 8) = 0;" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"& %\"aG6$F+\"\"#F,/F+;\"\"$\"\")\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 13 "is satisfied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "Sum(b[i]*a[i,2],i=3..8);\nev al(subs(Sum=add,%));\nsubs(ee,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#- %$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F*\"\"#F+/F*;\"\"$\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*&&%\"bG6#\"\"$\"\"\"&%\"aG6$F(\"\"#F)F)*& &F&6#\"\"%F)&F+6$F1F-F)F)*&&F&6#\"\"&F)&F+6$F7F-F)F)*&&F&6#\"\"'F)&F+6 $F=F-F)F)*&&F&6#\"\"(F)&F+6$FCF-F)F)*&&F&6#\"\")F)&F+6$FIF-F)F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 27 "The simplifying condition: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*c[i]*a[i,2],i = 3 \+ .. 8) = 0;" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cG6#F+F,&%\"aG6$F+\" \"#F,/F+;\"\"$\"\")\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 13 "is satisfied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "Sum(b[i]*c[i]*a[i,2],i=3..8);\neval(subs(Sum= add,%));\nsubs(ee,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*(&% \"bG6#%\"iG\"\"\"&%\"cGF)F+&%\"aG6$F*\"\"#F+/F*;\"\"$\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*(&%\"bG6#\"\"$\"\"\"&%\"cGF'F)&%\"aG6$F( \"\"#F)F)*(&F&6#\"\"%F)&F+F2F)&F-6$F3F/F)F)*(&F&6#\"\"&F)&F+F9F)&F-6$F :F/F)F)*(&F&6#\"\"'F)&F+F@F)&F-6$FAF/F)F)*(&F&6#\"\"(F)&F+FGF)&F-6$FHF /F)F)*(&F&6#\"\")F)&F+FNF)&F-6$FOF/F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The simplifying condition: " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "Sum(b[i]*c[i]^2*a[i,2],i = 3 .. 8) = 0;" "6#/-%$SumG 6$*(&%\"bG6#%\"iG\"\"\"*$&%\"cG6#F+\"\"#F,&%\"aG6$F+F1F,/F+;\"\"$\"\") \"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 13 "is satisfied." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "Sum(b[i]*c[i]^2*a[i,2],i=3..8);\neval(subs(Sum=add,%));\nsubs(ee ,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*(&%\"bG6#%\"iG\"\"\" )&%\"cGF)\"\"#F+&%\"aG6$F*F/F+/F*;\"\"$\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*(&%\"bG6#\"\"$\"\"\")&%\"cGF'\"\"#F)&%\"aG6$F(F-F)F) *(&F&6#\"\"%F))&F,F3F-F)&F/6$F4F-F)F)*(&F&6#\"\"&F))&F,F;F-F)&F/6$F " 0 "" {MPLTEXT 1 0 85 "Sum(b[i]*c[i]*Sum(a[i,j]*a[j,2],j=3..i-1),i=3..8);\neval(subs(Su m=add,%));\nsubs(ee,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*( &%\"bG6#%\"iG\"\"\"&%\"cGF)F+-F$6$*&&%\"aG6$F*%\"jGF+&F26$F4\"\"#F+/F4 ;\"\"$,&F*F+F+!\"\"F+/F*;F:\"\")" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,, **&%\"bG6#\"\"%\"\"\"&%\"cGF'F)&%\"aG6$F(\"\"$F)&F-6$F/\"\"#F)F)*(&F&6 #\"\"&F)&F+F5F),&*&&F-6$F6F/F)F0F)F)*&&F-6$F6F(F)&F-6$F(F2F)F)F)F)*(&F &6#\"\"'F)&F+FCF),(*&&F-6$FDF/F)F0F)F)*&&F-6$FDF(F)F?F)F)*&&F-6$FDF6F) &F-6$F6F2F)F)F)F)*(&F&6#\"\"(F)&F+FTF),**&&F-6$FUF/F)F0F)F)*&&F-6$FUF( F)F?F)F)*&&F-6$FUF6F)FPF)F)*&&F-6$FUFDF)&F-6$FDF2F)F)F)F)*(&F&6#\"\")F )&F+FboF),,*&&F-6$FcoF/F)F0F)F)*&&F-6$FcoF(F)F?F)F)*&&F-6$FcoF6F)FPF)F )*&&F-6$FcoFDF)F^oF)F)*&&F-6$FcoFUF)&F-6$FUF2F)F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "#---------------------------------------" }} {PARA 0 "" 0 "" {TEXT -1 101 "We can calculate the 2 norm of the princ ipal error, that is, the 2-norm of the principal error terms." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "errterms6_8 := PrincipalErr orTerms(6,8,'expanded'):\nsqrt(add(subs(ee,errterms6_8[i])^2,i=1.. nop s(errterms6_8))):\nevalf[10](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"+9SXx*)!#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "12 of the 48 principal error conditions are satisfied." } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 243 "RK6_8err_eqs := PrincipalE rrorConditions(6,8,'expanded'):\nexpand(subs(ee,RK6_8err_eqs)):\nmap(u ->`if`(lhs(u)=rhs(u),0,1),%);\nL := %: ind := []:\nfor ct to nops(L) d o\n if L[ct]=0 then ind := [op(ind),ct] end if:\nend do:\nnops(L);\n ind;\nnops(ind);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7R\"\"\"F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$\"\"!F$F$F$F$F$F%F$F%F%F$F$F$F$F%F$F$F%F$F$F $F%F%F$F%F$F%F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#[" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7.\"#?\"#E\"#G\"#H\"#M\"#P\"#T\"#U\"#W\"#Y\"#Z \"#[" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "These simple principal er ror conditions in abreviated form are as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "RK6_err_eqs := PrincipalErrorConditions(6): \nconvert([seq([ind[i],` `,RK6_err_eqs[ind[i]]],i=1..nops(ind))],matr ix);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7.7%\"#?%#~~G/*&% \"bG\"\"\"-%!G6#*(%\"aGF--F/6#*&F2F-%\"cGF-F--F/6#*&)F6\"\"#F-F2F-F-F- #F-\"$_#7%\"#EF)/*&F,F--F/6#*(F2F-F6F-)F3F;F-F-#F-\"$o\"7%\"#GF)/*&F,F -)F3\"\"$F-#F-\"#c7%\"#HF)/*&F,F--F/6#*(F2F-F:F-F7F-F-#F-\"$E\"7%\"#MF )/**F,F-F6F-F3F-F7F-#F-\"#U7%\"#PF)/*&F,F--F/6#*(F2F-)F6FMF-F3F-F-#F- \"#%)7%\"#TF)/*&F,F-)F7F;F-#F-\"#j7%FhnF)/*(F,F-F:F-FEF-#F-FI7%\"#WF)/ *(F,F-F`oF-F7F-#F-\"#@7%\"#YF)/*(F,F-)F6\"\"%F-F3F-#F-\"#97%\"#ZF)/*&F ,F--F/6#*&)F6\"\"&F-F2F-F-Fgn7%\"#[F)/*&F,F-)F6\"\"'F-#F-\"\"(Q(pprint 46\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 34 "#------------------ ---------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 27 "construction o f the scheme " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "We specify the nodes " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "c[2] = 1/9;" "6#/&%\"cG6#\"\"#*&\"\"\"F)\"\"*!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3] = 1/6;" "6#/&%\"cG6#\"\"$*&\"\" \"F)\"\"'!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 1/3;" "6#/&% \"cG6#\"\"%*&\"\"\"F)\"\"$!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5 ] = 1/2;" "6#/&%\"cG6#\"\"&*&\"\"\"F)\"\"#!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[6] = 2/3;" "6#/&%\"cG6#\"\"'*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7] = 5/6;" "6#/&%\"cG6#\"\"(*&\"\" &\"\"\"\"\"'!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[8]=1" "6#/&%\"c G6#\"\")\"\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 15 "and th e weight " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "b[2]=0" "6#/&%\"bG6#\"\" #\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "The scheme is designed so as to satisfy the " } {TEXT 260 7 "order 7" }{TEXT -1 23 " quadrature conditions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "li nalg[transpose](convert([QuadratureConditions(7,8,'expanded')],matrix) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7#/,2&%\"bG6#\"\" \"F-&F+6#\"\"#F-&F+6#\"\"$F-&F+6#\"\"%F-&F+6#\"\"&F-&F+6#\"\"'F-&F+6# \"\"(F-&F+6#\"\")F-F-7#/,0*&F.F-&%\"cGF/F-F-*&F1F-&FHF2F-F-*&F4F-&FHF5 F-F-*&F7F-&FHF8F-F-*&F:F-&FHF;F-F-*&F=F-&FHF>F-F-*&F@F-&FHFAF-F-#F-F07 #/,0*&F.F-)FGF0F-F-*&F1F-)FJF0F-F-*&F4F-)FLF0F-F-*&F7F-)FNF0F-F-*&F:F- )FPF0F-F-*&F=F-)FRF0F-F-*&F@F-)FTF0F-F-#F-F37#/,0*&F.F-)FGF3F-F-*&F1F- )FJF3F-F-*&F4F-)FLF3F-F-*&F7F-)FNF3F-F-*&F:F-)FPF3F-F-*&F=F-)FRF3F-F-* &F@F-)FTF3F-F-#F-F67#/,0*&F.F-)FGF6F-F-*&F1F-)FJF6F-F-*&F4F-)FLF6F-F-* &F7F-)FNF6F-F-*&F:F-)FPF6F-F-*&F=F-)FRF6F-F-*&F@F-)FTF6F-F-#F-F97#/,0* &F.F-)FGF9F-F-*&F1F-)FJF9F-F-*&F4F-)FLF9F-F-*&F7F-)FNF9F-F-*&F:F-)FPF9 F-F-*&F=F-)FRF9F-F-*&F@F-)FTF9F-F-#F-F<7#/,0*&F.F-)FGF " 0 " " {MPLTEXT 1 0 183 "Qeqs := QuadratureConditions(7,8,'expanded'):\ne1 \+ := \{c[3]=1/6,c[4]=1/3,c[5]=1/2,c[6]=2/3,c[7]=5/6,c[8]=1,b[2]=0\}:\nqu adeqns := subs(e1,Qeqs):\nnops(quadeqns);\nindets(quadeqns);\nnops(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<)&%\"bG6#\"\"\"&F%6#\"\"$&F%6#\"\"%&F%6#\"\"&&F%6#\"\" '&F%6#\"\")&F%6#\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"(" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "We have 7 linear equations for the 7 weights." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "e2 := solve(\{op(quadeqns)\}):\ninfolevel[solve] := 0 :\ne3 := `union`(e1,e2):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 27 "The weights are as follows." }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 31 "seq(b[i]=subs(e3,b[i]),i=1..8);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6*/&%\"bG6#\"\"\"#\"#T\"$S)/&F%6#\"\"#\"\"!/&F%6# \"\"$#\"\"*\"#N/&F%6#\"\"%#F5\"$!G/&F%6#\"\"&#\"#M\"$0\"/&F%6#\"\"'F;/ &F%6#\"\"(F4/&F%6#\"\")F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 186 "e3 := \{b[8] = 41/840, b[4] = 9/280, b[6] = 9/280, b [7] = 9/35, b[5] = 34/105, b[1] = 41/840, b[3] = 9/35, c[8] = 1, c[7] \+ = 5/6, b[2] = 0, c[6] = 2/3, c[3] = 1/6, c[4] = 1/3, c[5] = 1/2\}:" } {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 159 "We determine the linking coefficients by means of a system of equations that consists in part of the stage-order equati ons that ensure that stages 2 to 8 have " }{TEXT 260 13 "stage-order 3 " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 24 "We also incorporate t he " }{TEXT 260 22 "simplifying conditions" }{TEXT -1 2 ": " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j+1 .. \+ 8) = b[j]*(1-c[j]);" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+%\"jG F,/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 = 5;" "6#/% \"jG\"\"&" }{TEXT -1 15 ", 6, 7 ( with " }{XPPEDIT 18 0 "c[1]=0" "6#/ &%\"cG6#\"\"\"\"\"!" }{TEXT -1 55 " ), together with the further simp lifying conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*c[i]*a[i,2],i=3..8)=0" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"& %\"cG6#F+F,&%\"aG6$F+\"\"#F,/F+;\"\"$\"\")\"\"!" }{TEXT -1 8 ", \+ " }{XPPEDIT 18 0 "Sum(b[i]*c[i]*Sum(a[i,j]*a[j,2],j=3..i-1),i=3..8)=0 " "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cG6#F+F,-F%6$*&&%\"aG6$F+%\"jG F,&F46$F6\"\"#F,/F6;\"\"$,&F+F,F,!\"\"F,/F+;F<\"\")\"\"!" }{TEXT -1 8 ", " }{XPPEDIT 18 0 "Sum(b[i]*c[i]^2*a[i,2],i=3..8)=0" "6#/-%$Su mG6$*(&%\"bG6#%\"iG\"\"\"*$&%\"cG6#F+\"\"#F,&%\"aG6$F+F1F,/F+;\"\"$\" \")\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 26 "We use the s ingle order 6 " }{TEXT 260 15 "order condition" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b [i]*c[i]*Sum(a[i,j]*c[j]^3,j = 2 .. i-1),i = 3 .. 8) = 1/24" "6#/-%$Su mG6$*(&%\"bG6#%\"iG\"\"\"&%\"cG6#F+F,-F%6$*&&%\"aG6$F+%\"jGF,*$&F.6#F6 \"\"$F,/F6;\"\"#,&F+F,F,!\"\"F,/F+;F:\"\")*&F,F,\"#CF?" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "We in troduce the " }{TEXT 260 23 "additional requirements" }{TEXT -1 27 " t hat the coefficients of " }{XPPEDIT 18 0 "z^7;" "6#*$%\"zG\"\"(" } {TEXT -1 6 " and " }{XPPEDIT 18 0 "z^8" "6#*$%\"zG\"\")" }{TEXT -1 35 " in the stability polynomial are " }{XPPEDIT 18 0 "9/(8*`.`*7!) \+ = 1/4480;" "6#/*&\"\"*\"\"\"*(\"\")F&%\".GF&-%*factorialG6#\"\"(F&!\" \"*&F&F&\"%![%F." }{TEXT -1 7 " and " }{XPPEDIT 18 0 "1/(12*`.`*8!) \+ = 1/483840;" "6#/*&\"\"\"F%*(\"#7F%%\".GF%-%*factorialG6#\"\")F%!\"\"* &F%F%\"'SQ[F-" }{TEXT -1 16 " respectively. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "StabilityFun ction(6,8,'expanded');" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,4\"\"\"F$% \"zGF$*&\"\"#!\"\"F%F'F$*&\"\"'F(F%\"\"$F$*&\"#CF(F%\"\"%F$*&\"$?\"F(F %\"\"&F$*&\"$?(F(F%F*F$*&,&*0&%\"bG6#\"\"(F$&%\"aG6$F:F*F$&F<6$F*F1F$& F<6$F1F.F$&F<6$F.F+F$&F<6$F+F'F$&%\"cG6#F'F$F$*&&F86#\"\")F$,&*.&F<6$F LF*F$F>F$F@F$FBF$FDF$FFF$F$*&&F<6$FLF:F$,&*,&F<6$F:F1F$F@F$FBF$FDF$FFF $F$*&F;F$,&**&F<6$F*F.F$FBF$FDF$FFF$F$*&F>F$,&*(&F<6$F1F+F$FDF$FFF$F$* &F@F$,&*&&F<6$F.F'F$FFF$F$*&FBF$&FG6#F+F$F$F$F$F$F$F$F$F$F$F$F$F$)F%F: F$F$*4FJF$FRF$F;F$F>F$F@F$FBF$FDF$FFF$)F%FLF$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "Thus we specify that " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[7]*a[7,6]*a[6,5]*a[ 5,4]*a[4,3]*a[3,2]*c[2]+``" "6#,&*0&%\"bG6#\"\"(\"\"\"&%\"aG6$F(\"\"'F )&F+6$F-\"\"&F)&F+6$F0\"\"%F)&F+6$F3\"\"$F)&F+6$F6\"\"#F)&%\"cG6#F9F)F )%!GF)" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[8]*(a[8,6]*a[6,5]*a[5,4]*a[4,3]*a[3,2]*c[2]+a[8,7]*(a[7,5]*a[5 ,4]*a[4,3]*a[3,2]*c[2]+a[7,6]*(a[6,4]*a[4,3]*a[3,2]*c[2]+a[6,5]*(a[5,3 ]*a[3,2]*c[2]+a[5,4]*(a[4,2]*c[2]+a[4,3]*c[3]))))) = 1/4480;" "6#/*&&% \"bG6#\"\")\"\"\",&*.&%\"aG6$F(\"\"'F)&F-6$F/\"\"&F)&F-6$F2\"\"%F)&F-6 $F5\"\"$F)&F-6$F8\"\"#F)&%\"cG6#F;F)F)*&&F-6$F(\"\"(F),&*,&F-6$FBF2F)& F-6$F2F5F)&F-6$F5F8F)&F-6$F8F;F)&F=6#F;F)F)*&&F-6$FBF/F),&**&F-6$F/F5F )&F-6$F5F8F)&F-6$F8F;F)&F=6#F;F)F)*&&F-6$F/F2F),&*(&F-6$F2F8F)&F-6$F8F ;F)&F=6#F;F)F)*&&F-6$F2F5F),&*&&F-6$F5F;F)&F=6#F;F)F)*&&F-6$F5F8F)&F=6 #F8F)F)F)F)F)F)F)F)F)F)F)*&F)F)\"%![%!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "b[8]*a[8,7]*a[7,6]*a[6,5]*a[5,4]*a[4,3]*a[3,2]*c[2] = 1 /483840;" "6#/*2&%\"bG6#\"\")\"\"\"&%\"aG6$F(\"\"(F)&F+6$F-\"\"'F)&F+6 $F0\"\"&F)&F+6$F3\"\"%F)&F+6$F6\"\"$F)&F+6$F9\"\"#F)&%\"cG6#F " 0 "" {MPLTEXT 1 0 875 "SO_eqs := [op(RowSumConditi ons(8,'expanded')),op(StageOrderConditions(2,8,'expanded')),\n \+ op(StageOrderConditions(3,8,'expanded'))]:\nsimp_eqs := [add(b[i ]*a[i,1],i=2..8)=b[1],seq(add(b[i]*a[i,j],i=j+1..8)=b[j]*(1-c[j]),j=[5 ,6,7]),\n add(b[i]*c[i]*a[i,2],i=3..8)=0,add(b[i]*c[i]*ad d(a[i,j]*a[j,2],j=3..i-1),i=3..8)=0,\n add(b[i]*c[i]^2 *a[i,2],i=3..8)=0]:\nord_cdn := add(b[i]*c[i]*add(a[i,j]*c[j]^3,j=2..i -1),i=3..8)=1/24:\nextra_eqs := [b[7]*a[7,6]*a[6,5]*a[5,4]*a[4,3]*a[3, 2]*c[2]+b[8]*(a[8,6]*a[6,5]*a[5,4]*a[4,3]*a[3,2]*c[2]+\n \+ a[8,7]*(a[7,5]*a[5,4]*a[4,3]*a[3,2]*c[2]+a[7,6]*(a[6,4]*a[4,3]*a[3,2]* c[2]+\n a[6,5]*(a[5,3]*a[3,2]*c[2]+a[5,4]*(a[4,2]*c[2]+a[ 4,3]*c[3])))))=1/4480,\n b[8]*a[8,7]*a[7,6]*a[6,5]*a[5,4] *a[4,3]*a[3,2]*c[2]=1/483840]:\ncdns := [op(SO_eqs),op(simp_eqs),ord_c dn,op(extra_eqs)]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "eqns := subs(e3,cdns):\nnops(eqns);\nindets (eqns);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#F(&F%6$F1F>&F%6$F'F.&F%6$F1F;&F%6$F>F4&F%6$F>F;&F%6$F1F.&F %6$F;F4&F%6$F>F.&F%6$F>F+&%\"cG6#F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #\"#H" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "There are 29 equations and 29 unknown coefficients." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "e4 := solve(\{op(eqns)\}):\ninfolevel[sol ve] := 0:\ne5 := `union`(e3,e4):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 2 "e5" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 632 "e5 := \{b[8] = 41/840, b[4] = 9/280, b[6] = 9 /280, b[7] = 9/35, b[5] = 34/105, b[1] = 41/840, a[5,1] = -5/8, a[5,2] = 27/8, a[5,3] = -3, a[5,4] = 3/4, b[3] = 9/35, c[8] = 1, c[7] = 5/6, b[2] = 0, c[6] = 2/3, c[3] = 1/6, c[4] = 1/3, c[5] = 1/2, c[2] = 1/9, a[2,1] = 1/9, a[3,1] = 1/24, a[3,2] = 1/8, a[4,1] = 1/6, a[4,2] = -1/ 2, a[4,3] = 2/3, a[8,7] = 36/41, a[6,1] = 221/9, a[6,2] = -109, a[6,3] = 289/3, a[6,4] = -34/3, a[6,5] = 1/9, a[7,1] = -61/16, a[7,2] = 113/ 8, a[7,3] = -59/6, a[7,4] = -11/8, a[7,5] = 5/3, a[7,6] = 1/16, a[8,1] = 358/41, a[8,2] = -2079/82, a[8,3] = 501/41, a[8,4] = 417/41, a[8,5] = -227/41, a[8,6] = -9/82\}:" }}}{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 64 "The Butcher tableau in exact an d approximate form is as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "subs(e5,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(9 -i)],i=2..8),\n[``,seq(b[i],i=1..8)]]));\n``;\nevalf[8](%%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7*7+#\"\"\"\"\"*F(%!GF+F+F+F+F+F +7+#F)\"\"'#F)\"#C#F)\"\")F+F+F+F+F+F+7+#F)\"\"$F-#!\"\"\"\"##F8F5F+F+ F+F+F+7+#F)F8#!\"&F2#\"#FF2!\"$#F5\"\"%F+F+F+F+7+F9#\"$@#F*!$4\"#\"$*G F5#!#MF5F(F+F+F+7+#\"\"&F.#!#h\"#;#\"$8\"F2#!#fF.#!#6F2#FMF5#F)FPF+F+7 +F)#\"$e$\"#T#!%z?\"##)#\"$,&Ffn#\"$<%Ffn#!$F#Ffn#!\"*Fin#\"#OFfnF+7+F +#Ffn\"$S)\"\"!#F*\"#N#F*\"$!G#\"#M\"$0\"FjoFhoFeoQ(pprint96\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #K%'matrixG6#7*7+$\")6666!\")F(%!GF+F+F+F+F+F+7+$\")nmm;F*$\")nmmT!\"* $\")++]7F*F+F+F+F+F+F+7+$\")LLLLF*F-$!)+++]F*$\")nmmmF*F+F+F+F+F+7+$\" )+++]F*$!)++]iF*$\")++vL!\"($!\"$\"\"!$\")+++vF*F+F+F+F+7+F9$\")cbbC! \"'$!$4\"FE$\")LLL'*FK$!)LLL6FKF(F+F+F+7+$\")LLL$)F*$!)+]7QFB$\")+]79F K$!)LLL)*FB$!)++v8FB$F.FB$\")++]iF1F+F+7+$\"\"\"FE$\")tqJ()FB$!)fONDFK $\")7&>A\"FK$\")K2<5FK$!)aeObFB$!)5c(4\"F*$\")y[!y)F*F+7+F+$\")C&4)[F1 $FEFE$\")'G9d#F*$\")dG9KF1$\")_4QKF*FapF_pF\\pQ)pprint106\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "RK 6_8eqs := [op(RowSumConditions(8,'expanded')),op(OrderConditions(6,8,' expanded'))]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "simpl ify(subs(e5,RK6_8eqs)):\nmap(u->`if`(lhs(u)=rhs(u),0,1),%);\nnops(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7N\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#W" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 34 "#---------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 25 "absolute stability region" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "coefficients of the scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 550 "ee := \{c[2]=1/9, \nc[3]=1/6,\nc[4]=1/3,\nc[5]=1/2,\nc[6]=2/3,\nc[7]=5/6,\nc[8]=1,\n\na[ 2,1]=1/9,\na[3,1]=1/24,\na[3,2]=1/8,\na[4,1]=1/6,\na[4,2]=-1/2,\na[4,3 ]=2/3,\na[5,1]=-5/8,\na[5,2]=27/8,\na[5,3]=-3,\na[5,4]=3/4,\na[6,1]=22 1/9,\na[6,2]=-981/9,\na[6,3]=289/3,\na[6,4]=-34/3,\na[6,5]=1/9,\na[7,1 ]=-61/16,\na[7,2]=113/8,\na[7,3]=-59/6,\na[7,4]=-11/8,\na[7,5]=5/3,\na [7,6]=1/16,\na[8,1]=358/41,\na[8,2]=-2079/82,\na[8,3]=501/41,\na[8,4]= 417/41,\na[8,5]=-227/41,\na[8,6]=-9/82,\na[8,7]=36/41,\n\nb[1]=41/840, \nb[2]=0,\nb[3]=9/35,\nb[4]=9/280,\nb[5]=34/105,\nb[6]=9/280,\nb[7]=9/ 35,\nb[8]=41/840\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 77 "The stability function R for the 8 stage, order 6 scheme is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "subs(ee,StabilityFunction(6,8,'expanded')):\nR := una pply(%,z):\n'R(z)'=R(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"RG6#% \"zG,4\"\"\"F)F'F)*&#F)\"\"#F)*$)F'F,F)F)F)*&#F)\"\"'F)*$)F'\"\"$F)F)F )*&#F)\"#CF)*$)F'\"\"%F)F)F)*&#F)\"$?\"F)*$)F'\"\"&F)F)F)*&#F)\"$?(F)* $)F'F1F)F)F)*&#F)\"%![%F)*$)F'\"\"(F)F)F)*&#F)\"'SQ[F)*$)F'\"\")F)F)F) " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We \+ can find the point where the boundary of the stability region intersec ts the negative real axis by solving the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R(z) = -1;" "6#/-%\"RG6#%\"zG,$\"\" \"!\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 " z0 := newton(R(z)=-1,z=-3.8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0 G$!+QW-SQ!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 308 "z0 := newton(R(z)=-1,z=-3.8):\np1 := plot([R(z) ,-1],z=-4.29..0.49,color=[red,blue]):\np2 := plot([[[z0,-1]]$3],style= point,symbol=[circle,cross,diamond],color=black):\np3 := plot([[z0,0], [z0,-1]],linestyle=3,color=COLOR(RGB,0,.5,0)):\nplots[display]([p1,p2, p3],view=[-4.29..0.49,-1.47..1.47],font=[HELVETICA,9]);" }}{PARA 13 " " 1 "" {GLPLOT2D 360 253 253 {PLOTDATA 2 "6+-%'CURVESG6$7U7$$!3/++++++ !H%!#<$!3%=k:9J\\rJ#F*7$$!3;n;a,[!zB%F*$!3V%Hk*=-\"R6#F*7$$!3QLL3.'4e= %F*$!3]+KTyX+E>F*7$$!3`]iX\\=[STF*$!3!)**H3H!3Ux\"F*7$$!3ym\"He4a^4%F* $!3xB(oRlUEj\"F*7$$!3EL$el\\-K*RF*$!39U'*Hdye[8F*7$$!3QL$3f4v0*QF*$!3G ?6)Ri0c5\"F*7$$!3am\"z`YN%)y$F*$!3A(R$z%3<#4!*!#=7$$!35Le4`*QPp$F*$!33 =6t^wj)R(FP7$$!3-+Dm*>'o&f$F*$!3qU%*[WI%H)fFP7$$!3ELez,*zU\\$F*$!3cMcn y`H^ZFP7$$!35+D@:))>$R$F*$!3[JF>O$F*$!3u-Ee*e5')>#FP7$$!3a***\\YuXX4$F*$!3ST0Q*pd#)e \"FP7$$!3!)***\\ZIC5*HF*$!37?>6:0x!3\"FP7$$!3#)***\\`@i7*GF*$!3#=RFK4K mt'!#>7$$!3)H$ez!oo1!GF*$!3b)3FGIg\"4OFhp7$$!3gmm\"e?WHp#F*$!3_A#zJ823 d%!#?7$$!3HmmEXuo,EF*$\"3**pEYzdvI=Fhp7$$!3#)*\\i$QT`&\\#F*$\"37'R*z,s F*$\"3]_8 )4AQ$Q9FP7$$!3/LezXE(pz\"F*$\"3gehL%zHGi\"FP7$$!3\"**\\PG?L_q\"F*$\"3) \\u!y*)>^$z\"FP7$$!38L$e64_hg\"F*$\"3cZkjE0O\"*>FP7$$!3w****p2;z.:F*$ \"3!e(4gh#RN@#FP7$$!3p**\\_W=l.9F*$\"3!HBn7v-9X#FP7$$!3!)*\\iXZhnI\"F* $\"3.wmLUkn.FFP7$$!3)***\\2Q7=*>\"F*$\"3r45@i&yE,$FP7$$!3$pmm2h:D5\"F* $\"3'fA:^()>%>LFP7$$!3+(***\\2U/$***FP$\"3!GL\\j<14o$FP7$$!3WJ$eRU(zd! *FP$\"3kNg[*)*[?/%FP7$$!3O%***\\ecKN!)FP$\"3ko/))4_NxWFP7$$!3sl;H5%fK2 (FP$\"3LeA4]8dH\\FP7$$!3i(*\\7uHingFP$\"3W8Q\"*))y6^aFP7$$!3[mm;W&HW3& FP$\"3]L'Ql#*4V,'FP7$$!3u**\\iSr6bSFP$\"3WI6'>Objm'FP7$$!3)\\KL$fpwjIF P$\"3oyLl)p#4htFP7$$!3sGL3:#o*\\?FP$\"33(R(=`!*\\Y\")FP7$$!3&Hm\"HOUcW 5FP$\"3zB1b2093!*FP7$$!3e)*)***pR-27Fhp$\"3#fn&QMJ-!))*FP7$$\"33'RLeGv 9Q*Fhp$\"3CURrfiN)4\"F*7$$\"31ummCb>&)=FP$\"37.Nw\"pgu?\"F*7$$\"3[4]P; i%\\*GFP$\"3l\\@>gAvN8F*7$$\"351]ipkShQFP$\"3%Ho=*3;Hr9F*7$$\"3!****** *********[FP$\"3LMN*yI;Bj\"F*-%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!F\\ \\lF[\\l-F$6$7S7$F($!\"\"F\\\\l7$F3Fa\\l7$F=Fa\\l7$FBFa\\l7$FGFa\\l7$F LFa\\l7$FRFa\\l7$FWFa\\l7$FfnFa\\l7$F[oFa\\l7$F`oFa\\l7$FeoFa\\l7$FjoF a\\l7$F_pFa\\l7$FdpFa\\l7$FjpFa\\l7$F_qFa\\l7$FeqFa\\l7$FjqFa\\l7$F_rF a\\l7$FdrFa\\l7$FirFa\\l7$F^sFa\\l7$FcsFa\\l7$FhsFa\\l7$F]tFa\\l7$FbtF a\\l7$FgtFa\\l7$F\\uFa\\l7$FauFa\\l7$FfuFa\\l7$F[vFa\\l7$F`vFa\\l7$Fev Fa\\l7$FjvFa\\l7$F_wFa\\l7$FdwFa\\l7$FiwFa\\l7$F^xFa\\l7$FcxFa\\l7$Fhx Fa\\l7$F]yFa\\l7$FbyFa\\l7$FgyFa\\l7$F\\zFa\\l7$FazFa\\l7$FfzFa\\l7$F[ [lFa\\l7$F`[lFa\\l-Fe[l6&Fg[lF[\\lF[\\lFh[l-F$6&7#7$$!3$******zVC+%QF* Fa\\l-%'SYMBOLG6#%'CIRCLEG-Fe[l6&Fg[lF\\\\lF\\\\lF\\\\l-%&STYLEG6#%&PO INTG-F$6&Fg_l-F\\`l6#%&CROSSGF_`lFa`l-F$6&Fg_l-F\\`l6#%(DIAMONDGF_`lFa `l-F$6%7$7$Fi_lF[\\lFh_l-%&COLORG6&Fg[lF[\\l$\"\"&Fb\\lF[\\l-%*LINESTY LEG6#\"\"$-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6%Q\"z6\"Q!Febl-F]b l6#%(DEFAULTG-%%VIEWG6$;$!$H%!\"#$\"#\\F`cl;$!$Z\"F`cl$\"$Z\"F`cl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "The following picture shows the stability region. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1438 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/ 120*z^5+1/720*z^6+1/4480*z^7+1/483840*z^8:\npts := []: z0 := 0: tt := \+ 0: \nwhile tt<=201/20 do\n zz := newton(`R`(z)=exp(tt*Pi*I),z=z0):\n z0 := zz:\n if (17/20<=tt and tt<=29/20) or (171/20<=tt and tt<=1 83/20) then\n hh := 1/40\n else \n hh := 1/20\n end if; \n tt := tt+hh;\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np1 : = plot(pts,color=COLOR(RGB,.48,0,.1)):\np2 := plots[polygonplot]([seq( [pts[i-1],pts[i],[-1.9,0]],i=2..nops(pts))],\n style=patchnog rid,color=COLOR(RGB,.95,0,.2)):\npts := []: z0 := 1.6+4.3*I:\nfor ct f rom 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,col or=COLOR(RGB,.48,0,.1)):\np4 := plots[polygonplot]([seq([pts[i-1],pts[ i],[1.54,4.24]],i=2..nops(pts))],\n style=patchnogrid,color=C OLOR(RGB,.95,0,.2)):\npts := []: z0 := 1.6-4.3*I:\nfor ct from 0 to 40 do\n zz := newton(R(z)=exp(ct*Pi/20*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np5 := plot(pts,color=COLOR(RG B,.48,0,.1)):\np6 := plots[polygonplot]([seq([pts[i-1],pts[i],[1.54,-4 .24]],i=2..nops(pts))],\n style=patchnogrid,color=COLOR(RGB,. 95,0,.2)):\np7 := plot([[[-4.49,0],[2.09,0]],[[0,-4.79],[0,4.79]]],col or=black,linestyle=3):\nplots[display]([p||(1..7)],view=[-4.49..2.09,- 4.79..4.79],font=[HELVETICA,9],\n labels=[`Re(z)`,`Im(z)` ],axes=boxed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 381 565 565 {PLOTDATA 2 "6/-%'CURVESG6$7^y7$$\"\"!F)F(7$$\"3!******z<8/v\" !#G$\"3++++Fjzq:!#=7$$\"3M+++0RJ*Q%!#E$\"3\")******=FfTJF07$$\"3-+++@a k'3\"!#C$\"3;+++**)*Q7ZF07$$\"33+++Yz'G.\"!#B$\"3[*****H$G'F07$$\"3 %******p:^Nw&F@$\"3s*****41$*R&yF07$$\"3-+++!)Q1yA!#A$\"32+++i.xC%*F07 $$\"3p*****p&>/OqFK$\"35+++K&R&*4\"!#<7$$\"3%******HiHcz\"!#@$\"3#**** ***4&\\lD\"FS7$$\"37+++4US3RFW$\"33+++B,U89FS7$$\"3m*****4`!)GO(FW$\"3 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the intersection of the stability region with the real line. " }}{PARA 0 "" 0 "" {TEXT -1 59 "For this scheme the stability interva l is (approximately) " }{XPPEDIT 18 0 "[-3.8400, 0];" "6#7$,$-%&Float G6$\"&+%Q!\"%!\"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 143 "We can distort the boundary curve horizontally by taking the 11th root of the real part of points along the curve. In this way we see that the " }{TEXT 260 50 "largest inter val on the nonnegative imaginary axis" }{TEXT -1 83 " that contains th e origin and lies inside the stability region is (approximately) " } {XPPEDIT 18 0 "[0, 2.2];" "6#7$\"\"!-%&FloatG6$\"#A!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 334 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+1/7 20*z^6+1/4480*z^7+1/483840*z^8:\nDigits := 25:\npts := []: z0 := 0:\nf or ct from 0 to 80 do\n zz := newton(R(z)=exp(ct*Pi/100*I),z=z0):\n \+ z0 := zz:\n pts := [op(pts),[surd(Re(zz),11),Im(zz)]]:\nend do:\np lot(pts,color=COLOR(RGB,.95,0,.2),thickness=2,font=[HELVETICA,9]);\nDi gits := 10:" }}{PARA 13 "" 1 "" {GLPLOT2D 261 276 276 {PLOTDATA 2 "6(- %'CURVESG6#7]p7$$\"\"!F)F(7$$\":H:w-^D4J!>&fE$!#E$\":j#=+0o)*e`EfTJF-7 $$\":ybzxc\"*Hi-1kS&F-$\":(H!p7`7*=2`=$G'F-7$$\":y;JIy$H;z!\\(fsF-$\": 1%G-v08$4'zxC%*F-7$$\":g+'=F'=NE^(pZ*)F-$\":_N,x47c:1PmD\"!#D7$$\":YnM (*oje:*=>_5F?$\":b'z*zqJhtK'zq:F?7$$\":q#=v7Fqj=j0,7F?$\":8SJ5\\%R:%fb \\)=F?7$$\":Pzbp/libE7JM\"F?$\":NyeRN]%Hj[6*>#F?7$$\":7`g%y76)[RS&z9F? $\":8#\\t'>-&HPTF8DF?7$$\":6Z\"f^,N%\\R%=6;F?$\":!QQf&Ho`.ULu#GF?7$$\" :jZuhK-#HfGnQF?$\":3q*eBfsr.9\"*pPF?7$$\":4Z+MF*yP\"Gc 05#F?$\"::>\\LaE$G>32%3%F?7$$\":NWb**Qq^^Mk`@#F?$\":a$=Ow\\Og4.B)R%F?7 $$\":J2f\\kYcVyYwK#F?$\":e4Hte%*>&)*)*Q7ZF?7$$\":eB$[mK$enixvV#F?$\":? %\\@qwi=7'\\l-&F?7$$\":*G,X=1:OhRIXDF?$\":#Hd#ez1\"3x%42M&F?7$$\":*3U; +%oN!=5&4l#F?$\":&))*)G`mF(o^p[l&F?7$$\":Z'4%Gy7YX.EYv#F?$\":mVJK_#f&y uH!pfF?7$$\":V0@&=k!Q]C@k&GF?$\":slzMpMXE$G'F?7$$\":!zw2RX'G%>iTcHF ?$\":fJ^=C)))yr2N(f'F?7$$\":.0+Oy7iYrzY0$F?$\":cx%e&=[MH\\6:\"pF?7$$\" :qZ4w>EXG:r7:$F?$\":Y-R7`wkrBscA(F?7$$\":zrBm*o\\!)4=CYKF?$\":7R2\"[!G $4UG$)RvF?7$$\":ViX]O$QivdjRLF?$\":DM&G7i8IhI*R&yF?7$$\":Jj$>!GqvIt!\\ JMF?$\":!=P]\\PPoRD:o\")F?7$$\":nQOI\\>$e)eQ=_$F?$\":bV#eE=aE%y5B[)F?7 $$\":3%y$F?$\":^!Q:0UT4i.xC%*F?7$$\":cx +'Qi9-)y?'oQF?$\":JD1?K[6Hn9*Q(*F?7$$\":w$f#R#oq6#**R<&RF?$\":)*H\"yYE vV#=0`+\"!#C7$$\":azLLOWN#yWWLSF?$\":-'G$yV#4jdzrO5Ffu7$$\":Gbn/!yC^O_ t8TF?$\":E.U!)>!zZ=&H\"o5Ffu7$$\":(**)4OZ!*)G#Q4E>%F?$\":)=!)[8\"4hA`R &*4\"Ffu7$$\":?$p![VKPG8g+F%F?$\":G0?>;sY#*fZ48\"Ffu7$$\":'[m#os&*Rrtw gM%F?$\"::AU5\"pnCVKNi6Ffu7$$\":UVrmq#4=eVk?WF?$\":OO5]9PY9!fv$>\"Ffu7 $$\":0vg_m#G&f!Ru$\\%F?$\":,5*Q)po?N\"\\:D7Ffu7$$\":-^,_6T9!==NlXF?$\" :)>zbRl335&\\lD\"Ffu7$$\":rg7myr'*GyRaj%F?$\":yBEamL10!)QzG\"Ffu7$$\": t*4#o%*GY)*RuRq%F?$\":2!zH\"e@S-w@$>8Ffu7$$\":pgW>0:,iw;4x%F?$\":aG)y/ =ikAO[F?$\":_Mz=P$*[0%Q1#Q\"Ffu7$$\":J]EM0j ;(\\&Q)**[F?$\":c*))[,jVgA,U89Ffu7$$\":_a,E/?J9n2<'\\F?$\":&>:!>V%QnpV wW9Ffu7$$\":9e#Rn(fp@Qi<-&F?$\":!*o)4A[8S'o%4w9Ffu7$$\":$3pK\"*o?iAl#* z]F?$\":FeO(3s2Aj*3u]\"Ffu7$$\":>)4qVd^O'[8h8&F?$\":Hm7XUOXS'[qQ:Ffu7$ $\":\"p%y\"3*fv:pC->&F?$\":I.q\"Q6X\"\\\")z*p:Ffu7$$\":$zCZg=Gt*pZ@C&F ?$\":MVyMAR#*Q*4B,;Ffu7$$\":Qc5I<RO[!*z>ojaF?$\":Qnh8U-%zwK,d*oD58%p[0`&F?$\":5HQaWeHl`J!>=Ffu7$$\":@NWAL]l ].%3ebF?$\":3?iq9s\")Q/a`co5e&F?$\":W#)*\\E;/NNi #3)=Ffu7$$\":\"G!yfZL\\)>V*))f&F?$\":J\"\\$G)[FnMni6>Ffu7$$\":@Vgv=.p/ %fy5cF?$\":/5X!4B:T1`NU>Ffu7$$\":(>yG2Ce6O)Hdh&F?$\":(e#3([FDE/i+t>Ffu 7$$\":L\"4jLRZy?1O7cF?$\":a-b93&*\\.`tN+#Ffu7$$\":uw4(zt))e9%p()f&F?$ \":-h\\-')*=Ph70M?Ffu7$$\":)p_8sk3K:&e@d&F?$\":&4)p1;i2dFLW1#Ffu7$$\": )>)=YIJ\\Z\\k\"GbF?$\":.f!Q17Oq!Q8Z4#Ffu7$$\":pB?K5Ppz&*p$faF?$\":8R\" f.(*zFR`)[7#Ffu7$$\":$Hn#o)fa)pkP9N&F?$\":ludy?(>$\\*G%\\:#Ffu7$$\":EN 42[t^pVM*p^F?$\":?Hd5z4QUzz[=#Ffu7$$\":^I0(oq2\")GE[!y%F?$\":F*)pprV\" *e!)*o9AFfu7$$!:!GNs2U_/y//7YF?$\":Q(3C:gBYNnOWAFfu7$$!:$RD!y(=)['e*\\ /B&F?$\":,UHjp?z]V/RF#Ffu7$$!:3BK8#Hq8H=4UbF?$\":5XBS;*GR2oH.BFfu7$$!: %\\$yti5b@j!GodF?$\":tDHwUUIL!y`KBFfu7$$!:w\"HY8bfs+%yE&fF?$\":Ls\"Ql7 aK89ihBFfu7$$!:#z%oOW'[^qy!>6'F?$\":?_)Rld_\">lT0R#Ffu7$$!:'HJUqg0O6k7 aiF?$\":*>iS9p4\"e`#H>CFfu7$$!:%==sjbmI`Q&RQ'F?$\":3K*HmK.xb!oyW#Ffu7$ $!:OIavp]D&4JE/lF?$\":I>Le*\\H$G9iiZ#Ffu-%&COLORG6&%$RGBG$\"#&*!\"#F($ \"\"#!\"\"-%%FONTG6$%*HELVETICAG\"\"*-%*THICKNESSG6#F_el-%+AXESLABELSG 6$Q!6\"F\\fl-%%VIEWG6$%(DEFAULTGFafl" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 123 "The relevant intersection point of the b oundary curve with the imaginary axis can be determined more accuratel y as follows." }}{PARA 0 "" 0 "" {TEXT -1 86 "First we look for points on the boundary curve either side of the intersection point. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "Digits := 15:\nz0 := 2.2*I:\nfor ct from 70 to 73 do\n newton(R (z)=exp(ct*Pi/100*I),z=z0);\nend do;\nDigits := 10:" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#^$$\"0F[[!\\P_q!#=$\"05QUzz[=#!#9" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#^$$\"06z3A9)zH!#=$\"0W\"*e!)*o9A!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0h%\\n(z#3?!#=$\"0OiatmVC#!#9" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#^$$!0xV'o0[:!)!#=$\"0@z]V/RF#!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "Then we apply the bisect ion method to calculate the parameter value associated with the inters ection point." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 175 "real_part \+ := proc(u)\n Re(newton(R(z)=exp(u*Pi*I),z=2.2*I))\nend proc:\nDigits := 15:\nu0 := bisect('real_part'(u),u=0.7..0.73);\nnewton(R(z)=exp(u0 *Pi*I),z=2.2*I);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%# u0G$\"0>*=))*H?;(!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0PAKN@q?' !#H$\"0-*>mW6LA!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 4 "The " }{TEXT 260 112 "largest interval on the nonnegativ e imaginary axis that contains the origin and lies inside the stabilit y region" }{TEXT -1 5 " is " }{XPPEDIT 18 0 "[0, 2.2331];" "6#7$\"\"! -%&FloatG6$\"&JB#!\"%" }{TEXT -1 18 " (approximately)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 34 "#---------------------------------" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 23 "#======================" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 16 "Huta's scheme B " }}{PARA 0 "" 0 "" {TEXT -1 5 "See: " }}{PARA 0 "" 0 "" {TEXT -1 149 "1. Une am\351lioration de la m\351thode de Runge-Ku tta-Nystr\366m pour la r\351solution num\351rique des \351quations dif f\351rentielles du premi\350r ordre, by Anton Huta, " }}{PARA 0 "" 0 " " {TEXT -1 70 " Acta Fac. Nat. Univ. Comenian Math., Vol. 1, pages \+ 201-224 (1956)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 "2. Contribution \340 la formule de sixi\350me ordre dans la m\351thode de Runge-Kutta-Nystr\366m, by Anton Huta, " }}{PARA 0 " " 0 "" {TEXT -1 67 " Acta Fac.Nat. Univ. Comenian Math., Vol. 2, pa ges 21-24 (1957)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[1/9, 1/9, ``, ``, ``, ``, `` , ``, ``], [1/6, 1/24, 1/8, ``, ``, ``, ``, ``, ``], [1/3, 1/6, -1/2, \+ 2/3, ``, ``, ``, ``, ``], [1/2, 139/272, -945/544, 105/68, 99/544, ``, ``, ``, ``], [2/3, -53/3, 91/2, -52/3, -107/6, 8, ``, ``, ``], [5/6, \+ 55487/22824, -83/16, 2849/1902, 34601/15216, -640/2853, 107/2536, ``, \+ ``], [1, -101195/25994, 351/41, -35994/12997, -26109/25994, -10000/129 97, -36/12997, 36/41, ``], [``, 41/840, 0, 9/35, 9/280, 34/105, 9/280, 9/35, 41/840]])" "6#-%'matrixG6#7*7+*&\"\"\"F)\"\"*!\"\"*&F)F)F*F+%!G F-F-F-F-F-F-7+*&F)F)\"\"'F+*&F)F)\"#CF+*&F)F)\"\")F+F-F-F-F-F-F-7+*&F) F)\"\"$F+*&F)F)F0F+,$*&F)F)\"\"#F+F+*&F;F)F7F+F-F-F-F-F-7+*&F)F)F;F+*& \"$R\"F)\"$s#F+,$*&\"$X*F)\"$W&F+F+*&\"$0\"F)\"#oF+*&\"#**F)FEF+F-F-F- F-7+*&F;F)F7F+,$*&\"#`F)F7F+F+*&\"#\"*F)F;F+,$*&\"#_F)F7F+F+,$*&\"$2\" F)F0F+F+F4F-F-F-7+*&\"\"&F)F0F+*&\"&([bF)\"&CG#F+,$*&\"#$)F)\"#;F+F+*& \"%\\GF)\"%->F+*&\"&,Y$F)\"&;_\"F+,$*&\"$S'F)\"%`GF+F+*&FWF)\"%ODF+F-F -7+F),$*&\"'&>,\"F)\"&%*f#F+F+*&\"$^$F)\"#TF+,$*&\"&%*f$F)\"&(*H\"F+F+ ,$*&\"&4h#F)F\\pF+F+,$*&\"&++\"F)FcpF+F+,$*&\"#OF)FcpF+F+*&F\\qF)F_pF+ F-7+F-*&F_pF)\"$S)F+\"\"!*&F*F)\"#NF+*&F*F)\"$!GF+*&\"#MF)FGF+*&F*F)Fe qF+*&F*F)FcqF+*&F_pF)F`qF+" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 34 "#---------------- -----------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "checking the scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "coefficient s of the scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 613 "ee := \{c[2]=1/9,\nc[3]=1/6,\nc[4]=1/3,\nc[5] =1/2,\nc[6]=2/3,\nc[7]=5/6,\nc[8]=1,\n\na[2,1]=1/9,\na[3,1]=1/24,\na[3 ,2]=1/8,\na[4,1]=1/6,\na[4,2]=-1/2,\na[4,3]=2/3,\na[5,1]=139/272,\na[5 ,2]=-945/544,\na[5,3]=105/68,\na[5,4]=99/544,\na[6,1]=-53/3,\na[6,2]=9 1/2,\na[6,3]=-52/3,\na[6,4]=-107/6,\na[6,5]=8,\na[7,1]=55487/22824,\na [7,2]=-83/16,\na[7,3]=2849/1902,\na[7,4]=34601/15216,\na[7,5]=-640/285 3,\na[7,6]=107/2536,\na[8,1]=-101195/25994,\na[8,2]=351/41,\na[8,3]=-3 5994/12997,\na[8,4]=-26109/25994,\na[8,5]=-10000/12997,\na[8,6]=-36/12 997,\na[8,7]=36/41,\n\nb[1]=41/840,\nb[2]=0,\nb[3]=9/35,\nb[4]=9/280, \nb[5]=34/105,\nb[6]=9/280,\nb[7]=9/35,\nb[8]=41/840\}:" }}}{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 0 109 "subs(ee,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(9-i)],i=2..8 ),\n[``,seq(b[i],i=1..8)]]));\n``;\nevalf[8](%%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7*7+#\"\"\"\"\"*F(%!GF+F+F+F+F+F+7+#F)\"\"' #F)\"#C#F)\"\")F+F+F+F+F+F+7+#F)\"\"$F-#!\"\"\"\"##F8F5F+F+F+F+F+7+#F) F8#\"$R\"\"$s##!$X*\"$W&#\"$0\"\"#o#\"#**FAF+F+F+F+7+F9#!#`F5#\"#\"*F8 #!#_F5#!$2\"F.F2F+F+F+7+#\"\"&F.#\"&([b\"&CG##!#$)\"#;#\"%\\G\"%->#\"& ,Y$\"&;_\"#!$S'\"%`G#\"$2\"\"%ODF+F+7+F)#!'&>,\"\"&%*f##\"$^$\"#T#!&%* f$\"&(*H\"#!&4h#Fbo#!&++\"Fho#!#OFho#\"#OFeoF+7+F+#Feo\"$S)\"\"!#F*\"# N#F*\"$!G#\"#MFCFgpFepFbpQ)pprint156\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7*7+$\")6666! \")F(%!GF+F+F+F+F+F+7+$\")nmm;F*$\")nmmT!\"*$\")++]7F*F+F+F+F+F+F+7+$ \")LLLLF*F-$!)+++]F*$\")nmmmF*F+F+F+F+F+7+$\")+++]F*$\")TH5^F*$!)C8P=F*F+F+F+F+7+F9$!)nmmUF1F+F+7+$\"\"\"FS$!)Q,$*QFB$\")c(4c)FB$!)$ 3%pFFB$!)TU/5FB$!)K3%p(F*$!)+()pF!#5$\")y[!y)F*F+7+F+$\")C&4)[F1$FSFS$ \")'G9d#F*$\")dG9KF1$\")_4QKF*FepFcpF`pQ)pprint166\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "RK6_8eqs \+ := [op(RowSumConditions(8,'expanded')),op(OrderConditions(6,8,'expande d'))]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "simplify(subs(ee ,RK6_8eqs)):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7N\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#W" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "Next we set-up stage-order condtions to check for stage-orders \+ from 2 to 4 inclusive. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "f or ct from 2 to 4 do\n so||ct||_8 := StageOrderConditions(ct,8,'expa nded');\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "Stages 3 to 9 have the following respective stage-orders. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "[seq([seq(expand(subs( ee,so||i||_8[j])),i=2..4)],j=1..6)]:\nmap(proc(L) local i; for i to no ps(L) do if not evalb(L[i]) then break end if end do; i end proc,%):\n simplify(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(\"\"$F$F$F$F$F$" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "#------- ------------------------" }}{PARA 0 "" 0 "" {TEXT -1 27 "The simplifyi ng conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum (b[i]*a[i,j],i = j+1 .. 8) = b[j]*(1-c[j]);" "6#/-%$SumG6$*&&%\"bG6#% \"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,\"\")*&&F)6#F0F,,&F,F,&%\"cG6 #F0!\"\"F," }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" } {TEXT -1 7 " . . 7 " }}{PARA 0 "" 0 "" {TEXT -1 8 "(where " } {XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\"\"!" }{TEXT -1 17 " ) are s atisfied." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "[Sum(b[i]*a[i,1],i=2..8)=b[1],seq(Sum(b[i]*a[i,j] ,i=j+1..8)=b[j]*(1-c[j]),j=2..7)];\neval(subs(Sum=add,%)):\nsubs(ee,%) :\nmap(u->`if`(lhs(u)=rhs(u),0,1),%);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7)/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F,F-F-/F,;\"\" #\"\")&F*6#F-/-F&6$*&F)F-&F/6$F,F3F-/F,;\"\"$F4*&&F*6#F3F-,&F-F-&%\"cG FB!\"\"F-/-F&6$*&F)F-&F/6$F,F?F-/F,;\"\"%F4*&&F*6#F?F-,&F-F-&FEFRFFF-/ -F&6$*&F)F-&F/6$F,FOF-/F,;\"\"&F4*&&F*6#FOF-,&F-F-&FEFjnFFF-/-F&6$*&F) F-&F/6$F,FgnF-/F,;\"\"'F4*&&F*6#FgnF-,&F-F-&FEFhoFFF-/-F&6$*&F)F-&F/6$ F,FeoF-/F,;\"\"(F4*&&F*6#FeoF-,&F-F-&FEFfpFFF-/-F&6$*&F)F-&F/6$F,FcpF- /F,;F4F4*&&F*6#FcpF-,&F-F-&FEFcqFFF-" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#7)\"\"!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 simpl ifying condition: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,2],i = 3 .. 8) = 0;" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"& %\"aG6$F+\"\"#F,/F+;\"\"$\"\")\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 13 "is satisfied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "Sum(b[i]*a[i,2],i=3..8);\nev al(subs(Sum=add,%));\nsubs(ee,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#- %$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F*\"\"#F+/F*;\"\"$\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*&&%\"bG6#\"\"$\"\"\"&%\"aG6$F(\"\"#F)F)*& &F&6#\"\"%F)&F+6$F1F-F)F)*&&F&6#\"\"&F)&F+6$F7F-F)F)*&&F&6#\"\"'F)&F+6 $F=F-F)F)*&&F&6#\"\"(F)&F+6$FCF-F)F)*&&F&6#\"\")F)&F+6$FIF-F)F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 27 "The simplifying condition: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*c[i]*a[i,2],i = 3 \+ .. 8) = 0;" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cG6#F+F,&%\"aG6$F+\" \"#F,/F+;\"\"$\"\")\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 13 "is satisfied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "Sum(b[i]*c[i]*a[i,2],i=3..8);\neval(subs(Sum= add,%));\nsubs(ee,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*(&% \"bG6#%\"iG\"\"\"&%\"cGF)F+&%\"aG6$F*\"\"#F+/F*;\"\"$\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*(&%\"bG6#\"\"$\"\"\"&%\"cGF'F)&%\"aG6$F( \"\"#F)F)*(&F&6#\"\"%F)&F+F2F)&F-6$F3F/F)F)*(&F&6#\"\"&F)&F+F9F)&F-6$F :F/F)F)*(&F&6#\"\"'F)&F+F@F)&F-6$FAF/F)F)*(&F&6#\"\"(F)&F+FGF)&F-6$FHF /F)F)*(&F&6#\"\")F)&F+FNF)&F-6$FOF/F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The simplifying condition: " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "Sum(b[i]*c[i]^2*a[i,2],i = 3 .. 8) = 0;" "6#/-%$SumG 6$*(&%\"bG6#%\"iG\"\"\"*$&%\"cG6#F+\"\"#F,&%\"aG6$F+F1F,/F+;\"\"$\"\") \"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 13 "is satisfied." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "Sum(b[i]*c[i]^2*a[i,2],i=3..8);\neval(subs(Sum=add,%));\nsubs(ee ,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*(&%\"bG6#%\"iG\"\"\" )&%\"cGF)\"\"#F+&%\"aG6$F*F/F+/F*;\"\"$\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*(&%\"bG6#\"\"$\"\"\")&%\"cGF'\"\"#F)&%\"aG6$F(F-F)F) *(&F&6#\"\"%F))&F,F3F-F)&F/6$F4F-F)F)*(&F&6#\"\"&F))&F,F;F-F)&F/6$F " 0 "" {MPLTEXT 1 0 85 "Sum(b[i]*c[i]*Sum(a[i,j]*a[j,2],j=3..i-1),i=3..8);\neval(subs(Su m=add,%));\nsubs(ee,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*( &%\"bG6#%\"iG\"\"\"&%\"cGF)F+-F$6$*&&%\"aG6$F*%\"jGF+&F26$F4\"\"#F+/F4 ;\"\"$,&F*F+F+!\"\"F+/F*;F:\"\")" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,, **&%\"bG6#\"\"%\"\"\"&%\"cGF'F)&%\"aG6$F(\"\"$F)&F-6$F/\"\"#F)F)*(&F&6 #\"\"&F)&F+F5F),&*&&F-6$F6F/F)F0F)F)*&&F-6$F6F(F)&F-6$F(F2F)F)F)F)*(&F &6#\"\"'F)&F+FCF),(*&&F-6$FDF/F)F0F)F)*&&F-6$FDF(F)F?F)F)*&&F-6$FDF6F) &F-6$F6F2F)F)F)F)*(&F&6#\"\"(F)&F+FTF),**&&F-6$FUF/F)F0F)F)*&&F-6$FUF( F)F?F)F)*&&F-6$FUF6F)FPF)F)*&&F-6$FUFDF)&F-6$FDF2F)F)F)F)*(&F&6#\"\")F )&F+FboF),,*&&F-6$FcoF/F)F0F)F)*&&F-6$FcoF(F)F?F)F)*&&F-6$FcoF6F)FPF)F )*&&F-6$FcoFDF)F^oF)F)*&&F-6$FcoFUF)&F-6$FUF2F)F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "#---------------------------------------" }} {PARA 0 "" 0 "" {TEXT -1 101 "We can calculate the 2 norm of the princ ipal error, that is, the 2-norm of the principal error terms." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "errterms6_8 := PrincipalErr orTerms(6,8,'expanded'):\nsqrt(add(subs(ee,errterms6_8[i])^2,i=1.. nop s(errterms6_8))):\nevalf[10](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"++_&>^\"!#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "12 of the 48 principal error conditions are satisfied." } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 243 "RK6_8err_eqs := PrincipalE rrorConditions(6,8,'expanded'):\nexpand(subs(ee,RK6_8err_eqs)):\nmap(u ->`if`(lhs(u)=rhs(u),0,1),%);\nL := %: ind := []:\nfor ct to nops(L) d o\n if L[ct]=0 then ind := [op(ind),ct] end if:\nend do:\nnops(L);\n ind;\nnops(ind);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7R\"\"\"F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$\"\"!F$F$F$F$F$F%F$F%F%F$F$F$F$F%F$F$F%F$F$F $F%F%F$F%F$F%F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#[" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7.\"#?\"#E\"#G\"#H\"#M\"#P\"#T\"#U\"#W\"#Y\"#Z \"#[" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "These simple principal er ror conditions in abreviated form are as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "RK6_err_eqs := PrincipalErrorConditions(6): \nconvert([seq([ind[i],` `,RK6_err_eqs[ind[i]]],i=1..nops(ind))],matr ix);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7.7%\"#?%#~~G/*&% \"bG\"\"\"-%!G6#*(%\"aGF--F/6#*&F2F-%\"cGF-F--F/6#*&)F6\"\"#F-F2F-F-F- #F-\"$_#7%\"#EF)/*&F,F--F/6#*(F2F-F6F-)F3F;F-F-#F-\"$o\"7%\"#GF)/*&F,F -)F3\"\"$F-#F-\"#c7%\"#HF)/*&F,F--F/6#*(F2F-F:F-F7F-F-#F-\"$E\"7%\"#MF )/**F,F-F6F-F3F-F7F-#F-\"#U7%\"#PF)/*&F,F--F/6#*(F2F-)F6FMF-F3F-F-#F- \"#%)7%\"#TF)/*&F,F-)F7F;F-#F-\"#j7%FhnF)/*(F,F-F:F-FEF-#F-FI7%\"#WF)/ *(F,F-F`oF-F7F-#F-\"#@7%\"#YF)/*(F,F-)F6\"\"%F-F3F-#F-\"#97%\"#ZF)/*&F ,F--F/6#*&)F6\"\"&F-F2F-F-Fgn7%\"#[F)/*&F,F-)F6\"\"'F-#F-\"\"(Q)pprint 176\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 34 "#------------------ ---------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 27 "construction o f the scheme " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "We start in the same way as for the previous scheme by sp ecifying the nodes: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[2] = 1/9;" "6#/&%\"cG6#\"\"#*&\"\"\"F)\"\"*!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3] = 1/6;" "6#/&%\"cG6#\"\"$*&\"\"\"F)\"\"'!\" \"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 1/3;" "6#/&%\"cG6#\"\"%*& \"\"\"F)\"\"$!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5] = 1/2;" "6# /&%\"cG6#\"\"&*&\"\"\"F)\"\"#!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 " c[6] = 2/3;" "6#/&%\"cG6#\"\"'*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "c[7] = 5/6;" "6#/&%\"cG6#\"\"(*&\"\"&\"\"\"\"\"'!\" \"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[8]=1" "6#/&%\"cG6#\"\")\"\"\" " }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 15 "and the weight " }} {PARA 256 "" 0 "" {XPPEDIT 18 0 "b[2]=0" "6#/&%\"bG6#\"\"#\"\"!" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "The scheme is designed so as to satisfy the " }{TEXT 260 7 "order 7" }{TEXT -1 23 " quadrature conditions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "linalg[trans pose](convert([QuadratureConditions(7,8,'expanded')],matrix));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7#/,2&%\"bG6#\"\"\"F-&F+ 6#\"\"#F-&F+6#\"\"$F-&F+6#\"\"%F-&F+6#\"\"&F-&F+6#\"\"'F-&F+6#\"\"(F-& F+6#\"\")F-F-7#/,0*&F.F-&%\"cGF/F-F-*&F1F-&FHF2F-F-*&F4F-&FHF5F-F-*&F7 F-&FHF8F-F-*&F:F-&FHF;F-F-*&F=F-&FHF>F-F-*&F@F-&FHFAF-F-#F-F07#/,0*&F. F-)FGF0F-F-*&F1F-)FJF0F-F-*&F4F-)FLF0F-F-*&F7F-)FNF0F-F-*&F:F-)FPF0F-F -*&F=F-)FRF0F-F-*&F@F-)FTF0F-F-#F-F37#/,0*&F.F-)FGF3F-F-*&F1F-)FJF3F-F -*&F4F-)FLF3F-F-*&F7F-)FNF3F-F-*&F:F-)FPF3F-F-*&F=F-)FRF3F-F-*&F@F-)FT F3F-F-#F-F67#/,0*&F.F-)FGF6F-F-*&F1F-)FJF6F-F-*&F4F-)FLF6F-F-*&F7F-)FN F6F-F-*&F:F-)FPF6F-F-*&F=F-)FRF6F-F-*&F@F-)FTF6F-F-#F-F97#/,0*&F.F-)FG F9F-F-*&F1F-)FJF9F-F-*&F4F-)FLF9F-F-*&F7F-)FNF9F-F-*&F:F-)FPF9F-F-*&F= F-)FRF9F-F-*&F@F-)FTF9F-F-#F-F<7#/,0*&F.F-)FGF " 0 "" {MPLTEXT 1 0 183 "Qeqs := QuadratureConditions(7,8,'expanded'):\ne1 := \{c[3]=1/6,c[4]=1/3,c[5]=1/2,c[6]=2/3,c[7]=5/6,c[8]=1,b[2]=0\}:\nquad eqns := subs(e1,Qeqs):\nnops(quadeqns);\nindets(quadeqns);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<)&%\"bG6#\"\"\"&F%6#\"\"$&F%6#\"\"%&F%6#\"\"&&F%6#\"\"'&F%6#\" \")&F%6#\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "We have 7 linear equ ations for the 7 weights." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "e2 := solve(\{op(quadeqns)\}):\ninfolevel[solve] := 0:\ne3 := `union` (e1,e2):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The weights are as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "seq(b[i]=subs(e3,b[i]),i=1..8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*/&%\"bG6#\"\"\"#\"#T\"$S)/&F%6#\"\"#\"\"!/&F%6#\"\"$#\"\"*\"#N/& F%6#\"\"%#F5\"$!G/&F%6#\"\"&#\"#M\"$0\"/&F%6#\"\"'F;/&F%6#\"\"(F4/&F%6 #\"\")F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 186 "e 3 := \{b[8] = 41/840, b[4] = 9/280, b[6] = 9/280, b[7] = 9/35, b[5] = \+ 34/105, b[1] = 41/840, b[3] = 9/35, c[8] = 1, c[7] = 5/6, b[2] = 0, c[ 6] = 2/3, c[3] = 1/6, c[4] = 1/3, c[5] = 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 -1 159 "We determine the linking coefficients by means of a system of equ ations that consists in part of the stage-order equations that ensure \+ that stages 2 to 8 have " }{TEXT 260 13 "stage-order 3" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 24 "We also incorporate the " }{TEXT 260 22 "simplifying conditions" }{TEXT -1 2 ": " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j+1 .. 8) = b[j]*(1 -c[j]);" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F, F,F,\"\")*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "for " }{XPPEDIT 18 0 "j = 5;" "6#/%\"jG\"\"&" } {TEXT -1 15 ", 6, 7 ( with " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\" \"\"\"!" }{TEXT -1 55 " ), together with the further simplifying cond itions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]* c[i]*a[i,2],i=3..8)=0" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cG6#F+F,& %\"aG6$F+\"\"#F,/F+;\"\"$\"\")\"\"!" }{TEXT -1 8 ", " }{XPPEDIT 18 0 "Sum(b[i]*c[i]*Sum(a[i,j]*a[j,2],j=3..i-1),i=3..8)=0" "6#/-%$SumG 6$*(&%\"bG6#%\"iG\"\"\"&%\"cG6#F+F,-F%6$*&&%\"aG6$F+%\"jGF,&F46$F6\"\" #F,/F6;\"\"$,&F+F,F,!\"\"F,/F+;F<\"\")\"\"!" }{TEXT -1 8 ", " } {XPPEDIT 18 0 "Sum(b[i]*c[i]^2*a[i,2],i=3..8)=0" "6#/-%$SumG6$*(&%\"bG 6#%\"iG\"\"\"*$&%\"cG6#F+\"\"#F,&%\"aG6$F+F1F,/F+;\"\"$\"\")\"\"!" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 11 "We use the " }{TEXT 260 30 "single order 6 order condition" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*c[i]*Sum (a[i,j]*c[j]^3,j = 2 .. i-1),i = 3 .. 8) = 1/24" "6#/-%$SumG6$*(&%\"bG 6#%\"iG\"\"\"&%\"cG6#F+F,-F%6$*&&%\"aG6$F+%\"jGF,*$&F.6#F6\"\"$F,/F6; \"\"#,&F+F,F,!\"\"F,/F+;F:\"\")*&F,F,\"#CF?" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "We introduce th e " }{TEXT 260 22 "additional requirement" }{TEXT -1 26 " that the coe fficient of " }{XPPEDIT 18 0 "z^8" "6#*$%\"zG\"\")" }{TEXT -1 34 " i n the stability polynomial is " }{XPPEDIT 18 0 "10593/(10778*`.`*8!) \+ = 1177/48285440;" "6#/*&\"&$f5\"\"\"*(\"&y2\"F&%\".GF&-%*factorialG6# \"\")F&!\"\"*&\"%x6F&\")SaG[F." }{TEXT -1 16 " respectively. " }} {PARA 0 "" 0 "" {TEXT -1 25 "Thus the coefficient of " }{XPPEDIT 18 0 "z^8" "6#*$%\"zG\"\")" }{TEXT -1 14 " is close to " }{XPPEDIT 18 0 "1/8!" "6#*&\"\"\"F$-%*factorialG6#\"\")!\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "StabilityFunction(6,8,'expanded');" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,4\"\"\"F$%\"zGF$*&\"\"#!\"\"F%F'F$*&\"\"'F(F%\"\"$F$*&\"#CF(F% \"\"%F$*&\"$?\"F(F%\"\"&F$*&\"$?(F(F%F*F$*&,&*0&%\"bG6#\"\"(F$&%\"aG6$ F:F*F$&F<6$F*F1F$&F<6$F1F.F$&F<6$F.F+F$&F<6$F+F'F$&%\"cG6#F'F$F$*&&F86 #\"\")F$,&*.&F<6$FLF*F$F>F$F@F$FBF$FDF$FFF$F$*&&F<6$FLF:F$,&*,&F<6$F:F 1F$F@F$FBF$FDF$FFF$F$*&F;F$,&**&F<6$F*F.F$FBF$FDF$FFF$F$*&F>F$,&*(&F<6 $F1F+F$FDF$FFF$F$*&F@F$,&*&&F<6$F.F'F$FFF$F$*&FBF$&FG6#F+F$F$F$F$F$F$F $F$F$F$F$F$F$)F%F:F$F$*4FJF$FRF$F;F$F>F$F@F$FBF$FDF$FFF$)F%FLF$F$" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "Thus we s pecify that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[8]* a[8,7]*a[7,6]*a[6,5]*a[5,4]*a[4,3]*a[3,2]*c[2] = 1177/48285440;" "6#/* 2&%\"bG6#\"\")\"\"\"&%\"aG6$F(\"\"(F)&F+6$F-\"\"'F)&F+6$F0\"\"&F)&F+6$ F3\"\"%F)&F+6$F6\"\"$F)&F+6$F9\"\"#F)&%\"cG6#F " 0 "" {MPLTEXT 1 0 683 "SO6_8 := SimpleOrderConditions(6,8,'expanded '):\nord_cdns := [seq(SO6_8[i],i=[13,24,28,29])]:\nSO_eqs := [op(RowSu mConditions(8,'expanded')),op(StageOrderConditions(2,8,'expanded')),\n op(StageOrderConditions(3,8,'expanded'))]:\nsimp_eqs := \+ [add(b[i]*a[i,1],i=2..8)=b[1],seq(add(b[i]*a[i,j],i=j+1..8)=b[j]*(1-c[ j]),j=[5,6,7]),\n add(b[i]*c[i]*a[i,2],i=3..8)=0,add(b[i] *c[i]*add(a[i,j]*a[j,2],j=3..i-1),i=3..8)=0,\n add(b[i ]*c[i]^2*a[i,2],i=3..8)=0]:\nord_cdn := add(b[i]*c[i]*add(a[i,j]*c[j]^ 3,j=2..i-1),i=3..8)=1/24:\nextra_eq := b[8]*a[8,7]*a[7,6]*a[6,5]*a[5,4 ]*a[4,3]*a[3,2]*c[2]=1177/48285440:\ncdns := [op(SO_eqs),op(simp_eqs), ord_cdn,extra_eq]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "eqns := subs(e3,cdns):\nnops(eqns);\nindets (eqns) minus \{a[7,6]\};\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# \"#G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<>&%\"aG6$\"\")\"\"$&F%6$F'\" \"#&F%6$F(\"\"\"&F%6$\"\"(F(&F%6$F1\"\"%&F%6$F1F+&F%6$F'F4&F%6$F'\"\"& &F%6$F'\"\"'&F%6$F'F1&F%6$F+F.&F%6$F;F+&F%6$F;F(&F%6$F(F+&F%6$F4F(&F%6 $F;F.&F%6$F4F.&F%6$F4F+&F%6$F>F(&F%6$F'F.&F%6$F1F;&F%6$F>F4&F%6$F>F;&F %6$F1F.&F%6$F;F4&F%6$F>F.&F%6$F>F+&%\"cG6#F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#G" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 77 "Solving this system of equations gives the linking coef ficients in terms of " }{XPPEDIT 18 0 "a[7,6]" "6#&%\"aG6$\"\"(\"\"' " }{TEXT -1 17 " as a parameter." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "e4 := solve(\{op(eqns)\},indets(eqns) minus \{a[7,6] \}):\ne5 := `union`(e3,e4):\ninfolevel[solve] := 0:" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "indets(map (rhs,e5));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#&%\"aG6$\"\"(\"\"'" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Examples: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "a[6,5]=subs(e5,a[6,5]); \na[8,5]=subs(e5,a[8,5]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"\"'\"\"&,$*&#\"%JN\"&Ci)\"\"\"*(,&F(!\"\"*&\"#))F.&F%6$\"\"(F'F.F.F. F4F1,&F.F1*&\"#?F.F4F.F.F1F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"\")\"\"&,$*&#\"\"\"\"(%=NNF,*(,*\"'&ow%!\"\"*&\")ss(G$F,&F%6$ \"\"(\"\"'F,F,*&\"*c)\\UcF,)F4\"\"#F,F1*&\"+?2&**[\"F,)F4\"\"$F,F,F,F4 F1,&F,F1*&\"#?F,F4F,F,F1F,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 12 "Specifying " }{XPPEDIT 18 0 "a[6,5]=8" " 6#/&%\"aG6$\"\"'\"\"&\"\")" }{TEXT -1 33 " gives two possible values \+ for " }{XPPEDIT 18 0 "a[7,6]" "6#&%\"aG6$\"\"(\"\"'" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "subs(e5,a[6,5])=8;\nsolve( %,a[7,6]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*&#\"%JN\"&Ci)\"\"\"* (,&\"\"&!\"\"*&\"#))F)&%\"aG6$\"\"(\"\"'F)F)F)F0F-,&F)F-*&\"#?F)F0F)F) F-F)F)\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$#\"#L\"%)3\"#\"$2\"\"%O D" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "The Huta scheme has " }{XPPEDIT 18 0 "a[7,6]=107/2536" "6#/&%\"aG6$\"\"( \"\"'*&\"$2\"\"\"\"\"%OD!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "e6 := \{a[7, 6]=107/2536\}:\ne7 := `union`(e6,simplify(subs(e6,e5))):\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e7" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 697 "e7 := \{a[7,1] = 554 87/22824, a[7,2] = -83/16, a[7,3] = 2849/1902, a[7,4] = 34601/15216, a [7,5] = -640/2853, a[7,6] = 107/2536, a[8,1] = -101195/25994, a[8,2] = 351/41, a[8,3] = -35994/12997, a[8,4] = -26109/25994, a[8,5] = -10000 /12997, a[8,6] = -36/12997, b[8] = 41/840, b[4] = 9/280, b[6] = 9/280, b[7] = 9/35, b[5] = 34/105, b[1] = 41/840, b[3] = 9/35, c[8] = 1, c[7 ] = 5/6, b[2] = 0, c[6] = 2/3, c[3] = 1/6, c[4] = 1/3, c[5] = 1/2, a[6 ,4] = -107/6, a[6,5] = 8, a[6,3] = -52/3, c[2] = 1/9, a[2,1] = 1/9, a[ 3,1] = 1/24, a[3,2] = 1/8, a[4,1] = 1/6, a[4,2] = -1/2, a[4,3] = 2/3, \+ a[8,7] = 36/41, a[5,1] = 139/272, a[5,2] = -945/544, a[5,3] = 105/68, \+ a[5,4] = 99/544, a[6,1] = -53/3, a[6,2] = 91/2\}:" }{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 64 "The Butcher tableau in exact and approximate form is as follows." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "subs(e7,matrix([seq([c[i],s eq(a[i,j],j=1..i-1),``$(9-i)],i=2..8),\n[``,seq(b[i],i=1..8)]]));\n``; \nevalf[8](%%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7*7+#\" \"\"\"\"*F(%!GF+F+F+F+F+F+7+#F)\"\"'#F)\"#C#F)\"\")F+F+F+F+F+F+7+#F)\" \"$F-#!\"\"\"\"##F8F5F+F+F+F+F+7+#F)F8#\"$R\"\"$s##!$X*\"$W&#\"$0\"\"# o#\"#**FAF+F+F+F+7+F9#!#`F5#\"#\"*F8#!#_F5#!$2\"F.F2F+F+F+7+#\"\"&F.# \"&([b\"&CG##!#$)\"#;#\"%\\G\"%->#\"&,Y$\"&;_\"#!$S'\"%`G#\"$2\"\"%ODF +F+7+F)#!'&>,\"\"&%*f##\"$^$\"#T#!&%*f$\"&(*H\"#!&4h#Fbo#!&++\"Fho#!#O Fho#\"#OFeoF+7+F+#Feo\"$S)\"\"!#F*\"#N#F*\"$!G#\"#MFCFgpFepFbpQ)pprint 246\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7*7+$\")6666!\")F(%!GF+F+F+F+F+F+7+$\")nmm; F*$\")nmmT!\"*$\")++]7F*F+F+F+F+F+F+7+$\")LLLLF*F-$!)+++]F*$\")nmmmF*F +F+F+F+F+7+$\")+++]F*$\")TH5^F*$!)C8P=F*F+F+F+F +7+F9$!)nmmUF1F +F+7+$\"\"\"FS$!)Q,$*QFB$\")c(4c)FB$!)$3%pFFB$!)TU/5FB$!)K3%p(F*$!)+() pF!#5$\")y[!y)F*F+7+F+$\")C&4)[F1$FSFS$\")'G9d#F*$\")dG9KF1$\")_4QKF*F epFcpF`pQ)pprint256\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "RK6_8eqs := [op(RowSumConditions(8, 'expanded')),op(OrderConditions(6,8,'expanded'))]:" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "simplify(subs(e7,RK6_8eqs)):\nmap(u->`if` (lhs(u)=rhs(u),0,1),%);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7 N\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#W" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 34 "#------------------------------ ---" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 25 "absolute stability region " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "coefficients of the \+ scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 613 "ee := \{c[2]=1/9,\nc[3]=1/6,\nc[4]=1/3,\nc[5]=1/2,\n c[6]=2/3,\nc[7]=5/6,\nc[8]=1,\n\na[2,1]=1/9,\na[3,1]=1/24,\na[3,2]=1/8 ,\na[4,1]=1/6,\na[4,2]=-1/2,\na[4,3]=2/3,\na[5,1]=139/272,\na[5,2]=-94 5/544,\na[5,3]=105/68,\na[5,4]=99/544,\na[6,1]=-53/3,\na[6,2]=91/2,\na [6,3]=-52/3,\na[6,4]=-107/6,\na[6,5]=8,\na[7,1]=55487/22824,\na[7,2]=- 83/16,\na[7,3]=2849/1902,\na[7,4]=34601/15216,\na[7,5]=-640/2853,\na[7 ,6]=107/2536,\na[8,1]=-101195/25994,\na[8,2]=351/41,\na[8,3]=-35994/12 997,\na[8,4]=-26109/25994,\na[8,5]=-10000/12997,\na[8,6]=-36/12997,\na [8,7]=36/41,\n\nb[1]=41/840,\nb[2]=0,\nb[3]=9/35,\nb[4]=9/280,\nb[5]=3 4/105,\nb[6]=9/280,\nb[7]=9/35,\nb[8]=41/840\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "The stability f unction R for the 8 stage, order 6 scheme is given as follows." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "subs(ee,StabilityFunction(6, 8,'expanded')):\nR := unapply(%,z):\n'R(z)'=R(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"RG6#%\"zG,4\"\"\"F)F'F)*&#F)\"\"#F)*$)F'F,F)F)F)*& #F)\"\"'F)*$)F'\"\"$F)F)F)*&#F)\"#CF)*$)F'\"\"%F)F)F)*&#F)\"$?\"F)*$)F '\"\"&F)F)F)*&#F)\"$?(F)*$)F'F1F)F)F)*&#\"'Bl=\"*SM&=lF)*$)F'\"\"(F)F) F)*&#\"%x6\")SaG[F)*$)F'\"\")F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can find the point where the bounda ry of the stability region intersects the negative real axis by solvin g the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R (z) = -1;" "6#/-%\"RG6#%\"zG,$\"\"\"!\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "z0 := newton(R(z)=-1,z=-4);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$!+no)G/%!\"*" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 306 "z0 := new ton(R(z)=-1,z=-4):\np1 := plot([R(z),-1],z=-4.69..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(R GB,0,.5,0)):\nplots[display]([p1,p2,p3],view=[-4.69..0.49,-1.47..1.47] ,font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 253 253 {PLOTDATA 2 "6+-%'CURVESG6$7S7$$!3Q++++++!p%!#<$!3Y+2?719$4*) QFP7$$!33++lJx]%R$F*$!3W2CC*)p5sIFP7$$!3/++v4MK#G$F*$!3=r+G'e$*3P#FP7$ $!3_++NGI@uJF*$!3m\"*o'Q\\)e$z\"FP7$$!3)o;z[WQg2$F*$!3fL'fgne>M\"FP7$$ !3QLL[p$*HfHF*$!3^2=y!Ga#>))!#>7$$!3kLL$*fgSgGF*$!3O6YmJ+\\xaFhp7$$!3A +D6J'p`u#F*$!3?@h!)Q7C?@Fhp7$$!3lLLV**GaVEF*$\"3:[tBE74FY!#?7$$!39+D6O G#=`#F*$\"3;\"Q:O0oa'HFhp7$$!3J+vL;*QaU#F*$\"3o;obG.*G5&Fhp7$$!3NL$ekX QWJ#F*$\"3snqEMiF^rFhp7$$!3\\Lep$Q0D@#F*$\"3l)o(HeZ2F*)Fhp7$$!3&pmTSzc D5#F*$\"3IKyAoCoy5FP7$$!3'o;z)46N))>F*$\"3YAj&y+7:F\"FP7$$!3C+v3EZ$*)) =F*$\"3OwecL]Fl&f$FP7$$!3?'***\\)ozx6*FP$\"3+U5>67kfFP$\"3nX @O-Z?KbFP7$$!3e-]7G$*\\/[FP$\"33sJKu_/&='FP7$$!3Wgmms4>IPFP$\"3QQ:*R$* >l)oFP7$$!3$zm;M^b:j#FP$\"3w'on6?Bio(FP7$$!3NK$e/J " 0 "" {MPLTEXT 1 0 1314 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+1/ 720*z^6+186523/651853440*z^7+1177/48285440*z^8:\npts := []: z0 := 0:\n for ct from 0 to 250 do\n zz := newton(R(z)=exp(ct*Pi/25*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np1 := plo t(pts,color=COLOR(RGB,.48,.28,0)):\np2 := plots[polygonplot]([seq([pts [i-1],pts[i],[-2,0]],i=2..nops(pts))],\n style=patchnogrid,co lor=COLOR(RGB,.95,.55,0)):\npts := []: z0 := 2.3+4.3*I:\nfor ct from 0 to 40 do\n zz := newton(R(z)=exp(ct*Pi/20*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np3 := plot(pts,color=CO LOR(RGB,.48,.28,0)):\np4 := plots[polygonplot]([seq([pts[i-1],pts[i],[ 2.22,4.25]],i=2..nops(pts))],\n style=patchnogrid,color=COLOR (RGB,.95,.55,0)):\npts := []: z0 := 2.3-4.3*I:\nfor ct from 0 to 40 do \n zz := newton(R(z)=exp(ct*Pi/20*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np5 := plot(pts,color=COLOR(RGB,. 48,.28,0)):\np6 := plots[polygonplot]([seq([pts[i-1],pts[i],[2.22,-4.2 5]],i=2..nops(pts))],\n style=patchnogrid,color=COLOR(RGB,.95 ,.55,0)):\np7 := plot([[[-4.89,0],[2.69,0]],[[0,-4.89],[0,4.89]]],colo r=black,linestyle=3):\nplots[display]([p||(1..7)],view=[-4.89..2.69,-4 .89..4.89],font=[HELVETICA,9],\n labels=[`Re(z)`,`Im(z)`] ,axes=boxed,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 381 565 565 {PLOTDATA 2 "6/-%'CURVESG6$7gz7$$\"\"!F)F(7$F($\"3-+++iqjc7!#= 7$$\"3'******Heh4R\"!#E$\"3++++xTF8DF-7$$\"3%******zdi`_$!#D$\"3%***** **p?\"*pPF-7$$\"3)******z'**\\mM!#C$\"3b*****4_al-&F-7$$\"3>+++_``C?!# B$\"3o******RI@$G'F-7$$\"3m*****Rk(*))[)FC$\"35+++SB\"*RvF-7$$\"3%**** 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^oFh^v7%Fhev7$Fa[u$!+z,#)>VFc^oFh^v7%F\\fv7$F[[u$!+Pu#RK%Fc^oFh^v7%F`f v7$Fejt$!+;;XEVFc^oFh^v7%Fdfv7$F_jt$!+L$GuK%Fc^oFh^v7%Fhfv7$Fiit$!+ZD \"pK%Fc^oFh^v7%F\\gv7$Fcit$!+ET(\\K%Fc^oFh^v7%F`gv7$F]it$!+BZp@VFc^oFh ^v7%Fdgv7$Fght$!++d; " 0 "" {MPLTEXT 1 0 351 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/12 0*z^5+1/720*z^6+186523/651853440*z^7+1177/48285440*z^8:\nDigits := 25: \npts := []: z0 := 0:\nfor ct from 0 to 120 do\n zz := newton(R(z)=e xp(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,.85,.45,0),thickness= 2,font=[HELVETICA,9]);\nDigits := 10:" }}{PARA 13 "" 1 "" {GLPLOT2D 261 276 276 {PLOTDATA 2 "6(-%'CURVESG6#7er7$$\"\"!F)F(7$$\":nD'[.(fg5 \\4YX$!#E$\"::%4X3e+f`EfTJF-7$$\":AN@G1I+'**G\"*=dF-$\":N]'=CRM@2`=$G' F-7$$\":,O#ybi9d'f-Z?:U\"F?$\":%[hvj[h $)y[6*>#F?7$$\":m:F?$\":ue:(\\M[ywTF8DF?7$$\":-k&)Gqw.)op !fq\"F?$\":kkoFVM#>5NVFGF?7$$\"::nm)3+g'f@>8%=F?$\":KlLH3#\\D1HfTJF?7$ 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$$\":*o`$f3WI5o8c\"oF?$\":5;#)fH+z.=Tq(HFfu7$$\":4C!*)H;Ei>^NmmF?$\":G 9un!\\^pkIe'*HFfu7$$\":[b5G\"e,$Ffu7$$\": Y_!y%***oI+T.>hF?$\":\")))4<68!f\"yJZ.$Ffu7$$\":b\"3E#=-$o6%og7&F?$\": 6*R3)[@:1JTL0$Ffu7$$!:uL8cKOgy&=%3(fF?$\":as8.KD7/!GkrIFfu7$$!:\\Lkjdd 5\"))HfFkF?$\":$y,iWQ4[$oP'*3$Ffu7$$!:e#*e5l:8[R1;v(zTJFfu7$$!:#)=yq%HyF^yg,sF?$\":80;q0&*3zb\"eeJFfu7$$!:IDhR ,f4:JP.K(F?$\":0n\\\"HkIfVn1vJFfu7$$!:LLGKm6jv#R!fU(F?$\":cQMK.='[5^D \">$Ffu7$$!:'\\lep$=faij7_(F?$\":s:7o$=//Q&[r?$Ffu7$$!:8i6#pu^;WOW3wF? $\":>Xm*eU:T&)*[FA$Ffu7$$!:?:'4\\)=d_n(*))o(F?$\":=$[6&Q4zC\\e!QKFfu7$ $!:#[RSjJ1q)e0Pw(F?$\":6!p'Q;%3%)z\"zID$Ffu7$$!:i`;a`GE0$HpLyF?$\":7M% e*RQCyC8yE$Ffu7$$!:4JKM.*Qnag]**yF?$\":Xm69rOxX(HE#G$Ffu-%%FONTG6$%*HE LVETICAG\"\"*-%*THICKNESSG6#\"\"#-%+AXESLABELSG6$Q!6\"F[bm-%&COLORG6&% $RGBG$\"#&)!\"#$\"#XFcbmF(-%%VIEWG6$%(DEFAULTGFibm" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 123 "The relevant intersectio n point of the boundary curve with the imaginary axis can be determine d more accurately as follows." }}{PARA 0 "" 0 "" {TEXT -1 86 "First we look for points on the boundary curve either side of the intersection point. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "Digits := 15:\nz0 := 3.1*I:\nfor ct from 105 to 108 \+ do\n newton(R(z)=exp(ct*Pi/100*I),z=z0);\nend do;\nDigits := 10:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0b=:<+J]%!#<$\"08!f\"yJZ.$!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0A-#)3!Q@k!#=$\"0@:1JTL0$!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0,(y()fnQM!#<$\"0D7/!GkrI!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0[t=rxit(!#<$\"0%4[$oP'*3$!#9" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "Then we \+ apply the bisection method to calculate the parameter value associated with the intersection point." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 176 "real_part := proc(u)\n Re(newton(R(z)=exp(u*Pi*I),z=3.1*I))\n end proc:\nDigits := 15:\nu0 := bisect('real_part'(u),u=1.05..1.08);\n newton(R(z)=exp(u0*Pi*I),z=3.1*I);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"0rH)\\(4;1\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"0)o*y:3j0$!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 4 "The " }{TEXT 260 112 "largest interval on the nonne gative imaginary axis that contains the origin and lies inside the sta bility region" }{TEXT -1 5 " is " }{XPPEDIT 18 0 "[0, 3.0563];" "6#7$ \"\"!-%&FloatG6$\"&j0$!\"%" }{TEXT -1 18 " (approximately)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 34 "#---------------------------------" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 32 "#===============================" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 30 "a companion to Huta's scheme B" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matri x([[1/9, 1/9, ``, ``, ``, ``, ``, ``, ``], [1/6, 1/24, 1/8, ``, ``, `` , ``, ``, ``], [1/3, 1/6, -1/2, 2/3, ``, ``, ``, ``, ``], [1/2, 935/25 36, -2781/2536, 309/317, 321/1268, ``, ``, ``, ``], [2/3, -12710/951, \+ 8287/317, -40/317, -6335/317, 8, ``, ``, ``], [5/6, 5840285/3104064, - 7019/2536, -52213/86224, 1278709/517344, -433/2448, 33/1088, ``, ``], \+ [1, -5101675/1767592, 112077/25994, 334875/441898, -973617/883796, -14 21/1394, 333/5576, 36/41, ``], [``, 41/840, 0, 9/35, 9/280, 34/105, 9/ 280, 9/35, 41/840]])" "6#-%'matrixG6#7*7+*&\"\"\"F)\"\"*!\"\"*&F)F)F*F +%!GF-F-F-F-F-F-7+*&F)F)\"\"'F+*&F)F)\"#CF+*&F)F)\"\")F+F-F-F-F-F-F-7+ *&F)F)\"\"$F+*&F)F)F0F+,$*&F)F)\"\"#F+F+*&F;F)F7F+F-F-F-F-F-7+*&F)F)F; F+*&\"$N*F)\"%ODF+,$*&\"%\"y#F)FAF+F+*&\"$4$F)\"$<$F+*&\"$@$F)\"%o7F+F -F-F-F-7+*&F;F)F7F+,$*&\"&5F\"F)\"$^*F+F+*&\"%(G)F)FGF+,$*&\"#SF)FGF+F +,$*&\"%NjF)FGF+F+F4F-F-F-7+*&\"\"&F)F0F+*&\"(&GSeF)\"(kS5$F+,$*&\"%>q F)FAF+F+,$*&\"&8A&F)\"&Ci)F+F+*&\"(4(y7F)\"'Wt^F+,$*&\"$L%F)\"%[CF+F+* &\"#LF)\"%)3\"F+F-F-7+F),$*&\"(v;5&F)\"(#fn " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 34 "#---------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "checking the scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "coefficients of the scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 655 "ee := \+ \{c[2]=1/9,\nc[3]=1/6,\nc[4]=1/3,\nc[5]=1/2,\nc[6]=2/3,\nc[7]=5/6,\nc[ 8]=1,\n\na[2,1]=1/9,\na[3,1]=1/24,\na[3,2]=1/8,\na[4,1]=1/6,\na[4,2]=- 1/2,\na[4,3]=2/3,\na[5,1]=935/2536,\na[5,2]=-2781/2536,\na[5,3]=309/31 7,\na[5,4]=321/1268,\na[6,1]=-12710/951,\na[6,2]=8287/317,\na[6,3]=-40 /317,\na[6,4]=-6335/317,\na[6,5]=8,\na[7,1]=5840285/3104064,\na[7,2]=- 7019/2536,\na[7,3]=-52213/86224,\na[7,4]=1278709/517344,\na[7,5]=-433/ 2448,\na[7,6]=33/1088,\na[8,1]=-5101675/1767592,\na[8,2]=112077/25994, \na[8,3]=334875/441898,\na[8,4]=-973617/883796,\na[8,5]=-1421/1394,\na [8,6]=333/5576,\na[8,7]=36/41,\n\nb[1]=41/840,\nb[2]=0,\nb[3]=9/35,\nb [4]=9/280,\nb[5]=34/105,\nb[6]=9/280,\nb[7]=9/35,\nb[8]=41/840\}:" }}} {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 0 109 "subs(ee,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(9 -i)],i=2..8),\n[``,seq(b[i],i=1..8)]]));\n``;\nevalf[8](%%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7*7+#\"\"\"\"\"*F(%!GF+F+F+F+F+F +7+#F)\"\"'#F)\"#C#F)\"\")F+F+F+F+F+F+7+#F)\"\"$F-#!\"\"\"\"##F8F5F+F+ F+F+F+7+#F)F8#\"$N*\"%OD#!%\"y#F>#\"$4$\"$<$#\"$@$\"%o7F+F+F+F+7+F9#!& 5F\"\"$^*#\"%(G)FC#!#SFC#!%NjFCF2F+F+F+7+#\"\"&F.#\"(&GSe\"(kS5$#!%>qF >#!&8A&\"&Ci)#\"(4(y7\"'Wt^#!$L%\"%[C#\"#L\"%)3\"F+F+7+F)#!(v;5&\"(#fn <#\"'x?6\"&%*f##\"'v[L\"')*=W#!'9EFJ$!)(H=E\"F*$!)FU)*>FJ$\"\")\"\"!F+F+F+7+$ \")LLL$)F*$\")j\\\")=FB$!)XunFFB$!)m]bgF*$\")/orCFB$!)3zo5FB$\") I-sfF1$\")y[!y)F*F+7+F+$\")C&4)[F1$FSFS$\")'G9d#F*$\")dG9KF1$\")_4QKF* FdpFbpF_pQ)pprint316\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "RK6_8eqs := [op(RowSumConditions(8, 'expanded')),op(OrderConditions(6,8,'expanded'))]:" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "simplify(subs(ee,RK6_8eqs)):\nmap(u->lhs( u)-rhs(u),%);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7N\"\"!F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#W" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "Next we set-up stage-o rder condtions to check for stage-orders from 2 to 4 inclusive. " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "for ct from 2 to 4 do\n so ||ct||_8 := StageOrderConditions(ct,8,'expanded');\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "Stages 3 to 9 h ave the following respective stage-orders. " }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 173 "[seq([seq(expand(subs(ee,so||i||_8[j])),i=2..4)],j =1..6)]:\nmap(proc(L) local i; for i to nops(L) do if not evalb(L[i]) \+ then break end if end do; i end proc,%):\nsimplify(%); " }}{PARA 11 " " 1 "" {XPPMATH 20 "6#7(\"\"$F$F$F$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "#-------------------------------" }}{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 .. \+ 8) = b[j]*(1-c[j]);" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+%\"jG F,/F+;,&F0F,F,F,\"\")*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 3 " , " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 7 " . . 7 " }} {PARA 0 "" 0 "" {TEXT -1 8 "(where " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\" cG6#\"\"\"\"\"!" }{TEXT -1 17 " ) are satisfied." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "[Sum(b[i]* a[i,1],i=2..8)=b[1],seq(Sum(b[i]*a[i,j],i=j+1..8)=b[j]*(1-c[j]),j=2..7 )];\neval(subs(Sum=add,%)):\nsubs(ee,%):\nmap(u->`if`(lhs(u)=rhs(u),0, 1),%);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7)/-%$SumG6$*&&%\" bG6#%\"iG\"\"\"&%\"aG6$F,F-F-/F,;\"\"#\"\")&F*6#F-/-F&6$*&F)F-&F/6$F,F 3F-/F,;\"\"$F4*&&F*6#F3F-,&F-F-&%\"cGFB!\"\"F-/-F&6$*&F)F-&F/6$F,F?F-/ F,;\"\"%F4*&&F*6#F?F-,&F-F-&FEFRFFF-/-F&6$*&F)F-&F/6$F,FOF-/F,;\"\"&F4 *&&F*6#FOF-,&F-F-&FEFjnFFF-/-F&6$*&F)F-&F/6$F,FgnF-/F,;\"\"'F4*&&F*6#F gnF-,&F-F-&FEFhoFFF-/-F&6$*&F)F-&F/6$F,FeoF-/F,;\"\"(F4*&&F*6#FeoF-,&F -F-&FEFfpFFF-/-F&6$*&F)F-&F/6$F,FcpF-/F,;F4F4*&&F*6#FcpF-,&F-F-&FEFcqF FF-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7)\"\"!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 condition: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,2],i = 3 .. 8) = 0 ;" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+\"\"#F,/F+;\"\"$\"\")\" \"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 13 "is satisfied." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "Sum(b[i]*a[i,2],i=3..8);\neval(subs(Sum=add,%));\nsubs(ee,%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$ F*\"\"#F+/F*;\"\"$\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*&&%\"bG6 #\"\"$\"\"\"&%\"aG6$F(\"\"#F)F)*&&F&6#\"\"%F)&F+6$F1F-F)F)*&&F&6#\"\"& F)&F+6$F7F-F)F)*&&F&6#\"\"'F)&F+6$F=F-F)F)*&&F&6#\"\"(F)&F+6$FCF-F)F)* &&F&6#\"\")F)&F+6$FIF-F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The sim plifying condition: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*c[i]*a[i,2],i = 3 .. 8) = 0;" "6#/-%$SumG6$*(&%\"bG6#%\"iG \"\"\"&%\"cG6#F+F,&%\"aG6$F+\"\"#F,/F+;\"\"$\"\")\"\"!" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 13 "is satisfied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "Sum(b[i]*c[i ]*a[i,2],i=3..8);\neval(subs(Sum=add,%));\nsubs(ee,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cGF)F+&%\"aG6$F* \"\"#F+/F*;\"\"$\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*(&%\"bG6# \"\"$\"\"\"&%\"cGF'F)&%\"aG6$F(\"\"#F)F)*(&F&6#\"\"%F)&F+F2F)&F-6$F3F/ F)F)*(&F&6#\"\"&F)&F+F9F)&F-6$F:F/F)F)*(&F&6#\"\"'F)&F+F@F)&F-6$FAF/F) F)*(&F&6#\"\"(F)&F+FGF)&F-6$FHF/F)F)*(&F&6#\"\")F)&F+FNF)&F-6$FOF/F)F) " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The simplifying condition: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*c[i]^2*a[i,2 ],i = 3 .. 8) = 0;" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"*$&%\"cG6#F+\"\" #F,&%\"aG6$F+F1F,/F+;\"\"$\"\")\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 13 "is satisfied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "Sum(b[i]*c[i]^2*a[i,2],i=3.. 8);\neval(subs(Sum=add,%));\nsubs(ee,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*(&%\"bG6#%\"iG\"\"\")&%\"cGF)\"\"#F+&%\"aG6$F*F/F+/F*; \"\"$\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*(&%\"bG6#\"\"$\"\"\") &%\"cGF'\"\"#F)&%\"aG6$F(F-F)F)*(&F&6#\"\"%F))&F,F3F-F)&F/6$F4F-F)F)*( &F&6#\"\"&F))&F,F;F-F)&F/6$F " 0 "" {MPLTEXT 1 0 85 "Sum(b[i]*c[i]*Sum(a[i,j]*a[j,2],j=3 ..i-1),i=3..8);\neval(subs(Sum=add,%));\nsubs(ee,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cGF)F+-F$6$*&&%\"a G6$F*%\"jGF+&F26$F4\"\"#F+/F4;\"\"$,&F*F+F+!\"\"F+/F*;F:\"\")" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#,,**&%\"bG6#\"\"%\"\"\"&%\"cGF'F)&%\"a G6$F(\"\"$F)&F-6$F/\"\"#F)F)*(&F&6#\"\"&F)&F+F5F),&*&&F-6$F6F/F)F0F)F) *&&F-6$F6F(F)&F-6$F(F2F)F)F)F)*(&F&6#\"\"'F)&F+FCF),(*&&F-6$FDF/F)F0F) F)*&&F-6$FDF(F)F?F)F)*&&F-6$FDF6F)&F-6$F6F2F)F)F)F)*(&F&6#\"\"(F)&F+FT F),**&&F-6$FUF/F)F0F)F)*&&F-6$FUF(F)F?F)F)*&&F-6$FUF6F)FPF)F)*&&F-6$FU FDF)&F-6$FDF2F)F)F)F)*(&F&6#\"\")F)&F+FboF),,*&&F-6$FcoF/F)F0F)F)*&&F- 6$FcoF(F)F?F)F)*&&F-6$FcoF6F)FPF)F)*&&F-6$FcoFDF)F^oF)F)*&&F-6$FcoFUF) &F-6$FUF2F)F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "#-------------- -------------------------" }}{PARA 0 "" 0 "" {TEXT -1 101 "We can calc ulate the 2 norm of the principal error, that is, the 2-norm of the pr incipal error terms." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "err terms6_8 := PrincipalErrorTerms(6,8,'expanded'):\nsqrt(add(subs(ee,err terms6_8[i])^2,i=1.. nops(errterms6_8))):\nevalf[10](%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"+Xg?f`!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 54 "12 of the 48 principal error conditions \+ are satisfied." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 243 "RK6_8err_ eqs := PrincipalErrorConditions(6,8,'expanded'):\nexpand(subs(ee,RK6_8 err_eqs)):\nmap(u->`if`(lhs(u)=rhs(u),0,1),%);\nL := %: ind := []:\nfo r ct to nops(L) do\n if L[ct]=0 then ind := [op(ind),ct] end if:\nen d do:\nnops(L);\nind;\nnops(ind);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7 R\"\"\"F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$\"\"!F$F$F$F$F$F%F$F%F%F$F$ F$F$F%F$F$F%F$F$F$F%F%F$F%F$F%F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# \"#[" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7.\"#?\"#E\"#G\"#H\"#M\"#P\"#T \"#U\"#W\"#Y\"#Z\"#[" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "These sim ple principal error conditions in abreviated form are as follows." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "RK6_err_eqs := PrincipalErr orConditions(6):\nconvert([seq([ind[i],` `,RK6_err_eqs[ind[i]]],i=1.. nops(ind))],matrix);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7. 7%\"#?%#~~G/*&%\"bG\"\"\"-%!G6#*(%\"aGF--F/6#*&F2F-%\"cGF-F--F/6#*&)F6 \"\"#F-F2F-F-F-#F-\"$_#7%\"#EF)/*&F,F--F/6#*(F2F-F6F-)F3F;F-F-#F-\"$o \"7%\"#GF)/*&F,F-)F3\"\"$F-#F-\"#c7%\"#HF)/*&F,F--F/6#*(F2F-F:F-F7F-F- #F-\"$E\"7%\"#MF)/**F,F-F6F-F3F-F7F-#F-\"#U7%\"#PF)/*&F,F--F/6#*(F2F-) F6FMF-F3F-F-#F-\"#%)7%\"#TF)/*&F,F-)F7F;F-#F-\"#j7%FhnF)/*(F,F-F:F-FEF -#F-FI7%\"#WF)/*(F,F-F`oF-F7F-#F-\"#@7%\"#YF)/*(F,F-)F6\"\"%F-F3F-#F- \"#97%\"#ZF)/*&F,F--F/6#*&)F6\"\"&F-F2F-F-Fgn7%\"#[F)/*&F,F-)F6\"\"'F- #F-\"\"(Q)pprint326\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 34 "# ---------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 27 "construction of the scheme " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 79 "We start in the same way as for the previ ous schemes by specifying the nodes: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[2] = 1/9;" "6#/&%\"cG6#\"\"#*&\"\"\"F)\"\"*!\" \"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3] = 1/6;" "6#/&%\"cG6#\"\"$*& \"\"\"F)\"\"'!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 1/3;" "6# /&%\"cG6#\"\"%*&\"\"\"F)\"\"$!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 " c[5] = 1/2;" "6#/&%\"cG6#\"\"&*&\"\"\"F)\"\"#!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6] = 2/3;" "6#/&%\"cG6#\"\"'*&\"\"#\"\"\"\"\"$!\"\" " }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7] = 5/6;" "6#/&%\"cG6#\"\"(*&\" \"&\"\"\"\"\"'!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[8]=1" "6#/&% \"cG6#\"\")\"\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 15 "and the weight " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "b[2]=0" "6#/&%\"bG6# \"\"#\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "The scheme is designed so as to satisfy the " } {TEXT 260 7 "order 7" }{TEXT -1 23 " quadrature conditions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "li nalg[transpose](convert([QuadratureConditions(7,8,'expanded')],matrix) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7#/,2&%\"bG6#\"\" \"F-&F+6#\"\"#F-&F+6#\"\"$F-&F+6#\"\"%F-&F+6#\"\"&F-&F+6#\"\"'F-&F+6# \"\"(F-&F+6#\"\")F-F-7#/,0*&F.F-&%\"cGF/F-F-*&F1F-&FHF2F-F-*&F4F-&FHF5 F-F-*&F7F-&FHF8F-F-*&F:F-&FHF;F-F-*&F=F-&FHF>F-F-*&F@F-&FHFAF-F-#F-F07 #/,0*&F.F-)FGF0F-F-*&F1F-)FJF0F-F-*&F4F-)FLF0F-F-*&F7F-)FNF0F-F-*&F:F- )FPF0F-F-*&F=F-)FRF0F-F-*&F@F-)FTF0F-F-#F-F37#/,0*&F.F-)FGF3F-F-*&F1F- )FJF3F-F-*&F4F-)FLF3F-F-*&F7F-)FNF3F-F-*&F:F-)FPF3F-F-*&F=F-)FRF3F-F-* &F@F-)FTF3F-F-#F-F67#/,0*&F.F-)FGF6F-F-*&F1F-)FJF6F-F-*&F4F-)FLF6F-F-* &F7F-)FNF6F-F-*&F:F-)FPF6F-F-*&F=F-)FRF6F-F-*&F@F-)FTF6F-F-#F-F97#/,0* &F.F-)FGF9F-F-*&F1F-)FJF9F-F-*&F4F-)FLF9F-F-*&F7F-)FNF9F-F-*&F:F-)FPF9 F-F-*&F=F-)FRF9F-F-*&F@F-)FTF9F-F-#F-F<7#/,0*&F.F-)FGF " 0 " " {MPLTEXT 1 0 183 "Qeqs := QuadratureConditions(7,8,'expanded'):\ne1 \+ := \{c[3]=1/6,c[4]=1/3,c[5]=1/2,c[6]=2/3,c[7]=5/6,c[8]=1,b[2]=0\}:\nqu adeqns := subs(e1,Qeqs):\nnops(quadeqns);\nindets(quadeqns);\nnops(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<)&%\"bG6#\"\"\"&F%6#\"\"$&F%6#\"\"%&F%6#\"\"&&F%6#\"\" '&F%6#\"\")&F%6#\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"(" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "We have 7 linear equations for the 7 weights." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "e2 := solve(\{op(quadeqns)\}):\ninfolevel[solve] := 0 :\ne3 := `union`(e1,e2):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 27 "The weights are as follows." }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 31 "seq(b[i]=subs(e3,b[i]),i=1..8);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6*/&%\"bG6#\"\"\"#\"#T\"$S)/&F%6#\"\"#\"\"!/&F%6# \"\"$#\"\"*\"#N/&F%6#\"\"%#F5\"$!G/&F%6#\"\"&#\"#M\"$0\"/&F%6#\"\"'F;/ &F%6#\"\"(F4/&F%6#\"\")F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 186 "e3 := \{b[8] = 41/840, b[4] = 9/280, b[6] = 9/280, b [7] = 9/35, b[5] = 34/105, b[1] = 41/840, b[3] = 9/35, c[8] = 1, c[7] \+ = 5/6, b[2] = 0, c[6] = 2/3, c[3] = 1/6, c[4] = 1/3, c[5] = 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 -1 159 "We determine the linking coefficients by means of a system of equations that consists in part of the stage-order equati ons that ensure that stages 2 to 8 have " }{TEXT 260 13 "stage-order 3 " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 24 "We also incorporate t he " }{TEXT 260 22 "simplifying conditions" }{TEXT -1 2 ": " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j+1 .. \+ 8) = b[j]*(1-c[j]);" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+%\"jG F,/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 = 5;" "6#/% \"jG\"\"&" }{TEXT -1 15 ", 6, 7 ( with " }{XPPEDIT 18 0 "c[1]=0" "6#/ &%\"cG6#\"\"\"\"\"!" }{TEXT -1 55 " ), together with the further simp lifying conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*c[i]*a[i,2],i=3..8)=0" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"& %\"cG6#F+F,&%\"aG6$F+\"\"#F,/F+;\"\"$\"\")\"\"!" }{TEXT -1 8 ", \+ " }{XPPEDIT 18 0 "Sum(b[i]*c[i]*Sum(a[i,j]*a[j,2],j=3..i-1),i=3..8)=0 " "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cG6#F+F,-F%6$*&&%\"aG6$F+%\"jG F,&F46$F6\"\"#F,/F6;\"\"$,&F+F,F,!\"\"F,/F+;F<\"\")\"\"!" }{TEXT -1 8 ", " }{XPPEDIT 18 0 "Sum(b[i]*c[i]^2*a[i,2],i=3..8)=0" "6#/-%$Su mG6$*(&%\"bG6#%\"iG\"\"\"*$&%\"cG6#F+\"\"#F,&%\"aG6$F+F1F,/F+;\"\"$\" \")\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 11 "We use the \+ " }{TEXT 260 30 "single order 6 order condition" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b [i]*c[i]*Sum(a[i,j]*c[j]^3,j = 2 .. i-1),i = 3 .. 8) = 1/24" "6#/-%$Su mG6$*(&%\"bG6#%\"iG\"\"\"&%\"cG6#F+F,-F%6$*&&%\"aG6$F+%\"jGF,*$&F.6#F6 \"\"$F,/F6;\"\"#,&F+F,F,!\"\"F,/F+;F:\"\")*&F,F,\"#CF?" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "We in troduce the " }{TEXT 260 22 "additional requirement" }{TEXT -1 26 " th at the coefficient of " }{XPPEDIT 18 0 "z^8" "6#*$%\"zG\"\")" }{TEXT -1 34 " in the stability polynomial is " }{XPPEDIT 18 0 "10593/(1077 8*`.`*8!) = 1177/48285440;" "6#/*&\"&$f5\"\"\"*(\"&y2\"F&%\".GF&-%*fac torialG6#\"\")F&!\"\"*&\"%x6F&\")SaG[F." }{TEXT -1 16 " respectively. " }}{PARA 0 "" 0 "" {TEXT -1 25 "Thus the coefficient of " } {XPPEDIT 18 0 "z^8" "6#*$%\"zG\"\")" }{TEXT -1 14 " is close to " } {XPPEDIT 18 0 "1/8!" "6#*&\"\"\"F$-%*factorialG6#\"\")!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "StabilityFunction(6,8,'expanded');" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#,4\"\"\"F$%\"zGF$*&\"\"#!\"\"F%F'F$*&\"\"'F(F%\"\"$F$ *&\"#CF(F%\"\"%F$*&\"$?\"F(F%\"\"&F$*&\"$?(F(F%F*F$*&,&*0&%\"bG6#\"\"( F$&%\"aG6$F:F*F$&F<6$F*F1F$&F<6$F1F.F$&F<6$F.F+F$&F<6$F+F'F$&%\"cG6#F' F$F$*&&F86#\"\")F$,&*.&F<6$FLF*F$F>F$F@F$FBF$FDF$FFF$F$*&&F<6$FLF:F$,& *,&F<6$F:F1F$F@F$FBF$FDF$FFF$F$*&F;F$,&**&F<6$F*F.F$FBF$FDF$FFF$F$*&F> F$,&*(&F<6$F1F+F$FDF$FFF$F$*&F@F$,&*&&F<6$F.F'F$FFF$F$*&FBF$&FG6#F+F$F $F$F$F$F$F$F$F$F$F$F$F$)F%F:F$F$*4FJF$FRF$F;F$F>F$F@F$FBF$FDF$FFF$)F%F LF$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "Thus we specify that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[8]*a[8,7]*a[7,6]*a[6,5]*a[5,4]*a[4,3]*a[3,2]*c[2] = 1177/48285 440;" "6#/*2&%\"bG6#\"\")\"\"\"&%\"aG6$F(\"\"(F)&F+6$F-\"\"'F)&F+6$F0 \"\"&F)&F+6$F3\"\"%F)&F+6$F6\"\"$F)&F+6$F9\"\"#F)&%\"cG6#F " 0 "" {MPLTEXT 1 0 683 "SO6_8 := SimpleOrderConditi ons(6,8,'expanded'):\nord_cdns := [seq(SO6_8[i],i=[13,24,28,29])]:\nSO _eqs := [op(RowSumConditions(8,'expanded')),op(StageOrderConditions(2, 8,'expanded')),\n op(StageOrderConditions(3,8,'expanded') )]:\nsimp_eqs := [add(b[i]*a[i,1],i=2..8)=b[1],seq(add(b[i]*a[i,j],i=j +1..8)=b[j]*(1-c[j]),j=[5,6,7]),\n add(b[i]*c[i]*a[i,2],i =3..8)=0,add(b[i]*c[i]*add(a[i,j]*a[j,2],j=3..i-1),i=3..8)=0,\n \+ add(b[i]*c[i]^2*a[i,2],i=3..8)=0]:\nord_cdn := add(b[i]*c[i] *add(a[i,j]*c[j]^3,j=2..i-1),i=3..8)=1/24:\nextra_eq := b[8]*a[8,7]*a[ 7,6]*a[6,5]*a[5,4]*a[4,3]*a[3,2]*c[2]=1177/48285440:\ncdns := [op(SO_e qs),op(simp_eqs),ord_cdn,extra_eq]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "eqns := subs(e3,cdns):\nno ps(eqns);\nindets(eqns) minus \{a[7,6]\};\nnops(%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#\"#G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<>&%\"aG6$\" \")\"\"$&F%6$F'\"\"#&F%6$F(\"\"\"&F%6$\"\"(F(&F%6$F1\"\"%&F%6$F1F+&F%6 $F'F4&F%6$F'\"\"&&F%6$F'\"\"'&F%6$F'F1&F%6$F+F.&F%6$F;F+&F%6$F;F(&F%6$ F(F+&F%6$F4F(&F%6$F;F.&F%6$F4F.&F%6$F4F+&F%6$F>F(&F%6$F'F.&F%6$F1F;&F% 6$F>F4&F%6$F>F;&F%6$F1F.&F%6$F;F4&F%6$F>F.&F%6$F>F+&%\"cG6#F+" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"#G" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 77 "Solving this system of equations give s the linking coefficients in terms of " }{XPPEDIT 18 0 "a[7,6]" "6#& %\"aG6$\"\"(\"\"'" }{TEXT -1 17 " as a parameter." }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "e4 := solve(\{op(eqns)\},indets(eqns) minus \+ \{a[7,6]\}):\ne5 := `union`(e3,e4):\ninfolevel[solve] := 0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "in dets(map(rhs,e5));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<#&%\"aG6$\"\"( \"\"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e5" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2136 " e5 := \{b[8] = 41/840, b[4] = 9/280, b[6] = 9/280, b[7] = 9/35, b[5] = 34/105, b[1] = 41/840, b[3] = 9/35, a[8,1] = -1/3535184*(-139812824*a [7,6]+3102724256*a[7,6]^2-26561130752*a[7,6]^3+32778915840*a[7,6]^4+23 83425)/(-5+88*a[7,6])/a[7,6]/(-1+20*a[7,6]), c[8] = 1, c[7] = 5/6, b[2 ] = 0, c[6] = 2/3, c[3] = 1/6, c[4] = 1/3, c[5] = 1/2, a[7,2] = 1/8*(4 7+152*a[7,6])/(-5+88*a[7,6]), a[8,6] = -216/41*a[7,6]+9/41, a[8,4] = - 3/3535184*(-174345536*a[7,6]+4297271552*a[7,6]^2-38277937664*a[7,6]^3+ 65557831680*a[7,6]^4+2383425)/(-5+88*a[7,6])/a[7,6]/(-1+20*a[7,6]), a[ 5,1] = 1/8*(-11+136*a[7,6])/(-5+88*a[7,6]), a[8,5] = 1/3535184*(-47668 5+32877272*a[7,6]-564249856*a[7,6]^2+1489950720*a[7,6]^3)/a[7,6]/(-1+2 0*a[7,6]), a[7,4] = 1/1034688*(-46245008*a[7,6]+950374208*a[7,6]^2-732 0072704*a[7,6]^3+10926305280*a[7,6]^4+794475)/(-5+88*a[7,6])/a[7,6]/(- 1+20*a[7,6]), a[8,2] = -27/82*(23+248*a[7,6])/(-5+88*a[7,6]), c[2] = 1 /9, a[2,1] = 1/9, a[3,1] = 1/24, a[3,2] = 1/8, a[4,1] = 1/6, a[4,2] = \+ -1/2, a[4,3] = 2/3, a[8,7] = 36/41, a[6,5] = 3531/86224*(-5+88*a[7,6]) /a[7,6]/(-1+20*a[7,6]), a[5,2] = -27/8*(-1+8*a[7,6])/(-5+88*a[7,6]), a [7,5] = -1/3104064*(8832232*a[7,6]-133129856*a[7,6]^2+248325120*a[7,6] ^3-158895)/a[7,6]/(-1+20*a[7,6]), a[6,2] = (-67+200*a[7,6])/(-5+88*a[7 ,6]), a[5,4] = 3/2*(-1+20*a[7,6])/(-5+88*a[7,6]), a[8,3] = 3/3535184*( -161670608*a[7,6]-27542014976*a[7,6]^3+43705221120*a[7,6]^4+3561608384 *a[7,6]^2+2383425)/(-5+88*a[7,6])/a[7,6]/(-1+20*a[7,6]), a[6,4] = 1/86 224*(10442752*a[7,6]-114797312*a[7,6]^2+206937600*a[7,6]^3-264825)/(-5 +88*a[7,6])/a[7,6]/(-1+20*a[7,6]), a[5,3] = 3*(-1+8*a[7,6])/(-5+88*a[7 ,6]), a[6,1] = -1/258672*(-6045328*a[7,6]+15122368*a[7,6]^2+27591680*a [7,6]^3+264825)/(-5+88*a[7,6])/a[7,6]/(-1+20*a[7,6]), a[7,3] = -1/3448 96*(-16392208*a[7,6]+317251264*a[7,6]^2-1997637632*a[7,6]^3+2428067840 *a[7,6]^4+264825)/(-5+88*a[7,6])/a[7,6]/(-1+20*a[7,6]), a[7,1] = 1/310 4064*(-35768792*a[7,6]+639450464*a[7,6]^2-5018926592*a[7,6]^3+54631526 40*a[7,6]^4+794475)/(-5+88*a[7,6])/a[7,6]/(-1+20*a[7,6]), a[6,3] = -1/ 86224*(14840176*a[7,6]-214472256*a[7,6]^2+441466880*a[7,6]^3-264825)/( -5+88*a[7,6])/a[7,6]/(-1+20*a[7,6])\}:" }{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 9 "Examples: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "a[6,5]=subs(e5,a[6,5]); \na[8,5]=subs(e5,a[8,5]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$ \"\"'\"\"&,$*&#\"%JN\"&Ci)\"\"\"*(,&F(!\"\"*&\"#))F.&F%6$\"\"(F'F.F.F. F4F1,&F.F1*&\"#?F.F4F.F.F1F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"aG6$\"\")\"\"&,$*&#\"\"\"\"(%=NNF,*(,*\"'&ow%!\"\"*&\")ss(G$F,&F%6$ \"\"(\"\"'F,F,*&\"*c)\\UcF,)F4\"\"#F,F1*&\"+?2&**[\"F,)F4\"\"$F,F,F,F4 F1,&F,F1*&\"#?F,F4F,F,F1F,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 12 "Specifying " }{XPPEDIT 18 0 "a[6,5]=8" " 6#/&%\"aG6$\"\"'\"\"&\"\")" }{TEXT -1 33 " gives two possible values \+ for " }{XPPEDIT 18 0 "a[7,6]" "6#&%\"aG6$\"\"(\"\"'" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "subs(e5,a[6,5])=8;\nsolve( %,a[7,6]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*&#\"%JN\"&Ci)\"\"\"* (,&\"\"&!\"\"*&\"#))F)&%\"aG6$\"\"(\"\"'F)F)F)F0F-,&F)F-*&\"#?F)F0F)F) F-F)F)\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$#\"#L\"%)3\"#\"$2\"\"%O D" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "At \+ this stage we depart from the method of construction of the Huta schem e by specifying that " }{XPPEDIT 18 0 "a[7,6]=33/1088" "6#/&%\"aG6$\" \"(\"\"'*&\"#L\"\"\"\"%)3\"!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "e6 := \{a[7, 6]=33/1088\}:\ne7 := `union`(e6,simplify(subs(e6,e5))):\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e7" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 739 "e7 := \{a[7,6] = 33/10 88, b[8] = 41/840, b[4] = 9/280, b[6] = 9/280, b[7] = 9/35, b[5] = 34/ 105, b[1] = 41/840, b[3] = 9/35, c[8] = 1, c[7] = 5/6, b[2] = 0, c[6] \+ = 2/3, c[3] = 1/6, c[4] = 1/3, c[5] = 1/2, a[7,2] = -7019/2536, a[8,6] = 333/5576, a[6,5] = 8, a[8,4] = -973617/883796, a[5,1] = 935/2536, a [8,5] = -1421/1394, a[7,4] = 1278709/517344, a[8,2] = 112077/25994, a[ 5,2] = -2781/2536, c[2] = 1/9, a[2,1] = 1/9, a[3,1] = 1/24, a[3,2] = 1 /8, a[4,1] = 1/6, a[4,2] = -1/2, a[4,3] = 2/3, a[8,7] = 36/41, a[7,5] \+ = -433/2448, a[6,2] = 8287/317, a[5,4] = 321/1268, a[8,3] = 334875/441 898, a[6,4] = -6335/317, a[5,3] = 309/317, a[6,1] = -12710/951, a[7,3] = -52213/86224, a[7,1] = 5840285/3104064, a[8,1] = -5101675/1767592, \+ a[6,3] = -40/317\}:" }{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 64 "The Butcher tableau in ex act and approximate form is as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "subs(e7,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(9 -i)],i=2..8),\n[``,seq(b[i],i=1..8)]]));\n``;\nevalf[8](%%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7*7+#\"\"\"\"\"*F(%!GF+F+F+F+F+F +7+#F)\"\"'#F)\"#C#F)\"\")F+F+F+F+F+F+7+#F)\"\"$F-#!\"\"\"\"##F8F5F+F+ F+F+F+7+#F)F8#\"$N*\"%OD#!%\"y#F>#\"$4$\"$<$#\"$@$\"%o7F+F+F+F+7+F9#!& 5F\"\"$^*#\"%(G)FC#!#SFC#!%NjFCF2F+F+F+7+#\"\"&F.#\"(&GSe\"(kS5$#!%>qF >#!&8A&\"&Ci)#\"(4(y7\"'Wt^#!$L%\"%[C#\"#L\"%)3\"F+F+7+F)#!(v;5&\"(#fn <#\"'x?6\"&%*f##\"'v[L\"')*=W#!'9EFJ$!)(H=E\"F*$!)FU)*>FJ$\"\")\"\"!F+F+F+7+$ \")LLL$)F*$\")j\\\")=FB$!)XunFFB$!)m]bgF*$\")/orCFB$!)3zo5FB$\") I-sfF1$\")y[!y)F*F+7+F+$\")C&4)[F1$FSFS$\")'G9d#F*$\")dG9KF1$\")_4QKF* FdpFbpF_pQ)pprint296\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "RK6_8eqs := [op(RowSumConditions(8, 'expanded')),op(OrderConditions(6,8,'expanded'))]:" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "simplify(subs(e7,RK6_8eqs)):\nmap(u->`if` (lhs(u)=rhs(u),0,1),%);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7 N\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#W" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 34 "#------------------------------ ---" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 25 "absolute stability region " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "coefficients of the \+ scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 655 "ee := \{c[2]=1/9,\nc[3]=1/6,\nc[4]=1/3,\nc[5]=1/2,\n c[6]=2/3,\nc[7]=5/6,\nc[8]=1,\n\na[2,1]=1/9,\na[3,1]=1/24,\na[3,2]=1/8 ,\na[4,1]=1/6,\na[4,2]=-1/2,\na[4,3]=2/3,\na[5,1]=935/2536,\na[5,2]=-2 781/2536,\na[5,3]=309/317,\na[5,4]=321/1268,\na[6,1]=-12710/951,\na[6, 2]=8287/317,\na[6,3]=-40/317,\na[6,4]=-6335/317,\na[6,5]=8,\na[7,1]=58 40285/3104064,\na[7,2]=-7019/2536,\na[7,3]=-52213/86224,\na[7,4]=12787 09/517344,\na[7,5]=-433/2448,\na[7,6]=33/1088,\na[8,1]=-5101675/176759 2,\na[8,2]=112077/25994,\na[8,3]=334875/441898,\na[8,4]=-973617/883796 ,\na[8,5]=-1421/1394,\na[8,6]=333/5576,\na[8,7]=36/41,\n\nb[1]=41/840, \nb[2]=0,\nb[3]=9/35,\nb[4]=9/280,\nb[5]=34/105,\nb[6]=9/280,\nb[7]=9/ 35,\nb[8]=41/840\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 77 "The stability function R for the 8 stage, order 6 scheme is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "subs(ee,StabilityFunction(6,8,'expanded')):\nR := una pply(%,z):\n'R(z)'=R(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"RG6#% \"zG,4\"\"\"F)F'F)*&#F)\"\"#F)*$)F'F,F)F)F)*&#F)\"\"'F)*$)F'\"\"$F)F)F )*&#F)\"#CF)*$)F'\"\"%F)F)F)*&#F)\"$?\"F)*$)F'\"\"&F)F)F)*&#F)\"$?(F)* $)F'F1F)F)F)*&#\"&8(=\")!o\"[\")F)*$)F'\"\"(F)F)F)*&#\"%x6\")SaG[F)*$) F'\"\")F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can find the point where the boundary of the stabilit y region intersects the negative real axis by solving the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R(z) = 1;" "6#/-%\" RG6#%\"zG\"\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "z0 := newton(R(z)=1,z=-5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#z0G$!+h>)3-&!\"*" }}}{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=-5.39..0.49,color=[red,blue]):\np2 := plot([[[z0,1]]$3],styl e=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=[-5.39..0.49,-0.13..1.47],font=[HELVETICA,9]);" }}{PARA 13 " " 1 "" {GLPLOT2D 360 253 253 {PLOTDATA 2 "6+-%'CURVESG6$7Z7$$!3o****** *******Q&!#<$\"3Mf%R\\-4EO#F*7$$!3i**\\P$=ezN&F*$\"3q`!eX#H&\\?#F*7$$! 3e***\\nO;fK&F*$\"3GG/#=uCf0#F*7$$!3_**\\7]X(QH&F*$\"3#f^vo3N^\">F*7$$ !3[****\\LF$=E&F*$\"3Y;w]s*3Ay\"F*7$$!3r*\\io!R21_F*$\"3iJ\\6)>v&o:F*7 $$!3'***\\A!3:.:&F*$\"3!f'o[OC'eP\"F*7$$!3!**\\P![%3w3&F*$\"3)[JiA7N?= \"F*7$$!3%)***\\e\"=!\\-&F*$\"3^CK*ob/.,\"F*7$$!3s****RB&z<'\\F*$\"3=. 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%'SYMBOLG6#%'CIRCLEG-F^]l6&F`]lFe]lFe]lFe]l-%&STYLEG6#%&POINTG-F$6&F`a l-Feal6#%&CROSSGFhalFjal-F$6&F`al-Feal6#%(DIAMONDGFhalFjal-F$6%7$7$Fba lFd]lFaal-%&COLORG6&F`]lFd]l$\"\"&!\"\"Fd]l-%*LINESTYLEG6#\"\"$-%%FONT G6$%*HELVETICAG\"\"*-%+AXESLABELSG6%Q\"z6\"Q!F_dl-Fgcl6#%(DEFAULTG-%%V IEWG6$;$!$R&!\"#$\"#\\Fjdl;$!#8Fjdl$\"$Z\"Fjdl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curv e 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 50 "The following picture shows the stability region. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1312 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+1/720*z^6+1 8713/81481680*z^7+1177/48285440*z^8:\npts := []: z0 := 0:\nfor ct from 0 to 300 do\n zz := newton(R(z)=exp(ct*Pi/25*I),z=z0):\n z0 := zz :\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np1 := plot(pts,color =COLOR(RGB,.28,0,.48)):\np2 := plots[polygonplot]([seq([pts[i-1],pts[i ],[-2.5,0]],i=2..nops(pts))],\n style=patchnogrid,color=COLOR (RGB,.55,0,.95)):\npts := []: z0 := 2.2+4.5*I:\nfor ct from 0 to 40 do \n zz := newton(R(z)=exp(ct*Pi/20*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(zz),Im(zz)]]:\nend do:\np3 := plot(pts,color=COLOR(RGB,. 28,0,.48)):\np4 := plots[polygonplot]([seq([pts[i-1],pts[i],[2.11,4.5] ],i=2..nops(pts))],\n style=patchnogrid,color=COLOR(RGB,.55,0 ,.95)):\npts := []: z0 := 2.2-4.5*I:\nfor ct from 0 to 40 do\n zz := newton(R(z)=exp(ct*Pi/20*I),z=z0):\n z0 := zz:\n pts := [op(pts), [Re(zz),Im(zz)]]:\nend do:\np5 := plot(pts,color=COLOR(RGB,.28,0,.48)) :\np6 := plots[polygonplot]([seq([pts[i-1],pts[i],[2.11,-4.5]],i=2..no ps(pts))],\n style=patchnogrid,color=COLOR(RGB,.55,0,.95)):\n p7 := plot([[[-5.69,0],[2.59,0]],[[0,-5.09],[0,5.09]]],color=black,lin estyle=3):\nplots[display]([p||(1..7)],view=[-5.69..2.59,-5.09..5.09], font=[HELVETICA,9],\n labels=[`Re(z)`,`Im(z)`],axes=boxed ,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 381 565 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%F\\[pF[[x7%F_\\x7$F^`w$!+9\"yKZ%F\\[pF[[x7%Fc\\x7$Fh_w$!+=%>QY%F\\[pF [[x7%Fg\\x7$Fb_w$!+M>TbWF\\[pF[[x7%F[]x7$F\\_w$!+%G\\$[WF\\[pF[[x7%F_] x7$Ff^w$!+md$HW%F\\[pF[[x7%Fc]x7$F`^w$!+DOZRWF\\[pF[[x7%Fg]x7$Fj]w$!+5 ECQWF\\[pF[[x7%F[^x7$Fd]w$!+.2ZRWF\\[pF[[x7%F_^x7$F^]w$!+#G(HVWF\\[pF[ [x7%Fc^x7$Fh\\w$!+0St\\WF\\[pF[[x7%Fg^x7$Fb\\w$!+iNjeWF\\[pF[[x7%F[_x7 $F\\\\w$!+Q!z'pWF\\[pF[[x7%F__x7$Ff[w$!+7KS#[%F\\[pF[[x7%Fc_x7$F`[w$!+ yqB'\\%F\\[pF[[x7%Fg_x7$Fjjv$!+Z]d5XF\\[pF[[x7%F[`x7$Fdjv$!+m$Q[_%F\\[ pF[[x7%F_`x7$F^jv$!+V=_QXF\\[pF[[x7%Fc`x7$Fhiv$!+:\"=7b%F\\[pF[[x7%Fg` x7$Fbiv$!+5@iiXF\\[pF[[x7%F[ax7$F\\iv$!+$\\ADd%F\\[pF[[x7%F_ax7$Ffhv$! +yoy!e%F\\[pF[[x7%Fcax7$F`hv$!+;dM(e%F\\[pF[[x7%Fgax7$Fjgv$!++w<#f%F\\ [pF[[x7%F[bx7$Fdgv$!+3rH&f%F\\[pF[[x7%F_bx7$F^gv$!+f`u'f%F\\[pF[[x7%Fc bx7$Fhfv$!+8Ie'f%F\\[pF[[x7%Fgbx7$Fbfv$!+q_)[f%F\\[pF[[x7%F[cx7$F\\fv$ !+k%Q " 0 "" {MPLTEXT 1 0 348 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+1/720*z^6+18713/81481680* z^7+1177/48285440*z^8:\nDigits := 25:\npts := []: z0 := 0:\nfor ct fro m 0 to 120 do\n zz := newton(R(z)=exp(ct*Pi/100*I),z=z0):\n z0 := \+ zz:\n pts := [op(pts),[surd(Re(zz),11),Im(zz)]]:\nend do:\nplot(pts, color=COLOR(RGB,.45,0,.9),thickness=2,font=[HELVETICA,9]);\nDigits := \+ 10:" }}{PARA 13 "" 1 "" {GLPLOT2D 261 276 276 {PLOTDATA 2 "6(-%'CURVES G6#7er7$$\"\"!F)F(7$$\":'4_@l`ex^$Rw9$!#E$\":E$)yrv))*e`EfTJF-7$$\":aF bBRMdAL23@&F-$\":*\\rx#fk\">2`=$G'F-7$$\":1;A5]E5)frm(*pF-$\":,yyNN*\\ (4'zxC%*F-7$$\":fMx;551LHici)F-$\":\"**)3![*\\*ehqjc7!#D7$$\":\\Rz%RmH 429Z95F?$\":=p=i%\\W_Fjzq:F?7$$\":6`l&e7D2tq?e6F?$\":/p6bu!ev%fb\\)=F? 7$$\":!*Ri2%z#fjPqaH\"F?$\":7*o()*Q6D^'[6*>#F?7$$\":a%yLtw[&e55uU\"F?$ \":nc:*od?8UTF8DF?7$$\":QQvq57Cq**\\[b\"F?$\":mH-*)=AO=VLu#GF?7$$\":Cz C8oE\")HF1%y;F?$\":**f/$*z^2Su#fTJF?7$$\":<7DpoOEU9b&)z\"F?$\":uha%*RV KN4_dX$F?7$$\":>MM&3Mt#4?uc\">F?$\":7PHK0*>2-:\"*pPF?7$$\":KTYtT0ZT8p+ .#F?$\":6&)p8!\\Z&***42%3%F?7$$\":Ya5i,d%G:<*>9#F?$\":d?tWzPu#G1B)R%F? 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l^t\\HeKFfu7$$!:2_1B(yE(y'e=\"*pF?$\":.xgs2aab\"yAyKFfu7$$!:+I,Fav^YLt <7(F?$\":2=&==c9P]q%yH$Ffu7$$!:Z=Mo-#4]:**=PsF?$\":N&p$))y-D$pE:O!o(F?$\":/Q='f^o[7c)*3MFf u7$$!:/$perG>3l)z[qTEMFfu-%%FONTG6$%*HELVETICAG\"\"*- %*THICKNESSG6#\"\"#-%+AXESLABELSG6$Q!6\"F[bm-%&COLORG6&%$RGBG$\"#X!\"# F($Fcam!\"\"-%%VIEWG6$%(DEFAULTGFibm" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 123 "The relevant intersection point of the b oundary curve with the imaginary axis can be determined more accuratel y as follows." }}{PARA 0 "" 0 "" {TEXT -1 86 "First we look for points on the boundary curve either side of the intersection point. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "Digits := 15:\nz0 := 3.15*I:\nfor ct from 105 to 108 do\n newto n(R(z)=exp(ct*Pi/100*I),z=z0);\nend do;\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0fe9>A,*\\!#<$\"0\"oJcn7KJ!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0xWO7kN<#!#<$\"0*)z*))[$R:$!#9" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#^$$!0=8lU$e*p)!#=$\"08+HsJa<$!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0OC1x'HTT!#<$\"0Ol\\j;m>$!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "Then we apply the bis ection method to calculate the parameter value associated with the int ersection point." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 "real_pa rt := proc(u)\n Re(newton(R(z)=exp(u*Pi*I),z=3.15*I))\nend proc:\nDi gits := 15:\nu0 := bisect('real_part'(u),u=1.05..1.08);\nnewton(R(z)=e xp(u0*Pi*I),z=3.15*I);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"0EF,q;s1\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0$Q! y'z#QB&!#H$\"0H[5\")z%pJ!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 112 "largest interval on the no nnegative imaginary axis that contains the origin and lies inside the \+ stability region" }{TEXT -1 5 " is " }{XPPEDIT 18 0 "[0, 3.1695];" "6 #7$\"\"!-%&FloatG6$\"&&pJ!\"%" }{TEXT -1 18 " (approximately)." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 34 "#------------------------------ ---" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 " " {TEXT -1 58 "#====================================================== ===" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 62 "a scheme with the same nod es and weights as Huta's two schemes" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[1/9, 1/9, \+ ``, ``, ``, ``, ``, ``, ``], [1/6, 1/24, 1/8, ``, ``, ``, ``, ``, ``], [1/3, 1/6, -1/2, 2/3, ``, ``, ``, ``, ``], [1/2, 1747/9704, -2403/970 4, 267/1213, 843/2426, ``, ``, ``, ``], [2/3, -9535/47307, 587/1213, 4 673/15769, -7104/15769, 7/13, ``, ``, ``], [5/6, -359555/1135368, 4265 /9704, 117323/189228, -92449/189228, 79/468, 16/39, ``, ``], [1, 16519 07/1293058, -131355/99466, -627519/646529, 649308/646529, 85/41, -1035 /533, 36/41, ``], [``, 41/840, 0, 9/35, 9/280, 34/105, 9/280, 9/35, 41 /840]])" "6#-%'matrixG6#7*7+*&\"\"\"F)\"\"*!\"\"*&F)F)F*F+%!GF-F-F-F-F -F-7+*&F)F)\"\"'F+*&F)F)\"#CF+*&F)F)\"\")F+F-F-F-F-F-F-7+*&F)F)\"\"$F+ *&F)F)F0F+,$*&F)F)\"\"#F+F+*&F;F)F7F+F-F-F-F-F-7+*&F)F)F;F+*&\"%Zl\"F)\"(eIH\"F+,$*&\"'b88F)\"&m%**F+F+,$*&\"'>vi F)\"'HlkF+F+*&\"'3$\\'F)FfpF+*&\"#&)F)\"#TF+,$*&\"%N5F)\"$L&F+F+*&\"#O F)F[qF+F-7+F-*&F[qF)\"$S)F+\"\"!*&F*F)\"#NF+*&F*F)\"$!GF+*&\"#MF)\"$0 \"F+*&F*F)FiqF+*&F*F)FgqF+*&F[qF)FdqF+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "This scheme has the \+ same nodes and weights as the Huta schemes. " }}{PARA 0 "" 0 "" {TEXT -1 19 "However the nodes " }{XPPEDIT 18 0 "a[6,5]" "6#&%\"aG6$\"\"'\" \"&" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[7,6]" "6#&%\"aG6$\"\"(\" \"'" }{TEXT -1 49 " are chosen to minimize the principal error norm." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 34 "#---------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "checking the scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "coefficients of the scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 656 "ee := \{c[2]=1/9, \nc[3]=1/6,\nc[4]=1/3,\nc[5]=1/2,\nc[6]=2/3,\nc[7]=5/6,\nc[8]=1,\n\na[ 2,1]=1/9,\na[3,1]=1/24,\na[3,2]=1/8,\na[4,1]=1/6,\na[4,2]=-1/2,\na[4,3 ]=2/3,\na[5,1]=1747/9704,\na[5,2]=-2403/9704,\na[5,3]=267/1213,\na[5,4 ]=843/2426,\na[6,1]=-9535/47307,\na[6,2]=587/1213,\na[6,3]=4673/15769, \na[6,4]=-7104/15769,\na[6,5]=7/13,\na[7,1]=-359555/1135368,\na[7,2]=4 265/9704,\na[7,3]=117323/189228,\na[7,4]=-92449/189228,\na[7,5]=79/468 ,\na[7,6]=16/39,\na[8,1]=1651907/1293058,\na[8,2]=-131355/99466,\na[8, 3]=-627519/646529,\na[8,4]=649308/646529,\na[8,5]=85/41,\na[8,6]=-1035 /533,\na[8,7]=36/41,\n\nb[1]=41/840,\nb[2]=0,\nb[3]=9/35,\nb[4]=9/280, \nb[5]=34/105,\nb[6]=9/280,\nb[7]=9/35,\nb[8]=41/840\}:" }}}{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 0 109 "subs(ee,matrix([seq([c[i],seq(a[i,j],j=1..i-1),``$(9-i)],i=2..8 ),\n[``,seq(b[i],i=1..8)]]));\n``;\nevalf[8](%%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7*7+#\"\"\"\"\"*F(%!GF+F+F+F+F+F+7+#F)\"\"' #F)\"#C#F)\"\")F+F+F+F+F+F+7+#F)\"\"$F-#!\"\"\"\"##F8F5F+F+F+F+F+7+#F) F8#\"%Z<\"%/(*#!%.CF>#\"$n#\"%87#\"$V)\"%ECF+F+F+F+7+F9#!%N&*\"&2t%#\" $(eFC#\"%tY\"&pd\"#!%/rFO#\"\"(\"#8F+F+F+7+#\"\"&F.#!'b&f$\"(o`8\"#\"% lUF>#\"'Bt6\"'G#*=#!&\\C*Fin#\"#z\"$o%#\"#;\"#RF+F+7+F)#\"(2>l\"\"(eIH \"#!'b88\"&m%**#!'>vi\"'Hlk#\"'3$\\'F[p#\"#&)\"#T#!%N5\"$L&#\"#OF`pF+7 +F+#F`p\"$S)\"\"!#F*\"#N#F*\"$!G#\"#M\"$0\"F\\qFjpFgpQ)pprint356\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #K%'matrixG6#7*7+$\")6666!\")F(%!GF+F+F+F+F+F+7+$\")nmm;F*$\")nmmT!\"* $\")++]7F*F+F+F+F+F+F+7+$\")LLLLF*F-$!)+++]F*$\")nmmmF*F+F+F+F+F+7+$\" )+++]F*$\")&)G+=F*$!)%)HwCF*$\")U:,AF*$\")d&[Z$F*F+F+F+F+7+F9$!)!eb,#F *$\"):CR[F*$\")#4M'HF*$!):/0XF*$\")ah%Q&F*F+F+F+7+$\")LLL$)F*$!)(eo;$F *$\")[4&R%F*$\")n3+iF*$!)ye&)[F*$\")U.)o\"F*$\")Tc-TF*F+F+7+$\"\"\"\" \"!$\")'>vF\"!\"($!)?g?8F`o$!)$ofq*F*$\")$)H/5F`o$\")2F` o$\")y[!y)F*F+7+F+$\")C&4)[F1$F]oF]o$\")'G9d#F*$\")dG9KF1$\")_4QKF*Fcp FapF^pQ)pprint366\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "RK6_8eqs := [op(RowSumConditions(8,'expan ded')),op(OrderConditions(6,8,'expanded'))]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "simplify(subs(ee,RK6_8eqs)):\nmap(u->lhs(u) -rhs(u),%);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7N\"\"!F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#W" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "Next we set-up stage-or der condtions to check for stage-orders from 2 to 4 inclusive. " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "for ct from 2 to 4 do\n so ||ct||_8 := StageOrderConditions(ct,8,'expanded');\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "Stages 3 to 9 h ave the following respective stage-orders. " }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 173 "[seq([seq(expand(subs(ee,so||i||_8[j])),i=2..4)],j =1..6)]:\nmap(proc(L) local i; for i to nops(L) do if not evalb(L[i]) \+ then break end if end do; i end proc,%):\nsimplify(%); " }}{PARA 11 " " 1 "" {XPPMATH 20 "6#7(\"\"$F$F$F$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "#-------------------------------" }}{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 .. \+ 8) = b[j]*(1-c[j]);" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+%\"jG F,/F+;,&F0F,F,F,\"\")*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 3 " , " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" }{TEXT -1 7 " . . 7 " }} {PARA 0 "" 0 "" {TEXT -1 8 "(where " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\" cG6#\"\"\"\"\"!" }{TEXT -1 17 " ) are satisfied." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "[Sum(b[i]* a[i,1],i=2..8)=b[1],seq(Sum(b[i]*a[i,j],i=j+1..8)=b[j]*(1-c[j]),j=2..7 )];\neval(subs(Sum=add,%)):\nsubs(ee,%):\nmap(u->`if`(lhs(u)=rhs(u),0, 1),%);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7)/-%$SumG6$*&&%\" bG6#%\"iG\"\"\"&%\"aG6$F,F-F-/F,;\"\"#\"\")&F*6#F-/-F&6$*&F)F-&F/6$F,F 3F-/F,;\"\"$F4*&&F*6#F3F-,&F-F-&%\"cGFB!\"\"F-/-F&6$*&F)F-&F/6$F,F?F-/ F,;\"\"%F4*&&F*6#F?F-,&F-F-&FEFRFFF-/-F&6$*&F)F-&F/6$F,FOF-/F,;\"\"&F4 *&&F*6#FOF-,&F-F-&FEFjnFFF-/-F&6$*&F)F-&F/6$F,FgnF-/F,;\"\"'F4*&&F*6#F gnF-,&F-F-&FEFhoFFF-/-F&6$*&F)F-&F/6$F,FeoF-/F,;\"\"(F4*&&F*6#FeoF-,&F -F-&FEFfpFFF-/-F&6$*&F)F-&F/6$F,FcpF-/F,;F4F4*&&F*6#FcpF-,&F-F-&FEFcqF FF-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7)\"\"!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 condition: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,2],i = 3 .. 8) = 0 ;" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+\"\"#F,/F+;\"\"$\"\")\" \"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 13 "is satisfied." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "Sum(b[i]*a[i,2],i=3..8);\neval(subs(Sum=add,%));\nsubs(ee,%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$ F*\"\"#F+/F*;\"\"$\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*&&%\"bG6 #\"\"$\"\"\"&%\"aG6$F(\"\"#F)F)*&&F&6#\"\"%F)&F+6$F1F-F)F)*&&F&6#\"\"& F)&F+6$F7F-F)F)*&&F&6#\"\"'F)&F+6$F=F-F)F)*&&F&6#\"\"(F)&F+6$FCF-F)F)* &&F&6#\"\")F)&F+6$FIF-F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The sim plifying condition: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*c[i]*a[i,2],i = 3 .. 8) = 0;" "6#/-%$SumG6$*(&%\"bG6#%\"iG \"\"\"&%\"cG6#F+F,&%\"aG6$F+\"\"#F,/F+;\"\"$\"\")\"\"!" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 13 "is satisfied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "Sum(b[i]*c[i ]*a[i,2],i=3..8);\neval(subs(Sum=add,%));\nsubs(ee,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cGF)F+&%\"aG6$F* \"\"#F+/F*;\"\"$\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*(&%\"bG6# \"\"$\"\"\"&%\"cGF'F)&%\"aG6$F(\"\"#F)F)*(&F&6#\"\"%F)&F+F2F)&F-6$F3F/ F)F)*(&F&6#\"\"&F)&F+F9F)&F-6$F:F/F)F)*(&F&6#\"\"'F)&F+F@F)&F-6$FAF/F) F)*(&F&6#\"\"(F)&F+FGF)&F-6$FHF/F)F)*(&F&6#\"\")F)&F+FNF)&F-6$FOF/F)F) " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The simplifying condition: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*c[i]^2*a[i,2 ],i = 3 .. 8) = 0;" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"*$&%\"cG6#F+\"\" #F,&%\"aG6$F+F1F,/F+;\"\"$\"\")\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 13 "is satisfied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "Sum(b[i]*c[i]^2*a[i,2],i=3.. 8);\neval(subs(Sum=add,%));\nsubs(ee,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*(&%\"bG6#%\"iG\"\"\")&%\"cGF)\"\"#F+&%\"aG6$F*F/F+/F*; \"\"$\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*(&%\"bG6#\"\"$\"\"\") &%\"cGF'\"\"#F)&%\"aG6$F(F-F)F)*(&F&6#\"\"%F))&F,F3F-F)&F/6$F4F-F)F)*( &F&6#\"\"&F))&F,F;F-F)&F/6$F " 0 "" {MPLTEXT 1 0 85 "Sum(b[i]*c[i]*Sum(a[i,j]*a[j,2],j=3 ..i-1),i=3..8);\neval(subs(Sum=add,%));\nsubs(ee,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cGF)F+-F$6$*&&%\"a G6$F*%\"jGF+&F26$F4\"\"#F+/F4;\"\"$,&F*F+F+!\"\"F+/F*;F:\"\")" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#,,**&%\"bG6#\"\"%\"\"\"&%\"cGF'F)&%\"a G6$F(\"\"$F)&F-6$F/\"\"#F)F)*(&F&6#\"\"&F)&F+F5F),&*&&F-6$F6F/F)F0F)F) *&&F-6$F6F(F)&F-6$F(F2F)F)F)F)*(&F&6#\"\"'F)&F+FCF),(*&&F-6$FDF/F)F0F) F)*&&F-6$FDF(F)F?F)F)*&&F-6$FDF6F)&F-6$F6F2F)F)F)F)*(&F&6#\"\"(F)&F+FT F),**&&F-6$FUF/F)F0F)F)*&&F-6$FUF(F)F?F)F)*&&F-6$FUF6F)FPF)F)*&&F-6$FU FDF)&F-6$FDF2F)F)F)F)*(&F&6#\"\")F)&F+FboF),,*&&F-6$FcoF/F)F0F)F)*&&F- 6$FcoF(F)F?F)F)*&&F-6$FcoF6F)FPF)F)*&&F-6$FcoFDF)F^oF)F)*&&F-6$FcoFUF) &F-6$FUF2F)F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "#-------------- -------------------------" }}{PARA 0 "" 0 "" {TEXT -1 101 "We can calc ulate the 2 norm of the principal error, that is, the 2-norm of the pr incipal error terms." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "err terms6_8 := PrincipalErrorTerms(6,8,'expanded'):\nsqrt(add(subs(ee,err terms6_8[i])^2,i=1.. nops(errterms6_8))):\nevalf[10](%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"+h+3x%*!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "12 of the 48 principal error conditions are satisfied." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 243 "RK6_8err _eqs := PrincipalErrorConditions(6,8,'expanded'):\nexpand(subs(ee,RK6_ 8err_eqs)):\nmap(u->`if`(lhs(u)=rhs(u),0,1),%);\nL := %: ind := []:\nf or ct to nops(L) do\n if L[ct]=0 then ind := [op(ind),ct] end if:\ne nd do:\nnops(L);\nind;\nnops(ind);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# 7R\"\"\"F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$\"\"!F$F$F$F$F$F%F$F%F%F$F $F$F$F%F$F$F%F$F$F$F%F%F$F%F$F%F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #\"#[" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7.\"#?\"#E\"#G\"#H\"#M\"#P\"# T\"#U\"#W\"#Y\"#Z\"#[" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "These sim ple principal error conditions in abreviated form are as follows." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "RK6_err_eqs := PrincipalErr orConditions(6):\nconvert([seq([ind[i],` `,RK6_err_eqs[ind[i]]],i=1.. nops(ind))],matrix);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7. 7%\"#?%#~~G/*&%\"bG\"\"\"-%!G6#*(%\"aGF--F/6#*&F2F-%\"cGF-F--F/6#*&)F6 \"\"#F-F2F-F-F-#F-\"$_#7%\"#EF)/*&F,F--F/6#*(F2F-F6F-)F3F;F-F-#F-\"$o \"7%\"#GF)/*&F,F-)F3\"\"$F-#F-\"#c7%\"#HF)/*&F,F--F/6#*(F2F-F:F-F7F-F- #F-\"$E\"7%\"#MF)/**F,F-F6F-F3F-F7F-#F-\"#U7%\"#PF)/*&F,F--F/6#*(F2F-) F6FMF-F3F-F-#F-\"#%)7%\"#TF)/*&F,F-)F7F;F-#F-\"#j7%FhnF)/*(F,F-F:F-FEF -#F-FI7%\"#WF)/*(F,F-F`oF-F7F-#F-\"#@7%\"#YF)/*(F,F-)F6\"\"%F-F3F-#F- \"#97%\"#ZF)/*&F,F--F/6#*&)F6\"\"&F-F2F-F-Fgn7%\"#[F)/*&F,F-)F6\"\"'F- #F-\"\"(Q)pprint376\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 34 "# ---------------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 27 "construction of the scheme " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 78 "We start in the same way as for the previ ous schemes by specifying the nodes: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "c[2] = 1/9;" "6#/&%\"cG6#\"\"#*&\"\"\"F)\"\"*!\"\" " }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3] = 1/6;" "6#/&%\"cG6#\"\"$*&\" \"\"F)\"\"'!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 1/3;" "6#/& %\"cG6#\"\"%*&\"\"\"F)\"\"$!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[ 5] = 1/2;" "6#/&%\"cG6#\"\"&*&\"\"\"F)\"\"#!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[6] = 2/3;" "6#/&%\"cG6#\"\"'*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7] = 5/6;" "6#/&%\"cG6#\"\"(*&\"\" &\"\"\"\"\"'!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[8]=1" "6#/&%\"c G6#\"\")\"\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 15 "and th e weight " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "b[2]=0" "6#/&%\"bG6#\"\" #\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "The scheme is designed so as to satisfy the " } {TEXT 260 7 "order 7" }{TEXT -1 23 " quadrature conditions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "li nalg[transpose](convert([QuadratureConditions(7,8,'expanded')],matrix) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7)7#/,2&%\"bG6#\"\" \"F-&F+6#\"\"#F-&F+6#\"\"$F-&F+6#\"\"%F-&F+6#\"\"&F-&F+6#\"\"'F-&F+6# \"\"(F-&F+6#\"\")F-F-7#/,0*&F.F-&%\"cGF/F-F-*&F1F-&FHF2F-F-*&F4F-&FHF5 F-F-*&F7F-&FHF8F-F-*&F:F-&FHF;F-F-*&F=F-&FHF>F-F-*&F@F-&FHFAF-F-#F-F07 #/,0*&F.F-)FGF0F-F-*&F1F-)FJF0F-F-*&F4F-)FLF0F-F-*&F7F-)FNF0F-F-*&F:F- )FPF0F-F-*&F=F-)FRF0F-F-*&F@F-)FTF0F-F-#F-F37#/,0*&F.F-)FGF3F-F-*&F1F- )FJF3F-F-*&F4F-)FLF3F-F-*&F7F-)FNF3F-F-*&F:F-)FPF3F-F-*&F=F-)FRF3F-F-* &F@F-)FTF3F-F-#F-F67#/,0*&F.F-)FGF6F-F-*&F1F-)FJF6F-F-*&F4F-)FLF6F-F-* &F7F-)FNF6F-F-*&F:F-)FPF6F-F-*&F=F-)FRF6F-F-*&F@F-)FTF6F-F-#F-F97#/,0* &F.F-)FGF9F-F-*&F1F-)FJF9F-F-*&F4F-)FLF9F-F-*&F7F-)FNF9F-F-*&F:F-)FPF9 F-F-*&F=F-)FRF9F-F-*&F@F-)FTF9F-F-#F-F<7#/,0*&F.F-)FGF " 0 " " {MPLTEXT 1 0 183 "Qeqs := QuadratureConditions(7,8,'expanded'):\ne1 \+ := \{c[3]=1/6,c[4]=1/3,c[5]=1/2,c[6]=2/3,c[7]=5/6,c[8]=1,b[2]=0\}:\nqu adeqns := subs(e1,Qeqs):\nnops(quadeqns);\nindets(quadeqns);\nnops(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<)&%\"bG6#\"\"\"&F%6#\"\"$&F%6#\"\"%&F%6#\"\"&&F%6#\"\" '&F%6#\"\")&F%6#\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"(" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "We have 7 linear equations for the 7 weights." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "e2 := solve(\{op(quadeqns)\}):\ninfolevel[solve] := 0 :\ne3 := `union`(e1,e2):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 27 "The weights are as follows." }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 31 "seq(b[i]=subs(e3,b[i]),i=1..8);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6*/&%\"bG6#\"\"\"#\"#T\"$S)/&F%6#\"\"#\"\"!/&F%6# \"\"$#\"\"*\"#N/&F%6#\"\"%#F5\"$!G/&F%6#\"\"&#\"#M\"$0\"/&F%6#\"\"'F;/ &F%6#\"\"(F4/&F%6#\"\")F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 186 "e3 := \{b[8] = 41/840, b[4] = 9/280, b[6] = 9/280, b [7] = 9/35, b[5] = 34/105, b[1] = 41/840, b[3] = 9/35, c[8] = 1, c[7] \+ = 5/6, b[2] = 0, c[6] = 2/3, c[3] = 1/6, c[4] = 1/3, c[5] = 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 -1 159 "We determine the linking coefficients by means of a system of equations that consists in part of the stage-order equati ons that ensure that stages 2 to 8 have " }{TEXT 260 13 "stage-order 3 " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 24 "We also incorporate t he " }{TEXT 260 22 "simplifying conditions" }{TEXT -1 2 ": " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j+1 .. \+ 8) = b[j]*(1-c[j]);" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+%\"jG F,/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 = 5;" "6#/% \"jG\"\"&" }{TEXT -1 15 ", 6, 7 ( with " }{XPPEDIT 18 0 "c[1]=0" "6#/ &%\"cG6#\"\"\"\"\"!" }{TEXT -1 55 " ), together with the further simp lifying conditions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*c[i]*a[i,2],i=3..8)=0" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"& %\"cG6#F+F,&%\"aG6$F+\"\"#F,/F+;\"\"$\"\")\"\"!" }{TEXT -1 8 ", \+ " }{XPPEDIT 18 0 "Sum(b[i]*c[i]*Sum(a[i,j]*a[j,2],j=3..i-1),i=3..8)=0 " "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cG6#F+F,-F%6$*&&%\"aG6$F+%\"jG F,&F46$F6\"\"#F,/F6;\"\"$,&F+F,F,!\"\"F,/F+;F<\"\")\"\"!" }{TEXT -1 8 ", " }{XPPEDIT 18 0 "Sum(b[i]*c[i]^2*a[i,2],i=3..8)=0" "6#/-%$Su mG6$*(&%\"bG6#%\"iG\"\"\"*$&%\"cG6#F+\"\"#F,&%\"aG6$F+F1F,/F+;\"\"$\" \")\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 11 "We use the \+ " }{TEXT 260 30 "single order 6 order condition" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b [i]*c[i]*Sum(a[i,j]*c[j]^3,j = 2 .. i-1),i = 3 .. 8) = 1/24" "6#/-%$Su mG6$*(&%\"bG6#%\"iG\"\"\"&%\"cG6#F+F,-F%6$*&&%\"aG6$F+%\"jGF,*$&F.6#F6 \"\"$F,/F6;\"\"#,&F+F,F,!\"\"F,/F+;F:\"\")*&F,F,\"#CF?" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 595 "SO6_8 := SimpleOrderConditions(6,8,'expanded'):\nord_cdns := \+ [seq(SO6_8[i],i=[13,24,28,29])]:\nSO_eqs := [op(RowSumConditions(8,'ex panded')),op(StageOrderConditions(2,8,'expanded')),\n op( StageOrderConditions(3,8,'expanded'))]:\nsimp_eqs := [add(b[i]*a[i,1], i=2..8)=b[1],seq(add(b[i]*a[i,j],i=j+1..8)=b[j]*(1-c[j]),j=[5,6,7]),\n add(b[i]*c[i]*a[i,2],i=3..8)=0,add(b[i]*c[i]*add(a[i,j]* a[j,2],j=3..i-1),i=3..8)=0,\n add(b[i]*c[i]^2*a[i,2],i =3..8)=0]:\nord_cdn := add(b[i]*c[i]*add(a[i,j]*c[j]^3,j=2..i-1),i=3.. 8)=1/24:\ncdns := [op(SO_eqs),op(simp_eqs),ord_cdn]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "eqns := s ubs(e3,cdns):\nnops(eqns);\nindets(eqns) minus \{a[6,5],a[7,6]\};\nnop s(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#F" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<=&%\"aG6$\"\")\"\"$&F%6$F'\"\"#&F%6$F(\"\"\"&F%6$\"\"( F(&F%6$F1\"\"%&F%6$F1F+&F%6$F'F4&F%6$F'\"\"&&F%6$F'\"\"'&F%6$F'F1&F%6$ F+F.&F%6$F;F+&F%6$F;F(&F%6$F(F+&F%6$F4F(&F%6$F;F.&F%6$F4F.&F%6$F4F+&F% 6$F>F(&F%6$F'F.&F%6$F1F;&F%6$F>F4&F%6$F1F.&F%6$F;F4&F%6$F>F.&F%6$F>F+& %\"cG6#F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#F" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Solving this system of eq uations gives the linking coefficients in terms of " }{XPPEDIT 18 0 " a[6,5]" "6#&%\"aG6$\"\"'\"\"&" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a [7,6]" "6#&%\"aG6$\"\"(\"\"'" }{TEXT -1 16 " as parameters." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "e4 := solve(\{op(eqns)\},in dets(eqns) minus \{a[6,5],a[7,6]\}):\ne5 := `union`(e3,e4):\ninfolevel [solve] := 0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "indets(map(rhs,e5));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$&%\"aG6$\"\"(\"\"'&F%6$F(\"\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e5" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1490 "e5 := \{a[6,3] = (64-256*a [7,6]-15*a[6,5]+264*a[6,5]*a[7,6])/(-5+88*a[7,6]), a[7,3] = -1/4*(82-1 088*a[7,6]+1408*a[7,6]^2-15*a[6,5]+264*a[6,5]*a[7,6])/(-5+88*a[7,6]), \+ b[8] = 41/840, b[4] = 9/280, b[6] = 9/280, b[7] = 9/35, b[5] = 34/105, b[1] = 41/840, a[7,5] = 35/18-4*a[7,6]-1/4*a[6,5], a[6,1] = -1/3*(-38 +16*a[7,6]-15*a[6,5]+264*a[6,5]*a[7,6])/(-5+88*a[7,6]), b[3] = 9/35, c [8] = 1, c[7] = 5/6, b[2] = 0, c[6] = 2/3, c[3] = 1/6, c[4] = 1/3, c[5 ] = 1/2, a[7,2] = 1/8*(47+152*a[7,6])/(-5+88*a[7,6]), a[8,3] = 3/41*(9 02-14704*a[7,6]+25344*a[7,6]^2-135*a[6,5]+2376*a[6,5]*a[7,6])/(-5+88*a [7,6]), a[8,6] = -216/41*a[7,6]+9/41, a[7,1] = 1/72*(181-5504*a[7,6]+6 336*a[7,6]^2-90*a[6,5]+1584*a[6,5]*a[7,6])/(-5+88*a[7,6]), a[5,1] = 1/ 8*(-11+136*a[7,6])/(-5+88*a[7,6]), a[8,5] = 27/41*a[6,5]-284/41+864/41 *a[7,6], a[8,2] = -27/82*(23+248*a[7,6])/(-5+88*a[7,6]), c[2] = 1/9, a [2,1] = 1/9, a[3,1] = 1/24, a[3,2] = 1/8, a[4,1] = 1/6, a[4,2] = -1/2, a[4,3] = 2/3, a[8,7] = 36/41, a[5,2] = -27/8*(-1+8*a[7,6])/(-5+88*a[7 ,6]), a[6,4] = -(13-120*a[7,6]-15*a[6,5]+264*a[6,5]*a[7,6])/(-5+88*a[7 ,6]), a[8,1] = -1/82*(1297-28904*a[7,6]+38016*a[7,6]^2-270*a[6,5]+4752 *a[6,5]*a[7,6])/(-5+88*a[7,6]), a[6,2] = (-67+200*a[7,6])/(-5+88*a[7,6 ]), a[5,4] = 3/2*(-1+20*a[7,6])/(-5+88*a[7,6]), a[5,3] = 3*(-1+8*a[7,6 ])/(-5+88*a[7,6]), a[8,4] = -3/41*(1049-20296*a[7,6]+38016*a[7,6]^2-13 5*a[6,5]+2376*a[6,5]*a[7,6])/(-5+88*a[7,6]), a[7,4] = 1/12*(212-3928*a [7,6]+6336*a[7,6]^2-45*a[6,5]+792*a[6,5]*a[7,6])/(-5+88*a[7,6])\}:" } {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 9 "Examples:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "a[6,4]=subs(e5,a[6,4]);\na[8,4]=subs(e5,a[8,4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\"'\"\"%,$*&,*\"#8\"\"\"*&\"$?\"F-&F%6$\" \"(F'F-!\"\"*&\"#:F-&F%6$F'\"\"&F-F3*(\"$k#F-F6F-F0F-F-F-,&F8F3*&\"#)) F-F0F-F-F3F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6$\"\")\"\"%,$* &#\"\"$\"#T\"\"\"*&,,\"%\\5F.*&\"&'H?F.&F%6$\"\"(\"\"'F.!\"\"*&\"&;!QF .)F4\"\"#F.F.*&\"$N\"F.&F%6$F7\"\"&F.F8*(\"%wBF.F?F.F4F.F.F.,&FAF8*&\" #))F.F4F.F.F8F.F8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "We can obtain an expression for the principal error norm \+ in terms of " }{XPPEDIT 18 0 "a[6,5]" "6#&%\"aG6$\"\"'\"\"&" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[7,6]" "6#&%\"aG6$\"\"(\"\"'" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "errterms6_8 := P rincipalErrorTerms(6,8,'expanded'):\nsm := add(simplify(subs(e5,errter ms6_8[i]))^2,i=1.. nops(errterms6_8)):\nsm := simplify(sm);\n" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%#smG,$*&#\"\"\"\"0+#fMzrSmF(*&,J*&\" .S-#*y^h#F(&%\"aG6$\"\"(\"\"'F(!\"\"*&\"+++^uGF(&F/6$F2\"\"&F(F3*&\"0# zh`.=W!*F()F.\"\"$F(F3*&\"17d()HeY')\\F()F.\"\"%F(F(*&\"/?b()>]*3(F()F .\"\"#F(F(*(\",+SV=&QF(F6F(F.F(F(*(\".+/rM'HjF(F6F(FCF(F(*&\"1kCA%[,'e qF()F.F2F(F(*&\"1C'ec![=vvF()F.F8F(F3*(\"/KSwig`PF()F6FDF(F;F(F3*(\".o n2ugH\"F(FQF(FCF(F(*(\"0Ca&G))\\+wF(FQF(F?F(F(*(\"0gl5(o\"3\\#F(F6F(F; F(F3*(\"2+koZTpA#QF(FQF(FKF(F(*(\"1O$yc%p:'\\)F(FQF(FNF(F3*(\"2?noY-fp k\"F(F6F(FNF(F3*(\"1WTt$>hZU$F(F6F(F?F(F(*&\"*]:P$[F(FQF(F(*(\",g(=0ZM F(FQF(F.F(F3\",D``\"QTF(F(,&F8F3*&\"#))F(F.F(F(!\"%F(F(" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 170 "plot 3d(sm,a[6,5]=0.3..0.7,a[7,6]=0.2..0.6,axes=boxed,grid=[30,30],orientat ion=[35,50],\n labels=[`a[6,5]`,`a[7,6]`,``],font=[HELVETICA,9],co lor=red,lightmodel=light4);" }}{PARA 13 "" 1 "" {GLPLOT3D 461 373 373 {PLOTDATA 3 "6(-%%GRIDG6&;$\"\"$!\"\"$\"\"(F);$\"\"#F)$\"\"'F)X,%)anyt hingG6\"6\"[gl'!%\"!!#_cn\"?\"?3E6763FB2CD99B353E660856D489673A3E64C1E E283975543E638E9CF45907303E626D01150215D33E615C2B280BDEE83E605B7339BB7 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}{XPPEDIT 18 0 "a[6,5]" "6#&%\"aG6$\"\"'\"\"&" }{TEXT -1 7 " \+ and " }{XPPEDIT 18 0 "a[7,6]" "6#&%\"aG6$\"\"(\"\"'" }{TEXT -1 60 " \+ for which the principal error norm (squared) is a minimum." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 158 "mn := minimize(evalf(sm),a[6,5]=-1 0..10,a[7,6]=-1..1,'location');\ne6 := map(convert,op(1,op(op(2,[mn])) ),rational,4);\ne7 := `union`(e6,simplify(subs(e6,e5))):" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#mnG6$$\"+'e!\\\")*)!#=<#7$<$/&%\"aG6$\"\"(\" \"'$\"+'y?,5%!#5/&F.6$F1\"\"&$\"+r\\M'Q&F4F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e6G<$/&%\"aG6$\"\"(\"\"'#\"#;\"#R/&F(6$F+\"\"&#F*\"# 8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e7" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 740 "e 7 := \{b[8] = 41/840, b[4] = 9/280, b[6] = 9/280, b[7] = 9/35, b[5] = \+ 34/105, b[1] = 41/840, b[3] = 9/35, c[8] = 1, c[7] = 5/6, b[2] = 0, c[ 6] = 2/3, c[3] = 1/6, c[4] = 1/3, c[5] = 1/2, a[6,3] = 4673/15769, a[7 ,5] = 79/468, a[7,3] = 117323/189228, a[8,5] = 85/41, a[8,3] = -627519 /646529, a[8,6] = -1035/533, a[7,2] = 4265/9704, a[6,1] = -9535/47307, a[5,2] = -2403/9704, a[8,2] = -131355/99466, a[5,1] = 1747/9704, a[7, 1] = -359555/1135368, a[6,4] = -7104/15769, a[5,4] = 843/2426, a[5,3] \+ = 267/1213, a[6,2] = 587/1213, a[8,1] = 1651907/1293058, a[7,4] = -924 49/189228, a[8,4] = 649308/646529, a[7,6] = 16/39, c[2] = 1/9, a[2,1] \+ = 1/9, a[3,1] = 1/24, a[3,2] = 1/8, a[4,1] = 1/6, a[4,2] = -1/2, a[4,3 ] = 2/3, a[8,7] = 36/41, a[6,5] = 7/13\}:" }{TEXT -1 0 "" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "The Butc her tableau in exact and approximate form is as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "subs(e7,matrix([seq([c[i],seq(a[i, j],j=1..i-1),``$(9-i)],i=2..8),\n[``,seq(b[i],i=1..8)]]));\n``;\nevalf [8](%%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7*7+#\"\"\"\" \"*F(%!GF+F+F+F+F+F+7+#F)\"\"'#F)\"#C#F)\"\")F+F+F+F+F+F+7+#F)\"\"$F-# !\"\"\"\"##F8F5F+F+F+F+F+7+#F)F8#\"%Z<\"%/(*#!%.CF>#\"$n#\"%87#\"$V)\" %ECF+F+F+F+7+F9#!%N&*\"&2t%#\"$(eFC#\"%tY\"&pd\"#!%/rFO#\"\"(\"#8F+F+F +7+#\"\"&F.#!'b&f$\"(o`8\"#\"%lUF>#\"'Bt6\"'G#*=#!&\\C*Fin#\"#z\"$o%# \"#;\"#RF+F+7+F)#\"(2>l\"\"(eIH\"#!'b88\"&m%**#!'>vi\"'Hlk#\"'3$\\'F[p #\"#&)\"#T#!%N5\"$L&#\"#OF`pF+7+F+#F`p\"$S)\"\"!#F*\"#N#F*\"$!G#\"#M\" $0\"F\\qFjpFgpQ)pprint336\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7*7+$\")6666!\")F(%!GF+F+F +F+F+F+7+$\")nmm;F*$\")nmmT!\"*$\")++]7F*F+F+F+F+F+F+7+$\")LLLLF*F-$!) +++]F*$\")nmmmF*F+F+F+F+F+7+$\")+++]F*$\")&)G+=F*$!)%)HwCF*$\")U:,AF*$ \")d&[Z$F*F+F+F+F+7+F9$!)!eb,#F*$\"):CR[F*$\")#4M'HF*$!):/0XF*$\")ah%Q &F*F+F+F+7+$\")LLL$)F*$!)(eo;$F*$\")[4&R%F*$\")n3+iF*$!)ye&)[F*$\")U.) o\"F*$\")Tc-TF*F+F+7+$\"\"\"\"\"!$\")'>vF\"!\"($!)?g?8F`o$!)$ofq*F*$\" )$)H/5F`o$\")2F`o$\")y[!y)F*F+7+F+$\")C&4)[F1$F]oF]o$\") 'G9d#F*$\")dG9KF1$\")_4QKF*FcpFapF^pQ)pprint346\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "RK6_8eqs := \+ [op(RowSumConditions(8,'expanded')),op(OrderConditions(6,8,'expanded') )]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Ch eck: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "simplify(subs(e7,RK 6_8eqs)):\nmap(u->`if`(lhs(u)=rhs(u),0,1),%);\nnops(%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#7N\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#W" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 34 "#----- ----------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 25 "a bsolute stability region" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "coefficients of the scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 656 "ee := \{c[2]=1/9,\nc[3]=1/6 ,\nc[4]=1/3,\nc[5]=1/2,\nc[6]=2/3,\nc[7]=5/6,\nc[8]=1,\n\na[2,1]=1/9, \na[3,1]=1/24,\na[3,2]=1/8,\na[4,1]=1/6,\na[4,2]=-1/2,\na[4,3]=2/3,\na [5,1]=1747/9704,\na[5,2]=-2403/9704,\na[5,3]=267/1213,\na[5,4]=843/242 6,\na[6,1]=-9535/47307,\na[6,2]=587/1213,\na[6,3]=4673/15769,\na[6,4]= -7104/15769,\na[6,5]=7/13,\na[7,1]=-359555/1135368,\na[7,2]=4265/9704, \na[7,3]=117323/189228,\na[7,4]=-92449/189228,\na[7,5]=79/468,\na[7,6] =16/39,\na[8,1]=1651907/1293058,\na[8,2]=-131355/99466,\na[8,3]=-62751 9/646529,\na[8,4]=649308/646529,\na[8,5]=85/41,\na[8,6]=-1035/533,\na[ 8,7]=36/41,\n\nb[1]=41/840,\nb[2]=0,\nb[3]=9/35,\nb[4]=9/280,\nb[5]=34 /105,\nb[6]=9/280,\nb[7]=9/35,\nb[8]=41/840\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "The stability f unction R for the 8 stage, order 6 scheme is given as follows." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "subs(ee,StabilityFunction(6, 8,'expanded')):\nR := unapply(%,z):\n'R(z)'=R(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"RG6#%\"zG,4\"\"\"F)F'F)*&#F)\"\"#F)*$)F'F,F)F)F)*& #F)\"\"'F)*$)F'\"\"$F)F)F)*&#F)\"#CF)*$)F'\"\"%F)F)F)*&#F)\"$?\"F)*$)F '\"\"&F)F)F)*&#F)\"$?(F)*$)F'F1F)F)F)*&#\"'*ef\"\"*Sqe&))F)*$)F'\"\"(F )F)F)*&#\"$\"G\"(l[A*F)*$)F'\"\")F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can find the point where the b oundary of the stability region intersects the negative real axis by s olving the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R(z) = 1;" "6#/-%\"RG6#%\"zG\"\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "z0 := newton(R(z)=1,z=-4);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$!+mbG3R!\"*" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 302 "z0 := new ton(R(z)=1,z=-4):\np1 := plot([R(z),1],z=-4.39..0.49,color=[red,blue]) :\np2 := plot([[[z0,1]]$3],style=point,symbol=[circle,cross,diamond],c olor=black):\np3 := plot([[z0,0],[z0,1]],linestyle=3,color=COLOR(RGB,0 ,.5,0)):\nplots[display]([p1,p2,p3],view=[-4.39..0.49,-0.13..1.47],fon t=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 253 253 {PLOTDATA 2 "6+-%'CURVESG6$7X7$$!3o*************Q%!#<$\"3@#\\a/*RC9DF*7$$!3lLL$Q %\\\"oL%F*$\"3?Yv!z'=?!G#F*7$$!3ummm())HOG%F*$\"33-D/$4Ee1#F*7$$!3\")* *\\naQNPUF*$\"3S!o[>8!=%*=F*7$$!3'GL$o@y2\">%F*$\"3V\\H8')fWNk]c$G\"F*7$$!3_mmOE*=A)RF*$\"3E+i[hJ1e6F*7$$!3]**\\# \\q!3IRF*$\"3rCE&>\\*RW5F*7$$!3#HL$[$[Uz(QF*$\"3!)p*GaIX3T*!#=7$$!3Amm hs[E\"y$F*$\"3NABW0U$>u(Fin7$$!3U***\\j!3;\"o$F*$\"3$o&)\\R9e.J'Fin7$$ !3Imm\"y.Lwd$F*$\"3eTsM^?r*4&Fin7$$!3a***\\TGPWZ$F*$\"3/QM22YdBTFin7$$ !3)GLLrw(GoLF*$\"3\"Rb'p`*>4K$Fin7$$!3Cmmw97zuKF*$\"3c[%>$4.1cFFin7$$! 3W****RTi`pJF*$\"355%3nExLD#Fin7$$!3')*****43\\Q1$F*$\"3_S7z=r4n=Fin7$ $!3m****f=***>'HF*$\"3o^8&fy`ye\"Fin7$$!3Amm\"=76&pGF*$\"3-y3w)po,S\"F in7$$!3'GLLn*H`fFF*$\"3#z(Q%>Xw[C\"Fin7$$!3yKL$*)4njm#F*$\"3%HP.\">Pkh 6Fin7$$!3!)***\\:,$*zb#F*$\"3`MIS$p,86\"Fin7$$!35LL$4ej?Y#F*$\"3Tk*>EV u?5\"Fin7$$!3g***\\:z8oN#F*$\"3&)=l(yC\"[C6Fin7$$!3S***\\E5\"fcAF*$\"3 OG%p>QtL<\"Fin7$$!3'HLL3C>?:#F*$\"3_$yCbYC)\\7Fin7$$!3sKL)yi*)f0#F*$\" 3.b(4l_l4M\"Fin7$$!3?mm;<(3C&>F*$\"3)**z8aON/Y\"Fin7$$!3\\mm\"=E<[%=F* $\"3[?BJ7'>rg\"Fin7$$!3m***\\Oee6v\"F*$\"3;(p8H\\\\Nv\"Fin7$$!3OmmOXY+ ];F*$\"3gZVOp9!=$>Fin7$$!3))****>HF]X:F*$\"3R4X[4>#*Q@Fin7$$!3b*****=* zEV9F*$\"3+N@Fq6klBFin7$$!3$)***\\ni]VM\"F*$\"3b&)y]yf[4EFin7$$!3y**** pd(>XB\"F*$\"3#o+exx35\"HFin7$$!3MLL$p$=$e8\"F*$\"36Gz75sA7KFin7$$!3r* ****pTh/.\"F*$\"3#e*3!y&>voNFin7$$!3qlm;a:!)\\$*Fin$\"3'oq*G$3(4ERFin7 $$!3r$****f;RfI)Fin$\"35H^x=6(zN%Fin7$$!3$RLL)3guBtFin$\"3&oxKp(oo2[Fi n7$$!3o&***\\`62(H'Fin$\"35;=voq[F`Fin7$$!3=HLLB)3LH&Fin$\"3CJ/cG?***) eFin7$$!36(***\\(oiCC%Fin$\"3g@*)\\(*yiUlFin7$$!3jcmmiHPIKFin$\"3!GXf! 3'o%RsFin7$$!3_jmmR]O&>#Fin$\"3`^]Q*[3*G!)Fin7$$!3kBL$[]F*o6Fin$\"3)oT u'eg!o*))Fin7$$!3%)=****>,QdA!#>$\"3YOQT13zw(*Fin7$$\"3Ozmmcej_&)Ffz$ \"3'GN&4t-H*3\"F*7$$\"3-TLL,V7A=Fin$\"37=q.b!p)*>\"F*7$$\"330+]%[**H&G Fin$\"3$eD " 0 "" {MPLTEXT 1 0 1449 "R := z -> \+ 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+1/720*z^6+159589/885587040*z^7+ 281/9224865*z^8:\npts := []: z0 := 0: tt := 0: \nwhile tt<=241/20 do\n zz := newton(`R`(z)=exp(tt*Pi*I),z=z0):\n z0 := zz:\n if (13/20 <=tt and tt<=29/20) or (211/20<=tt and tt<=227/20) then\n hh := 1 /40\n else \n hh := 1/20\n end if;\n tt := tt+hh;\n pts : = [op(pts),[Re(zz),Im(zz)]]:\nend do:\np1 := plot(pts,color=COLOR(RGB, 0,.38,.1)):\np2 := plots[polygonplot]([seq([pts[i-1],pts[i],[-2,0]],i= 2..nops(pts))],\n style=patchnogrid,color=COLOR(RGB,0,.75,.2) ):\npts := []: z0 := 2.3+4.8*I:\nfor ct from 0 to 40 do\n zz := newt on(R(z)=exp(ct*Pi/20*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(z z),Im(zz)]]:\nend do:\np3 := plot(pts,color=COLOR(RGB,0,.38,.1)):\np4 \+ := plots[polygonplot]([seq([pts[i-1],pts[i],[2.23,4.82]],i=2..nops(pts ))],\n style=patchnogrid,color=COLOR(RGB,0,.75,.2)):\npts := \+ []: z0 := 2.3-4.8*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:\np5 := plot(pts,color=COLOR(RGB,0,.38,.1)):\np6 := plots[po lygonplot]([seq([pts[i-1],pts[i],[2.23,-4.82]],i=2..nops(pts))],\n \+ style=patchnogrid,color=COLOR(RGB,0,.75,.2)):\np7 := plot([[[-4. 59,0],[2.59,0]],[[0,-5.19],[0,5.19]]],color=black,linestyle=3):\nplots [display]([p||(1..7)],view=[-4.59..2.59,-5.19..5.19],font=[HELVETICA,9 ],\n labels=[`Re(z)`,`Im(z)`],axes=boxed,scaling=constrai ned);" }}{PARA 13 "" 1 "" {GLPLOT2D 381 565 565 {PLOTDATA 2 "6/-%'CURV ESG6$7^\\l7$$\"\"!F)F(7$F($\"3#******fK'zq:!#=7$$!39+++vE*p=#!#E$\"3t* ****Rg#fTJF-7$$!3)******pEc;O&!#D$\"3:+++N#)Q7ZF-7$$!3!*******=\">T-&! #C$\"3C+++w2=$G'F-7$$!3%)*****RzZ%\\F!#B$\"3]+++tg'R&yF-7$$!3++++SJ3f5 !#A$\"3:+++DkuC%*F-7$$!3I+++!3vE;$FI$\"3#******>nb&*4\"!#<7$$!3?+++t<< @xFI$\"3#******pw^mD\"FQ7$$!31+++8Lj#e\"!#@$\"3&******>x\"z89FQ7$$!3#* ******\\enKFFZ$\"31+++14.r:FQ7$$!3F+++RcRVQFZ$\"35+++7zXG')=FQ7$$!37+++jJ'['[FC$\"3-+++&HfV/#FQ7$$\"3-+++YQM. 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3')*************e%FQF(7$$\"3')*************e#FQF(-%'COLOURG6&F_boF)F)F )-%*LINESTYLEG6#\"\"$-F$6%7$7$F($!3Q++++++!>&FQ7$F($\"3Q++++++!>&FQFfi wFiiw-%(SCALINGG6#%,CONSTRAINEDG-%%FONTG6$%*HELVETICAG\"\"*-%*AXESSTYL EG6#%$BOXG-%+AXESLABELSG6%%&Re(z)G%&Im(z)G-F[[x6#%(DEFAULTG-%%VIEWG6$; $!$f%Fbbo$\"$f#Fbbo;$!$>&Fbbo$\"$>&Fbbo" 1 2 0 1 10 0 2 9 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 30 "interval of absolute stability" }{TEXT -1 89 " (or stability interval) is the in tersection of the stability region with the real line." }}{PARA 0 "" 0 "" {TEXT -1 59 "For this scheme the stability interval is (approxima tely) " }{XPPEDIT 18 0 "[-3.9083, 0];" "6#7$,$-%&FloatG6$\"&$3R!\"%! \"\"\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 114 "We can distort the boundary curve horizontally b y taking the 11th root of the real part of points along the curve." }} {PARA 0 "" 0 "" {TEXT -1 33 "In this way we see that there is " } {TEXT 260 11 "no interval" }{TEXT -1 97 " on the nonnegative imaginary axis that contains the origin and lies inside the stability region." }}{PARA 0 "" 0 "" {TEXT -1 119 "However the stability region intersect s the nonnegative imaginary axis in an interval that does not contain \+ the origin." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 348 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+1 /720*z^6+159589/885587040*z^7+281/9224865*z^8:\nDigits := 25:\npts := \+ []: z0 := 0:\nfor ct from 0 to 180 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,0,.75,.2),thickness=2,font=[HE LVETICA,9]);\nDigits := 10:" }}{PARA 13 "" 1 "" {GLPLOT2D 261 276 276 {PLOTDATA 2 "6(-%'CURVESG6#7av7$$\"\"!F)F(7$$!:*fC%Q\"*)\\X'e=w1$!#E$ \":7A=/$Q(*e`EfTJF-7$$!:Gq6!p\"4()=c%)z2&F-$\":fe_D\\dsrI&=$G'F-7$$!:, H0BJlav06&=oF-$\":`oKH1:]1'zxC%*F-7$$!:-i9mj8<,@nMS)F-$\":&p5T&G>Z81Pm D\"!#D7$$!:$fSR^2hH]9L\"))*F-$\":%R#QxKZvjK'zq:F?7$$!:)y`$*)4hz$[&\\y7 \"F?$\":/@d'RIpm!fb\\)=F?7$$!:Wl'*=u')f]DM6E\"F?$\":)*zO_hc$>`[6*>#F?7 $$!:g^3a:8NYn\"4*Q\"F?$\":\"4r=x!4M?6uK^#F?7$$!:L%>=-9_())y4D^\"F?$\": Di\\os/GROLu#GF?7$$!:v%4ZfwHk$fu>j\"F?$\":rMC'H3Du.EfTJF?7$$!:%*pL'*GO ZZ(*>pq>F?$\":9l:d@+L&e,2%3%F?7$$!:u!3j=t'3Ij,!y?F?$ \":(p/$G'H,&*R#H#)R%F?7$$!://C%p4@A:l'G=#F?$\":6ZJ*z@h_M#)Q7ZF?7$$!:a# *)*fXz2zTXaG#F?$\":H)eBeO#>T6Zl-&F?7$$!:;bh)H^\"yCpreQ#F?$\":SaIrx!z'f %eqS`F?7$$!:fwpsr)eR=!eU[#F?$\":X3J_VSFLRk[l&F?7$$!:%f*\\d\")pSV,+2e#F ?$\":4!Ry7KIo;F-pfF?7$$!:Ym\">^&[mBhD& f\\6pF?7$$!:g^KR72@l**z%[HF?$\":&*oRu3(fm:IlDsF?7$$!:@EIDQ\\#ow@?OIF?$ \":Q>'e\"z`%*fr4)RvF?7$$!:Gnz1mc]20;B7$F?$\":nY!=@gn#H2mR&yF?7$$!:._[d )foA)4]o?$F?$\":g&RXI'f-(R@7o\")F?7$$!:9W^f;HU6*p#)*G$F?$\":`Fac,/RW,y A[)F?7$$!:JvQjm!*=*HSErLF?$\":C_a%z_X*H&QV'z)F?7$$!:oUGh$4=YxM<^MF?$\" :GUv(G_6y$))*e5\"*F?7$$!:L#oE\\n)>U!GcHNF?$\":012NBg,YUYZU*F?7$$!:j(=& )=J;9`[V1OF?$\":93Hgq'>c,R!*Q(*F?7$$!:)='=0/fq-$zy\"o$F?$\":uGHWp!y+(G 1`+\"!#C7$$!:*>tCwi1%R#ehbPF?$\":#o(R7y!ez.CsO5Ffu7$$!:x:U'ow>'Hv2z#QF ?$\":=D#\\+d-0H)Q\"o5Ffu7$$!:A+!yF7bmT#[')*QF?$\":B[)[,.=vrcb*4\"Ffu7$ $!:CARX*Go/W8M59m?-J85%F?$\":N7$G?Xtei,\"Q>\"Ffu7$$!:[H1&>e \\GD\"3c;%F?$\":r'H]Q%3nwFI_A\"Ffu7$$!:NdZ'fz/CPc8Ffu7$$!:^hX'>TE%fLj]S%F?$\":>Y3W5)Rvlw#4N\"Ffu 7$$!:PsN+qYrnp(=gWF?$\":#\\DJhm=Nh!eBQ\"Ffu7$$!:Hi`9L!)z%prC8XF?$\":pq I!owogr))e#HN3Y)eYF?$\":ep>DTp))fY= \"3:Ffu7$$!:Xii%QNAW4iM-ZF?$\":p`5g-S9^Rr&R:Ffu7$$!:*e%R0HfJ2s.Iu%F?$ \":$\\3uLYq&f!4.r:Ffu7$$!:T'z'H5N$*yHj0y%F?$\":;Y&*yi#p6Cy\\-;Ffu7$$!: yE*zjt#)QEVr9[F?$\":'))ytUw**))HI(Rj\"Ffu7$$!:#>3B[B41WS2X[F?$\":_IHO= FhZXdam\"Ffu7$$!:A0(QC[#Q'zN;r[F?$\":on7Rj>xj2_pp\"Ffu7$$!:4i(QXQdGBHP #*[F?$\":%4@.[(on<\"zXG!pDG)ol5jI< \\F?$\":Mb1ZXd)*3E`I#=Ffu7$$!:>8cbe3X.F&f2\\F?$\":n$3T_b)oTd?>L3P%)[F?$\":t:X(\\fZ,QB>')=Ffu7$$!:?!*H54yh!QQRU[F?$\":\"p.#= K!e(of(y<>Ffu7$$!:,\\hi/!RYY.8sZF?$\":2SJ87n8JA,%\\>Ffu7$$!:I%4edpITdh +`YF?$\":6a\\]-<3l*R.\")>Ffu7$$!:l\\6@*z?ifWQBWF?$\":>]DOd5,Ra'o7?Ffu7 $$!:taWUuIw\"G\\c)G$F?$\":Ur.r([oV&HfV/#Ffu7$$\":`Wq\"HYJyM6g)[%F?$\": E@`gRr!QMC0w?Ffu7$$\":OZ(H%\\%)Qw+iw$[F?$\":)4@#3iPE*Rew2@Ffu7$$\":q-0 G9B]jUgE2&F?$\":k&))))[:$GC-*\\R@Ffu7$$\":r!)>5tlq0ai*f_F?$\":(*QkeFQc 00^7<#Ffu7$$\":tLRg3;_.bQ1U&F?$\":GCI*yB,)*p/-.AFfu7$$\":X!)QdkUcg*>;k bF?$\":%4!=*>PiO8_![B#Ffu7$$\":S!*)[x]YfOKh&p&F?$\":(ex75K-L#Ffu7$$\":y2\\\"oUfwQl#H/'F?$\":9jF*eGLix)G?O#Ffu7$$\":Yu8v)4 u4zylZhF?$\":QgP'[IXCR/$QR#Ffu7$$\":ABvSPa**)\\:A[iF?$\":-\"3a3,+p:ghD CFfu7$$\":]wSY2)[NOt5XjF?$\":7S=]$\\/EFhPdCFfu7$$\":j/-Tn2MhIx'QkF?$\" :m%**Ho.S=k(*4*[#Ffu7$$\":,`\"y\")\\eX6#*>HlF?$\":QS')3J8@HHu2_#Ffu7$$ \":+FtK8#)o7Fpoh'F?$\":aN;M)zG!oQ&Q_DFfu7$$\":]lS#z`8**o#G=q'F?$\":tp; ^R>G)*)p\"Re#Ffu7$$\":i)fbj9zlTL<%y'F?$\":xT[k\"oR!HK^`h#Ffu7$$\":+CPY mung%p'R'oF?$\":U*Rb&\\N&fT*omk#Ffu7$$\":y>G3?PAe&RCTpF?$\":3b\"G\\t(Q L&)[yn#Ffu7$$\":(evPX%Hj2^M2]qRFFfu7$$\":t;:tA%*o+fF!erF?$\":u40\"GoU(>]L.x#F fu7$$\":n,:X*\\;KtRADsF?$\":s%)e)p;*\\V[H2!GFfu7$$\":CNEK9(yo$>W)*G(F? $\":ahlP![rX\\#o3$GFfu7$$\":zkkzm#faxx&=N(F?$\":d\"p'RI2t\\`D2'GFfu7$$ \":s\"3O'fpu3iN7T(F?$\":p')yWi,Yh(zF!*GFfu7$$\":X0'[$f%3FNC&zY(F?$\":P <3%[YxfVN]>HFfu7$$\":k?+#)y8C>t))>_(F?$\":$z*)Qtnu_S>Q[HFfu7$$\":'eE'G Yo(p4BLtvF?$\":2\"oT%*HSNQ\\*o(HFfu7$$\":5/?1R#=]K#z>i(F?$\":+[:Jh\\.X jE]+$Ffu7$$\":i\\#eGi1Y0V$zm(F?$\":Gc[&H!*4AAOwKIFfu7$$\":r$G52>8_%367 r(F?$\":(\\*p.S[A\"p]4gIFfu7$$\":o1n$e)Q1iZJ=v(F?$\":$)4As$eu\"yq7q3$F fu7$$\":&pQYHt')G+]#)*y(F?$\":,Eq^m.gvu5N6$Ffu7$$\":$*3\\i$\\)*y'zF_#y F?$\":B9dPSK++s&eRJFfu7$$\":^CW&y![t(f93eyF?$\":#[>8'>]#[uiBlJFfu7$$\" :U7GqP:N\\xJ%))yF?$\":&y7&\\Y1D!RHY!>$Ffu7$$\":=!*4Kmq(pFuK;zF?$\":0s. 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 201 "Digits := 2 5:\nz0 := 2.05*I:\nfor ct from 64 to 67 do\n newton(R(z)=exp(ct*Pi/1 00*I),z=z0);\nend do;``;\nz0 := 3.9*I:\nfor ct from 143 to 146 do\n \+ newton(R(z)=exp(ct*Pi/100*I),z=z0);\nend do;\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!:o`Ez3tLdSK&o7!#G$\":>]DOd5,Ra'o7?!#C" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!:Sl<7r%zkiJ'['[!#I$\":Ur.r([oV&Hf V/#!#C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\":;m#4v\"e@ps>,\\\"!#G$ \":E@`gRr!QMC0w?!#C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\":XFFa>E#4_ .='R$!#G$\":)4@#3iPE*Rew2@!#C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\":![f\")RR#3aA?&y5!#E$\":KVk:Cnb $=4$3\"R!#C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\":')e)[[hRX*)QY\"*[ !#F$\":c:u:G]bIDGG#R!#C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!:Yg5.Ak rd3Iy9\"!#F$\":'Qy2O))y+oOdMR!#C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$ $!:)o2G.#o`X**3vK(!#F$\":V\"=FL]TO!ypg%R!#C" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 "Then we apply the bisection me thod to calculate the parameter value associated with each intersectio n point." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 330 "Digits := 15:\n real_part := proc(u)\n Re(newton(R(z)=exp(u*Pi*I),z=2.05*I))\nend pr oc:\nu0 := bisect('real_part'(u),u=0.64..0.67);\nnewton(R(z)=exp(u0*Pi *I),z=2.05*I);``;\nreal_part := proc(u)\n Re(newton(R(z)=exp(u*Pi*I) ,z=3.9*I))\nend proc:\nu0 := bisect('real_part'(u),u=1.43..1.46);\nnew ton(R(z)=exp(u0*Pi*I),z=3.9*I);\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"0*4$>CKN]'!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"02R6t#o>U!#H$\"0b#>E%ya/#!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"00p]T<\"[9!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0@4%zxc@\\!#H$\"0mC@i\"QKR!#9" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " } {TEXT 260 73 "stability region intersects the nonegative imaginary axi s in the interval" }{TEXT -1 3 " " }{XPPEDIT 18 0 "[2.0455, 3.9324]; " "6#7$-%&FloatG6$\"&b/#!\"%-F%6$\"&C$RF(" }{TEXT -1 18 " (approximat ely)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 34 "#----------------- ----------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 58 "#======================================== =================" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 54 "a stage orde r 3 scheme with small principal error norm" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 211 "The nodes of this scheme have \+ been chosen in an attempt to minimize the principal error norm while a voiding the coefficients becoming excessively large in magnitude and r etaining reasonable stability properties." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 34 "#------------------ ---------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "checking the s cheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "coefficient s of the scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1949 "ee := \{c[2]=8/61,\nc[3]=12/61,\nc[4]=5/11, \nc[5]=51/92,\nc[6]=15/26,\nc[7]=67/75,\nc[8]=1,\n\na[2,1]=8/61,\na[3, 1]=3/61,\na[3,2]=9/61,\na[4,1]=136115/383328,\na[4,2]=-163175/127776, \na[4,3]=263825/191664,\na[5,1]=13026271485399/595711769169920,\na[5,2 ]=24327752855319/119142353833984,\na[5,3]=893431094000355/103058136066 39616,\na[5,4]=778084111317249/3220566752074880,\na[6,1]=-851829955294 98464014347399693/7004909900459502878327360312576,\na[6,2]=78222725941 6638225/2244072319191569152,\na[6,3]=-27274609047101126288010672520453 5/7948306442641975339499742161731456,\na[6,4]=463772636939896507985393 214891/1859455376317587715620654574952,\na[6,5]=5038901505362209573185 30/19830001765309846198610093,\na[7,1]=-229280166783601280181251392551 413373945146341/1922297216818279856585611944540531553417968750,\na[7,2 ]=-24373261255896329981/410397455685628125000,\na[7,3]=138494760410464 278481114601908086654010742317231861/157045350340868651125869860911967 647339719140625000,\na[7,4]=131599723108496952429397863063516581455035 46532/116418149577273037997050298350513445647705078125,\na[7,5]=-14888 57601208597223792690051882764869796820192/2444881546159637720593910707 10204344775390625,\na[7,6]=68233720896675821496674286908/1106474067978 0824085205078125,\na[8,1]=107265271705456718353000251694440587/5083431 2477772792379562256248860860,\na[8,2]=-3757510716191/5740230741960,\na [8,3]=-429828001838285896192739582712833397915761611/99266983907642719 853800424641670872644481800,\na[8,4]=-18544773327337271450067235223504 981052/26732856022073197552661912017781844625,\na[8,5]=956839075473456 74985937655053437329632/1692820548292066101564044986300227975,\na[8,6] =-10290287110810609832644/193430985482051515125,\na[8,7]=1261738278281 25000/101457352642065253,\n\nb[1]=75887971/1291626000,\nb[2]=0,\nb[3]= 48970167350504861/164785679704694160,\nb[4]=103793987429/697356166500, \nb[5]=-37821393208832/68552327940705,\nb[6]=450669502712/553751407125 ,\nb[7]=2682266748046875/12493923579891488,\nb[8]=1819/98784\}:" }}} {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 64 "The Butcher tableau in exact and approximate form is as follows." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 408 "subs(ee,matrix([[c[2],a[2,1 ],``],\n[c[3],a[3,1],a[3,2]],\n[c[4],a[4,1],a[4,2]],[``$2,a[4,3]],\n[c [5],a[5,1],a[5,2]],[``,a[5,3],a[5,4]],\n[c[6],a[6,1],a[6,2]],[``,a[6,3 ],a[6,4]],[``$2,a[6,5]],\n[c[7],a[7,1],a[7,2]],[``,a[7,3],a[7,4]],[``, a[7,5],a[7,6]],\n[c[8],a[8,1],a[8,2]],[``,a[8,3],a[8,4]],[``,a[8,5],a[ 8,6]],[``$2,a[8,7]],\n[``,`____________`$2],\n[``,b[1],b[2]],[``,b[3], b[4]],[``,b[5],b[6]],[``,b[7],b[8]]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#777%#\"\")\"#hF(%!G7%#\"#7F*#\"\"$F*#\"\"*F*7%#\"\"& \"#6#\"':h8\"'GLQ#!'vJ;\"'wx77%F+F+#\"'DQE\"'k;>7%#\"#^\"##*#\"/*R&[ri -8\"0?*p\"p`&GvFV#\"0%)R$QNU\">\"7%F+#\"0b.+%4JM*)\"2;'Rmg8eI 5#\"0\\sJ6T3y(\"1!)[2_nc?K7%#\"#:\"#E#!>$p*RZV,k%)\\Hb*H=&)\"@wDJgtKyG ]f/!*4\\+(#\"3D#QmTfsA#y\"4_\"p:>>B2WA7%F+#!BNX?Dn5!)GE65Z!4YFF\"Cc9th @u*\\R`(>kUkI[z#\"?\"*[@$R&)z]'*)RpjsPY\"@_\\da1i:xeY)4`w,+$)>7%#\"#n\"#v#!NTj9XRP89b#R^7=!G,Oym,GH# \"O](ozT`:`SX%>h&ec)z#=o@(HA>#!5\")*Hj*eDhKPC\"6+]7GcobuR5%7%F+#\"Th=B g96[yUY5/w%\\Q\"\"T+]iS\">(Rtkn>\"4')pe7^'o3M]`/d\"#\"PKla.b 9e;N1jyRHC&p\\3Js*fJ\"\"QD\"y]qZcW80N)H]q*z.tsd\\\"=k67%F+#!O#>?ozp[w# )=0!p#zBsf37gd))[\"\"ND1RvZM/-rq5Rf?xjfha\")[W##\">3pGum\\@en'*3sL#o\" >D\"y]?&3C3yz1uk5\"7%\"\"\"#\"E(eSWp^-+`$=nX0;2^dP\".g>uI-u&7%F+#!N6;w:zRLGr#eRF>'*eGQ=+G)H%\"M+=[WE(3nTYU +Q&)>Fk2R)pE**#!G_5)\\]B_Bn+XrsLFLxW&=\"GDY%=ym_v>t?-cGtE7%F+#\"G K'HtV`]lPf)\\nXta2Ro&*\"FvzA+j)\\/k:5m?H[0#Gp\"#!8WE$)41\"36(G!H5\"6D^ ^^?[&)4V$>7%F+F+#\"3+]7Gy#Q=\"&%y)*Q)pprint126\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "subs(ee,matrix([seq( [c[i],seq(a[i,j],j=1..i-1),``$(9-i)],i=2..8),\n[``,seq(b[i],i=1..8)]]) ):\nevalf[8](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7*7+$ \")aZ68!\")F(%!GF+F+F+F+F+F+7+$\")J@n>F*$\")G.=\\!\"*$\"))4aZ\"F*F+F+F +F+F+F+7+$\")XXXXF*$\")b(3b$F*$!)&RqF\"!\"($\")u\\w8F;F+F+F+F+F+7+$\") $yMa&F*$\")Nn'=#F1$\")j!>/#F*$\")_>p')F1$\")])fT#F*F+F+F+F+7+$\")3BpdF *$!)q/;7F1$\")*[d[$F*$!)%*\\JMF1$\");8%\\#F*$\")%\\5a#F1F+F+F+7+$\")LL L*)F*$!)0u#>\"F*$!)1%*QfF1$\")]x=))F*$\")cSI6F*$!)Ap*3'F;$\")4xmhF;F+F +7+$\"\"\"\"\"!$\")e45@F;$!)G#fa'F*$!))>+L%F;$!):2PpF*$\")hL_c!\"'$!)_ ()>`F]p$\")WhV7F;F+7+F+$\")FQveF1$FboFbo$\")!\\<(HF*$\")GR)[\"F*$!)n:< bF*$\")0[Q\")F*$\")q&o9#F*$\")8RT=F1Q)pprint146\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "RK6_8eqs := \+ [op(RowSumConditions(8,'expanded')),op(OrderConditions(6,8,'expanded') )]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Ch eck: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "simplify(subs(ee,RK 6_8eqs)):\nmap(u->lhs(u)-rhs(u),%);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7N\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#W" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "Next we set-up stage-order condtions to check for stage-orders \+ from 2 to 4 inclusive. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "f or ct from 2 to 4 do\n so||ct||_8 := StageOrderConditions(ct,8,'expa nded');\nend do:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "Stages 3 to 9 have the following respective stage-orders. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "[seq([seq(expand(subs( ee,so||i||_8[j])),i=2..4)],j=1..6)]:\nmap(proc(L) local i; for i to no ps(L) do if not evalb(L[i]) then break end if end do; i end proc,%):\n simplify(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(\"\"$F$F$F$F$F$" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "#------- ------------------------" }}{PARA 0 "" 0 "" {TEXT -1 27 "The simplifyi ng conditions:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum (b[i]*a[i,j],i = j+1 .. 8) = b[j]*(1-c[j]);" "6#/-%$SumG6$*&&%\"bG6#% \"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F,F,F,\"\")*&&F)6#F0F,,&F,F,&%\"cG6 #F0!\"\"F," }{TEXT -1 3 ", " }{XPPEDIT 18 0 "j=1" "6#/%\"jG\"\"\"" } {TEXT -1 7 " . . 7 " }}{PARA 0 "" 0 "" {TEXT -1 8 "(where " } {XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\"\"\"\"!" }{TEXT -1 17 " ) are s atisfied." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "[Sum(b[i]*a[i,1],i=2..8)=b[1],seq(Sum(b[i]*a[i,j] ,i=j+1..8)=b[j]*(1-c[j]),j=2..7)];\neval(subs(Sum=add,%)):\nsubs(ee,%) :\nmap(u->`if`(lhs(u)=rhs(u),0,1),%);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7)/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F,F-F-/F,;\"\" #\"\")&F*6#F-/-F&6$*&F)F-&F/6$F,F3F-/F,;\"\"$F4*&&F*6#F3F-,&F-F-&%\"cG FB!\"\"F-/-F&6$*&F)F-&F/6$F,F?F-/F,;\"\"%F4*&&F*6#F?F-,&F-F-&FEFRFFF-/ -F&6$*&F)F-&F/6$F,FOF-/F,;\"\"&F4*&&F*6#FOF-,&F-F-&FEFjnFFF-/-F&6$*&F) F-&F/6$F,FgnF-/F,;\"\"'F4*&&F*6#FgnF-,&F-F-&FEFhoFFF-/-F&6$*&F)F-&F/6$ F,FeoF-/F,;\"\"(F4*&&F*6#FeoF-,&F-F-&FEFfpFFF-/-F&6$*&F)F-&F/6$F,FcpF- /F,;F4F4*&&F*6#FcpF-,&F-F-&FEFcqFFF-" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#7)\"\"!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 simpl ifying condition: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,2],i = 3 .. 8) = 0;" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"& %\"aG6$F+\"\"#F,/F+;\"\"$\"\")\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 13 "is satisfied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "Sum(b[i]*a[i,2],i=3..8);\nev al(subs(Sum=add,%));\nsubs(ee,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#- %$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F*\"\"#F+/F*;\"\"$\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*&&%\"bG6#\"\"$\"\"\"&%\"aG6$F(\"\"#F)F)*& &F&6#\"\"%F)&F+6$F1F-F)F)*&&F&6#\"\"&F)&F+6$F7F-F)F)*&&F&6#\"\"'F)&F+6 $F=F-F)F)*&&F&6#\"\"(F)&F+6$FCF-F)F)*&&F&6#\"\")F)&F+6$FIF-F)F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 27 "The simplifying condition: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*c[i]*a[i,2],i = 3 \+ .. 8) = 0;" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cG6#F+F,&%\"aG6$F+\" \"#F,/F+;\"\"$\"\")\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 13 "is satisfied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "Sum(b[i]*c[i]*a[i,2],i=3..8);\neval(subs(Sum= add,%));\nsubs(ee,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*(&% \"bG6#%\"iG\"\"\"&%\"cGF)F+&%\"aG6$F*\"\"#F+/F*;\"\"$\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*(&%\"bG6#\"\"$\"\"\"&%\"cGF'F)&%\"aG6$F( \"\"#F)F)*(&F&6#\"\"%F)&F+F2F)&F-6$F3F/F)F)*(&F&6#\"\"&F)&F+F9F)&F-6$F :F/F)F)*(&F&6#\"\"'F)&F+F@F)&F-6$FAF/F)F)*(&F&6#\"\"(F)&F+FGF)&F-6$FHF /F)F)*(&F&6#\"\")F)&F+FNF)&F-6$FOF/F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The simplifying condition: " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "Sum(b[i]*c[i]^2*a[i,2],i = 3 .. 8) = 0;" "6#/-%$SumG 6$*(&%\"bG6#%\"iG\"\"\"*$&%\"cG6#F+\"\"#F,&%\"aG6$F+F1F,/F+;\"\"$\"\") \"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 13 "is satisfied." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "Sum(b[i]*c[i]^2*a[i,2],i=3..8);\neval(subs(Sum=add,%));\nsubs(ee ,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*(&%\"bG6#%\"iG\"\"\" )&%\"cGF)\"\"#F+&%\"aG6$F*F/F+/F*;\"\"$\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*(&%\"bG6#\"\"$\"\"\")&%\"cGF'\"\"#F)&%\"aG6$F(F-F)F) *(&F&6#\"\"%F))&F,F3F-F)&F/6$F4F-F)F)*(&F&6#\"\"&F))&F,F;F-F)&F/6$F " 0 "" {MPLTEXT 1 0 85 "Sum(b[i]*c[i]*Sum(a[i,j]*a[j,2],j=3..i-1),i=3..8);\neval(subs(Su m=add,%));\nsubs(ee,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*( &%\"bG6#%\"iG\"\"\"&%\"cGF)F+-F$6$*&&%\"aG6$F*%\"jGF+&F26$F4\"\"#F+/F4 ;\"\"$,&F*F+F+!\"\"F+/F*;F:\"\")" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,, **&%\"bG6#\"\"%\"\"\"&%\"cGF'F)&%\"aG6$F(\"\"$F)&F-6$F/\"\"#F)F)*(&F&6 #\"\"&F)&F+F5F),&*&&F-6$F6F/F)F0F)F)*&&F-6$F6F(F)&F-6$F(F2F)F)F)F)*(&F &6#\"\"'F)&F+FCF),(*&&F-6$FDF/F)F0F)F)*&&F-6$FDF(F)F?F)F)*&&F-6$FDF6F) &F-6$F6F2F)F)F)F)*(&F&6#\"\"(F)&F+FTF),**&&F-6$FUF/F)F0F)F)*&&F-6$FUF( F)F?F)F)*&&F-6$FUF6F)FPF)F)*&&F-6$FUFDF)&F-6$FDF2F)F)F)F)*(&F&6#\"\")F )&F+FboF),,*&&F-6$FcoF/F)F0F)F)*&&F-6$FcoF(F)F?F)F)*&&F-6$FcoF6F)FPF)F )*&&F-6$FcoFDF)F^oF)F)*&&F-6$FcoFUF)&F-6$FUF2F)F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "#---------------------------------------" }} {PARA 0 "" 0 "" {TEXT -1 101 "We can calculate the 2 norm of the princ ipal error, that is, the 2-norm of the principal error terms." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "errterms6_8 := PrincipalErr orTerms(6,8,'expanded'):\nsqrt(add(subs(ee,errterms6_8[i])^2,i=1.. nop s(errterms6_8))):\nevalf[10](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"+CS!>:)!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "12 of the 48 principal error conditions are satisfied." } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 243 "RK6_8err_eqs := PrincipalE rrorConditions(6,8,'expanded'):\nexpand(subs(ee,RK6_8err_eqs)):\nmap(u ->`if`(lhs(u)=rhs(u),0,1),%);\nL := %: ind := []:\nfor ct to nops(L) d o\n if L[ct]=0 then ind := [op(ind),ct] end if:\nend do:\nnops(L);\n ind;\nnops(ind);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7R\"\"\"F$F$F$F$F$ F$F$F$F$F$F$F$F$F$F$F$F$F$\"\"!F$F$F$F$F$F%F$F%F%F$F$F$F$F%F$F$F%F$F$F $F%F%F$F%F$F%F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#[" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7.\"#?\"#E\"#G\"#H\"#M\"#P\"#T\"#U\"#W\"#Y\"#Z \"#[" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "These simple principal er ror conditions in abreviated form are as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "RK6_err_eqs := PrincipalErrorConditions(6): \nconvert([seq([ind[i],` `,RK6_err_eqs[ind[i]]],i=1..nops(ind))],matr ix);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7.7%\"#?%#~~G/*&% \"bG\"\"\"-%!G6#*(%\"aGF--F/6#*&F2F-%\"cGF-F--F/6#*&)F6\"\"#F-F2F-F-F- #F-\"$_#7%\"#EF)/*&F,F--F/6#*(F2F-F6F-)F3F;F-F-#F-\"$o\"7%\"#GF)/*&F,F -)F3\"\"$F-#F-\"#c7%\"#HF)/*&F,F--F/6#*(F2F-F:F-F7F-F-#F-\"$E\"7%\"#MF )/**F,F-F6F-F3F-F7F-#F-\"#U7%\"#PF)/*&F,F--F/6#*(F2F-)F6FMF-F3F-F-#F- \"#%)7%\"#TF)/*&F,F-)F7F;F-#F-\"#j7%FhnF)/*(F,F-F:F-FEF-#F-FI7%\"#WF)/ *(F,F-F`oF-F7F-#F-\"#@7%\"#YF)/*(F,F-)F6\"\"%F-F3F-#F-\"#97%\"#ZF)/*&F ,F--F/6#*&)F6\"\"&F-F2F-F-Fgn7%\"#[F)/*&F,F-)F6\"\"'F-#F-\"\"(Q)pprint 176\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 34 "#------------------ ---------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 27 "construction o f the scheme " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "We specify the nodes " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "c[2] = 8/61;" "6#/&%\"cG6#\"\"#*&\"\")\"\"\"\"#h!\" \"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[3] = 12/61;" "6#/&%\"cG6#\"\"$ *&\"#7\"\"\"\"#h!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4] = 5/11; " "6#/&%\"cG6#\"\"%*&\"\"&\"\"\"\"#6!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "c[5] = 51/92;" "6#/&%\"cG6#\"\"&*&\"#^\"\"\"\"##*!\"\" " }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[6] = 15/26;" "6#/&%\"cG6#\"\"'*& \"#:\"\"\"\"#E!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[7] = 67/75;" "6#/&%\"cG6#\"\"(*&\"#n\"\"\"\"#v!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[8]=1" "6#/&%\"cG6#\"\")\"\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 15 "and the weight " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 " b[2]=0" "6#/&%\"bG6#\"\"#\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "The scheme is designed so as to satisfy the " }{TEXT 260 7 "order 7" }{TEXT -1 23 " quadrature \+ conditions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 74 "linalg[transpose](convert([QuadratureConditions(7,8 ,'expanded')],matrix));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6 #7)7#/,2&%\"bG6#\"\"\"F-&F+6#\"\"#F-&F+6#\"\"$F-&F+6#\"\"%F-&F+6#\"\"& F-&F+6#\"\"'F-&F+6#\"\"(F-&F+6#\"\")F-F-7#/,0*&F.F-&%\"cGF/F-F-*&F1F-& FHF2F-F-*&F4F-&FHF5F-F-*&F7F-&FHF8F-F-*&F:F-&FHF;F-F-*&F=F-&FHF>F-F-*& F@F-&FHFAF-F-#F-F07#/,0*&F.F-)FGF0F-F-*&F1F-)FJF0F-F-*&F4F-)FLF0F-F-*& F7F-)FNF0F-F-*&F:F-)FPF0F-F-*&F=F-)FRF0F-F-*&F@F-)FTF0F-F-#F-F37#/,0*& F.F-)FGF3F-F-*&F1F-)FJF3F-F-*&F4F-)FLF3F-F-*&F7F-)FNF3F-F-*&F:F-)FPF3F -F-*&F=F-)FRF3F-F-*&F@F-)FTF3F-F-#F-F67#/,0*&F.F-)FGF6F-F-*&F1F-)FJF6F -F-*&F4F-)FLF6F-F-*&F7F-)FNF6F-F-*&F:F-)FPF6F-F-*&F=F-)FRF6F-F-*&F@F-) FTF6F-F-#F-F97#/,0*&F.F-)FGF9F-F-*&F1F-)FJF9F-F-*&F4F-)FLF9F-F-*&F7F-) FNF9F-F-*&F:F-)FPF9F-F-*&F=F-)FRF9F-F-*&F@F-)FTF9F-F-#F-F<7#/,0*&F.F-) FGF " 0 "" {MPLTEXT 1 0 100 "c_3 := 12/61: c_4 := 5/11: \+ c_5 := 51/92: c_6 := 15/26: c_7 := 67/75:\nGam7 := 100/103: Gam8 := 27 /29:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 192 "Qeqs := QuadratureConditions(7,8,'expanded'):\ne1 := \{c[3]=12/61,c[4]=5/11,c[5]=51/92,c[6]=15/26,c[7]=67/75,c[8]=1,b[2]=0 \}:\nquadeqns := subs(e1,Qeqs):\nnops(quadeqns);\nindets(quadeqns);\nn ops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<)&%\"bG6#\"\"'&F%6#\"\")&F%6#\"\"(&F%6#\"\"\"&F%6#\"\" $&F%6#\"\"%&F%6#\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"(" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "We have 7 linear equations for the 7 weights." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "e2 := solve(\{op(quadeqns)\}):\ninfolevel[solve] := 0 :\ne3 := `union`(e1,e2):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 27 "The weights are as follows." }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 31 "seq(b[i]=subs(e3,b[i]),i=1..8);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6*/&%\"bG6#\"\"\"#\")rz)e(\"++gi\"H\"/&F%6#\"\"#\" \"!/&F%6#\"\"$#\"2h[]]t;q*[\"3gTp/(zcyk\"/&F%6#\"\"%#\"-Hu)Rz.\"\"-+l; ctp/&F%6#\"\"&#!/K)3KR@y$\"/02%zK_&o/&F%6#\"\"'#\"-7F]p1X\"-DrS^Pb/&F% 6#\"\"(#\"1vo/[nE#o#\"2)[\"*)zN#R\\7/&F%6#\"\")#\"%>=\"&%y)*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 338 "e3 := \{b[4 ] = 103793987429/697356166500, b[5] = -37821393208832/68552327940705, \+ b[6] = 450669502712/553751407125, c[3] = 12/61, c[7] = 67/75, b[2] = 0 , c[8] = 1, c[4] = 5/11, b[8] = 1819/98784, c[5] = 51/92, b[7] = 26822 66748046875/12493923579891488, b[1] = 75887971/1291626000, c[6] = 15/2 6, b[3] = 48970167350504861/164785679704694160\}:" }{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 -1 159 "We determine the linking coefficients by means of a system of equ ations that consists in part of the stage-order equations that ensure \+ that stages 2 to 8 have " }{TEXT 260 13 "stage-order 3" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 24 "We also incorporate the " }{TEXT 260 22 "simplifying conditions" }{TEXT -1 2 ": " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*a[i,j],i = j+1 .. 8) = b[j]*(1 -c[j]);" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"&%\"aG6$F+%\"jGF,/F+;,&F0F, F,F,\"\")*&&F)6#F0F,,&F,F,&%\"cG6#F0!\"\"F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "for " }{XPPEDIT 18 0 "j = 5;" "6#/%\"jG\"\"&" } {TEXT -1 15 ", 6, 7 ( with " }{XPPEDIT 18 0 "c[1]=0" "6#/&%\"cG6#\"\" \"\"\"!" }{TEXT -1 55 " ), together with the further simplifying cond itions: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]* c[i]*a[i,2],i=3..8)=0" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cG6#F+F,& %\"aG6$F+\"\"#F,/F+;\"\"$\"\")\"\"!" }{TEXT -1 8 ", " }{XPPEDIT 18 0 "Sum(b[i]*c[i]*Sum(a[i,j]*a[j,2],j=3..i-1),i=3..8)=0" "6#/-%$SumG 6$*(&%\"bG6#%\"iG\"\"\"&%\"cG6#F+F,-F%6$*&&%\"aG6$F+%\"jGF,&F46$F6\"\" #F,/F6;\"\"$,&F+F,F,!\"\"F,/F+;F<\"\")\"\"!" }{TEXT -1 8 ", " } {XPPEDIT 18 0 "Sum(b[i]*c[i]^2*a[i,2],i=3..8)=0" "6#/-%$SumG6$*(&%\"bG 6#%\"iG\"\"\"*$&%\"cG6#F+\"\"#F,&%\"aG6$F+F1F,/F+;\"\"$\"\")\"\"!" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 26 "We use the single order 6 " }{TEXT 260 15 "order condition" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*c[i]*Sum(a[ i,j]*c[j]^3,j = 2 .. i-1),i = 3 .. 8) = 1/24" "6#/-%$SumG6$*(&%\"bG6#% \"iG\"\"\"&%\"cG6#F+F,-F%6$*&&%\"aG6$F+%\"jGF,*$&F.6#F6\"\"$F,/F6;\"\" #,&F+F,F,!\"\"F,/F+;F:\"\")*&F,F,\"#CF?" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "We introduce the " } {TEXT 260 23 "additional requirements" }{TEXT -1 27 " that the coeffic ients of " }{XPPEDIT 18 0 "z^7;" "6#*$%\"zG\"\"(" }{TEXT -1 6 " and \+ " }{XPPEDIT 18 0 "z^8" "6#*$%\"zG\"\")" }{TEXT -1 35 " in the stabili ty polynomial are " }{XPPEDIT 18 0 "100/(103*`.`*7!) = 5/25956;" "6#/ *&\"$+\"\"\"\"*(\"$.\"F&%\".GF&-%*factorialG6#\"\"(F&!\"\"*&\"\"&F&\"& cf#F." }{TEXT -1 7 " and " }{XPPEDIT 18 0 "27/(29*`.`*8!) = 3/129920 ;" "6#/*&\"#F\"\"\"*(\"#HF&%\".GF&-%*factorialG6#\"\")F&!\"\"*&\"\"$F& \"'?*H\"F." }{TEXT -1 16 " respectively. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "StabilityFunction( 6,8,'expanded');" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,4\"\"\"F$%\"zGF$* &\"\"#!\"\"F%F'F$*&\"\"'F(F%\"\"$F$*&\"#CF(F%\"\"%F$*&\"$?\"F(F%\"\"&F $*&\"$?(F(F%F*F$*&,&*0&%\"bG6#\"\"(F$&%\"aG6$F:F*F$&F<6$F*F1F$&F<6$F1F .F$&F<6$F.F+F$&F<6$F+F'F$&%\"cG6#F'F$F$*&&F86#\"\")F$,&*.&F<6$FLF*F$F> F$F@F$FBF$FDF$FFF$F$*&&F<6$FLF:F$,&*,&F<6$F:F1F$F@F$FBF$FDF$FFF$F$*&F; F$,&**&F<6$F*F.F$FBF$FDF$FFF$F$*&F>F$,&*(&F<6$F1F+F$FDF$FFF$F$*&F@F$,& *&&F<6$F.F'F$FFF$F$*&FBF$&FG6#F+F$F$F$F$F$F$F$F$F$F$F$F$F$)F%F:F$F$*4F JF$FRF$F;F$F>F$F@F$FBF$FDF$FFF$)F%FLF$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "27/(29*`.`*8!);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"$\"\"\"\"'?*H\"!\"\"%\".GF(F& " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "Thus we specify that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " b[7]*a[7,6]*a[6,5]*a[5,4]*a[4,3]*a[3,2]*c[2]+``" "6#,&*0&%\"bG6#\"\"( \"\"\"&%\"aG6$F(\"\"'F)&F+6$F-\"\"&F)&F+6$F0\"\"%F)&F+6$F3\"\"$F)&F+6$ F6\"\"#F)&%\"cG6#F9F)F)%!GF)" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[8]*(a[8,6]*a[6,5]*a[5,4]*a[4,3]*a[3,2 ]*c[2]+a[8,7]*(a[7,5]*a[5,4]*a[4,3]*a[3,2]*c[2]+a[7,6]*(a[6,4]*a[4,3]* a[3,2]*c[2]+a[6,5]*(a[5,3]*a[3,2]*c[2]+a[5,4]*(a[4,2]*c[2]+a[4,3]*c[3] ))))) = 5/25956;" "6#/*&&%\"bG6#\"\")\"\"\",&*.&%\"aG6$F(\"\"'F)&F-6$F /\"\"&F)&F-6$F2\"\"%F)&F-6$F5\"\"$F)&F-6$F8\"\"#F)&%\"cG6#F;F)F)*&&F-6 $F(\"\"(F),&*,&F-6$FBF2F)&F-6$F2F5F)&F-6$F5F8F)&F-6$F8F;F)&F=6#F;F)F)* &&F-6$FBF/F),&**&F-6$F/F5F)&F-6$F5F8F)&F-6$F8F;F)&F=6#F;F)F)*&&F-6$F/F 2F),&*(&F-6$F2F8F)&F-6$F8F;F)&F=6#F;F)F)*&&F-6$F2F5F),&*&&F-6$F5F;F)&F =6#F;F)F)*&&F-6$F5F8F)&F=6#F8F)F)F)F)F)F)F)F)F)F)F)*&F2F)\"&cf#!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "b[8]*a[8,7]*a[7,6]*a[6,5]*a[5,4]*a[4, 3]*a[3,2]*c[2] = 3/129920;" "6#/*2&%\"bG6#\"\")\"\"\"&%\"aG6$F(\"\"(F) &F+6$F-\"\"'F)&F+6$F0\"\"&F)&F+6$F3\"\"%F)&F+6$F6\"\"$F)&F+6$F9\"\"#F) &%\"cG6#F " 0 "" {MPLTEXT 1 0 876 "SO_eqs := [ op(RowSumConditions(8,'expanded')),op(StageOrderConditions(2,8,'expand ed')),\n op(StageOrderConditions(3,8,'expanded'))]:\nsimp _eqs := [add(b[i]*a[i,1],i=2..8)=b[1],seq(add(b[i]*a[i,j],i=j+1..8)=b[ j]*(1-c[j]),j=[5,6,7]),\n add(b[i]*c[i]*a[i,2],i=3..8)=0, add(b[i]*c[i]*add(a[i,j]*a[j,2],j=3..i-1),i=3..8)=0,\n \+ add(b[i]*c[i]^2*a[i,2],i=3..8)=0]:\nord_cdn := add(b[i]*c[i]*add(a[i, j]*c[j]^3,j=2..i-1),i=3..8)=1/24:\nextra_eqs := [b[7]*a[7,6]*a[6,5]*a[ 5,4]*a[4,3]*a[3,2]*c[2]+b[8]*(a[8,6]*a[6,5]*a[5,4]*a[4,3]*a[3,2]*c[2]+ \n a[8,7]*(a[7,5]*a[5,4]*a[4,3]*a[3,2]*c[2]+a[7,6]*(a[6,4 ]*a[4,3]*a[3,2]*c[2]+\n a[6,5]*(a[5,3]*a[3,2]*c[2]+a[5,4] *(a[4,2]*c[2]+a[4,3]*c[3])))))=5/25956,\n b[8]*a[8,7]*a[7 ,6]*a[6,5]*a[5,4]*a[4,3]*a[3,2]*c[2]=3/129920]:\ncdns := [op(SO_eqs),o p(simp_eqs),ord_cdn,op(extra_eqs)]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "eqns := subs(e3,cdns):\nno ps(eqns);\nindets(eqns);\nnops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# \"#H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# " 0 "" {MPLTEXT 1 0 22 "infolevel[solve] := 4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "e4 := solve(\{op(eqns)\}) :\ninfolevel[solve] := 0:\ne5 := `union`(e3,e4):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "e5" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2033 "e5 := \{b[4] = 10379398742 9/697356166500, a[6,2] = 782227259416638225/2244072319191569152, a[3,2 ] = 9/61, b[5] = -37821393208832/68552327940705, b[6] = 450669502712/5 53751407125, a[8,2] = -3757510716191/5740230741960, a[6,4] = 463772636 939896507985393214891/1859455376317587715620654574952, c[3] = 12/61, a [2,1] = 8/61, a[8,3] = -429828001838285896192739582712833397915761611/ 99266983907642719853800424641670872644481800, a[8,5] = 956839075473456 74985937655053437329632/1692820548292066101564044986300227975, a[4,2] \+ = -163175/127776, a[5,3] = 893431094000355/10305813606639616, a[8,7] = 126173827828125000/101457352642065253, a[7,3] = 138494760410464278481 114601908086654010742317231861/157045350340868651125869860911967647339 719140625000, c[7] = 67/75, a[7,5] = -14888576012085972237926900518827 64869796820192/244488154615963772059391070710204344775390625, a[8,6] = -10290287110810609832644/193430985482051515125, b[2] = 0, c[8] = 1, a [5,4] = 778084111317249/3220566752074880, c[4] = 5/11, a[5,1] = 130262 71485399/595711769169920, a[7,1] = -2292801667836012801812513925514133 73945146341/1922297216818279856585611944540531553417968750, b[8] = 181 9/98784, a[4,3] = 263825/191664, a[8,1] = 1072652717054567183530002516 94440587/50834312477772792379562256248860860, a[5,2] = 24327752855319/ 119142353833984, a[7,4] = 13159972310849695242939786306351658145503546 532/116418149577273037997050298350513445647705078125, a[7,2] = -243732 61255896329981/410397455685628125000, a[4,1] = 136115/383328, a[6,3] = -272746090471011262880106725204535/7948306442641975339499742161731456 , c[5] = 51/92, b[7] = 2682266748046875/12493923579891488, b[1] = 7588 7971/1291626000, a[8,4] = -18544773327337271450067235223504981052/2673 2856022073197552661912017781844625, c[6] = 15/26, a[6,1] = -8518299552 9498464014347399693/7004909900459502878327360312576, b[3] = 4897016735 0504861/164785679704694160, a[7,6] = 68233720896675821496674286908/110 64740679780824085205078125, a[6,5] = 503890150536220957318530/19830001 765309846198610093, a[3,1] = 3/61, c[2] = 8/61\}:" }{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 64 "The Butcher tableau in exact and approximate form is as follows." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 408 "subs(e5,matrix([[c[2],a[2,1 ],``],\n[c[3],a[3,1],a[3,2]],\n[c[4],a[4,1],a[4,2]],[``$2,a[4,3]],\n[c [5],a[5,1],a[5,2]],[``,a[5,3],a[5,4]],\n[c[6],a[6,1],a[6,2]],[``,a[6,3 ],a[6,4]],[``$2,a[6,5]],\n[c[7],a[7,1],a[7,2]],[``,a[7,3],a[7,4]],[``, a[7,5],a[7,6]],\n[c[8],a[8,1],a[8,2]],[``,a[8,3],a[8,4]],[``,a[8,5],a[ 8,6]],[``$2,a[8,7]],\n[``,`____________`$2],\n[``,b[1],b[2]],[``,b[3], b[4]],[``,b[5],b[6]],[``,b[7],b[8]]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#777%#\"\")\"#hF(%!G7%#\"#7F*#\"\"$F*#\"\"*F*7%#\"\"& \"#6#\"':h8\"'GLQ#!'vJ;\"'wx77%F+F+#\"'DQE\"'k;>7%#\"#^\"##*#\"/*R&[ri -8\"0?*p\"p`&GvFV#\"0%)R$QNU\">\"7%F+#\"0b.+%4JM*)\"2;'Rmg8eI 5#\"0\\sJ6T3y(\"1!)[2_nc?K7%#\"#:\"#E#!>$p*RZV,k%)\\Hb*H=&)\"@wDJgtKyG ]f/!*4\\+(#\"3D#QmTfsA#y\"4_\"p:>>B2WA7%F+#!BNX?Dn5!)GE65Z!4YFF\"Cc9th @u*\\R`(>kUkI[z#\"?\"*[@$R&)z]'*)RpjsPY\"@_\\da1i:xeY)4`w,+$)>7%#\"#n\"#v#!NTj9XRP89b#R^7=!G,Oym,GH# \"O](ozT`:`SX%>h&ec)z#=o@(HA>#!5\")*Hj*eDhKPC\"6+]7GcobuR5%7%F+#\"Th=B g96[yUY5/w%\\Q\"\"T+]iS\">(Rtkn>\"4')pe7^'o3M]`/d\"#\"PKla.b 9e;N1jyRHC&p\\3Js*fJ\"\"QD\"y]qZcW80N)H]q*z.tsd\\\"=k67%F+#!O#>?ozp[w# )=0!p#zBsf37gd))[\"\"ND1RvZM/-rq5Rf?xjfha\")[W##\">3pGum\\@en'*3sL#o\" >D\"y]?&3C3yz1uk5\"7%\"\"\"#\"E(eSWp^-+`$=nX0;2^dP\".g>uI-u&7%F+#!N6;w:zRLGr#eRF>'*eGQ=+G)H%\"M+=[WE(3nTYU +Q&)>Fk2R)pE**#!G_5)\\]B_Bn+XrsLFLxW&=\"GDY%=ym_v>t?-cGtE7%F+#\"G K'HtV`]lPf)\\nXta2Ro&*\"FvzA+j)\\/k:5m?H[0#Gp\"#!8WE$)41\"36(G!H5\"6D^ ^^?[&)4V$>7%F+F+#\"3+]7Gy#Q=\"&%y)*Q(pprint86\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "subs(e5,matrix([seq( [c[i],seq(a[i,j],j=1..i-1),``$(9-i)],i=2..8),\n[``,seq(b[i],i=1..8)]]) ):\nevalf[8](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7*7+$ \")aZ68!\")F(%!GF+F+F+F+F+F+7+$\")J@n>F*$\")G.=\\!\"*$\"))4aZ\"F*F+F+F +F+F+F+7+$\")XXXXF*$\")b(3b$F*$!)&RqF\"!\"($\")u\\w8F;F+F+F+F+F+7+$\") $yMa&F*$\")Nn'=#F1$\")j!>/#F*$\")_>p')F1$\")])fT#F*F+F+F+F+7+$\")3BpdF *$!)q/;7F1$\")*[d[$F*$!)%*\\JMF1$\");8%\\#F*$\")%\\5a#F1F+F+F+7+$\")LL L*)F*$!)0u#>\"F*$!)1%*QfF1$\")]x=))F*$\")cSI6F*$!)Ap*3'F;$\")4xmhF;F+F +7+$\"\"\"\"\"!$\")e45@F;$!)G#fa'F*$!))>+L%F;$!):2PpF*$\")hL_c!\"'$!)_ ()>`F]p$\")WhV7F;F+7+F+$\")FQveF1$FboFbo$\")!\\<(HF*$\")GR)[\"F*$!)n:< bF*$\")0[Q\")F*$\")q&o9#F*$\")8RT=F1Q)pprint116\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "RK6_8eqs := \+ [op(RowSumConditions(8,'expanded')),op(OrderConditions(6,8,'expanded') )]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Ch eck: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "simplify(subs(e5,RK 6_8eqs)):\nmap(u->`if`(lhs(u)=rhs(u),0,1),%);\nnops(%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#7N\"\"!F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F $F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#W" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 34 "#----- ----------------------------" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 25 "a bsolute stability region" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "coefficients of the scheme" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1949 "ee := \{c[2]=8/61,\nc[3]=1 2/61,\nc[4]=5/11,\nc[5]=51/92,\nc[6]=15/26,\nc[7]=67/75,\nc[8]=1,\n\na [2,1]=8/61,\na[3,1]=3/61,\na[3,2]=9/61,\na[4,1]=136115/383328,\na[4,2] =-163175/127776,\na[4,3]=263825/191664,\na[5,1]=13026271485399/5957117 69169920,\na[5,2]=24327752855319/119142353833984,\na[5,3]=893431094000 355/10305813606639616,\na[5,4]=778084111317249/3220566752074880,\na[6, 1]=-85182995529498464014347399693/7004909900459502878327360312576,\na[ 6,2]=782227259416638225/2244072319191569152,\na[6,3]=-2727460904710112 62880106725204535/7948306442641975339499742161731456,\na[6,4]=46377263 6939896507985393214891/1859455376317587715620654574952,\na[6,5]=503890 150536220957318530/19830001765309846198610093,\na[7,1]=-22928016678360 1280181251392551413373945146341/19222972168182798565856119445405315534 17968750,\na[7,2]=-24373261255896329981/410397455685628125000,\na[7,3] =138494760410464278481114601908086654010742317231861/15704535034086865 1125869860911967647339719140625000,\na[7,4]=13159972310849695242939786 306351658145503546532/116418149577273037997050298350513445647705078125 ,\na[7,5]=-1488857601208597223792690051882764869796820192/244488154615 963772059391070710204344775390625,\na[7,6]=682337208966758214966742869 08/11064740679780824085205078125,\na[8,1]=1072652717054567183530002516 94440587/50834312477772792379562256248860860,\na[8,2]=-3757510716191/5 740230741960,\na[8,3]=-429828001838285896192739582712833397915761611/9 9266983907642719853800424641670872644481800,\na[8,4]=-1854477332733727 1450067235223504981052/26732856022073197552661912017781844625,\na[8,5] =95683907547345674985937655053437329632/169282054829206610156404498630 0227975,\na[8,6]=-10290287110810609832644/193430985482051515125,\na[8, 7]=126173827828125000/101457352642065253,\n\nb[1]=75887971/1291626000, \nb[2]=0,\nb[3]=48970167350504861/164785679704694160,\nb[4]=1037939874 29/697356166500,\nb[5]=-37821393208832/68552327940705,\nb[6]=450669502 712/553751407125,\nb[7]=2682266748046875/12493923579891488,\nb[8]=1819 /98784\}:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "The stability function R for the 8 stage, order 6 sche me is given as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "s ubs(ee,StabilityFunction(6,8,'expanded')):\nR := unapply(%,z):\n'R(z)' =R(z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"RG6#%\"zG,4\"\"\"F)F'F) *&#F)\"\"#F)*$)F'F,F)F)F)*&#F)\"\"'F)*$)F'\"\"$F)F)F)*&#F)\"#CF)*$)F' \"\"%F)F)F)*&#F)\"$?\"F)*$)F'\"\"&F)F)F)*&#F)\"$?(F)*$)F'F1F)F)F)*&#F@ \"&cf#F)*$)F'\"\"(F)F)F)*&#F4\"'?*H\"F)*$)F'\"\")F)F)F)" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 124 "We can find the p oint where the boundary of the stability region intersects the negativ e real axis by solving the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R(z) = 1;" "6#/-%\"RG6#%\"zG\"\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "z0 := newton(R(z)=1,z=- 4.3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G$!+?36PV!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 302 "z 0 := newton(R(z)=1,z=-4):\np1 := plot([R(z),1],z=-4.69..0.49,color=[re d,blue]):\np2 := plot([[[z0,1]]$3],style=point,symbol=[circle,cross,di amond],color=black):\np3 := plot([[z0,0],[z0,1]],linestyle=3,color=COL OR(RGB,0,.5,0)):\nplots[display]([p1,p2,p3],view=[-4.69..0.49,-0.07..1 .47],font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 253 253 {PLOTDATA 2 "6+-%'CURVESG6$7Z7$$!3Q++++++!p%!#<$\"3,q2oR_&*GGF*7$$!3An TN&osG.+Qb#F*7$$!3++D1c !=`g%F*$\"3;6bgr1ODCF*7$$!3#om;9u!4xXF*$\"3]EkO+4q-BF*7$$!3J]7Lq)pz_%F *$\"3uDO.[`O-@F*7$$!3!Q$eC***[)yWF*$\"3YXmF*7$$!3.^(=BW2OU%F*$\" 3Y1vIB#)\\F 1<)!#=7$$!3H+DTEYePRF*$\"3q4e)*zl.plFho7$$!3Cn\"zeW#pFQF*$\"3(3gn]WQoA &Fho7$$!3X+D'4p_\"=PF*$\"3?4!H%\\SzaTFho7$$!3pLL)))exag$F*$\"3eBw!o=#y \"G$Fho7$$!3!omTwHLi]$F*$\"3[-dK_g6tEFho7$$!33++lJx]%R$F*$\"3:$*)4PKPj 8#Fho7$$!3/++v4MK#G$F*$\"3*pkw)yIlF7$$!3A+D6J'p`u#F*$\"3u#4(Qo=P p$*F`s7$$!3lLLV**GaVEF*$\"35Yx5fvNX#*F`s7$$!39+D6OG#=`#F*$\"3v,7xfU(yU *F`s7$$!3J+vL;*QaU#F*$\"3=YAwswSn)*F`s7$$!3NL$ekXQWJ#F*$\"3Q#R>*G[#o0 \"Fho7$$!3\\Lep$Q0D@#F*$\"3e0liQQ'49\"Fho7$$!3&pmTSzcD5#F*$\"3iP.TRml^ 7Fho7$$!3'o;z)46N))>F*$\"39dnW)*['zQ\"Fho7$$!3C+v3EZ$*))=F*$\"3%[m)RI \"HW_\"Fho7$$!3#pm\"*zIi:y\"F*$\"3uxiV.%)3\"p\"Fho7$$!3?++q$4O1n\"F*$ \"33.&>S'4^&)=Fho7$$!3W+]-Mk6i:F*$\"3m%*>`KZI*4#Fho7$$!3O+DJ$3=rX\"F*$ \"3@OH$***=QIBFho7$$!3-+]d;``S8F*$\"30rkAd1qfFho$\"3%=?AnXFA`&Fho7$$!3e-]7G$*\\/[Fho$\"3?!))R`_]]='Fho7$$!3Wgm ms4>IPFho$\"3E\"Q*pA3_')oFho7$$!3$zm;M^b:j#Fho$\"3I[?3yKA'o(Fho7$$!3NK $e/JB;Bj\"F*-%'COLOURG6&%$RGBG $\"*++++\"!\")$\"\"!Fd]lFc]l-F$6$7S7$F($\"\"\"Fd]l7$F=Fi]l7$FGFi]l7$FQ Fi]l7$FenFi]l7$F_oFi]l7$FdoFi]l7$FjoFi]l7$F_pFi]l7$FdpFi]l7$FipFi]l7$F ^qFi]l7$FcqFi]l7$FhqFi]l7$F]rFi]l7$FbrFi]l7$FgrFi]l7$F\\sFi]l7$FbsFi]l 7$FgsFi]l7$F\\tFi]l7$FatFi]l7$FftFi]l7$F[uFi]l7$F`uFi]l7$FeuFi]l7$FjuF i]l7$F_vFi]l7$FdvFi]l7$FivFi]l7$F^wFi]l7$FcwFi]l7$FhwFi]l7$F]xFi]l7$Fb xFi]l7$FgxFi]l7$F\\yFi]l7$FayFi]l7$FfyFi]l7$F[zFi]l7$F`zFi]l7$FezFi]l7 $FjzFi]l7$F_[lFi]l7$Fd[lFi]l7$Fi[lFi]l7$F^\\lFi]l7$Fc\\lFi]l7$Fh\\lFi] l-F]]l6&F_]lFc]lFc]lF`]l-F$6&7#7$$!3:+++jEzVTF*Fi]l-%'SYMBOLG6#%'CIRCL EG-F]]l6&F_]lFd]lFd]lFd]l-%&STYLEG6#%&POINTG-F$6&F_al-Fdal6#%&CROSSGFg alFial-F$6&F_al-Fdal6#%(DIAMONDGFgalFial-F$6%7$7$FaalFc]lF`al-%&COLORG 6&F_]lFc]l$\"\"&!\"\"Fc]l-%*LINESTYLEG6#\"\"$-%+AXESLABELSG6%Q\"z6\"Q! Ficl-%%FONTG6#%(DEFAULTG-F\\dl6$%*HELVETICAG\"\"*-%%VIEWG6$;$!$p%!\"#$ \"#\\Fidl;$!\"(Fidl$\"$Z\"Fidl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "The following picture shows the stability region. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1304 "R := z -> 1+z+1/2*z^2+1/6*z^3+1/24*z^4+1/120*z^5+1/720*z^6+5/259 56*z^7+3/129920*z^8:\npts := []: z0 := 0:\nfor ct from 0 to 300 do\n \+ zz := newton(R(z)=exp(ct*Pi/25*I),z=z0):\n z0 := zz:\n pts := [op (pts),[Re(zz),Im(zz)]]:\nend do:\np1 := plot(pts,color=COLOR(RGB,0,.25 ,.48)):\np2 := plots[polygonplot]([seq([pts[i-1],pts[i],[-2.1,0]],i=2. .nops(pts))],\n style=patchnogrid,color=COLOR(RGB,0,.55,.95)) :\npts := []: z0 := 2.25+4.7*I:\nfor ct from 0 to 40 do\n zz := newt on(R(z)=exp(ct*Pi/20*I),z=z0):\n z0 := zz:\n pts := [op(pts),[Re(z z),Im(zz)]]:\nend do:\np3 := plot(pts,color=COLOR(RGB,0,.25,.48)):\np4 := plots[polygonplot]([seq([pts[i-1],pts[i],[1.94,4.75]],i=2..nops(pt s))],\n style=patchnogrid,color=COLOR(RGB,0,.55,.95)):\npts : = []: z0 := 2.25-4.7*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:\np5 := plot(pts,color=COLOR(RGB,0,.25,.48)):\np6 := plot s[polygonplot]([seq([pts[i-1],pts[i],[1.94,-4.75]],i=2..nops(pts))],\n style=patchnogrid,color=COLOR(RGB,0,.55,.95)):\np7 := plot([ [[-5.09,0],[2.39,0]],[[0,-5.19],[0,5.19]]],color=black,linestyle=3):\n plots[display]([p||(1..7)],view=[-5.09..2.39,-5.19..5.19],font=[HELVET ICA,9],\n labels=[`Re(z)`,`Im(z)`],axes=boxed,scaling=con strained);" }}{PARA 13 "" 1 "" {GLPLOT2D 381 565 565 {PLOTDATA 2 "6/-% 'CURVESG6$7i]l7$$\"\"!F)F(7$F($\"3%******41PmD\"!#=7$$!3D+++lHd=i!#G$ \"3?+++>TF8DF-7$$!32+++D3\\7:!#E$\"3s*****z76*pPF-7$$!33+++%4P)*R\"!#D $\"3a*****>&yaE]F-7$$!3P*****\\'Q`?vF=$\"3Y+++5O=$G'F-7$$!30+++!R&e=G! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 148 "W e 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 th ere is " }{TEXT 260 53 "no largest interval on the nonnegative imagina ry axis" }{TEXT -1 64 " that contains the origin and lies inside the s tability region. " }}{PARA 0 "" 0 "" {TEXT -1 119 "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 0 336 "R := z -> 1+z+1/2*z^2+1/6*z ^3+1/24*z^4+1/120*z^5+1/720*z^6+5/25956*z^7+3/129920*z^8:\nDigits := 2 5:\npts := []: z0 := 0:\nfor ct from 0 to 120 do\n zz := newton(R(z) =exp(ct*Pi/100*I),z=z0):\n z0 := zz:\n pts := [op(pts),[surd(Re(zz ),11),Im(zz)]]:\nend do:\nplot(pts,color=COLOR(RGB,0,.5,.95),thickness =2,font=[HELVETICA,9]);\nDigits := 10:" }}{PARA 13 "" 1 "" {GLPLOT2D 261 276 276 {PLOTDATA 2 "6(-%'CURVESG6#7er7$$\"\"!F)F(7$$!:_@yxL%eIR] \"=h#!#E$\":6?)ezv(*e`EfTJF-7$$!:_@C(4R$[c#f8BVF-$\":1R^C$et<2`=$G'F-7 $$!:;#\\![Zqo_*==/eF-$\":p<1\\uVJ2'zxC%*F-7$$!:pwid,3x&y31_rF-$\":wi4* HEwShqjc7!#D7$$!:2B(\\7)G')e9xyS)F-$\":Uqh\"=53mEjzq:F?7$$!:&)=9&3H4L3 i$Rf*F-$\":Fl,0@(pn\"fb\\)=F?7$$!:'\\'of-YoI\\-C2\"F?$\":iW[`<7Ah&[6*> #F?7$$!:g8%yUH^+Y%Q2=\"F?$\":%fxW)G5m$>TF8DF?7$$!:cv #e'**>$>F?$\":)ygOYx0I_yaE]F?7$$!:fVOtSO)))*HN[,#F?$\":4V!\\8m$oK#pqS` F?7$$!:G&*GeA%z[y9f&4#F?$\":0,lvxO&\\?f'[l&F?7$$!:_6J&[l())yd>V<#F?$\" :Q_VKHr&)Q#[-pfF?7$$!:cXDX5Gr%\\d0^AF?$\":q+#>8vH.5O=$G'F?7$$!:zag3]u7 sAAeK#F?$\":,u&HI%*)H@DUtf'F?7$$!:T(*)H/8e]W\"G')R#F?$\"::\"QG;2] 6pF?7$$!:8d&HgD\\>;*p%pCF?$\":oI39LSa*z*ecA(F?7$$!:ol>D9I\"Q:7LQDF?$\" :LtrE^?y`*p\")RvF?7$$!::2Q*>z(**[a#=0EF?$\":IQqgD%=`EZ(R&yF?7$$!:qk=Vj ycSP!)*pEF?$\":VDlMPdg58K\"o\")F?7$$!:$)=nr$pfUydmKFF?$\":*\\_6O,n_zdW'z)F?7$$!:HIq8z$yRGTP^GF? $\":9w,!3o_TG>g5\"*F?7$$!:aGR:sNfCf!=2HF?$\":-T\\&eIG0hvvC%*F?7$$!:ixI gi7*Q(*F?7$$!:$\\E/g())G[KY4,$F?$\":e_8tD.4q q1`+\"!#C7$$!:DTUy>flKu!\\eIF?$\":RfWs'efY'=x#=@$F ?$\":\">_a2@b9&y#Qi6Ffu7$$!:d9P0U!*zz&p2QKF?$\":))4u!o%Q^%=xz$>\"Ffu7$ $!:4n?H3UVR/ZvD$F?$\":!*4/Xpe+Nf(3 _(ygX>8Ffu7$$!:BH5a/\\evR)y[Tdz@JF?$\":xK\"*>L&>_xSG#Q\"Ffu7$$!:O=6')[!RSc,X)*GF?$\" :(['3P\"=S#pj(p89Ffu7$$\":Io/gMD8$3jG\\GF?$\":-)HRgaoV;36X9Ffu7$$\":$G +j$[E&eam8OKF?$\":,[z'p4j**HN_w9Ffu7$$\":LLRh'\\aHzDK[MF?$\":gq=B@vI-n Nz]\"Ffu7$$\":ZhAlFxL24o0h$F?$\":?J*fa4*QT5Z$R:Ffu7$$\":Y(*3'\\Z*QtOZ$ [PF?$\":=yE@.TN\"owvq:Ffu7$$\":$[rMp#*oCc4UrQF?$\":2'GB(Q-2C;n@g\"Ffu7 $$\":bYy.q9tG3#e%)RF?$\":'*RUS7#=_W`dL;Ffu7$$\":MRiijBW/gR04%F?$\":Zl \"4]FMBA>)\\m\"Ffu7$$\"::st%4\"z.u%3(4>%F?$\":(zKMivLUXlQ'p\"Ffu7$$\": \"4[\\?Oc'RB$)pG%F?$\":3Y%fy*[Erz)yFV$zVF?$\":35tx &[ue$=)=fNcwC')*\\$f(>#=Ffu7$$\":fs'e)y0X0]E$RYF?$\":mh%\\[O.aCG O`=Ffu7$$\":*zR0>)QcX$HL@ZF?$\":Z?oYU[22(Qu%)=Ffu7$$\":5'R[7rY*y!)e8![ F?$\":(>*)G#fQu)4!=h\">Ffu7$$\":I$))HeWc3@Maz[F?$\":=f(o\"o-!e>S[Z>Ffu 7$$\":*[lU>oZ3(z%*f&\\F?$\":U$y)R%G%H5]S)y>Ffu7$$\":vpt*=aT'o8'zI]F?$ \":VUc!z>#Ff u7$$\":e%obQi=ny8o6bF?$\":v&*GMr/2Q*y5HAFfu7$$\":Xz4Wei`?x5Wd&F?$\":p! >**3D/lKoFgAFfu7$$\":l4&Hu)QJBG6cj&F?$\":FyzOwt:W;39H#Ffu7$$\":7EKEuO1 6u]_p&F?$\":Q)4?WsQLZv_hF?$\":nQA*))oMtu%=%*f#Ffu7$ $\":4'F?$\":wAz#\\%Hf.*etHEFfu7$$\":-'o:WzB1![m9B'F?$\":$e *)Q&e%>_zC$*fEFfu7$$\":(p![pMm`0(e+niF?$\":R0FL&))3\"4+****o#Ffu7$$\": C0)=\"R$eKz#*y*H'F?$\":q>W'z93#e,E*>FFfu7$$\":9(>r%[%>T6HjHjF?$\":$)o4 dgc'HURq\\FFfu7$$\":[&G5>7R%f-GjN'F?$\":<'>nb\">QC6B$zFFfu7$$\":@O(3s. B#\\JN'zjF?$\":ROJ;?'ocJQx3GFfu7$$\":`Rc&\\/6B-uF*R'F?$\":Di1(G,9#yVY! QGFfu7$$\":FHw_\\(>/5(G\\T'F?$\":%\\-m)pQ&og88nGFfu7$$\":lh1mXldSc,iU' F?$\":H-s!)RlWp<>g*GFfu7$$\":PbWAxyF%*zEEV'F?$\":SxkY^Z[Em+Z#HFfu7$$\" :xh2$)osIp5EOV'F?$\":b$4)pJ@$zRo;`HFfu7$$\":D7'[KnR\\z:ZGkF?$\":Wu)yn3 zl;!49)HFfu7$$\":P$3!fufgUoCiT'F?$\"::i1u'*>&)4#)=%4IFfu7$$\":'eEPrJu- )pScR'F?$\":Wz.V.J-bA)=PIFfu7$$\":DSel'H0'o17]O'F?$\":&e#[2UB(fl&4Z1$F fu7$$\":B0<%*[>Jdh'*=K'F?$\":x1Ge)>aG\\b(>4$Ffu7$$\":-^)f'>U,#p1'F?$\":AJ%)[3q()H/u@<$Ffu7$$\":gf&frqxn$Ffu7$$\":5;'RU7U(yCI!4cF?$\":%Q,!>Yp?$3eCCKFfu7$$\":'*Q1 )=5NS8K?XZF?$\":'z,lRb9_$)p%)\\KFfu7$$!:&3*[!*>TeTG]<\\&F?$\":jd=#Qyx@ =<:vKFfu7$$!:L8Ws4]hBeJU$fF?$\":f-b6(3=Ob(>>&oF?$\":b3V4;R\\^.70U$Ffu7$$!:empS8es&f#yu'pF ?$\":6YV:aVUD%\\iVMFfu7$$!:$)*RZVGH-%fHL2(F?$\":&=x\"e_V\\(=1TmMFfu7$$ !:ll7f6*p>$Q;8<(F?$\":o%p8*4GO)Qh'))[$Ffu7$$!:\">(4Eb!o74wvisF?$\":A[. 4LTmby))4^$Ffu7$$!:7c`lyd$o!*>j[tF?$\":6fikm(ylegxKNFfu7$$!:QW^2`#Gfvu oHuF?$\":V0oP7)e4'oDUb$Ffu7$$!:;pw&\\I#RS%*4l](F?$\"::Q[A$3^(ojN`d$Ffu -%&COLORG6&%$RGBGF($\"\"&!\"\"$\"#&*!\"#-%%FONTG6$%*HELVETICAG\"\"*-%* THICKNESSG6#\"\"#-%+AXESLABELSG6$Q!6\"Febm-%%VIEWG6$%(DEFAULTGFjbm" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 "The rele vant intersection points of the boundary curve with the imaginary axis can be determined as follows." }}{PARA 0 "" 0 "" {TEXT -1 87 "First w e look for points on the boundary curve either side of each intersecti on point. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 201 "Digits := 15:\nz0 := 1.4*I:\nfor ct from 44 to 47 \+ do\n newton(R(z)=exp(ct*Pi/100*I),z=z0);\nend do;``;\nz0 := 3.25*I: \nfor ct from 105 to 108 do\n newton(R(z)=exp(ct*Pi/100*I),z=z0);\ne nd do;\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0]$Q!37Wu #!#?$\"0&>_xSG#Q\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!0Biox(*G@ \"!#?$\"0-Cpj(p89!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0w*y.-%[+ \"!#?$\"0&oV;36X9!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0gK=Xul2% !#?$\"0J'**HN_w9!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"0_nS*H))G " 0 "" {MPLTEXT 1 0 332 "Digits := 15:\nreal_part := proc(u)\n Re(newton(R(z)=exp(u*Pi *I),z=1.42*I))\nend proc:\nu0 := bisect('real_part'(u),u=0.44..0.47); \nnewton(R(z)=exp(u0*Pi*I),z=1.42*I);``;\nreal_part := proc(u)\n Re( newton(R(z)=exp(u*Pi*I),z=3.25*I))\nend proc:\nu0 := bisect('real_part '(u),u=1.05..1.08);\nnewton(R(z)=exp(u0*Pi*I),z=3.25*I);\nDigits := 10 :" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"0M(z]W*)eX!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"0wQvI)>K9!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"0mDU*\\ vh5!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^#$\"04!)>`4VD$!#9" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " } {TEXT 260 73 "stability region intersects the nonegative imaginary axi s in the interval" }{TEXT -1 2 " " }{XPPEDIT 18 0 "[1.4322,3.2543]" " 6#7$-%&FloatG6$\"&AV\"!\"%-F%6$\"&VD$F(" }{TEXT -1 18 " (approximatel y)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 34 "#----------------- ----------------" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {PARA 0 "" 0 "" {TEXT -1 58 "#======================================== =================" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "#======================" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "Abreviated calculations " }}{PARA 257 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 28 "Set up order conditions etc." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 968 "SO6_8 := SimpleOr derConditions(6,8,'expanded'):\ncdns := [seq(SO6_8[i],i=[1,2,4,8,16,29 ,32]),add(b[i]*c[i]^6,i=2..8)=1/7]:\nSO_eqs := [op(RowSumConditions(8, 'expanded')),op(StageOrderConditions(2,8,'expanded')),\n op( StageOrderConditions(3,8,'expanded'))]:\nsimp_eqs := [add(b[i]*a[i,1], i=2..8)=b[1],seq(add(b[i]*a[i,j],i=j+1..8)=b[j]*(1-c[j]),j=[5,6,7]),\n add(b[i]*c[i]*a[i,2],i=3..8)=0,add(b[i]*c[i]*add(a[i,j]* a[j,2],j=3..i-1),i=3..8)=0,\n add(b[i]*c[i]^2*a[i,2],i= 3..8)=0]:\ncdns := [op(cdns),op(simp_eqs),op(SO_eqs)]:\n##`SO5_9*` := \+ subs(b=`b*`,SimpleOrderConditions(5,9,'expanded')):\n##`ord_cdns*` := \+ [seq(`SO5_9*`[i],i=[1,2,4,5,7,8,12,13,15,16])]:\n##`cdns*` := [op(`ord _cdns*`),add(a[9,i],i=1..8)=c[9],add(a[9,j]*c[j],j=2..8)=1/2*c[9]^2]: \n\nerrterms6_8 := PrincipalErrorTerms(6,8,'expanded'):\n##`errterms6_ 9*` :=subs(b=`b*`,PrincipalErrorTerms(6,9,'expanded')):\n##`errterms5_ 9*` := subs(b=`b*`,PrincipalErrorTerms(5,9,'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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2526 " calc_RKcoeffs := proc()\n local cdns1,eqns,mn,`eqns*`,sm,ct,Rz,stb6, stb5,nmB,snmB,dnB,sdnB,nmC,snmC,B_7,C_7;\n global e1,e2,e3,e4,e5,e6; \n\n if nargs>0 and args[1]=2 then\n e1 := \{c[3]=c_3,c[4]=c_4 ,c[5]=c_5,c[6]=c_6,c[7]=c_7,c[8]=1,b[2]=0\};\n cdns1 := [op(cdns ),b[8]*a[8,7]*a[7,6]*a[6,5]*a[5,4]*a[4,3]*a[3,2]*c[2]=Gam8/8!,\n \+ b[7]*a[7,6]*a[6,5]*a[5,4]*a[4,3]*a[3,2]*c[2]+b[8]*(a[8,6]*a[6,5]*a[5 ,4]*a[4,3]*a[3,2]*c[2]+\n a[8,7]*(a[7,5]*a[5,4]*a[4,3]*a[ 3,2]*c[2]+a[7,6]*(a[6,4]*a[4,3]*a[3,2]*c[2]+\n a[6,5]*(a[ 5,3]*a[3,2]*c[2]+a[5,4]*(a[4,2]*c[2]+a[4,3]*c[3])))))=Gam7/7!];\n \+ eqns := subs(e1,cdns1):\n e2 := solve(\{op(eqns)\}):\n e5 : = `union`(e1,e2):\n sm := evalf[14](add(subs(e5,errterms6_8[i])^2 ,i=1.. nops(errterms6_8))):\n elif nargs>0 and args[1]=1 then\n \+ e1 := \{c[3]=c_3,c[4]=c_4,c[5]=c_5,c[6]=c_6,c[7]=c_7,c[8]=1,b[2]=0,a[ 7,6]=a7_6\};\n cdns1 := [op(cdns),b[8]*a[8,7]*a[7,6]*a[6,5]*a[5,4 ]*a[4,3]*a[3,2]*c[2]-Gam8/8!];\n eqns := subs(e1,cdns1):\n e 2 := solve(\{op(eqns)\}):\n e5 := `union`(e1,e2):\n sm := ev alf[14](add(subs(e5,errterms6_8[i])^2,i=1.. nops(errterms6_8))):\n e lif nargs>0 and args[1]='min_err_norm' then\n e1 := \{c[3]=c_3,c[ 4]=c_4,c[5]=c_5,c[6]=c_6,c[7]=c_7,c[8]=1,b[2]=0\};\n eqns := s ubs(e1,cdns):\n e2 := solve(\{op(eqns)\},indets(eqns) minus \{a[6 ,5],a[7,6]\}):\n e3 := `union`(e1,e2):\n sm := add(simplify( subs(e3,errterms6_8[i]))^2,i=1.. nops(errterms6_8)):\n sm := simp lify(sm);\n mn := minimize(evalf(sm),a[7,6]=-1..1,a[6,5]=-10..10, 'location');\n e4 := map(convert,op(1,op(op(2,[mn]))),rational,4) ;\n e5 := `union`(e4,simplify(subs(e4,e3)));\n sm := eval(sm ,e4);\n else\n e1 := \{c[3]=c_3,c[4]=c_4,c[5]=c_5,c[6]=c_6,c[7] =c_7,a[6,5]=a6_5,a[7,6]=a7_6,c[8]=1,b[2]=0\};\n eqns := subs(e1,c dns):\n e2 := solve(\{op(eqns)\}):\n e5 := `union`(e1,e2):\n sm := add(evalf[14](subs(e5,errterms6_8[i]))^2,i=1.. nops(errter ms6_8)):\n end if;\n Rz := subs(e5,StabilityFunction(6,8,'expanded '));\n stb6 := evalf[14](max(fsolve(Rz=1,z=-10..-1e-7),fsolve(Rz=-1, z=-10..-1e-7)));\n stb6 := evalf[8](stb6);\n print(`nodes:`,c[2]=s ubs(e5,c[2]),c[3]=c_3,c[4]=c_4,c[5]=c_5,c[6]=c_6,c[7]=c_7);\n print( `weights:`,seq(b[i]=evalf[6](subs(e5,b[i])),i=[1,$3..8]));\n print(i nfinity*`-norm of linking coeffs`=evalf(max(seq(seq(subs(e5,abs(a[i,j] )),j=1..i-1),i=2..8))));\n print(`2-norm of principal error` = evalf [10](sqrt(sm)));\n print(`stability interval` = [stb6,0]);\nend proc :" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 13 "pr in_err_norm" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 789 "prin_err_norm := proc(c3,c4,c5,c6) \n local e1,cdns1,eqns,e2,sm;\n global e3;\n \n e1 := \{c[3]=c onvert(c3,rational,10),c[4]=convert(c4,rational,10),c[5]=convert(c5,ra tional,10),\n c[6]=convert(c6,rational,10),c[7]=c_7,c[8]=1,b[2]=0\}; \n cdns1 := [op(cdns),b[8]*a[8,7]*a[7,6]*a[6,5]*a[5,4]*a[4,3]*a[3,2 ]*c[2]=Gam8/8!,\n b[7]*a[7,6]*a[6,5]*a[5,4]*a[4,3]*a[3,2]*c[2]+ b[8]*(a[8,6]*a[6,5]*a[5,4]*a[4,3]*a[3,2]*c[2]+\n a[8,7]*( a[7,5]*a[5,4]*a[4,3]*a[3,2]*c[2]+a[7,6]*(a[6,4]*a[4,3]*a[3,2]*c[2]+\n \+ a[6,5]*(a[5,3]*a[3,2]*c[2]+a[5,4]*(a[4,2]*c[2]+a[4,3]*c[3 ])))))=Gam7/7!];\n eqns := subs(e1,cdns1):\n e2 := solve(\{op(eqns )\}):\n e3 := `union`(e1,e2):\n sm := evalf[14](add(subs(e3,errter ms6_8[i])^2,i=1.. nops(errterms6_8))):\n evalf[10](sqrt(sm));\nend p roc:" }}}{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 35 "m inimizing the principal error norm" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 97 "Sample calculations that attempt to minim ize the principal error norm with respect to the nodes " }{XPPEDIT 18 0 "c[3]" "6#&%\"cG6#\"\"$" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[4]" "6#&%\"cG6#\"\"%" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "c[5]" "6#&%\"cG6# \"\"&" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "c[6]" "6#&%\"cG6#\"\"'" } {TEXT -1 29 " by a simple cycling method." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 666 "Digits := 10:\nc_3 := .1929713218: c_4 := .4352850 138: c_5 := .5416481301: c_6 := .5931884373: c_7 := 67/75:\nGam7 := 10 0/103: Gam8 := 27/29:\nfor ii to 20 do \n c_6 := findmin('prin_err_n orm'(c_3,c_4,c_5,c6),c6=0.56..0.62,convergence=value)[1];\n c_5 := f indmin('prin_err_norm'(c_3,c_4,c5,c_6),c5=0.49..0.59,convergence=value )[1];\n c_4 := findmin('prin_err_norm'(c_3,c4,c_5,c_6),c4=0.37..0.47 ,convergence=value)[1];\n c_3 := findmin('prin_err_norm'(c3,c_4,c_5, c_6),c3=0.15..0.23,convergence=value)[1];\n print(c[3]=c_3,c[4]=c_4, c[5]=c_5,c[6]=c_6);\n sm := evalf[14](add(subs(e3,errterms6_8[i])^2, i=1.. nops(errterms6_8)));\n print(evalf[10](sqrt(sm)));\nend do:" } }{PARA 11 "" 1 "" {XPPMATH 20 "6&/&%\"cG6#\"\"$$\"+mD]J>!#5/&F%6#\"\"% $\"+f]khVF*/&F%6#\"\"&$\"+Y\\kBaF*/&F%6#\"\"'$\"+7*o<#fF*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+x[<9#)!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6&/&%\"cG6#\"\"$$\"+%fuJ$>!#5/&F%6#\"\"%$\"+St))pVF*/&F%6#\"\"&$\"+duD IaF*/&F%6#\"\"'$\"+odZ7fF*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+*f5 \"4#)!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&/&%\"cG6#\"\"$$\"+\"3SZ$>! #5/&F%6#\"\"%$\"+(fFwP%F*/&F%6#\"\"&$\"+V)ojV&F*/&F%6#\"\"'$\"+8!=R!fF *" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+G4_/#)!#9" }}{PARA 256 "" 0 " " {TEXT -1 1 ":" }}{PARA 256 "" 0 "" {TEXT -1 1 ":" }}{PARA 11 "" 1 " " {XPPMATH 20 "6&/&%\"cG6#\"\"$$\"+nCm]>!#5/&F%6#\"\"%$\"+#*)HzX%F*/&F %6#\"\"&$\"+B[\\$\\&F*/&F%6#\"\"'$\"+T0zDeF*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+(G6s;)!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&/&%\"c G6#\"\"$$\"+&3\"H^>!#5/&F%6#\"\"%$\"+=7 " 0 "" {MPLTEXT 1 0 650 "c_3 := .1951291085: c_4 := \+ .4461171218: c_5 := .5495549073: c_6 := .5823063164: c_7 := 67/75:\nGa m7 := 100/103: Gam8 := 27/29:\nfor ii to 20 do \n c_6 := findmin('pr in_err_norm'(c_3,c_4,c_5,c6),c6=0.56..0.62,convergence=value)[1];\n \+ c_5 := findmin('prin_err_norm'(c_3,c_4,c5,c_6),c5=0.5..0.6,convergence =value)[1];\n c_4 := findmin('prin_err_norm'(c_3,c4,c_5,c_6),c4=0.39 ..0.49,convergence=value)[1];\n c_3 := findmin('prin_err_norm'(c3,c_ 4,c_5,c_6),c3=0.15..0.23,convergence=value)[1];\n print(c[3]=c_3,c[4 ]=c_4,c[5]=c_5,c[6]=c_6);\n sm := evalf[14](add(subs(e3,errterms6_8[ i])^2,i=1.. nops(errterms6_8)));\n print(evalf[10](sqrt(sm)));\nend \+ do:" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&/&%\"cG6#\"\"$$ \"+A;*=&>!#5/&F%6#\"\"%$\"+VWFkWF*/&F%6#\"\"&$\"+\\$*\\(\\&F*/&F%6#\" \"'$\"+]l[?eF*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+tJ8l\")!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6&/&%\"cG6#\"\"$$\"+7bY_>!#5/&F%6#\"\"%$ \"+DcCnWF*/&F%6#\"\"&$\"+dWN*\\&F*/&F%6#\"\"'$\"+?%\\!=eF*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+cu?k\")!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&/&%\"cG6#\"\"$$\"+VV,`>!#5/&F%6#\"\"%$\"+)R#4qWF*/&F%6#\"\"&$\"+ +!>6]&F*/&F%6#\"\"'$\"+a3u:eF*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+ #=\\L;)!#9" }}{PARA 256 "" 0 "" {TEXT -1 1 ":" }}{PARA 256 "" 0 "" {TEXT -1 1 ":" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&/&%\"cG6#\"\"$$\"+Fh) *e>!#5/&F%6#\"\"%$\"+oTS,XF*/&F%6#\"\"&$\"+$z&o>bF*/&F%6#\"\"'$\"+<:1# z&F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+vv#e:)!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&/&%\"cG6#\"\"$$\"+M2Ff>!#5/&F%6#\"\"%$\"+&39H]%F*/ &F%6#\"\"&$\"+9\\a?bF*/&F%6#\"\"'$\"+,;+\"z&F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+_obb\")!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&/&%\" cG6#\"\"$$\"+]Uaf>!#5/&F%6#\"\"%$\"+BoO/XF*/&F%6#\"\"&$\"+m`O@bF*/&F%6 #\"\"'$\"+Q&*)**y&F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+DYIb\")!#9 " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 651 "c_3 := .1959544250: c_4 := .4504366823: c_5 := .5521 365366: c_6 := .5789989538: c_7 := 67/75:\nGam7 := 100/103: Gam8 := 27 /29:\nfor ii to 200 do \n c_6 := findmin('prin_err_norm'(c_3,c_4,c_5 ,c6),c6=0.56..0.62,convergence=value)[1];\n c_5 := findmin('prin_err _norm'(c_3,c_4,c5,c_6),c5=0.5..0.6,convergence=value)[1];\n c_4 := f indmin('prin_err_norm'(c_3,c4,c_5,c_6),c4=0.39..0.49,convergence=value )[1];\n c_3 := findmin('prin_err_norm'(c3,c_4,c_5,c_6),c3=0.15..0.23 ,convergence=value)[1];\n print(c[3]=c_3,c[4]=c_4,c[5]=c_5,c[6]=c_6) ;\n sm := evalf[14](add(subs(e3,errterms6_8[i])^2,i=1.. nops(errterm s6_8)));\n print(evalf[10](sqrt(sm)));\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&/&%\"cG6#\"\"$$\"+/z!)f>!#5/&F%6#\"\"%$\"+Tkw0XF*/&F%6# \"\"&$\"+ap:AbF*/&F%6#\"\"'$\"+];-*y&F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+8$p]:)!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&/&%\"cG6#\"\"$$ \"+a<1g>!#5/&F%6#\"\"%$\"+'[:r]%F*/&F%6#\"\"&$\"+M)>H_&F*/&F%6#\"\"'$ \"+4v3)y&F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+%e\\[:)!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6&/&%\"cG6#\"\"$$\"+DiIg>!#5/&F%6#\"\"%$ \"+OhT3XF*/&F%6#\"\"&$\"+xukBbF*/&F%6#\"\"'$\"+i`>(y&F*" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"+]Wka\")!#9" }}{PARA 256 "" 0 "" {TEXT -1 1 " :" }}{PARA 256 "" 0 "" {TEXT -1 1 ":" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6&/&%\"cG6#\"\"$$\"+`DHn>!#5/&F%6#\"\"%$\"+Dz;YXF*/&F%6#\"\"&$\"+hi3Wb F*/&F%6#\"\"'$\"+=ezjdF*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+H)f;:) !#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&/&%\"cG6#\"\"$$\"+EIHn>!#5/&F%6 #\"\"%$\"+O2!#5/&F%6#\"\"%$\"+OK " 0 "" {MPLTEXT 1 0 108 "[c[3] = .1967293026, c[4] = .4546170736, c[5] = .5544086935, c[6] = .5763795234];\nmap(convert,%,rational,4); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&/&%\"cG6#\"\"$$\"+EIHn>!#5/&F&6# \"\"%$\"+O2 " 0 "" {MPLTEXT 1 0 108 "[c[3] = .1967293477, c[4] = .4546173236, c[5] = .5544089952, c[6] = .5763793535];\nmap(con vert,%,rational,4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&/&%\"cG6#\"\" $$\"+xMHn>!#5/&F&6#\"\"%$\"+OK " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "#---------------------------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Huta scheme A o btained by specifying two linking coefficients" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "c_3 := 1/6: c_4 := 1/3: c_5 := 1/2: c_6 := 2/3: c_7 := 5/6:\na6_5 := 1/9: a7_6 := 1/16:\ncalc_RKcoeffs();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%'nodes:G/&%\"cG6#\"\"##\"\"\"\"\"*/&F&6#\" \"$#F*\"\"'/&F&6#\"\"%#F*F//&F&6#\"\"&#F*F(/&F&6#F1#F(F//&F&6#\"\"(#F: F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%)weights:G/&%\"bG6#\"\"\"$\"'&4 )[!\"(/&F&6#\"\"$$\"'VrD!\"'/&F&6#\"\"%$\"'H9KF+/&F&6#\"\"&$\"'5QKF2/& F&6#\"\"'F7/&F&6#\"\"(F0/&F&6#\"\")F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-norm~of~linking~coeffsGF&$\"$4\"\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%:2-norm~of~principal~errorG$\"+4SXx* )!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%3stability~intervalG7$$!)W-S Q!\"(\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "Huta scheme A obtained by specifying that the coefficients of \+ " }{XPPEDIT 18 0 "z^7" "6#*$%\"zG\"\"(" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "z^8" "6#*$%\"zG\"\")" }{TEXT -1 35 " in the stability \+ polynomial are " }{XPPEDIT 18 0 "9/(8*`.`*7!)" "6#*&\"\"*\"\"\"*(\"\" )F%%\".GF%-%*factorialG6#\"\"(F%!\"\"" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "1/(12*`.`*8!)" "6#*&\"\"\"F$*(\"#7F$%\".GF$-%*factorial G6#\"\")F$!\"\"" }{TEXT -1 15 " respectively." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "c_3 := 1/6: c_4 := 1/3: c_5 := 1/2: c_6 := 2/3: c_7 := 5/6:\nGam7 := 9/8: Gam8 := 1/12:\ncalc_RKcoeffs(2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%'nodes:G/&%\"cG6#\"\"##\"\"\"\"\"*/&F&6#\" \"$#F*\"\"'/&F&6#\"\"%#F*F//&F&6#\"\"&#F*F(/&F&6#F1#F(F//&F&6#\"\"(#F: F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%)weights:G/&%\"bG6#\"\"\"$\"'&4 )[!\"(/&F&6#\"\"$$\"'VrD!\"'/&F&6#\"\"%$\"'H9KF+/&F&6#\"\"&$\"'5QKF2/& F&6#\"\"'F7/&F&6#\"\"(F0/&F&6#\"\")F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-norm~of~linking~coeffsGF&$\"$4\"\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%:2-norm~of~principal~errorG$\"+5SXx* )!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%3stability~intervalG7$$!)W-S Q!\"(\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Huta scheme B" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "c_3 := 1/6: c_4 := 1/3: c_5 := 1/2: c_6 := 2/3: c_7 := 5/6:\na6_5 := 8: a 7_6 := 107/2536:\ncalc_RKcoeffs();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6) %'nodes:G/&%\"cG6#\"\"##\"\"\"\"\"*/&F&6#\"\"$#F*\"\"'/&F&6#\"\"%#F*F/ /&F&6#\"\"&#F*F(/&F&6#F1#F(F//&F&6#\"\"(#F:F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%)weights:G/&%\"bG6#\"\"\"$\"'&4)[!\"(/&F&6#\"\"$$\"'Vr D!\"'/&F&6#\"\"%$\"'H9KF+/&F&6#\"\"&$\"'5QKF2/&F&6#\"\"'F7/&F&6#\"\"(F 0/&F&6#\"\")F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\" %8-norm~of~linking~coeffsGF&$\"++++]X!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%:2-norm~of~principal~errorG$\"+-_&>^\"!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%3stability~intervalG7$$!)p)G/%!\"(\"\"!" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "compani on to Huta scheme B" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "c_3 \+ := 1/6: c_4 := 1/3: c_5 := 1/2: c_6 := 2/3: c_7 := 5/6:\na6_5 := 8: a7 _6 := 33/1088:\ncalc_RKcoeffs();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%' nodes:G/&%\"cG6#\"\"##\"\"\"\"\"*/&F&6#\"\"$#F*\"\"'/&F&6#\"\"%#F*F//& F&6#\"\"&#F*F(/&F&6#F1#F(F//&F&6#\"\"(#F:F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%)weights:G/&%\"bG6#\"\"\"$\"'&4)[!\"(/&F&6#\"\"$$\"'Vr D!\"'/&F&6#\"\"%$\"'H9KF+/&F&6#\"\"&$\"'5QKF2/&F&6#\"\"'F7/&F&6#\"\"(F 0/&F&6#\"\")F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\" %8-norm~of~linking~coeffsGF&$\"+%e&>9E!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%:2-norm~of~principal~errorG$\"+Yg?f`!#8" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/%3stability~intervalG7$$!)?)3-&!\"(\"\"!" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "a scheme \+ with the same nodes as Huta scheme B and having a large imaginary axis inclusion" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "c_3 := 1/6: c_ 4 := 1/3: c_5 := 1/2: c_6 := 2/3: c_7 := 5/6:\nGam7 := 1: Gam8 := 1:\n calc_RKcoeffs(2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%'nodes:G/&%\"cG6 #\"\"##\"\"\"\"\"*/&F&6#\"\"$#F*\"\"'/&F&6#\"\"%#F*F//&F&6#\"\"&#F*F(/ &F&6#F1#F(F//&F&6#\"\"(#F:F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%)weig hts:G/&%\"bG6#\"\"\"$\"'&4)[!\"(/&F&6#\"\"$$\"'VrD!\"'/&F&6#\"\"%$\"'H 9KF+/&F&6#\"\"&$\"'5QKF2/&F&6#\"\"'F7/&F&6#\"\"(F0/&F&6#\"\")F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-norm~of~linking ~coeffsGF&$\"+X55LB!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%:2-norm~o f~principal~errorG$\"+a7CZH!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%3s tability~intervalG7$$!)si8V!\"(\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 71 "a scheme with the same nodes and weig hts as the Huta schemes but with " }{XPPEDIT 18 0 "a[6,5]" "6#&%\"aG6 $\"\"'\"\"&" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a[7,6]" "6#&%\"aG6$ \"\"(\"\"'" }{TEXT -1 45 " chosen to minimize the principal error nor m" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "c_3 := 1/6: c_4 := 1/3: c_5 := 1/2: c_6 := 2/3: c_7 := 5/6:\ncalc_RKcoeffs(min_err_norm);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6)%'nodes:G/&%\"cG6#\"\"##\"\"\"\"\"*/&F &6#\"\"$#F*\"\"'/&F&6#\"\"%#F*F//&F&6#\"\"&#F*F(/&F&6#F1#F(F//&F&6#\" \"(#F:F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%)weights:G/&%\"bG6#\"\"\" $\"'&4)[!\"(/&F&6#\"\"$$\"'VrD!\"'/&F&6#\"\"%$\"'H9KF+/&F&6#\"\"&$\"'5 QKF2/&F&6#\"\"'F7/&F&6#\"\"(F0/&F&6#\"\")F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-norm~of~linking~coeffsGF&$\"+K2< t?!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%:2-norm~of~principal~error G$\"+h+3x%*!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%3stability~interva lG7$$!)cG3R!\"(\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 " a stage order 3 scheme with small principal error nor m" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "c_3 := 12/61: c_4 := 5 /11: c_5 := 51/92: c_6 := 15/26: c_7 := 67/75:\nGam7 := 100/103: Gam8 \+ := 27/29:\ncalc_RKcoeffs(2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%'node s:G/&%\"cG6#\"\"##\"\")\"#h/&F&6#\"\"$#\"#7F+/&F&6#\"\"%#\"\"&\"#6/&F& 6#F7#\"#^\"##*/&F&6#\"\"'#\"#:\"#E/&F&6#\"\"(#\"#n\"#v" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%)weights:G/&%\"bG6#\"\"\"$\"'Qve!\"(/&F&6#\"\"$$ \"'vrH!\"'/&F&6#\"\"%$\"'R)[\"F2/&F&6#\"\"&$!';:)!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%3stability~interva lG7$$!)36PV!\"(\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "another stage order 3 scheme with small principal erro r norm" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "c_3 := 8/41: c_4 \+ := 29/65: c_5 := 11/20: c_6 := 46/79: c_7 := 67/75:\nGam7 := 100/103: \+ Gam8 := 27/29:\ncalc_RKcoeffs(2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)% 'nodes:G/&%\"cG6#\"\"##\"#;\"$B\"/&F&6#\"\"$#\"\")\"#T/&F&6#\"\"%#\"#H \"#l/&F&6#\"\"&#\"#6\"#?/&F&6#\"\"'#\"#Y\"#z/&F&6#\"\"(#\"#n\"#v" }} {PARA 11 "" 1 "" {XPPMATH 20 "6*%)weights:G/&%\"bG6#\"\"\"$\"'LTe!\"(/ &F&6#\"\"$$\"'!f$H!\"'/&F&6#\"\"%$\"'4^9F2/&F&6#\"\"&$!'n^LF2/&F&6#\" \"'$\"'uagF2/&F&6#\"\"($\"'VR@F2/&F&6#\"\")$\"'#Q'=F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-norm~of~linking~coeffsGF&$ \"+SPRVT!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%:2-norm~of~principal ~errorG$\"+AJ1m\")!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%3stability~ intervalG7$$!)36PV!\"(\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "############################" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "c_3 := 1/ 6: c_4 := 1/3: c_5 := 1/2: c_6 := 2/3: c_7 := 5/6:\nGam7 := 1: Gam8 := 9/10:\ncalc_RKcoeffs(2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%'nodes:G /&%\"cG6#\"\"##\"\"\"\"\"*/&F&6#\"\"$#F*\"\"'/&F&6#\"\"%#F*F//&F&6#\" \"&#F*F(/&F&6#F1#F(F//&F&6#\"\"(#F:F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%)weights:G/&%\"bG6#\"\"\"$\"'&4)[!\"(/&F&6#\"\"$$\"'VrD!\"'/&F&6# \"\"%$\"'H9KF+/&F&6#\"\"&$\"'5QKF2/&F&6#\"\"'F7/&F&6#\"\"(F0/&F&6#\"\" )F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-norm~of~ linking~coeffsGF&$\"+21z?B!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%:2 -norm~of~principal~errorG$\"+j\")3%)H!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%3stability~intervalG7$$!)Mp!\\%!\"(\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "c_3 := 1/6: c_4 := 1/3: c_5 := 1/2: c_6 := 2/3: c_7 := 5/6:\nGam7 := 11/10: Gam8 \+ := 9/10:\ncalc_RKcoeffs(2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%'nodes :G/&%\"cG6#\"\"##\"\"\"\"\"*/&F&6#\"\"$#F*\"\"'/&F&6#\"\"%#F*F//&F&6# \"\"&#F*F(/&F&6#F1#F(F//&F&6#\"\"(#F:F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%)weights:G/&%\"bG6#\"\"\"$\"'&4)[!\"(/&F&6#\"\"$$\"'VrD!\"'/&F& 6#\"\"%$\"'H9KF+/&F&6#\"\"&$\"'5QKF2/&F&6#\"\"'F7/&F&6#\"\"(F0/&F&6#\" \")F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infinityG\"\"\"%8-norm~o f~linking~coeffsGF&$\"+YV " 0 "" {MPLTEXT 1 0 115 "c_3 := 1/6: c_4 := 1/3: c_5 := 1/2: c_6 := 2/3: c_7 := 5/6:\nGam7 := 9709 /10000: Gam8 := 931/1000:\ncalc_RKcoeffs(2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)%'nodes:G/&%\"cG6#\"\"##\"\"\"\"\"*/&F&6#\"\"$#F*\"\"'/ &F&6#\"\"%#F*F//&F&6#\"\"&#F*F(/&F&6#F1#F(F//&F&6#\"\"(#F:F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%)weights:G/&%\"bG6#\"\"\"$\"'&4)[!\"(/&F&6# \"\"$$\"'VrD!\"'/&F&6#\"\"%$\"'H9KF+/&F&6#\"\"&$\"'5QKF2/&F&6#\"\"'F7/ &F&6#\"\"(F0/&F&6#\"\")F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%)infi nityG\"\"\"%8-norm~of~linking~coeffsGF&$\"+ln2$3#!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%:2-norm~of~principal~errorG$\"+)4$3&3\"!#8" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%3stability~intervalG7$$!)SCPV!\"(\" \"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 23 "#======================" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "#=================================" }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "Test-bed procedures for the examp les" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 7 "RK6step" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2662 "rk6step := proc(x_rk6step::realco ns)\n local c2,c3,c4,c5,c6,c7,a21,a31,a32,a41,a42,a43,a51,a52,a53,a5 4,\n a61,a62,a63,a64,a65,a71,a72,a73,a74,a75,a76,\n f1,f2,f3,f4,f5 ,f6,f7,b1,b2,b3,b4,b5,b6,b7,\n xk,yk,t,jF,jM,jS,n,h,data,fn,xx,ys,sa veDigits;\n options `Copyright 2004 by Peter Stone`;\n \n data : = SOLN_;\n\n saveDigits := Digits;\n Digits := max(trunc(evalhf(Di gits)),Digits+5);\n\n # procedure to evaluate the slope field\n fn := proc(X_,Y_)\n local val; \n val := traperror(evalf(FXY_) );\n if val=lasterror or not type(val,numeric) then\n err or \"evaluation of slope field failed at %1\",evalf([X_,Y_],saveDigits );\n end if;\n val;\n end proc;\n\n xx := evalf(x_rk6 step);\n n := nops(data);\n\n if (data[1,1]data [n,1] or xxdata[1,1])) then\n error \"independent variable is outsi de the interpolation interval: %1\",evalf(data[1,1])..evalf(data[n,1]) ;\n end if;\n\n c2 := c2_; c3 := c3_; c4 := c4_; c5 := c5_; c6 := \+ c6_; c7 := c7_; \n a21 := c2; a31 := a31_; a32 := a32_; a41 := a41_; a42 := a42_; a43 := a43_;\n a51 := a51_; a52 := a52_; a53 := a53_; \+ a54 := a54_;\n a61 := a61_; a62 := a62_; a63 := a63_; a64 := a64_; a 65 := a65_;\n a71 := a71_; a72 := a72_; a73 := a73_; a74 := a74_; a7 5 := a75_; a76 := a76_;\n b1 := b1_; b2 := b2_; b3 := b3_; b4 := b4_ ; b5 := b5_; b6 := b6_; b7 := b7_;\n # Perform a binary search for t he interval containing x.\n n := nops(data);\n jF := 0;\n jS := \+ n+1;\n\n if data[1,1]1 do\n \+ jM := trunc((jF+jS)/2);\n if xx>=data[jM,1] then jF := jM els e jS := jM end if;\n end do;\n if jM = n then jF := n-1; jS \+ := n end if;\n else\n while jS-jF> 1 do\n jM := trunc((j F+jS)/2);\n if xx<=data[jM,1] then jF := jM else jS := jM end i f;\n end do;\n if jM = n then jF := n-1; jS := n end if;\n \+ end if;\n \n # Get the data needed from the list.\n xk := data[j F,1];\n yk := data[jF,2];\n\n # Do one step with step-size ..\n \+ h := xx-xk;\n f1 := fn(xk,yk);\n t := a21*f1;\n f2 := fn(xk + c2 *h,yk + t*h);\n t := a31*f1 + a32*f2;\n f3 := fn(xk + c3*h,yk + t* h);\n t := a41*f1 + a42*f2 + a43*f3;\n f4 := fn(xk + c4*h,yk + t*h );\n t := a51*f1 + a52*f2 + a53*f3 + a54*f4;\n f5 := fn(xk + c5*h, yk + t*h);\n t := a61*f1 + a62*f2 + a63*f3 + a64*f4 + a65*f5;\n f6 := fn(xk + c6*h,yk + t*h);\n t := a71*f1 + a72*f2 + a73*f3 + a74*f4 + a75*f5 + a76*f6;\n f7 := fn(xk + c7*h,yk + t*h);\n \n ys \+ := yk + (b1*f1 + b2*f2 + b3*f3 + b4*f4 + b5*f5 + b6*f6 + b7*f7)*h;\n\n evalf[saveDigits](ys);\nend proc: # of rk7_6step" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 23 "RK6_1 Huta's scheme A " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2797 "RK6_1 := proc(fxy,x,y,xx,yy,h,stps,bb)\n local c2,c3,c4,c5,c6,c7,a2 1,a31,a32,a41,a42,a43,\n a51,a52,a53,a54,a61,a62,a63,a64,a65,a71,a72 ,a73,a74,a75,a76,\n b1,b2,b3,b4,b5,b6,b7,f1,f2,f3,f4,f5,f6,f7,t,k,fn ,xk,yk,soln,\n eqns,A,saveDigits;\n \n saveDigits := Digits;\n \+ Digits := max(trunc(evalhf(Digits)),Digits+5);\n\n fn := unapply(fx y,x,y);\n \n A := matrix([[1/9,1/9,0,0,0,0,0,0,0],\n [1/ 6,1/24,1/8,0,0,0,0,0,0],\n [1/3,1/6,-1/2,2/3,0,0,0,0,0],\n \+ [1/2,-5/8,27/8,-3,3/4,0,0,0,0],\n [2/3,221/9,-109,289/ 3,-34/3,1/9,0,0,0],\n [5/6,-61/16,113/8,-59/6,-11/8,5/3,1/16, 0,0],\n [1,358/41,-2079/82,501/41,417/41,-227/41,-9/82,36/41, 0],\n [0,41/840,0,9/35,9/280,34/105,9/280,9/35,41/840]]);\n \+ \n c2 := evalf(A[1,1]);\n c3 := evalf(A[2,1]);\n c4 := evalf(A[3 ,1]);\n c5 := evalf(A[4,1]);\n c6 := evalf(A[5,1]);\n c7 := eval f(A[6,1]);\n a21 := c2;\n a31 := evalf(A[2,2]);\n a32 := evalf(A [2,3]);\n a41 := evalf(A[3,2]);\n a42 := evalf(A[3,3]);\n a43 := evalf(A[3,4]);\n a51 := evalf(A[4,2]);\n a52 := evalf(A[4,3]);\n \+ a53 := evalf(A[4,4]);\n a54 := evalf(A[4,5]);\n a61 := evalf(A[5 ,2]);\n a62 := evalf(A[5,3]);\n a63 := evalf(A[5,4]);\n a64 := e valf(A[5,5]);\n a65 := evalf(A[5,6]);\n a71 := evalf(A[6,2]);\n \+ a72 := evalf(A[6,3]);\n a73 := evalf(A[6,4]);\n a74 := evalf(A[6,5 ]);\n a75 := evalf(A[6,6]);\n a76 := evalf(A[6,7]);\n b1 := eval f(A[7,2]);\n b2 := evalf(A[7,3]);\n b3 := evalf(A[7,4]);\n b4 := evalf(A[7,5]);\n b5 := evalf(A[7,6]);\n b6 := evalf(A[7,7]);\n \+ b7 := evalf(A[7,8]);\n\n xk := evalf(xx);\n yk := evalf(yy);\n s oln := [xk,yk]; \n for k from 1 to stps do\n f1 := fn(xk,yk);\n t := a21*f1;\n f2 := fn(xk + c2*h,yk + t*h);\n t := a3 1*f1 + a32*f2;\n f3 := fn(xk + c3*h,yk + t*h);\n t := a41*f1 + a42*f2 + a43*f3;\n f4 := fn(xk + c4*h,yk + t*h);\n t := a 51*f1 + a52*f2 + a53*f3 + a54*f4;\n f5 := fn(xk + c5*h,yk + t*h); \n t := a61*f1 + a62*f2 + a63*f3 + a64*f4 + a65*f5;\n f6 := \+ fn(xk + c6*h,yk + t*h);\n t := a71*f1 + a72*f2 + a73*f3 + a74*f4 \+ + a75*f5 + a76*f6;\n f7 := fn(xk + c7*h,yk + t*h);\n \n \+ yk := yk + (b1*f1 + b2*f2 + b3*f3 + b4*f4 + b5*f5 + b6*f6 + b7*f7)*h ;\n xk := xk + h:\n soln := soln,[xk,yk];\n end do;\n if bb=true then\n eqns := \{SOLN_=[soln],FXY_=fxy,X_=x,Y_=y,c2_=c2, c3_=c3,\n c4_=c4,c5_=c5,c6_=c6,c7_=c7,a31_=a31,a32_=a32,a41_=a 41,\n a42_=a42,a43_=a43,a51_=a51,a52_=a52,a53_=a53,a54_=a54,\n a61_=a61,a62_=a62,a63_=a63,a64_=a64,a65_=a65,\n a71_= a71,a72_=a72,a73_=a73,a74_=a74,a75_=a75,a76_=a76,\n b1_=b1,b2_ =b2,b3_=b3,b4_=b4,b5_=b5,b6_=b6,b7_=b7\};\n return subs(eqns,eval (rk6step)); \n else \n return evalf[saveDigits]([soln]);\n \+ end if;\nend proc: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 23 "RK6_2 Huta's scheme B " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2869 "RK6_2 := proc(fxy,x,y,xx,y y,h,stps,bb)\n local c2,c3,c4,c5,c6,c7,a21,a31,a32,a41,a42,a43,\n \+ a51,a52,a53,a54,a61,a62,a63,a64,a65,a71,a72,a73,a74,a75,a76,\n b1,b2 ,b3,b4,b5,b6,b7,f1,f2,f3,f4,f5,f6,f7,t,k,fn,xk,yk,soln,\n eqns,A,sav eDigits,SQRT5;\n \n saveDigits := Digits;\n Digits := max(trunc( evalhf(Digits)),Digits+5);\n\n fn := unapply(fxy,x,y);\n \n A : = matrix([[1/9,1/9,0,0,0,0,0,0,0],\n [1/6,1/24,1/8,0,0,0,0,0, 0],\n [1/3,1/6,-1/2,2/3,0,0,0,0,0],\n [1/2,139/272,- 945/544,105/68,99/544,0,0,0,0],\n [2/3,-53/3,91/2,-52/3,-107/ 6,8,0,0,0],\n [5/6,55487/22824,-83/16,2849/1902,34601/15216,- 640/2853,107/2536,0,0],\n [1,-101195/25994,351/41,-35994/1299 7,-26109/25994,-10000/12997,-36/12997,36/41,0],\n [0,41/840,0 ,9/35,9/280,34/105,9/280,9/35,41/840]]) ;\n \n c2 := evalf(A[1,1]); \n c3 := evalf(A[2,1]);\n c4 := evalf(A[3,1]);\n c5 := evalf(A[4 ,1]);\n c6 := evalf(A[5,1]);\n c7 := evalf(A[6,1]);\n a21 := c2; \n a31 := evalf(A[2,2]);\n a32 := evalf(A[2,3]);\n a41 := evalf( A[3,2]);\n a42 := evalf(A[3,3]);\n a43 := evalf(A[3,4]);\n a51 : = evalf(A[4,2]);\n a52 := evalf(A[4,3]);\n a53 := evalf(A[4,4]);\n a54 := evalf(A[4,5]);\n a61 := evalf(A[5,2]);\n a62 := evalf(A[ 5,3]);\n a63 := evalf(A[5,4]);\n a64 := evalf(A[5,5]);\n a65 := \+ evalf(A[5,6]);\n a71 := evalf(A[6,2]);\n a72 := evalf(A[6,3]);\n \+ a73 := evalf(A[6,4]);\n a74 := evalf(A[6,5]);\n a75 := evalf(A[6, 6]);\n a76 := evalf(A[6,7]);\n b1 := evalf(A[7,2]);\n b2 := eval f(A[7,3]);\n b3 := evalf(A[7,4]);\n b4 := evalf(A[7,5]);\n b5 := evalf(A[7,6]);\n b6 := evalf(A[7,7]);\n b7 := evalf(A[7,8]);\n\n \+ xk := evalf(xx);\n yk := evalf(yy);\n soln := [xk,yk]; \n for \+ k from 1 to stps do\n f1 := fn(xk,yk);\n t := a21*f1;\n \+ f2 := fn(xk + c2*h,yk + t*h);\n t := a31*f1 + a32*f2;\n f3 \+ := fn(xk + c3*h,yk + t*h);\n t := a41*f1 + a42*f2 + a43*f3;\n \+ f4 := fn(xk + c4*h,yk + t*h);\n t := a51*f1 + a52*f2 + a53*f3 + a54*f4;\n f5 := fn(xk + c5*h,yk + t*h);\n t := a61*f1 + a62 *f2 + a63*f3 + a64*f4 + a65*f5;\n f6 := fn(xk + c6*h,yk + t*h);\n t := a71*f1 + a72*f2 + a73*f3 + a74*f4 + a75*f5 + a76*f6;\n \+ f7 := fn(xk + c7*h,yk + t*h);\n \n yk := yk + (b1*f1 + b2* f2 + b3*f3 + b4*f4 + b5*f5 + b6*f6 + b7*f7)*h;\n xk := xk + h:\n \+ soln := soln,[xk,yk];\n end do;\n if bb=true then\n eqns := \{SOLN_=[soln],FXY_=fxy,X_=x,Y_=y,c2_=c2,c3_=c3,\n c4_=c4, c5_=c5,c6_=c6,c7_=c7,a31_=a31,a32_=a32,a41_=a41,\n a42_=a42,a4 3_=a43,a51_=a51,a52_=a52,a53_=a53,a54_=a54,\n a61_=a61,a62_=a6 2,a63_=a63,a64_=a64,a65_=a65,\n a71_=a71,a72_=a72,a73_=a73,a74 _=a74,a75_=a75,a76_=a76,\n b1_=b1,b2_=b2,b3_=b3,b4_=b4,b5_=b5, b6_=b6,b7_=b7\};\n return subs(eqns,eval(rk6step)); \n else \+ \n return evalf[saveDigits]([soln]);\n end if;\nend proc: " }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "RK6_3 a companion to H uta's scheme B " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 2895 "RK6_3 := proc(fxy,x,y,xx,yy,h,stps,bb)\n l ocal c2,c3,c4,c5,c6,c7,a21,a31,a32,a41,a42,a43,\n a51,a52,a53,a54,a6 1,a62,a63,a64,a65,a71,a72,a73,a74,a75,a76,\n b1,b2,b3,b4,b5,b6,b7,f1 ,f2,f3,f4,f5,f6,f7,t,k,fn,xk,yk,soln,\n eqns,A,saveDigits;\n \n \+ saveDigits := Digits;\n Digits := max(trunc(evalhf(Digits)),Digits+5 );\n\n fn := unapply(fxy,x,y);\n\n A := matrix([[1/9,1/9,0,0,0,0,0 ,0,0],\n [1/6,1/24,1/8,0,0,0,0,0,0],\n [1/3,1/6,-1/2 ,2/3,0,0,0,0,0],\n [1/2,935/2536,-2781/2536,309/317,321/1268, 0,0,0,0],\n [2/3,-12710/951,8287/317,-40/317,-6335/317,8,0,0, 0],\n [5/6,5840285/3104064,-7019/2536,-52213/86224,1278709/51 7344,-433/2448,33/1088,0,0],\n [1,-5101675/1767592,112077/259 94,334875/441898,-973617/883796,-1421/1394,333/5576,36/41,0],\n \+ [0,41/840,0,9/35,9/280,34/105,9/280,9/35,41/840]]);\n\n c2 := eva lf(A[1,1]);\n c3 := evalf(A[2,1]);\n c4 := evalf(A[3,1]);\n c5 : = evalf(A[4,1]);\n c6 := evalf(A[5,1]);\n c7 := evalf(A[6,1]);\n \+ 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 b1 := evalf(A[7,2]);\n \+ b2 := evalf(A[7,3]);\n b3 := evalf(A[7,4]);\n b4 := evalf(A[7,5] );\n b5 := evalf(A[7,6]);\n b6 := evalf(A[7,7]);\n b7 := evalf(A [7,8]);\n\n xk := evalf(xx);\n yk := evalf(yy);\n soln := [xk,yk ]; \n for k from 1 to stps do\n f1 := fn(xk,yk);\n t := a2 1*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 + a 43*f3;\n f4 := fn(xk + c4*h,yk + t*h);\n t := a51*f1 + a52*f 2 + 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 + a7 6*f6;\n f7 := fn(xk + c7*h,yk + t*h);\n \n yk := yk + \+ (b1*f1 + b2*f2 + b3*f3 + b4*f4 + b5*f5 + b6*f6 + b7*f7)*h;\n xk : = xk + h:\n soln := soln,[xk,yk];\n end do;\n if bb=true then \n eqns := \{SOLN_=[soln],FXY_=fxy,X_=x,Y_=y,c2_=c2,c3_=c3,\n \+ c4_=c4,c5_=c5,c6_=c6,c7_=c7,a31_=a31,a32_=a32,a41_=a41,\n \+ a42_=a42,a43_=a43,a51_=a51,a52_=a52,a53_=a53,a54_=a54,\n a61_ =a61,a62_=a62,a63_=a63,a64_=a64,a65_=a65,\n a71_=a71,a72_=a72, a73_=a73,a74_=a74,a75_=a75,a76_=a76,\n b1_=b1,b2_=b2,b3_=b3,b4 _=b4,b5_=b5,b6_=b6,b7_=b7\};\n return subs(eqns,eval(rk6step));\n else\n return evalf[saveDigits]([soln]);\n end if;\nend proc : " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 80 "RK6_4 an \+ efficient scheme with the same nodes and weights as Huta's two schemes " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2896 "RK6_4 := proc(fxy,x,y,xx,yy,h,stps,bb)\n local c2 ,c3,c4,c5,c6,c7,a21,a31,a32,a41,a42,a43,\n a51,a52,a53,a54,a61,a62,a 63,a64,a65,a71,a72,a73,a74,a75,a76,\n b1,b2,b3,b4,b5,b6,b7,f1,f2,f3, f4,f5,f6,f7,t,k,fn,xk,yk,soln,\n eqns,A,saveDigits;\n \n saveDig its := Digits;\n Digits := max(trunc(evalhf(Digits)),Digits+5);\n\n \+ fn := unapply(fxy,x,y);\n\n A := matrix([[1/9,1/9,0,0,0,0,0,0,0], \n [1/6,1/24,1/8,0,0,0,0,0,0],\n [1/3,1/6,-1/2,2/3,0 ,0,0,0,0],\n [1/2,1747/9704,-2403/9704,267/1213,843/2426,0,0, 0,0],\n [2/3,-9535/47307,587/1213,4673/15769,-7104/15769,7/13 ,0,0,0],\n [5/6,-359555/1135368,4265/9704,117323/189228,-9244 9/189228,79/468,16/39,0,0],\n [1,1651907/1293058,-131355/9946 6,-627519/646529,649308/646529,85/41,-1035/533,36/41,0],\n [0 ,41/840,0,9/35,9/280,34/105,9/280,9/35,41/840]]);\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 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 := ev alf(A[5,3]);\n a63 := evalf(A[5,4]);\n a64 := evalf(A[5,5]);\n a 65 := 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 := eval f(A[6,6]);\n a76 := evalf(A[6,7]);\n b1 := evalf(A[7,2]);\n b2 : = evalf(A[7,3]);\n b3 := evalf(A[7,4]);\n b4 := evalf(A[7,5]);\n \+ b5 := evalf(A[7,6]);\n b6 := evalf(A[7,7]);\n b7 := evalf(A[7,8]) ;\n\n xk := evalf(xx);\n yk := evalf(yy);\n soln := [xk,yk]; \n \+ for k from 1 to stps do\n f1 := fn(xk,yk);\n t := a21*f1; \n f2 := fn(xk + c2*h,yk + t*h);\n t := a31*f1 + a32*f2;\n \+ f3 := fn(xk + c3*h,yk + t*h);\n t := a41*f1 + a42*f2 + a43*f3 ;\n f4 := fn(xk + c4*h,yk + t*h);\n t := a51*f1 + a52*f2 + a 53*f3 + a54*f4;\n f5 := fn(xk + c5*h,yk + t*h);\n t := a61*f 1 + a62*f2 + a63*f3 + a64*f4 + a65*f5;\n f6 := fn(xk + c6*h,yk + \+ t*h);\n t := a71*f1 + a72*f2 + a73*f3 + a74*f4 + a75*f5 + a76*f6; \n f7 := fn(xk + c7*h,yk + t*h);\n \n yk := yk + (b1*f 1 + b2*f2 + b3*f3 + b4*f4 + b5*f5 + b6*f6 + b7*f7)*h;\n xk := xk \+ + h:\n soln := soln,[xk,yk];\n end do;\n if bb=true then\n \+ eqns := \{SOLN_=[soln],FXY_=fxy,X_=x,Y_=y,c2_=c2,c3_=c3,\n \+ c4_=c4,c5_=c5,c6_=c6,c7_=c7,a31_=a31,a32_=a32,a41_=a41,\n a42_ =a42,a43_=a43,a51_=a51,a52_=a52,a53_=a53,a54_=a54,\n a61_=a61, a62_=a62,a63_=a63,a64_=a64,a65_=a65,\n a71_=a71,a72_=a72,a73_= a73,a74_=a74,a75_=a75,a76_=a76,\n b1_=b1,b2_=b2,b3_=b3,b4_=b4, b5_=b5,b6_=b6,b7_=b7\};\n return subs(eqns,eval(rk6step));\n el se\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 61 "RK6_5 a stage order \+ 3 scheme with small principal error norm" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4665 "RK6_5 := proc(fxy, x,y,xx,yy,h,stps,bb)\n local c2,c3,c4,c5,c6,c7,a21,a31,a32,a41,a42,a 43,\n a51,a52,a53,a54,a61,a62,a63,a64,a65,a71,a72,a73,a74,a75,a76,\n b1,b2,b3,b4,b5,b6,b7,f1,f2,f3,f4,f5,f6,f7,t,k,fn,xk,yk,soln,\n eq ns,A,saveDigits;\n \n saveDigits := Digits;\n Digits := max(trun c(evalhf(Digits)),Digits+5);\n\n fn := unapply(fxy,x,y);\n\n A := \+ matrix([[8/61,8/61,0,0,0,0,0,0,0],\n [12/61,3/61,9/61,0,0,0,0 ,0,0],\n [5/11,136115/383328,-163175/127776,263825/191664,0,0 ,0,0,0],\n [51/92,13026271485399/595711769169920,243277528553 19/119142353833984,\n 893431094000355/1030581360663 9616,778084111317249/3220566752074880,0,0,0,0],\n [15/26,-851 82995529498464014347399693/7004909900459502878327360312576,\n \+ 782227259416638225/2244072319191569152,\n \+ -272746090471011262880106725204535/7948306442641975339499742161731456 ,\n 463772636939896507985393214891/1859455376317587 715620654574952,\n 503890150536220957318530/1983000 1765309846198610093,0,0,0],\n [67/75,-22928016678360128018125 1392551413373945146341/\n 1922297216818279856585611 944540531553417968750,\n -24373261255896329981/41039 7455685628125000,\n 1384947604104642784811146019080 86654010742317231861/\n 157045350340868651125869860 911967647339719140625000,\n 13159972310849695242939 786306351658145503546532/\n 11641814957727303799705 0298350513445647705078125,\n -1488857601208597223792 690051882764869796820192/\n 24448815461596377205939 1070710204344775390625,\n 6823372089667582149667428 6908/11064740679780824085205078125,0,0],\n [1,107265271705456 718353000251694440587/50834312477772792379562256248860860,\n \+ -3757510716191/5740230741960,\n -4298280018 38285896192739582712833397915761611/\n 992669839076 42719853800424641670872644481800,\n -185447733273372 71450067235223504981052/26732856022073197552661912017781844625,\n \+ 95683907547345674985937655053437329632/1692820548292066 101564044986300227975,\n -10290287110810609832644/19 3430985482051515125,\n 126173827828125000/101457352 642065253,0],\n [0,75887971/1291626000,0,48970167350504861/16 4785679704694160,\n 103793987429/697356166500,-3782 1393208832/68552327940705,\n 450669502712/553751407 125,2682266748046875/12493923579891488,1819/98784]]);\n\n c2 := eval f(A[1,1]);\n c3 := evalf(A[2,1]);\n c4 := evalf(A[3,1]);\n c5 := evalf(A[4,1]);\n c6 := evalf(A[5,1]);\n c7 := evalf(A[6,1]);\n \+ a21 := c2;\n a31 := evalf(A[2,2]);\n a32 := evalf(A[2,3]);\n a41 := evalf(A[3,2]);\n a42 := evalf(A[3,3]);\n a43 := evalf(A[3,4]); \n a51 := evalf(A[4,2]);\n a52 := evalf(A[4,3]);\n a53 := evalf( A[4,4]);\n a54 := evalf(A[4,5]);\n a61 := evalf(A[5,2]);\n a62 : = evalf(A[5,3]);\n a63 := evalf(A[5,4]);\n a64 := evalf(A[5,5]);\n a65 := evalf(A[5,6]);\n a71 := evalf(A[6,2]);\n a72 := evalf(A[ 6,3]);\n a73 := evalf(A[6,4]);\n a74 := evalf(A[6,5]);\n a75 := \+ evalf(A[6,6]);\n a76 := evalf(A[6,7]);\n b1 := evalf(A[7,2]);\n \+ b2 := evalf(A[7,3]);\n b3 := evalf(A[7,4]);\n b4 := evalf(A[7,5]); \n b5 := evalf(A[7,6]);\n b6 := evalf(A[7,7]);\n b7 := evalf(A[7 ,8]);\n\n xk := evalf(xx);\n yk := evalf(yy);\n soln := [xk,yk]; \n for k from 1 to stps do\n f1 := fn(xk,yk);\n t := a21* f1;\n f2 := fn(xk + c2*h,yk + t*h);\n t := a31*f1 + a32*f2; \n f3 := fn(xk + c3*h,yk + t*h);\n t := a41*f1 + a42*f2 + a4 3*f3;\n f4 := fn(xk + c4*h,yk + t*h);\n t := a51*f1 + a52*f2 + a53*f3 + a54*f4;\n f5 := fn(xk + c5*h,yk + t*h);\n t := a 61*f1 + a62*f2 + a63*f3 + a64*f4 + a65*f5;\n f6 := fn(xk + c6*h,y k + t*h);\n t := a71*f1 + a72*f2 + a73*f3 + a74*f4 + a75*f5 + a76 *f6;\n f7 := fn(xk + c7*h,yk + t*h);\n \n yk := yk + ( b1*f1 + b2*f2 + b3*f3 + b4*f4 + b5*f5 + b6*f6 + b7*f7)*h;\n xk := xk + h:\n soln := soln,[xk,yk];\n end do;\n if bb=true then \n eqns := \{SOLN_=[soln],FXY_=fxy,X_=x,Y_=y,c2_=c2,c3_=c3,\n \+ c4_=c4,c5_=c5,c6_=c6,c7_=c7,a31_=a31,a32_=a32,a41_=a41,\n \+ a42_=a42,a43_=a43,a51_=a51,a52_=a52,a53_=a53,a54_=a54,\n a61_ =a61,a62_=a62,a63_=a63,a64_=a64,a65_=a65,\n a71_=a71,a72_=a72, a73_=a73,a74_=a74,a75_=a75,a76_=a76,\n b1_=b1,b2_=b2,b3_=b3,b4 _=b4,b5_=b5,b6_=b6,b7_=b7\};\n return subs(eqns,eval(rk6step));\n else\n return evalf[saveDigits]([soln]);\n end if;\nend proc : " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{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 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "Test 1 of 8 stage order 6 Runge-Kutta methods " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " dy/dx=12*x*cos(4*x)*exp(-x)*y" "6#/*&%#dyG\"\"\"%#dxG!\"\"*,\"#7F&%\"x GF&-%$cosG6#*&\"\"%F&F+F&F&-%$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,i c\},y(x)):\ny(x)=simplify(numer(rhs(%))/convert(denom(rhs(%)),exp));\n f := unapply(rhs(%),x):\nplot(f(x),x=0..5,0..1.45,font=[HELVETICA,9],l abels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diff G6$-%\"yG6#%\"xGF,,$*,\"#7\"\"\"F,F0-%$cosG6#,$*&\"\"%F0F,F0F0F0-%$exp G6#,$F,!\"\"F0F)F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6 #\"\"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%$expG6 #,,*&#\"#7\"#<\"\"\"*(F'F0-%$cosG6#,$*&\"\"%F0F'F0F0F0-F)6#,$F'!\"\"F0 F0F;*&#\"$!=\"$*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@K i8!#>$\"3Fk>e\"G.6+\"!#<7$$\"3ALL$3FWYs#F/$\"3!H*fm:2P/5F27$$\"3%)*** \\iSmp3%F/$\"3Qn()\\Dat45F27$$\"3WmmmT&)G\\aF/$\"34$Q7t`Dr,\"F27$$\"3m ****\\7G$R<)F/$\"3S2-*\\9jw.\"F27$$\"3GLLL3x&)*3\"!#=$\"3U([#>C\\El5F2 7$$\"3))**\\i!R(*Rc\"FJ$\"3>&=^@[0u7\"F27$$\"3umm\"H2P\"Q?FJ$\"3k\\#o# G?)=?\"F27$$\"3!***\\PMnNrDFJ$\"3s_j<)f!R*G\"F27$$\"3MLL$eRwX5$FJ$\"37 '\\4u:c`O\"F27$$\"3_LLe*[`HP$FJ$\"3[!\\'y0#yNR\"F27$$\"3rLLL$eI8k$FJ$ \"3N\"Ha_9o@T\"F27$$\"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$$\"3emm\"z%4\\Y_FJ$\" 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`OlFJ7$$\"3GLLeR%p\")Q$F2$\"3#o,C;(=8foFJ7$$\"3B++](=]@W$F2$\"3#G%=QV$ \\;4(FJ7$$\"3C$ekyZ2mY$F2$\"3u,m uc\"4C(FJ7$$\"3hTgx.2vFNF2$\"3/^M\"Q[;lC(FJ7$$\"35L$e*[$z*RNF2$\"3=wJ% fi2nC(FJ7$$\"3)*\\PMFwrmNF2$\"3R[i&\\xl(GsFJ7$$\"3%o;Hd!fX$f$F2$\"3IEK i0hy'=(FJ7$$\"3r$e9T=%>?OF2$\"3(>gS`&3dArFJ7$$\"3e++]iC$pk$F2$\"3ma\\o RiHQqFJ7$$\"3ILe*[t\\sp$F2$\"3'e9/wG(3MoFJ7$$\"3[m;H2qcZPF2$\"3CYQ8S*3 be'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%QCu nR#[FJ7$$\"3yLLL$)*pp;%F2$\"3g*yCm#3E'p%FJ7$$\"3)QL3-$H**>UF2$\"3$*o:W ?mr0YFJ7$$\"3)RL$3xe,tUF2$\"3!\\Bp&*))oXb%FJ7$$\"3h+v=n(*fDVF2$\"3kIpK $)H$3a%FJ7$$\"3Cn;HdO=yVF2$\"3u&G6!oNOhXFJ7$$\"3MMe9\"z-lU%F2$\"3kC\"> #=Lu2YFJ7$$\"3a+++D>#[Z%F2$\"3w_(eqj7vn%FJ7$$\"3SnmT&G!e&e%F2$\"3W>T$> g**p!\\FJ7$$\"3#RLLL)Qk%o%F2$\"3'yDBP_q:;&FJ7$$\"37+]iSjE!z%F2$\"3J;fP @m(pV&FJ7$$\"3a+]P40O\"*[F2$\"3!>+$=fU-gcFJ7$$\"\"&F)$\"3h(Q0fOqh\"eFJ -%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABE LSG6$%\"xG%%y(x)G-%%VIEWG6$;F(F]am;F($\"$X\"!\"#" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The following code constr ucts a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " based on eac h of the methods and gives the " }{TEXT 260 22 "root mean square error " }{TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 763 "F := (x,y) -> 12*x*cos(4*x)*exp(-x)*y: hh := 0.01: n umsteps := 500: x0 := 0: y0 := 1:\nmatrix([[`slope field: `,F(x,y)], [`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: \+ `,numsteps]]);``;\nmthds := [`Huta's scheme A`,`Huta's scheme B`,`a \+ companion to Huta's scheme B`,`an efficient scheme with the same nodes and weights as Huta's two schemes`,`a stage order 3 scheme with small principal error norm`]: errs := []:\nDigits := 25:\nfor ct to 5 do\n \+ Fn_RK6_||ct := RK6_||ct(F(x,y),x,y,x0,y0,hh,numsteps,false);\n sm \+ := 0: numpts := nops(Fn_RK6_||ct):\n for ii to numpts do\n sm : = sm+(Fn_RK6_||ct[ii,2]-f(Fn_RK6_||ct[ii,1]))^2;\n end do:\n errs \+ := [op(errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpo se]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrix G6#7&7$%0slope~field:~~~G,$*,\"#7\"\"\"%\"xGF,-%$cosG6#,$*&\"\"%F,F-F, F,F,-%$expG6#,$F-!\"\"F,%\"yGF,F,7$%0initial~point:~G-%!G6$\"\"!F,7$%/ step~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$+&Q(pprint16\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'ma trixG6#7'7$%0Huta's~scheme~AG$\"+n&f6J)!#97$%0Huta's~scheme~BG$\"+lG]z GF+7$%?a~companion~to~Huta's~scheme~BG$\"+t(*Gy7F+7$%doan~efficient~sc heme~with~the~same~nodes~and~weights~as~Huta's~two~schemesG$\"+)GY/A$! #:7$%Wa~stage~order~3~scheme~with~small~principal~error~normG$\"+r'RRY $F8Q(pprint26\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code constructs " }{TEXT 260 20 "numerical \+ procedures" }{TEXT -1 56 " for solutions based on each of the Runge-Ku tta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value ob tained by each of the methods at the point where " }{XPPEDIT 18 0 "x \+ = 4.999;" "6#/%\"xG-%&FloatG6$\"%**\\!\"$" }{TEXT -1 16 " is also giv en." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 694 "F := (x,y) -> 12*x*c os(4*x)*exp(-x)*y: hh := 0.01: numsteps := 500: x0 := 0: y0 := 1:\nmat rix([[`slope field: `,F(x,y)],[`initial point: `,``(x0,y0)],[`step w idth: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Huta's scheme A`,`Huta's scheme B`,`a companion to Huta's scheme B`,`an effi cient scheme with the same nodes and weights as Huta's two schemes`,`a stage order 3 scheme with small principal error norm`]: errs := []:\n Digits := 25:\nfor ct to 5 do\n fn_RK6_||ct := RK6_||ct(F(x,y),x,y,x 0,y0,hh,numsteps,true);\nend do:\nxx := 4.999: fxx := evalf(f(xx)):\nf or ct to 5 do\n errs := [op(errs),abs(fn_RK6_||ct(xx)-fxx)];\nend do :\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,$*,\"#7\"\" \"%\"xGF,-%$cosG6#,$*&\"\"%F,F-F,F,F,-%$expG6#,$F-!\"\"F,%\"yGF,F,7$%0 initial~point:~G-%!G6$\"\"!F,7$%/step~width:~~~G$F,!\"#7$%1no.~of~step s:~~~G\"$+&Q(pprint36\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%0Huta's~scheme~AG$\"+ cdS'4\"!#87$%0Huta's~scheme~BG$\"+cq8rP!#97$%?a~companion~to~Huta's~sc heme~BG$\"+Z \+ " 0 "" {MPLTEXT 1 0 438 "mthds := [`Huta's scheme A`,`Huta's scheme B` ,`a companion to Huta's scheme B`,`an efficient scheme with the same n odes and weights as Huta's two schemes`,`a stage order 3 scheme with s mall principal error norm`]: errs := []:\nDigits := 20:\nfor ct to 5 d o\n sm := NCint((f(x)-'fn_RK6_||ct'(x))^2,x=0..5,adaptive=false,nump oints=7,factor=200);\n errs := [op(errs),sqrt(sm/5)];\nend do:\nDigi ts := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#K%'matrixG6#7'7$%0Huta's~scheme~AG$\"+_2D4$)!#97$%0Hu ta's~scheme~BG$\"+oE!*yGF+7$%?a~companion~to~Huta's~scheme~BG$\"+_$4!y 7F+7$%doan~efficient~scheme~with~the~same~nodes~and~weights~as~Huta's~ two~schemesG$\"+5\\i>K!#:7$%Wa~stage~order~3~scheme~with~small~princip al~error~normG$\"+\\?fnMF8Q(pprint56\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following error graphs are const ructed using the numerical procedures for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 522 "evalf[20](plot([f(x)-'fn_RK6_1'(x) ,f(x)-'fn_RK6_2'(x),f(x)-'fn_RK6_3'(x),f(x)-'fn_RK6_4'(x),f(x)-'fn_RK6 _5'(x)],\nx=0..5,color=[COLOR(RGB,.95,0,.2),COLOR(RGB,1,.55,0),COLOR(R GB,.5,0,.95),COLOR(RGB,0,.75,.2),COLOR(RGB,0,.55,.95)],\nlegend=[`Huta 's scheme A`,`Huta's scheme B`,`a companion to Huta's scheme B`,`an ef ficient scheme with the same nodes and weights as Huta's two schemes`, `a stage order 3 scheme with small principal error norm`],font=[HELVET ICA,9],title=`error curves for 8 stage order 6 Runge-Kutta methods`)); " }}{PARA 13 "" 1 "" {GLPLOT2D 1047 629 629 {PLOTDATA 2 "6+-%'CURVESG6 %7es7$$\"\"!F)F(7$$\"5MLLLL3x&)*3\"!#?$!/_(*pCP'\\*!#>7$$\"5+++]i!R(*R c\"F-$!07'yzi)el\"F07$$\"5nmmm\"H2P\"Q?F-$!0obe4(*yC#F07$$\"5++]7G))>W r@F-$!0HBNEqcM#F07$$\"5MLLek.pu/BF-$!0nIII)48CF07$$\"5nm;/,>=0QCF-$!0_ 3l2wuV#F07$$\"5+++]PMnNrDF-$!0E)pS9m?CF07$$\"5nmmT5ll'z$GF-$!0F07$$\"5MLLLe*[`HP$F-$!0T(=a8&4k\"F 07$$\"5MLLLL$eI8k$F-$!0Sdih)e38F07$$\"5MLL$3-8>bx$F-$!0B6&pTO$=\"F07$$ \"5MLLL3xwq4RF-$!0Txx8js3\"F07$$\"5MLL$eRA'*Q/%F-$!0T3qG5#p5F07$$\"5NL LL$3x%3yTF-$!0r()=o-'36F07$$\"5-++]PfyG7ZF-$!0lI;8xX)>F07$$\"5ommm\"z% 4\\Y_F-$!05dOxahc$F07$$\"5++++v$fl)p;`F-7$$\"5qmmm\"HKkIz(F-$\"1-#f >jV1/\"F-7$$\"50+++Dc\"[#e!)F-$\"18M!3TooH#F-7$$\"5OLLLe*)>VB$)F-$\"1! f<\\hU0m$F-7$$\"5qmmmTg()4_))F-$\"11:^WbE(\\&F-7$$\"5.+++DJbw!Q*F-$\"1 o1Tl0t_mF-7$$\"5qmmT&)3\\m_'*F-$\"1If_1,,FpF-7$$\"5NLL$ekGkX#**F-$\"1G 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5omT5lDho;FF0$!0$z*=#\\?[8F-7$$\"5ML$3F>HT'HFF0$!0Low(y4L8F-7$$\"5,+DJ ?ekfUFF0$!0#**Gs6jD8F-7$Ffjl$!0))pXoQ@K\"F-7$$\"5,+]7.d>Y\"y#F0$!0N;$) \\kjK\"F-7$F[[m$!029iChsM\"F-7$F`[m$!0zdfuBZW\"F-7$Fe[m$!0wUFF4Tg\"F-7 $Fj[m$!0o1r;(=1@F-7$F_\\m$!0%yJl)>ss#F-7$Fd\\m$!0d:\"*4')zM$F-7$$\"5ML L$3_NJOG$F0$!01cTZSKa$F-7$Fi\\m$!0uVEPF4n$F-7$$\"5,++]il!z6O$F0$!0Q\"* zxRpq$F-7$F^]m$!0sPoI`bt$F-7$$\"5,++DcEYm,MF0$!0:C0v[\"RPF-7$$\"5omm;a 8)f^T$F0$!0hG&[V1[PF-7$$\"5MLL3_+]lGMF0$!0d\"*)QabXPF-7$Fc]m$!0^_$yikU PF-7$F]^m$!0,Np7.vr$F-7$Fa_m$!0>/xBsbn$F-7$F_am$!07d5*)\\ua$F-7$Fiam$! 0qipDm(pLF-7$F^bm$!0)\\%G*zb>JF-7$Fcbm$!0%*\\H`r:%GF-7$Fhbm$!0--xnTvd# F-7$F]cm$!0vb=&yJwCF-7$Fbcm$!0.wmZZoR#F-7$Fgcm$!0A$>'=0*RBF-7$F\\dm$!0 g[.i[#4BF-7$Fadm$!0v9V^_FI#F-7$Ffdm$!0E@8e!)*=BF-7$F[em$!0IMFd48N#F-7$ F`em$!0yfZv#>)R#F-7$Feem$!05*=o$*eUDF-7$Fjem$!0%p;y>@(o#F-7$F_fm$!0wMx 1dG#GF-7$Fdfm$!0\"paL2w9HF-7$Fifm$!0^/DWUS'HF--F^gm6&F`gmF(FcgmF]io-Fg gm6#%Wa~stage~order~3~scheme~with~small~principal~error~normG-%&TITLEG 6#%Uerror~curves~for~8~stage~order~6~Runge-Kutta~methodsG-%%FONTG6$%*H ELVETICAG\"\"*-%+AXESLABELSG6$Q\"x6\"Q!Fhgs-%%VIEWG6$;F(Fifm%(DEFAULTG " 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Huta's sche me B" "a companion to Huta's scheme B" "an efficient scheme with the s ame nodes and weights as Huta's two schemes" "a stage order 3 scheme w ith small principal error norm" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 46 "Test 2 of 8 stage order 6 Runge-Kutta methods " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=x/y" "6#/*&%#dy G\"\"\"%#dxG!\"\"*&%\"xGF&%\"yGF(" }{TEXT -1 10 ", " } {XPPEDIT 18 0 "y(0)=1" "6#/-%\"yG6#\"\"!\"\"\"" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y=sqrt(1+x^2)" "6#/%\"yG-%%sqrtG6#,&\"\"\"F)*$% \"xG\"\"#F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The following code constructs a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " based on each of the methods and gives the " }{TEXT 260 22 "root mean square error" }{TEXT -1 18 " of \+ each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 766 "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 := [`Huta's scheme A `,`Huta's scheme B`,`a companion to Huta's scheme B`,`an efficient sch eme with the same nodes and weights as Huta's two schemes`,`a stage or der 3 scheme with small principal error norm`]: errs := []:\nDigits := 20:\ng := x -> sqrt(1+x^2):\nfor ct to 5 do\n Gn_RK6_||ct := RK6_|| ct(G(x,y),x,y,x0,y0,hh,numsteps,false);\n sm := 0: numpts := nops(Gn _RK6_||ct):\n for ii to numpts do\n sm := sm+(Gn_RK6_||ct[ii,2] -g(Gn_RK6_||ct[ii,1]))^2;\n end do:\n errs := [op(errs),sqrt(sm/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(pprint06\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrix G6#7'7$%0Huta's~scheme~AG$\"+.EW*R%!#97$%0Huta's~scheme~BG$\"+T;B+:F+7 $%?a~companion~to~Huta's~scheme~BG$\"+c`T-v!#:7$%doan~efficient~scheme ~with~the~same~nodes~and~weights~as~Huta's~two~schemesG$\"+3Z " 0 "" {MPLTEXT 1 0 696 "G := (x,y) -> x/y: hh := 0.05: numsteps \+ := 200: x0 := 0: y0 := 1:\nmatrix([[`slope field: `,G(x,y)],[`initia l point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,nums teps]]);``;\nmthds := [`Huta's scheme A`,`Huta's scheme B`,`a companio n to Huta's scheme B`,`an efficient scheme with the same nodes and wei ghts as Huta's two schemes`,`a stage order 3 scheme with small princip al error norm`]: errs := []:\nDigits := 20:\nfor ct to 5 do\n gn_RK6 _||ct := RK6_||ct(G(x,y),x,y,x0,y0,hh,numsteps,true);\nend do:\ng := x -> sqrt(1+x^2):\nxx := 9.99: gxx := evalf(g(xx)):\nfor ct to 5 do\n \+ errs := [op(errs),abs(gn_RK6_||ct(xx)-gxx)];\nend do:\nDigits := 10: \nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G*&%\"xG\"\"\"%\"yG!\" \"7$%0initial~point:~G-%!G6$\"\"!F+7$%/step~width:~~~G$\"\"&!\"#7$%1no .~of~steps:~~~G\"$+#Q(pprint26\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%! G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%0Huta's~scheme~A G$\"+IF-$)>!#97$%0Huta's~scheme~BG$\"+$e3\"eo!#:7$%?a~companion~to~Hut a's~scheme~BG$\"+#**o6[$F07$%doan~efficient~scheme~with~the~same~nodes ~and~weights~as~Huta's~two~schemesG$\"+9`0K5F07$%Wa~stage~order~3~sche me~with~small~principal~error~normG$\"+uzx^h!#;Q(pprint36\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square error" }{TEXT -1 20 " over the interval " } {XPPEDIT 18 0 "[0, 10]" "6#7$\"\"!\"#5" }{TEXT -1 82 " of each Runge- Kutta method is estimated as follows using the special procedure " } {TEXT 0 5 "NCint" }{TEXT -1 98 " to perform numerical integration by \+ the 7 point Newton-Cotes method over 100 equal subintervals." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 463 "mthds := [`Huta's scheme A` ,`Huta's scheme B`,`a companion to Huta's scheme B`,`an efficient sche me with the same nodes and weights as Huta's two schemes`,`a stage ord er 3 scheme with small principal error norm`]: errs := []:\nDigits := \+ 20:\ng := x -> sqrt(1+x^2):\nfor ct to 5 do\n sm := NCint((g(x)-'gn_ RK6_||ct'(x))^2,x=0..10,adaptive=false,numpoints=7,factor=100);\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$%0Huta's~scheme~AG$\"+Au'[R%!#97$%0Huta's~scheme~BG$\"+erl)\\\"F +7$%?a~companion~to~Huta's~scheme~BG$\"+())HX\\(!#:7$%doan~efficient~s cheme~with~the~same~nodes~and~weights~as~Huta's~two~schemesG$\"+]m'QB# F47$%Wa~stage~order~3~scheme~with~small~principal~error~normG$\"+L4%)f 7F4Q(pprint46\"" }}}{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 0 523 "evalf[20](plot([g(x)-'gn_RK6_1'(x),g(x)-'gn_RK6_2'(x ),g(x)-'gn_RK6_3'(x),g(x)-'gn_RK6_4'(x),g(x)-'gn_RK6_5'(x)],\nx=0..10, color=[COLOR(RGB,.95,0,.2),COLOR(RGB,1,.55,0),COLOR(RGB,.5,0,.95),COLO R(RGB,0,.75,.2),COLOR(RGB,0,.55,.95)],\nlegend=[`Huta's scheme A`,`Hut a's scheme B`,`a companion to Huta's scheme B`,`an efficient scheme wi th the same nodes and weights as Huta's two schemes`,`a stage order 3 \+ scheme with small principal error norm`],font=[HELVETICA,9],title=`err or curves for 8 stage order 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 1013 571 571 {PLOTDATA 2 "6+-%'CURVESG6%7[p7$$\"\"!F)F( 7$$\"5KLLL$3FWYs#!#@$!,f3Ao:*!#>7$$\"5lmmmmT&)G\\aF-$!.%>c:iL5F07$$\"5 ILLL3x1h6oF-$!-#p[mv%fF07$$\"5++++]7G$R<)F-$\".(\\aUI%G\"F07$$\"5lmmm \"z%\\DO&*F-$\".(3C#>-a&F07$$\"5LLLLL3x&)*3\"!#?$\".a34r3y(F07$$\"5+++ +]ilyM;FH$\"/5(o>Q!GEF07$$\"5mmmmm;arz@FH$\"/\\'GK3!*R&F07$$\"5LLL$e*) 4bQl#FH$\"/f&pN9Cx)F07$$\"5++++D\"y%*z7$FH$\"0M3(z]Uw7F07$$\"5mmm;ajW8 -OFH$\"0k'>(zlQs\"F07$$\"5LLLL$e9ui2%FH$\"0*4/.R;0AF07$$\"5mmm;H2Q\\4Y FH$\"0?O-/n_q#F07$$\"5++++voMrU^FH$\"03^i@D3@$F07$$\"5NLL$3-8Lfn&FH$\" 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The following code c onstructs a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " based o n each of the methods and gives the " }{TEXT 260 22 "root mean square \+ error" }{TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 766 "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 := [`Huta's scheme A`,`Huta's scheme B`,`a companion to Huta's scheme B`,`an efficient scheme with the same nodes and weights as Hut a's two schemes`,`a stage order 3 scheme with small principal error no rm`]: errs := []:\nDigits := 20:\nh := x -> exp(-x^2/2):\nfor ct to 5 \+ do\n Hn_RK6_||ct := RK6_||ct(H(x,y),x,y,x0,y0,hh,numsteps,false);\n \+ sm := 0: numpts := nops(Hn_RK6_||ct):\n for ii to numpts do\n \+ sm := sm+(Hn_RK6_||ct[ii,2]-h(Hn_RK6_||ct[ii,1]))^2;\n end do:\n \+ errs := [op(errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[tr anspose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'m atrixG6#7&7$%0slope~field:~~~G,$*&%\"xG\"\"\"%\"yGF,!\"\"7$%0initial~p oint:~G-%!G6$\"\"!F,7$%/step~width:~~~G$F,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$%0Huta's~scheme~AG$\"+O$*o<=!#87$% 0Huta's~scheme~BG$\"+C>#4I'!#97$%?a~companion~to~Huta's~scheme~BG$\"+= d!)[EF07$%doan~efficient~scheme~with~the~same~nodes~and~weights~as~Hut a's~two~schemesG$\"+Z/&3L'!#:7$%Wa~stage~order~3~scheme~with~small~pri ncipal~error~normG$\"+$e2!)R*F9Q(pprint66\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code constructs \+ " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for solutions ba sed on each of the Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each of the methods at the poin t where " }{XPPEDIT 18 0 "9.99;" "6#-%&FloatG6$\"$***!\"#" }{TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 696 "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 widt h: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Huta's sc heme A`,`Huta's scheme B`,`a companion to Huta's scheme B`,`an efficie nt scheme with the same nodes and weights as Huta's two schemes`,`a st age order 3 scheme with small principal error norm`]: errs := []:\nDig its := 20:\nfor ct to 5 do\n hn_RK6_||ct := RK6_||ct(H(x,y),x,y,x0,y 0,hh,numsteps,true);\nend do:\nh := x -> exp(-x^2/2):\nxx := 9.99: hxx := evalf(h(xx)):\nfor ct to 5 do\n errs := [op(errs),abs(hn_RK6_||c t(xx)-hxx)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(e rrs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~fie ld:~~~G,$*&%\"xG\"\"\"%\"yGF,!\"\"7$%0initial~point:~G-%!G6$\"\"!F,7$% /step~width:~~~G$F,F.7$%1no.~of~steps:~~~G\"$+\"Q(pprint76\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'ma trixG6#7'7$%0Huta's~scheme~AG$\"+0iSJ@!#J7$%0Huta's~scheme~BG$\"+k!>)o " 0 "" {MPLTEXT 1 0 462 "mthds := [`Huta's scheme A`,`Huta's scheme B`,`a com panion to Huta's scheme B`,`an efficient scheme with the same nodes an d weights as Huta's two schemes`,`a stage order 3 scheme with small pr incipal error norm`]: errs := []:\nDigits := 20:\nh := x -> exp(-x^2/2 ):\nfor ct to 5 do\n sm := NCint((h(x)-'hn_RK6_||ct'(x))^2,x=0..10,a daptive=false,numpoints=7,factor=50);\n errs := [op(errs),sqrt(sm/10 )];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%0Huta's~scheme~AG$ \"+j[;\\ " 0 "" {MPLTEXT 1 0 537 "evalf[20](plot(['h n_RK6_1'(x)-h(x),'hn_RK6_2'(x)-h(x),'hn_RK6_3'(x)-h(x),\n'hn_RK6_4'(x) -h(x),'hn_RK6_5'(x)-h(x)],x=0..6,numpoints=100,\ncolor=[COLOR(RGB,.95, 0,.2),COLOR(RGB,1,.55,0),COLOR(RGB,.5,0,.95),COLOR(RGB,0,.75,.2),COLOR (RGB,0,.55,.95)],\nlegend=[`Huta's scheme A`,`Huta's scheme B`,`a comp anion to Huta's scheme B`,`an efficient scheme with the same nodes and weights as Huta's two schemes`,`a stage order 3 scheme with small pri ncipal error norm`],font=[HELVETICA,9],title=`error curves for 8 stage order 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 1026 457 457 {PLOTDATA 2 "6+-%'CURVESG6%7\\z7$$\"\"!F)F(7$$\"5.......*4M'!# @$!/uwqp&4p#!#?7$$\"5!======@'*4*F-$!0E\\tibX9\"F07$$\"5111111_#e=\"F0 $!0-z]p&)3d\"F07$$\"5.....`O%4M\"F0$!0h$Rm4;v5F07$$\"5++++++@1'\\\"F0$ \"/K'e$>A[7F07$$\"5(pppppa!=^;F0$\"0A\")=jsaJ#F07$$\"5%RRRRR**)H1=F0$ \"02S$))*3=w&F07$$\"5IIIIIIBP%)=F0$\"0Btre>#\\!)F07$$\"5nmmmmmcWi>F0$ \"1=WE8Ieu5F07$$\"5/.....!>0/#F0$\"1diuCm))=7F07$$\"5SRRRRRBf=@F0$\"1) Hx]B)f@7F07$$\"5777777!RZF#F0$\"1**>sui\"GF\"F07$$\"5&[[[[[o&)3V#F0$\" 1-%o&zg&)Q9F07$$\"5aaaaaaZpTFF0$\"18J&>Ll!eBF07$$\"5CCCCCCQ]_IF0$\"1]W 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with~small~principal~error~normG-%&TITLEG6#%Uerror~curves~for~8~stage~ order~6~Runge-Kutta~methodsG-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6 $Q\"x6\"Q!F[e]l-%%VIEWG6$;F(Fdfo%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Huta's scheme A" "Huta's scheme B" " a companion to Huta's scheme B" "an efficient scheme with the same nod es and weights as Huta's two schemes" "a stage order 3 scheme with sma ll principal error norm" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 543 "evalf[20](plot(['hn_RK6_1'(x)/h(x) -1,'hn_RK6_2'(x)/h(x)-1,'hn_RK6_3'(x)/h(x)-1,\n'hn_RK6_4'(x)/h(x)-1,'h n_RK6_5'(x)/h(x)-1],x=0..10,\ncolor=[COLOR(RGB,.95,0,.2),COLOR(RGB,1,. 55,0),COLOR(RGB,.5,0,.95),COLOR(RGB,0,.75,.2),COLOR(RGB,0,.55,.95)],\n legend=[`Huta's scheme A`,`Huta's scheme B`,`a companion to Huta's sch eme B`,`an efficient scheme with the same nodes and weights as Huta's \+ two schemes`,`a stage order 3 scheme with small principal error norm`] ,font=[HELVETICA,9],title=`relative error curves for 8 stage order 6 R unge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 1036 625 625 {PLOTDATA 2 "6+-%'CURVESG6%7S7$$\"\"!F)F(7$$\"5mmmmm;arz@!#?$\"09$RzDo h7!#>7$$\"5LLLL$e9ui2%F-$\"0Z6WH,,\\)F07$$\"5mmmmm\"z_\"4iF-$\"1T`7&)[ @6?F07$$\"5lmmmmT&phN)F-$\"1;7\")o6itKF07$$\"5LLLLe*=)H\\5F0$\"1%)Hw<` BdSF07$$\"5mmmm\"z/3uC\"F0$\"1ZwHBHY&e$F07$$\"5++++DJ$RDX\"F0$\"0S)RTq no#)F07$$\"5mmmm\"zR'ok;F0$!25&e$3*>*3*eF-7$$\"5++++D1J:w=F0$!3`Zj*R* \\Yg=F-7$$\"5LLLLL3En$4#F0$!3RX,+Fnw+UF-7$$\"5mmmm;/RE&G#F0$!3Prq3CL!z #oF-7$$\"5+++++D.&4]#F0$!47'4v\\([k%y6F-7$$\"5+++++vB_F-7$$\"5LLLLeR\"3Gy%F0 $!5i#Q)Gt$z\\;N#F-7$$\"5mmmm;/T1&*\\F0$!5cjALrP<@YGF-7$$\"5mmmm\"zRQb@ &F0$!5k%*Q%z5Ti^N$F-7$$\"5*******\\(=>Y2aF0$!5#QDE3N8L3'QF-7$$\"5mmmm; zXu9cF0$!5<$Hg:(=!GbR%F-7$$\"5**********\\y))GeF0$!5&f>)pHWL%R&\\F-7$$ \"5********\\i_QQgF0$!5Ep8;\\hD1EbF-7$$\"5*******\\7y%3TiF0$!5xUD<)=s, Z4'F-7$$\"5********\\P![hY'F0$!5B?VTP>&p#*p'F-7$$\"5KLLLL$Qx$omF0$!5#o 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\"5++v$f$=.5K**F0$\"5Lnss?s-^onF07$Fbhl$\"5QJaa,JI7'y'F07$Fghl$\"5RW8_ GCVYSoF07$F\\il$\"5+e2D:h?]tpF07$Fail$\"5iO7PU!f,bD(F07$Fbz$\"5(eHM-cb 5nz(F0-Fgz6&FizF(F\\jlFjz-Fa[l6#%Wa~stage~order~3~scheme~with~small~pr incipal~error~normG-%&TITLEG6#%hnrelative~error~curves~for~8~stage~ord er~6~Runge-Kutta~methodsG-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6$Q \"x6\"Q!F^\\o-%%VIEWG6$;F(Fbz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Huta's scheme A" "Huta's scheme B" " a companion to Huta's scheme B" "an efficient scheme with the same nod es and weights as Huta's two schemes" "a stage order 3 scheme with sma ll principal error norm" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 46 "Test 4 of 8 stage order 6 Runge-Kutta methods " }}{PARA 0 "" 0 "" {TEXT -1 81 "F. G. Lether: Mathematics of Computation, Vol. \+ 20, no. 95, (July 1966) page 381. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = -32*x*y*ln(2) ;" "6#/*&%#dyG\"\"\"%#dxG!\"\",$**\"#KF&%\"xGF&%\"yGF&-%#lnG6#\"\"#F&F (" }{TEXT -1 6 ", " }{XPPEDIT 18 0 "y(-1) = 1/8;" "6#/-%\"yG6#,$\" \"\"!\"\"*&F(F(\"\")F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y = 2 ^(13-6*x^2);" "6#/%\"yG)\"\"#,&\"#8\"\"\"*&\"\"'F)*$%\"xGF&F)!\"\"" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 191 "de := diff(y(x),x)=-32*x*y(x)*ln(2);\nic := y(- 1)=1/8;\ndsolve(\{de,ic\},y(x)):\ny(x)=2^simplify(log[2](rhs(%)));\nk \+ := unapply(rhs(%),x):\nplot(k(x),x=-1..1,font=[HELVETICA,9],labels=[`x `,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG 6#%\"xGF,,$**\"#K\"\"\"F,F0F)F0-%#lnG6#\"\"#F0!\"\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#icG/-%\"yG6#!\"\"#\"\"\"\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG)\"\"#,&\"#8\"\"\"*&\"#;F,)F'F)F,!\"\"" } }{PARA 13 "" 1 "" {GLPLOT2D 577 286 286 {PLOTDATA 2 "6&-%'CURVESG6$7eo 7$$!\"\"\"\"!$\"3+++++++]7!#=7$$!3ommm;p0k&*F-$\"3!Hg[[\"[$)=KF-7$$!3w KL$3$3(F-7$$!3mmmmT%p\"e()F-$\"3!=E-TWD`l\"!#<7$$!3 :mmm\"4m(G$)F-$\"3M\"fONp()[t$F=7$$!3\"QLL3i.9!zF-$\"3A!e'4A%*Rg!)F=7$ $!3\"ommT!R=0vF-$\"3%z2Mbncie\"!#;7$$!3u****\\P8#\\4(F-$\"3C>dT>$)H#3$ FM7$$!3+nm;/siqmF-$\"3gp%*z`g)4*eFM7$$!3[++](y$pZiF-$\"3%R6L-Y$zz5!#:7 $$!33LLL$yaE\"eF-$\"3xvp\"p)==K>Fgn7$$!3hmmm\">s%HaF-$\"3dBW_P%Gb6$Fgn 7$$!3Q+++]$*4)*\\F-$\"3e;N4:OF ap7$$!3]++]PYx\"\\#F-$\"37]-4,Tp9TFap7$$!3QnmTNz>&H#F-$\"3y(*QMk^JnXFa p7$$!3EMLLL7i)4#F-$\"3yCPsPtXE]Fap7$$!3#pm;aVXH)=F-$\"3_cYryDpGbFap7$$ !3c****\\P'psm\"F-$\"38i9x*[!p=gFap7$$!3s*****\\F&*=Y\"F-$\"3K`b3X@Jjk Fap7$$!3')****\\74_c7F-$\"3co2Qfx9woFap7$$!3ZmmT5VBU5F-$\"3E!>K?nVAE(F ap7$$!3)3LLL3x%z#)!#>$\"3C'Q/NU&H#f(Fap7$$!3gKL$e9d;J'Fft$\"3@erx_1&z$ yFap7$$!3KMLL3s$QM%Fft$\"3<%pUH&HNA!)Fap7$$!3'ym;aQdDG$Fft$\"3%eWuwq(o %4)Fap7$$!3T,+]ivF@AFft$\"3[vW[$G&HZ\")Fap7$$!3=o;/^wj!p\"Fft$\"3]3j^O K2m\")Fap7$$!3'\\L$eRx**f6Fft$\"355#oX4%yz\")Fap7$$!3S<+D\"GyNH'!#?$\" 3#QE)R9AS)=)Fap7$$!3]^omm;zr)*!#@$\"3;#)*eHY6>>)Fap7$$\"3o'H$3x\"yY_%F jv$\"3Q>*>/AS,>)Fap7$$\"3&yK$3_Nl.5Fft$\"3eqLS$Q`G=)Fap7$$\"3/E$ekGR[b \"Fft$\"3-l$f@nl+<)Fap7$$\"3@CL$3-Dg5#Fft$\"3kJX?)*G!=:)Fap7$$\"3e?Le* ['R3KFft$\"3E'yGoI5!*4)Fap7$$\"3%pJL$ezw5VFft$\"3U_-I6(**[-)Fap7$$\"3L `mmmJ+IiFft$\"3%pB(\\hv&o%yFap7$$\"3s*)***\\PQ#\\\")Fft$\"3!QM&=wHL5wF ap7$$\"3ilm\"z\\1A-\"F-$\"3#*[#H(\\2i&H(Fap7$$\"3GKLLe\"*[H7F-$\"3))\\ \\;@heFpFap7$$\"3ylm;HCjV9F-$\"3)e+$\\9-Y,lFap7$$\"3I*******pvxl\"F-$ \"3S%z:5s)zRgFap7$$\"3g)***\\7JFn=F-$\"31))p(30[[c&Fap7$$\"3#z****\\_q n2#F-$\"3ae5F\"zuv2&Fap7$$\"3=)**\\P/q%zAF-$\"3ZUhzOe!Rg%Fap7$$\"3U)** *\\i&p@[#F-$\"3r&f%4uLbOTFap7$$\"3L)**\\(=GB2FF-$\"3WV]5@%**Rj$Fap7$$ \"3B)****\\2'HKHF-$\"3ul]=$GLo:$Fap7$$\"3uJL$3UDX8$F-$\"3sKZjodBbFFap7 $$\"3ElmmmZvOLF-$\"3!>\\-t_7IQ#Fap7$$\"3i******\\2goPF-$\"3Q>G9F7l&p\" Fap7$$\"3UKL$eR<*fTF-$\"3?\"Fap7$$\"3m******\\)Hxe%F-$\"3V-? C_;$p$zFgn7$$\"3ckm;H!o-*\\F-$\"31MiF2c]v^Fgn7$$\"3y)***\\7k.6aF-$\"3# pB[/J``=$Fgn7$$\"3#emmmT9C#eF-$\"3&*=.D]9+3>Fgn7$$\"33****\\i!*3`iF-$ \"3%HX+j$our5Fgn7$$\"3%QLLL$*zym'F-$\"3!o4*yfd(\\\"fFM7$$\"3wKLL3N1#4( F-$\"3!\\\\K5**)='4$FM7$$\"3Nmm;HYt7vF-$\"3%o[)olFVm:FM7$$\"3Y*******p (G**yF-$\"3)3H-pcT.4)F=7$$\"3]mmmT6KU$)F-$\"35omE\\#[Ck$F=7$$\"3fKLLLb dQ()F-$\"3TxwT%Qu%>ei< " 0 "" {MPLTEXT 1 0 773 "K := (x,y) -> -32*x*y(x)*ln (2): hh := 0.01: numsteps := 200: x0 := -1: y0 := 1/8:\nmatrix([[`slop e field: `,K(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,h h],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Huta's scheme A`, `Huta's scheme B`,`a companion to Huta's scheme B`,`an efficient schem e with the same nodes and weights as Huta's two schemes`,`a stage orde r 3 scheme with small principal error norm`]: errs := []:\nDigits := 2 0:\nfor ct to 5 do\n Kn_RK6_||ct := RK6_||ct(evalf(K(x,y)),x,y,x0,ev alf(y0),hh,numsteps,false);\n sm := 0: numpts := nops(Kn_RK6_||ct): \n for ii to numpts do\n sm := sm+(Kn_RK6_||ct[ii,2]-k(Kn_RK6_| |ct[ii,1]))^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\nen d do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,$** \"#K\"\"\"%\"xGF,-%\"yG6#F-F,-%#lnG6#\"\"#F,!\"\"7$%0initial~point:~G- %!G6$F5#F,\"\")7$%/step~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$+#Q)p print176\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%0Huta's~scheme~AG$\"+[Kyq6!\"(7$%0Huta 's~scheme~BG$\"+yHY^N!\")7$%?a~companion~to~Huta's~scheme~BG$\"+_j!y!> F07$%doan~efficient~scheme~with~the~same~nodes~and~weights~as~Huta's~t wo~schemesG$\"+$G(*H;(!\"*7$%Wa~stage~order~3~scheme~with~small~princi pal~error~normG$\"+np()>6F0Q)pprint186\"" }}}{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 = 0;" "6#/%\"xG\"\"!" }{TEXT -1 20 ".995 is a lso given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 704 "K := (x,y) -> -32*x*y(x)*ln(2): hh := 0.01: numsteps := 200: x0 := -1: y0 := 1/8:\n matrix([[`slope field: `,K(x,y)],[`initial point: `,``(x0,y0)],[`ste p width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Hut a's scheme A`,`Huta's scheme B`,`a companion to Huta's scheme B`,`an e fficient scheme with the same nodes and weights as Huta's two schemes` ,`a stage order 3 scheme with small principal error norm`]: errs := [] :\nDigits := 20:\nfor ct to 5 do\n kn_RK6_||ct := RK6_||ct(evalf(K(x ,y)),x,y,x0,evalf(y0),hh,numsteps,true);\nend do:\nxx := 0.995: kxx := evalf(k(xx)):\nfor ct to 5 do\n errs := [op(errs),abs(kn_RK6_||ct(x x)-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~p oint:~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$%0Huta's~scheme~AG$\"+N)H\"*o\" !#77$%0Huta's~scheme~BG$\"+)f6'R*)!#87$%?a~companion~to~Huta's~scheme~ BG$\"+%))G_F$F07$%doan~efficient~scheme~with~the~same~nodes~and~weight s~as~Huta's~two~schemesG$\"+y,Nx " 0 "" {MPLTEXT 1 0 439 "mthds := [`Huta's scheme A`,`Huta's sche me B`,`a companion to Huta's scheme B`,`an efficient scheme with the s ame nodes and weights as Huta's two schemes`,`a stage order 3 scheme w ith small principal error norm`]: errs := []:\nDigits := 20:\nfor ct t o 5 do\n sm := NCint((k(x)-'kn_RK6_||ct'(x))^2,x=-1..1,adaptive=fals e,numpoints=7,factor=100);\n errs := [op(errs),sqrt(sm/2)];\nend do: \nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%0Huta's~scheme~AG$\"+Ghqt6!\"(7 $%0Huta's~scheme~BG$\"+#eN.c$!\")7$%?a~companion~to~Huta's~scheme~BG$ \"+mPd7>F07$%doan~efficient~scheme~with~the~same~nodes~and~weights~as~ Huta's~two~schemesG$\"+]:)3=(!\"*7$%Wa~stage~order~3~scheme~with~small ~principal~error~normG$\"+a&pE7\"F0Q)pprint216\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following error graph s are constructed using the numerical procedures for the solutions." } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 523 "evalf[20](plot(['kn_RK6_1' (x)-k(x),'kn_RK6_2'(x)-k(x),'kn_RK6_3'(x)-k(x),'kn_RK6_4'(x)-k(x),'kn_ RK6_5'(x)-k(x)],\nx=-1..1,color=[COLOR(RGB,.95,0,.2),COLOR(RGB,1,.55,0 ),COLOR(RGB,.5,0,.95),COLOR(RGB,0,.75,.2),COLOR(RGB,0,.55,.95)],\nlege nd=[`Huta's scheme A`,`Huta's scheme B`,`a companion to Huta's scheme \+ B`,`an efficient scheme with the same nodes and weights as Huta's two \+ schemes`,`a stage order 3 scheme with small principal error norm`],fon t=[HELVETICA,9],title=`error curves for 8 stage order 6 Runge-Kutta me thods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 1072 628 628 {PLOTDATA 2 "6+-% 'CURVESG6%7eo7$$!\"\"\"\"!$F*F*7$$!5nmmmm;p0k&*!#?$\"3l7c>$**GUf\"F/7$ $!5LLLL$37$$!5nmmmm\"4m(G$)F/$\"3/[#oi7[%)G'F<7$$!5LLLL$3i.9!zF/$\"4)RZTbc!>di \"F<7$$!5mmmm;/R=0vF/$\"3\"\\I3`4c[e$!#=7$$!5++++]P8#\\4(F/$\"3l[tv_:# ok(FL7$$!5mmmm;/siqmF/$\"4%G1>fU1Fo:FL7$$!5++++](y$pZiF/$\"3NdIT\\^`TI !#<7$$!5LLLLL$yaE\"eF/$\"3-!*4#f+I#3dFfn7$$!5mmmmm\">s%HaF/$\"3t\">m'[ y)\\[*Ffn7$$!5+++++]$*4)*\\F/$\"43nr%>#=qxg\"Ffn7$$!5+++++]_&\\c%F/$\" 4eJ'*zGDEGf#Ffn7$$!5+++++]1aZTF/$\"3y,8(4$R^MR!#;7$$!5mmmm;/#)[oPF/$\" 3wL(>48j'QbF`p7$$!5LLLLL$=exJ$F/$\"3p_REh-^[zF`p7$$!5LLLLLeW%o7$F/$\"3 LK36&\\,#G\"*F`p7$$!5LLLLLL2$f$HF/$\"48,C$3L*H%R5F`p7$$!5mmmmT&o_Qr#F/ $\"4>\"pm+sPC'>\"F`p7$$!5********\\PYx\"\\#F/$\"4,'Hz`0\"G5O\"F`p7$$!5 mmmmTNz>&H#F/$\"4\"\\RN#=%*48^\"F`p7$$!5LLLLLL7i)4#F/$\"47[?$>:wXj;F`p 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L7$Fc`l$\"2(Hy2%*H.(G#FL7$Fh`l$\"3z!\\FR/'*Q1\"F<7$F]al$\"2MF9C:^z9%F< 7$Fbal$\"2E=kk#*RSh\"F<7$Fgal$\"2aN;Kf&ym^F/7$F\\bl$\"2>R#foa0b8F/7$Fa bl$\"0,QBG/p)oF/-Ffbl6&FhblF+$\"#vF[clF\\cl-F_cl6#%doan~efficient~sche me~with~the~same~nodes~and~weights~as~Huta's~two~schemesG-F$6%7eoF'7$F -$\"2'zh;]Y\"*>$\"2[4#z !G5,`'F<7$FC$\"3@6p*)fzdq;F<7$FH$\"2xUQ#\\'HRk$FL7$FN$\"2g@z_Nb!)p(FL7 $FS$\"3x]c5nu;k:FL7$FX$\"2t[W>o&H2IFfn7$Fhn$\"2_vtZC;&)f&Ffn7$F]o$\"2m Db$o]X\\#*Ffn7$Fbo$\"3][WiYTwe:Ffn7$Fgo$\"3x)y)Hq*\\F]#Ffn7$F\\p$\"26= 2ItEWy$F`p7$Fbp$\"25_0)\\mE9`F`p7$Fgp$\"2=ej;LY2h(F`p7$F\\q$\"2cwgyq\" HN()F`p7$Faq$\"2WB=9e4C%**F`p7$Ffq$\"32#=$>nhxV6F`p7$F[r$\"3If%y%on.,8 F`p7$F`r$\"34Qgi;icW9F`p7$Fer$\"3G\"[p+.\"***e\"F`p7$Fjr$\"3qtrZkS()[< F`p7$F_s$\"3AM$4O?9P!>F`p7$Fds$\"3Yh11M([S/#F`p7$Fis$\"3E]D*=f/U<#F`p7 $F^t$\"3_cI2dkz&H#F`p7$Fct$\"3jQJ6))4g*R#F`p7$Fht$\"3O9R_9axwCF`p7$F^u $\"3'[Z@^B=Y`#F`p7$Fcu$\"3Kdp*e#>EdDF`p7$Fhu$\"3`MkQz**otDF`p7$F]v$\"3 Xk&Hn#R_zDF`p7$Fbv$\"3F:-*HY#y$e#F`p7$Fgv$\"3ce'[:)\\V'e#F`p7$F]w$\"3d )\\fr'eY(e#F`p7$Fbw$\"3/+[<=L'oe#F`p7$Fgw$\"3Y/[T*eoWe#F`p7$F\\x$\"3^M yjuVU!e#F`p7$Fax$\"3Dq6a;oeuDF`p7$Ffx$\"3UV`JU3'yb#F`p7$F[y$\"3YRG\"[` CW`#F`p7$F`y$\"3\\kD'z')*=yCF`p7$Fey$\"3[TV)o@^NS#F`p7$Fjy$\"30,1PJ#oU I#F`p7$F_z$\"3=)p61Mr\")=#F`p7$Fdz$\"3+G1Kadt`?F`p7$Fiz$\"3c%R9MBk!3>F `p7$F^[l$\"3&fgX6Ui\"eFL7$F^`l$\"3kIaD!*><:5FL7$Fc`l$\"2@C(p#\\I/C&FL7 $Fh`l$\"3O*)pardCtFF<7$F]al$\"3TFIFO1x)G\"F<7$Fbal$\"2#o818edHjF<7$Fga l$\"3:07KR/\"R(GF/7$F\\bl$\"3mz)RceidJ\"F/7$Fabl$\"2&G0n%p!pxbF/-Ffbl6 &FhblF+F_amFibl-F_cl6#%Wa~stage~order~3~scheme~with~small~principal~er ror~normG-%&TITLEG6#%Uerror~curves~for~8~stage~order~6~Runge-Kutta~met hodsG-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6$Q\"x6\"Q!Fe\\p-%%VIEWG 6$;F(Fabl%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Huta's scheme A" "Huta's scheme B" "a companion to Huta 's scheme B" "an efficient scheme with the same nodes and weights as H uta's two schemes" "a stage order 3 scheme with small principal error \+ norm" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 46 "Test 5 of \+ 8 stage order 6 Runge-Kutta methods " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=16/((16*x+1)*y)" "6#/*&%#dyG\"\"\"%#dxG!\"\"* &\"#;F&*&,&*&F*F&%\"xGF&F&F&F&F&%\"yGF&F(" }{TEXT -1 10 ", " } {XPPEDIT 18 0 "y(0)=1" "6#/-%\"yG6#\"\"!\"\"\"" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y=sqrt (2*ln(16*x+1)+1)" "6#/%\"yG-%%sqrtG6#,&* &\"\"#\"\"\"-%#lnG6#,&*&\"#;F+%\"xGF+F+F+F+F+F+F+F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "de := diff(y(x),x)=16/((16*x+1)*y(x));\nic := y(0)=1;\ndsolve( \{de,ic\},y(x));\ns := unapply(rhs(%),x):\nplot(s(x),x=0..0.5,0..2.6,f ont=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,$*(\"#;\"\"\",&*&F/F0F,F0F0F0F0 !\"\"F)F3F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!\" \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG*$,&*&\"\"#\"\" \"-%#lnG6#,&*&\"#;F,F'F,F,F,F,F,F,F,F,#F,F+" }}{PARA 13 "" 1 "" {GLPLOT2D 503 318 318 {PLOTDATA 2 "6&-%'CURVESG6$7U7$$\"\"!F)$\"\"\"F) 7$$\"3WmmmT&)G\\a!#?$\"3EP,(Qyl.3\"!#<7$$\"3ILLL3x&)*3\"!#>$\"3?25A!pa &\\6F27$$\"3-+]i!R(*Rc\"F6$\"3oz*p77wF?\"F27$$\"3umm\"H2P\"Q?F6$\"3]_v ibZz]7F27$$\"3MLL$eRwX5$F6$\"3!Qb+fY'3W8F27$$\"3CLL$3x%3yTF6$\"31#\\\\ E7=EU\"F27$$\"3=mm\"z%4\\Y_F6$\"3s(e$4OUg*[\"F27$$\"3)HL$eR-/PiF6$\"3. fPtw=4W:F27$$\"3A***\\il'pisF6$\"3/07@a`R%f\"F27$$\"3`KLe*)>VB$)F6$\"3 K!\\`od36k\"F27$$\"3!))**\\7`l2Q*F6$\"3#HUv\"fmC$o\"F27$$\"3smm;/j$o/ \"!#=$\"3:'H!f>cuAjU6Fco$\"3K$o8QC!za=F27$$\" 3)*****\\P[6j9Fco$\"39iuo+OIZ=F27$$\"3KL$e*[z(yb\"Fco$\"3Q:]fA\\>F27$ $\"3))**\\iSj0x=Fco$\"3-5Hbh&QF%>F27$$\"3Wmmm\"pW`(>Fco$\"3So#znsrC'>F 27$$\"35+]i!f#=$3#Fco$\"3w)>Y)R!pI)>F27$$\"3/+](=xpe=#Fco$\"3?*eB@.[<+ #F27$$\"3smm\"H28IH#Fco$\"3/Fyh^(\\.-#F27$$\"3km;zpSS\"R#Fco$\"3)4US+% ypO?F27$$\"3GLL3_?`(\\#Fco$\"3#4Cj+a0O0#F27$$\"3#HLe*)>pxg#Fco$\"3ab\\ mG7Vq?F27$$\"3u**\\Pf4t.FFco$\"3Cx7m@=^%3#F27$$\"32LLe*Gst!GFco$\"3Q

>IFco $\"3&ocGC'[]F@F27$$\"3h**\\i!RU07$Fco$\"3HCH$Q\")f.9#F27$$\"3b***\\(=S 2LKFco$\"3C`wrWc9a@F27$$\"3Kmmm\"p)=MLFco$\"3;=S,IA7m@F27$$\"3!*****\\ (=]@W$Fco$\"3w4%eC\"p]y@F27$$\"35L$e*[$z*RNFco$\"3UyOr,.R*=#F27$$\"3#* ****\\iC$pk$Fco$\"3wIdFs1%4?#F27$$\"39m;H2qcZPFco$\"3Qbx\"QY%\\6AF27$$ \"3q**\\7.\"fF&QFco$\"3f+!e(oz@AAF27$$\"3Ymm;/OgbRFco$\"36qG(yA8CB#F27 $$\"3y**\\ilAFjSFco$\"3v.zLgjzUAF27$$\"3YLLL$)*pp;%Fco$\"3IImU*yHDD#F2 7$$\"3?LL3xe,tUFco$\"3I%R!fhiAiAF27$$\"3em;HdO=yVFco$\"3?ogo1xfrAF27$$ \"3))*****\\#>#[Z%Fco$\"3EO%fx0/+G#F27$$\"3immT&G!e&e%Fco$\"3)zsS%e\"3 %*G#F27$$\"3;LLL$)Qk%o%Fco$\"35e8d5)>wH#F27$$\"37+]iSjE!z%Fco$\"3e%4h. zwhI#F27$$\"35+]P40O\"*[Fco$\"3Nwd2,K=9BF27$$\"3++++++++]Fco$\"3m'>()) [`fABF2-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%%FONTG6$%*HELVETICAG\"\"*-%+A XESLABELSG6$%\"xG%%y(x)G-%%VIEWG6$;F($\"\"&Fj[l;F($\"#EFj[l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The following cod e constructs a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " base d on each of the methods and gives the " }{TEXT 260 22 "root mean squa re error" }{TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 756 "S := (x,y) -> 16/((16*x+1)*y): hh := 0.005: numste ps := 100: x0 := 0: y0 := 1:\nmatrix([[`slope field: `,S(x,y)],[`ini tial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,n umsteps]]);``;\nmthds := [`Huta's scheme A`,`Huta's scheme B`,`a compa nion to Huta's scheme B`,`an efficient scheme with the same nodes and \+ weights as Huta's two schemes`,`a stage order 3 scheme with small prin cipal error norm`]: errs := []:\nDigits := 20:\nfor ct to 5 do\n Sn_ RK6_||ct := RK6_||ct(S(x,y),x,y,x0,y0,hh,numsteps,false);\n sm := 0: numpts := nops(Sn_RK6_||ct):\n for ii to numpts do\n sm := sm+ (Sn_RK6_||ct[ii,2]-s(Sn_RK6_||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,$*(\"#;\"\"\",&*&F+F,%\"xGF,F,F,F,!\"\"%\"yGF0F,7 $%0initial~point:~G-%!G6$\"\"!F,7$%/step~width:~~~G$\"\"&!\"$7$%1no.~o f~steps:~~~G\"$+\"Q)pprint226\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%0Huta's~scheme~AG $\"+hiH(G$!#87$%0Huta's~scheme~BG$\"+E@(*f:F+7$%?a~companion~to~Huta's ~scheme~BG$\"+hhG65F+7$%doan~efficient~scheme~with~the~same~nodes~and~ weights~as~Huta's~two~schemesG$\"+6]Q/F!#97$%Wa~stage~order~3~scheme~w ith~small~principal~error~normG$\"+cG)*)*HF8Q)pprint236\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following cod e constructs " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for solutions based on each of the Runge-Kutta schemes." }}{PARA 0 "" 0 " " {TEXT -1 75 "The error in the value obtained by each of the methods \+ at the point where " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"!" }{TEXT -1 21 ".4995 is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 688 "S := (x,y) -> 16/((16*x+1)*y): hh := 0.005: numsteps := 100: x0 : = 0: y0 := 1:\nmatrix([[`slope field: `,S(x,y)],[`initial point: `,` `(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);``; \nmthds := [`Huta's scheme A`,`Huta's scheme B`,`a companion to Huta's scheme B`,`an efficient scheme with the same nodes and weights as Hut a's two schemes`,`a stage order 3 scheme with small principal error no rm`]: errs := []:\nDigits := 20:\nfor ct to 5 do\n sn_RK6_||ct := RK 6_||ct(S(x,y),x,y,x0,y0,hh,numsteps,true);\nend do:\nxx := 0.4995: sxx := evalf(s(xx)):\nfor ct to 5 do\n errs := [op(errs),abs(sn_RK6_||c t(xx)-sxx)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(e rrs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~fie ld:~~~G,$*(\"#;\"\"\",&*&F+F,%\"xGF,F,F,F,!\"\"%\"yGF0F,7$%0initial~po int:~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$%0Huta's~scheme~AG$\"+yHGvG! #87$%0Huta's~scheme~BG$\"+]%=vN\"F+7$%?a~companion~to~Huta's~scheme~BG $\"+I>&*y()!#97$%doan~efficient~scheme~with~the~same~nodes~and~weights ~as~Huta's~two~schemesG$\"+?n:^BF47$%Wa~stage~order~3~scheme~with~smal l~principal~error~normG$\"+,t\"*)f#F4Q)pprint256\"" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "roo t mean square error" }{TEXT -1 110 " over the interval [0, 0.5] of e ach Runge-Kutta method is estimated as follows using the special proce dure " }{TEXT 0 5 "NCint" }{TEXT -1 97 " to perform numerical integr ation by the 7 point Newton-Cotes method over 50 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 441 "mthds := [`Huta's scheme \+ A`,`Huta's scheme B`,`a companion to Huta's scheme B`,`an efficient sc heme with the same nodes and weights as Huta's two schemes`,`a stage o rder 3 scheme with small principal error norm`]: errs := []:\nDigits : = 20:\nfor ct to 5 do\n sm := NCint((s(x)-'sn_RK6_||ct'(x))^2,x=0..0 .5,adaptive=false,numpoints=7,factor=50);\n errs := [op(errs),sqrt(s m/0.5)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs) ]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%0Huta's~scheme ~AG$\"+c$z\"*G$!#87$%0Huta's~scheme~BG$\"+\"=;2c\"F+7$%?a~companion~to ~Huta's~scheme~BG$\"+CSr65F+7$%doan~efficient~scheme~with~the~same~nod es~and~weights~as~Huta's~two~schemesG$\"+%=$f0F!#97$%Wa~stage~order~3~ scheme~with~small~principal~error~normG$\"+Xc.+IF8Q)pprint266\"" }}} {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 0 524 "evalf[20]( plot(['sn_RK6_1'(x)-s(x),'sn_RK6_2'(x)-s(x),'sn_RK6_3'(x)-s(x),'sn_RK6 _4'(x)-s(x),'sn_RK6_5'(x)-s(x)],\nx=0..0.5,color=[COLOR(RGB,.95,0,.2), COLOR(RGB,1,.55,0),COLOR(RGB,.5,0,.95),COLOR(RGB,0,.75,.2),COLOR(RGB,0 ,.55,.95)],\nlegend=[`Huta's scheme A`,`Huta's scheme B`,`a companion \+ to Huta's scheme B`,`an efficient scheme with the same nodes and weigh ts as Huta's two schemes`,`a stage order 3 scheme with small principal error norm`],font=[HELVETICA,9],title=`error curves for 8 stage order 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 1016 503 503 {PLOTDATA 2 "6+-%'CURVESG6%7ip7$$\"\"!F)F(7$$\"5ommmTN@Ki8!#A$\"/s\\a2 bRL!#>7$$\"5NLLL$3FWYs#F-$\"08CQ$\\xYFF07$$\"5-++vo/[AlIF-$\"0T()>.vo$ RF07$$\"5omm;aQ`!eS$F-$\"0uv?1&\\NaF07$$\"5NLLeRseQYPF-$\"0Tmpda2G(F07 $$\"5-+++D1k'p3%F-$\"0=D9\")H9^*F07$$\"5pmmT5SpaFWF-$\"1[CjB@p;7F07$$ \"5OLL$eRZF\"oZF-$\"1%yaFw\"F07$$\" 5SLLL3x1h6oF-$\"1L()*f]Hxx\"F07$$\"5qmm;zWF07$$\"5SLL$3-)Q4b))F-$\"1dv>tW$>;#F07$$\" 5qmmm\"z%\\DO&*F-$\"1CCheS*\\X#F07$$\"5NLL3x\"[No()*F-$\"1)H!=k&4Gk#F0 7$$\"5+++Dc,;u@5!#@$\"1HPbS;'>r#F07$$\"5nm;z%\\l*zb5Fhq$\"18yi%\\-Jq#F 07$$\"5MLLLL3x&)*3\"Fhq$\"1$**G@FVjp#F07$$\"5nm\"z>'o^7\\6Fhq$\"1Wn$[O XKp#F07$$\"5++]i!*GER37Fhq$\"1&3$>gN*oq#F07$$\"5ML3F>*3gwE\"Fhq$\"10_; 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" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 169 "de := diff(y(x),x)=(1+2*(x+1)*sin(3*x))*exp(-y(x));\nic := y(0) =0;\ndsolve(\{de,ic\},y(x));\nu := unapply(rhs(%),x):\nplot(u(x),x=0.. 5,font=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,*&,&\"\"\"F/*(\"\"#F/,& F,F/F/F/F/-%$sinG6#,$*&\"\"$F/F,F/F/F/F/F/-%$expG6#,$F)!\"\"F/" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%#lnG6#,,F'\"\"\"*&#\"\"#\"\"*F,-%$ sinG6#,$*&\"\"$F,F'F,F,F,F,*&#F/F6F,*&F'F,-%$cosGF3F,F,!\"\"*&#F/F6F,F :F,F<#\"\"&F6F," }}{PARA 13 "" 1 "" {GLPLOT2D 503 318 318 {PLOTDATA 2 "6&-%'CURVESG6$7bp7$$\"\"!F)F(7$$\"3GLLL3x&)*3\"!#=$\"3QWK+t!=.P\"F-7$ $\"3umm\"H2P\"Q?F-$\"3pUCE&GmM$HF-7$$\"3MLL$eRwX5$F-$\"3l!G\"yWq,6\\F- 7$$\"33ML$3x%3yTF-$\"3dz%)zauhMpF-7$$\"3emm\"z%4\\Y_F-$\"3,G5kQO>C))F- 7$$\"3`LLeR-/PiF-$\"36YrjIBvP5!#<7$$\"3]***\\il'pisF-$\"3wGtPF*HL<\"FI 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/Rz()F-7$$\"3w3F>RL3GTFI$\"3JeP:9JjA()F-7$$\"3t]i!RbX59%FI$\"3mH1#H$\\ k'o)F-7$$\"3#=z>'ox+aTFI$\"3oLr_-o*=n)F-7$$\"3yLLL$)*pp;%FI$\"3A7j1wip y')F-7$$\"3!Q3_+sD-=%FI$\"32pcM,k23()F-7$$\"3#Q$3xc9[$>%FI$\"3Gri,**=4 g()F-7$$\"3'Qe*[$>Pn?%FI$\"3se,X+?^M))F-7$$\"3)QL3-$H**>UFI$\"3Z**e,OD #4$*)F-7$$\"3#R$ek.W]YUFI$\"3i#fiyx0s=*F-7$$\"3)RL$3xe,tUFI$\"3[2R[)*e VA&*F-7$$\"3Cn;HdO=yVFI$\"3#)>Y<=$f\\9\"FI7$$\"3MMe9\"z-lU%FI$\"3)4DVD mlMD\"FI7$$\"3a+++D>#[Z%FI$\"3qZKS'GmoO\"FI7$$\"3TM$3_5,-`%FI$\"3CFB-G n\\(\\\"FI7$$\"3SnmT&G!e&e%FI$\"3t\\(p9r/Xi\"FI7$$\"3m+]P%37^j%FI$\"3_ eaMDR_K " 0 "" {MPLTEXT 1 0 768 "U := (x,y) -> (1+2*(x+1)*sin(3*x))*exp(-y): hh := 0. 01: numsteps := 500: x0 := 0: y0 := 0:\nmatrix([[`slope field: `,U(x ,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of st eps: `,numsteps]]);``;\nmthds := [`Huta's scheme A`,`Huta's scheme B `,`a companion to Huta's scheme B`,`an efficient scheme with the same \+ nodes and weights as Huta's two schemes`,`a stage order 3 scheme with \+ small principal error norm`]: errs := []:\nDigits := 25:\nfor ct to 5 \+ do\n Un_RK6_||ct := RK6_||ct(U(x,y),x,y,x0,y0,hh,numsteps,false);\n \+ sm := 0: numpts := nops(Un_RK6_||ct):\n for ii to numpts do\n \+ sm := sm+(Un_RK6_||ct[ii,2]-u(Un_RK6_||ct[ii,1]))^2;\n end do:\n \+ errs := [op(errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[tr anspose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'m atrixG6#7&7$%0slope~field:~~~G*&,&\"\"\"F+*(\"\"#F+,&%\"xGF+F+F+F+-%$s inG6#,$*&\"\"$F+F/F+F+F+F+F+-%$expG6#,$%\"yG!\"\"F+7$%0initial~point:~ G-%!G6$\"\"!FA7$%/step~width:~~~G$F+!\"#7$%1no.~of~steps:~~~G\"$+&Q)pp rint276\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%0Huta's~scheme~AG$\"+*f*p\\7!#97$%0Hut a's~scheme~BG$\"+@%)4SS!#:7$%?a~companion~to~Huta's~scheme~BG$\"+K()Hp >F07$%doan~efficient~scheme~with~the~same~nodes~and~weights~as~Huta's~ two~schemesG$\"+==j+g!#;7$%Wa~stage~order~3~scheme~with~small~principa l~error~normG$\"+HDn)G&F9Q)pprint286\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code constructs " } {TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for solutions based on each of the Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "T he error in the value obtained by each of the methods at the point whe re " }{XPPEDIT 18 0 "x = 4.999;" "6#/%\"xG-%&FloatG6$\"%**\\!\"$" } {TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 699 "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 := [`Huta's scheme A`,`Huta's scheme B`,`a compan ion to Huta's scheme B`,`an efficient scheme with the same nodes and w eights as Huta's two schemes`,`a stage order 3 scheme with small princ ipal error norm`]: errs := []:\nDigits := 25:\nfor ct to 5 do\n un_R K6_||ct := RK6_||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_RK6_||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$%0Huta's~scheme~AG$\"+SelE?!#97$%0Huta's~scheme~BG$ \"+x\\j^j!#:7$%?a~companion~to~Huta's~scheme~BG$\"+*p#>PGF07$%doan~eff icient~scheme~with~the~same~nodes~and~weights~as~Huta's~two~schemesG$ \"+'**>*Q))!#;7$%Wa~stage~order~3~scheme~with~small~principal~error~no rmG$\"+\"y5=W#F9Q)pprint306\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square erro r" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 5];" "6#7$ \"\"!\"\"&" }{TEXT -1 82 " of each Runge-Kutta method is estimated as follows using the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perform numerical integration by the 7 point Newton-Cotes met hod over 200 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 438 "mthds := [`Huta's scheme A`,`Huta's scheme B`,`a companion to Huta's scheme B`,`an efficient scheme with the same nodes and weights as Huta's two schemes`,`a stage order 3 scheme with small principal e rror norm`]: errs := []:\nDigits := 20:\nfor ct to 5 do\n sm := NCin t((u(x)-'un_RK6_||ct'(x))^2,x=0..5,adaptive=false,numpoints=7,factor=2 00);\n errs := [op(errs),sqrt(sm/5)];\nend do:\nDigits := 10:\nlinal g[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# K%'matrixG6#7'7$%0Huta's~scheme~AG$\"+pHXY7!#97$%0Huta's~scheme~BG$\"+ ')owGS!#:7$%?a~companion~to~Huta's~scheme~BG$\"+YY0k>F07$%doan~efficie nt~scheme~with~the~same~nodes~and~weights~as~Huta's~two~schemesG$\"+(4 'y%)f!#;7$%Wa~stage~order~3~scheme~with~small~principal~error~normG$\" +;1#RH&F9Q)pprint316\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following error graphs are constructed using the numerical procedures for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 522 "evalf[20](plot([u(x)-'un_RK6_1'(x),u(x)-'un_RK6_2'(x ),u(x)-'un_RK6_3'(x),u(x)-'un_RK6_4'(x),u(x)-'un_RK6_5'(x)],\nx=0..5,c olor=[COLOR(RGB,.95,0,.2),COLOR(RGB,1,.55,0),COLOR(RGB,.5,0,.95),COLOR (RGB,0,.75,.2),COLOR(RGB,0,.55,.95)],\nlegend=[`Huta's scheme A`,`Huta 's scheme B`,`a companion to Huta's scheme B`,`an efficient scheme wit h the same nodes and weights as Huta's two schemes`,`a stage order 3 s cheme with small principal error norm`],font=[HELVETICA,9],title=`erro r curves for 8 stage order 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 1134 615 615 {PLOTDATA 2 "6+-%'CURVESG6%7^v7$$\"\"!F)F(7$ 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" }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "y = 1/3" "6#/%\"yG*&\"\"\"F&\"\"$!\"\"" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "exp(-4/3*sin(3*x)+8/3*sin(3/2*x)*cos(3/2*x))+2/3" "6 #,&-%$expG6#,&*(\"\"%\"\"\"\"\"$!\"\"-%$sinG6#*&F+F*%\"xGF*F*F,**\"\") F*F+F,-F.6#*(F+F*\"\"#F,F1F*F*-%$cosG6#*(F+F*F7F,F1F*F*F*F**&F7F*F+F,F *" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(-4/3*sin(3*x)-x)" "6#-%$expG6#, &*(\"\"%\"\"\"\"\"$!\"\"-%$sinG6#*&F*F)%\"xGF)F)F+F0F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 181 "de := diff(y(x),x)=-(1+4*cos(3*x))*(y(x)-1/3);\nic : = y(0)=1;\nsimplify(dsolve(\{de,ic\},y(x)));\nv := unapply(rhs(%),x): \nplot(v(x),x=0..5,0..1.1,font=[HELVETICA,9],labels=[`x`,`y(x)`]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,$*&,& \"\"\"F0*&\"\"%F0-%$cosG6#,$*&\"\"$F0F,F0F0F0F0F0,&F)F0#F0F8!\"\"F0F; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*&#\"\"\"\"\"$F+-%$exp G6#,&*&#\"\"%F,F+-%$sinG6#,$*&F,F+F'F+F+F+!\"\"*&#\"\")F,F+*&-F56#,$*( F,F+\"\"#F9F'F+F+F+-%$cosGF?F+F+F+F+F+*&#FBF,F+-F.6#,&F'F9*&#F3F,F+F4F +F9F+F+" }}{PARA 13 "" 1 "" {GLPLOT2D 503 318 318 {PLOTDATA 2 "6&-%'CU RVESG6$7ap7$$\"\"!F)$\"\"\"F)7$$\"3gmmTN@Ki8!#>$\"3W+7cSy5h&*!#=7$$\"3 ALL$3FWYs#F/$\"3KtP[t*Q;:*F27$$\"3%)***\\iSmp3%F/$\"3g.\"H>f!3q()F27$$ \"3WmmmT&)G\\aF/$\"36p*p.:G\\T)F27$$\"3m****\\7G$R<)F/$\"3a?glh]$zx(F2 7$$\"3GLLL3x&)*3\"F2$\"3IM[S(o-#HsF27$$\"3em\"z%\\v#pK\"F2$\"3i=)H'*Q$ =:oF27$$\"3))**\\i!R(*Rc\"F2$\"3w,'pRB0LX'F27$$\"3&edVF27$$\"3%QL$3_DG1qF2$\"3'fN^ hMLe*)>VB$)F2$\"3DB(Rfp)*\\j%F27$$\"3Y++DJbw!Q*F2$\"3%Gs Cu$*)zK]F27$$\"3+N$ekGkX#**F2$\"3u>+\\,YW?`F27$$\"3%ommTIOo/\"!#<$\"3q ]2x8ZEqcF27$$\"3E+]7GTt%4\"Fgt$\"39b$=$pWlHgF27$$\"3YLL3_>jU6Fgt$\"3nw Ydkc=KkF27$$\"3ym;HdNb'>\"Fgt$\"3l[hQOW]BpF27$$\"37++]i^Z]7Fgt$\"3IVnF )*yXIuF27$$\"35+++v\"=YI\"Fgt$\"3ahS!3L%e=zF27$$\"33++](=h(e8Fgt$\"3l& QV-<82M)F27$$\"3&*****\\7!Q4T\"Fgt$\"3^]H\"3wS2k)F27$$\"3/++]P[6j9Fgt$ \"3ur)[IAj$)z)F27$$\"3'=HKkAg\\Z\"Fgt$\"33z^;ogY6))F27$$\"3W$ek`h0o[\" Fgt$\"3h=q?g>u:))F27$$\"3/voH/5l)\\\"Fgt$\"3p\\\\U!)G36))F27$$\"3%o;HK R'\\5:Fgt$\"3%G4&GMdV(z)F27$$\"3-]P4rr=M:Fgt$\"3Erd.MaCV()F27$$\"3UL$e *[z(yb\"Fgt$\"3m)))[\\1qQl)F27$$\"34+Dc,#>Uh\"Fgt$\"3(fTb\\\\y3J)F27$$ \"3wmm;a/cq;Fgt$\"3-!y\"yF27$$\"3\"pm;a)))G=BtF 27$$\"3%ommmJFgt$\"3%RlX>.MR=&F2 7$$\"3gmmm\"pW`(>Fgt$\"3+6YS9:C2[F27$$\"3dLe9TOEH?Fgt$\"3!eWte3T%oWF27 $$\"3K+]i!f#=$3#Fgt$\"3:XZ<;2j,UF27$$\"3?+](=xpe=#Fgt$\"3E#Q(H44MbQF27 $$\"37nm\"H28IH#Fgt$\"3MH4)f2==l$F27$$\"3um;zpSS\"R#Fgt$\"3wpxg#Fgt$\"37l*=e[ EHY$F27$$\"33+]Pf4t.FFgt$\"35!4Ne]qiX$F27$$\"3uLLe*Gst!GFgt$\"3U+pq))z 7kMF27$$\"30+++DRW9HFgt$\"37'z:1TS%*[$F27$$\"3:++DJE>>IFgt$\"3N!o4Joz] `$F27$$\"3F+]i!RU07$Fgt$\"3=,?;D0\"Qg$F27$$\"3+++v=S2LKFgt$\"3wRH=fZn5 PF27$$\"3Jmmm\"p)=MLFgt$\"3RsXuk([b#QF27$$\"3B++](=]@W$Fgt$\"3%4[=*QOM SRF27$$\"3mm\"H#oZ1\"\\$Fgt$\"3QK??D+QyRF27$$\"35L$e*[$z*RNFgt$\"3UAxt ;S)>+%F27$$\"3%o;Hd!fX$f$Fgt$\"3+h91z&\\y+%F27$$\"3e++]iC$pk$Fgt$\"3eI Rs#H!Q\"*RF27$$\"3ILe*[t\\sp$Fgt$\"3m\"Rx)H&*[cRF27$$\"3[m;H2qcZPFgt$ \"3w)))[$RF!f!RF27$$\"3O+]7.\"fF&QFgt$\"3+Efp,iIqPF27$$\"3Ymm;/OgbRFgt $\"3W-Tml[`MOF27$$\"3w**\\ilAFjSFgt$\"3&zNMj#[Z%Fgt$\"3ADU\\K%G5O$F27$$\"3SnmT&G!e&e% Fgt$\"35gRzc#\\LF27$$\"\"&F) $\"3Ii#4)y!3AN$F2-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%%FONTG6$%*HELVETICA G\"\"*-%+AXESLABELSG6$%\"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 follo wing code constructs a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " based on each of the methods and gives the " }{TEXT 260 22 "root \+ mean square error" }{TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 763 "V := (x,y) -> -(1+4*cos(3*x))*(y-1/3): hh \+ := 0.02: numsteps := 250: x0 := 0: y0 := 1:\nmatrix([[`slope field: \+ `,V(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. \+ of steps: `,numsteps]]);``;\nmthds := [`Huta's scheme A`,`Huta's sch eme B`,`a companion to Huta's scheme B`,`an efficient scheme with the \+ same nodes and weights as Huta's two schemes`,`a stage order 3 scheme \+ with small principal error norm`]: errs := []:\nDigits := 30:\nfor ct \+ to 5 do\n Vn_RK6_||ct := RK6_||ct(V(x,y),x,y,x0,y0,hh,numsteps,false );\n sm := 0: numpts := nops(Vn_RK6_||ct):\n for ii to numpts do\n sm := sm+(Vn_RK6_||ct[ii,2]-v(Vn_RK6_||ct[ii,1]))^2;\n end do: \n errs := [op(errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlina lg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #K%'matrixG6#7&7$%0slope~field:~~~G,$*&,&\"\"\"F,*&\"\"%F,-%$cosG6#,$* &\"\"$F,%\"xGF,F,F,F,F,,&%\"yGF,#F,F4!\"\"F,F97$%0initial~point:~G-%!G 6$\"\"!F,7$%/step~width:~~~G$\"\"#!\"#7$%1no.~of~steps:~~~G\"$]#Q)ppri nt326\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%0Huta's~scheme~AG$\"+J(zB'[!#87$%0Huta 's~scheme~BG$\"+g^,v " 0 "" {MPLTEXT 1 0 694 "V := (x,y) -> -(1+4*cos(3*x))*(y-1/3): hh := 0.02: numsteps := \+ 250: x0 := 0: y0 := 1:\nmatrix([[`slope field: `,V(x,y)],[`initial p oint: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numstep s]]);``;\nmthds := [`Huta's scheme A`,`Huta's scheme B`,`a companion t o Huta's scheme B`,`an efficient scheme with the same nodes and weight s as Huta's two schemes`,`a stage order 3 scheme with small principal \+ error norm`]: errs := []:\nDigits := 30:\nfor ct to 5 do\n vn_RK6_|| ct := RK6_||ct(V(x,y),x,y,x0,y0,hh,numsteps,true);\nend do:\nxx := 4.9 99: vxx := evalf(v(xx)):\nfor ct to 5 do\n errs := [op(errs),abs(vn_ RK6_||ct(xx)-vxx)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds, evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0sl ope~field:~~~G,$*&,&\"\"\"F,*&\"\"%F,-%$cosG6#,$*&\"\"$F,%\"xGF,F,F,F, F,,&%\"yGF,#F,F4!\"\"F,F97$%0initial~point:~G-%!G6$\"\"!F,7$%/step~wid th:~~~G$\"\"#!\"#7$%1no.~of~steps:~~~G\"$]#Q)pprint346\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrix G6#7'7$%0Huta's~scheme~AG$\"+Z'4V$>!#97$%0Huta's~scheme~BG$\"+at,`p!#: 7$%?a~companion~to~Huta's~scheme~BG$\"+?z(oQ$F07$%doan~efficient~schem e~with~the~same~nodes~and~weights~as~Huta's~two~schemesG$\"+z4C+\"*!#; 7$%Wa~stage~order~3~scheme~with~small~principal~error~normG$\"++R%\\I) !#=Q)pprint356\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 5];" "6#7$\"\"!\"\"&" } {TEXT -1 82 " of each Runge-Kutta method is estimated as follows usin g the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perfo rm numerical integration by the 7 point Newton-Cotes method over 100 e qual subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 438 "mthds := [`Huta's scheme A`,`Huta's scheme B`,`a companion to Huta's scheme B`,`an efficient scheme with the same nodes and weights as Huta's two schemes`,`a stage order 3 scheme with small principal error norm`]: e rrs := []:\nDigits := 20:\nfor ct to 5 do\n sm := NCint((v(x)-'vn_RK 6_||ct'(x))^2,x=0..5,adaptive=false,numpoints=7,factor=100);\n errs \+ := [op(errs),sqrt(sm/5)];\nend do:\nDigits := 10:\nlinalg[transpose]([ mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7' 7$%0Huta's~scheme~AG$\"+')o_l[!#87$%0Huta's~scheme~BG$\"+<&\\hx\"F+7$% ?a~companion~to~Huta's~scheme~BG$\"+U'Hgy)!#97$%doan~efficient~scheme~ with~the~same~nodes~and~weights~as~Huta's~two~schemesG$\"+P\\D=BF47$%W a~stage~order~3~scheme~with~small~principal~error~normG$\"+YS]l:!#:Q)p print366\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following error graphs are constructed using the numerical \+ procedures for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 522 "evalf[20](plot(['vn_RK6_1'(x)-v(x),'vn_RK6_2'(x)-v(x),'vn_RK6_3 '(x)-v(x),'vn_RK6_4'(x)-v(x),'vn_RK6_5'(x)-v(x)],\nx=0..5,color=[COLOR (RGB,.95,0,.2),COLOR(RGB,1,.55,0),COLOR(RGB,.5,0,.95),COLOR(RGB,0,.75, .2),COLOR(RGB,0,.55,.95)],\nlegend=[`Huta's scheme A`,`Huta's scheme B `,`a companion to Huta's scheme B`,`an efficient scheme with the same \+ nodes and weights as Huta's two schemes`,`a stage order 3 scheme with \+ small principal error norm`],font=[HELVETICA,9],title=`error curves fo r 8 stage order 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 1047 534 534 {PLOTDATA 2 "6+-%'CURVESG6%7gr7$$\"\"!F)F(7$$\" 5NLLL$3FWYs#!#@$!2LFSbS)Qc5!#?7$$\"5qmmmmT&)G\\aF-$!2/oKCBSd7#F07$$\"5 0+++]7G$R<)F-$!2-?wiF(\\sJF07$$\"5MLLLL3x&)*3\"F0$!2'fwluzRXNF07$$\"5n mm\"z%\\v#pK\"F0$!2YGk5,&e))QF07$$\"5+++]i!R(*Rc\"F0$!2go!o$))fY=%F07$ $\"5n;H#o27JOf\"F0$!2U9U%e1%)=VF07$$\"5MLe9\"4&[EB;F0$!2-(G*QuH6J%F07$ $\"5+](oa5e)*Gl\"F0$!2rO!\\F.YhUF07$$\"5nm;z>6B`#o\"F0$!2acn=(*y(GUF07 $$\"5M$e9T8/m@r\"F0$!2_wUE'*)3AUF07$$\"5++vV[r(*zThXGK%F07$$\"5MLL3xJs1,=F0$!2\\dJp([ySWF07$$\"5nm \"Hd?pM.'=F0$!2&)f_cL5sL%F07$$\"5++]PM_@g>>F0$!2D&Q-$G-;I%F07$$\"5n;zp [#)eB\\>F0$!2,OZ)=r.HVF07$$\"5ML3-j7'p)y>F0$!2%pj#GP:sR%F07$$\"5+]PMxU L]3?F0$!27HiOe5)eWF07$$\"5nmmm\"H2P\"Q?F0$!2E%*)pvS%\\S%F07$$\"5+]7yv, LYr?F0$!2_uQe*pQcVF07$$\"5MLe*)fI&*y/@F0$!2s#[(4ia*HVF07$$\"5n;/,Wfd6Q @F0$!2mLWr@F0$!2dxRW\"ep#Q%F07$$\"5M$eRAr@oZ?#F 0$!2QPR<3V]X%F07$$\"5nmTN'fW%4QAF0$!2kP%=W0.'R%F07$$\"5+](o/[n?9F#F0$! 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_ks)yFgil7$$\"5-++]iSjE!z%F^v$!1`r)GM2HS(Fgil7$$\"5-++]P40O\"*[F^v$!18 8yeh=\"y(Fgil7$$\"\"&F)$!1Db9MP1:\"*Fgil-%&COLORG6&%$RGBGF($\"#v!\"#$ \"\"#!\"\"-%'LEGENDG6#%doan~efficient~scheme~with~the~same~nodes~and~w eights~as~Huta's~two~schemesG-F$6%7_uF'7$$\"5SLLL3x1h6oFgil$!.WHu**fS* F07$$\"5ommmTN@Ki8F-$!/lNFF%G:$F07$$\"5NLL3FpE!Hq\"F-$!/NcGlmk=F07$$\" 5-++]7.K[V?F-$\"/0U$)>_eIF07$$\"5omm\"zptjSQ#F-$\"/^+D$f/\"GF07$F+$\"/ x>y$e/$>F07$$\"5-++vo/[AlIF-$\".U)ouV3fF07$$\"5omm;aQ`!eS$F-$!.S'Gd/%* GF07$$\"5NLLeRseQYPF-$\".d9s\\Lg*F07$$\"5-+++D1k'p3%F-$\"/?6\"on\\m%F0 7$$\"5OLL$eRZF\"oZF-$\"/S#3L9tUF07$FF$!/8$*[:(RY*F07$FP$!0[6\\l'H!G\"F07$FZ$!0ep #f$Hbn\"F07$Fbp$!0)pI:O0Z>F07$F\\q$!0[`K\\%)G;#F07$$\"5MLLLL$eI8k$F0$! 0MNJ3vPj#F07$Faq$!0g03-t*GHF07$Ffq$!0'e6ZcB\"4$F07$F[r$!0@wBl#3pKF07$F _s$!1g&o#*4+&4NF-7$Fct$!02j3Rc!pPF07$Fht$!0YE\"fBvYQF07$F]u$!0neh:Q\"4 RF07$$\"5qmmmTg()4_))F0$!04VTrL_t$F07$Fbu$!08\"*=lm@K$F07$$\"5qmmT&)3 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" }}{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=[HELVETI CA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/ -%%diffG6$-%\"yG6#%\"xGF,*(F,\"\"\",&\"\"*F.*$)F,\"\"#F.!\"\"F.,&F.F.* $)F)F3F.F.F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!F) " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*&\"\"#!\"\",(*&\" \"$\"\"\")F'\"\"%F/F+*&\"#aF/)F'F*F/F/*$,*\"#kF/*&\"\"*F/)F'\"\")F/F/* &\"$C$F/)F'\"\"'F/F+*&\"%;HF/F0F/F/#F/F*F/#F/F.F/*&F*F/F,#F+F.F+" }} {PARA 13 "" 1 "" {GLPLOT2D 503 318 318 {PLOTDATA 2 "6&-%'CURVESG6$7io7 $$\"\"!F)F(7$$\"3()=*=*=*Qx#G!#>$\"3_LLtbH2)f$!#?7$$\"3uPy$y$yZbcF-$\" 3ZF^'eEW*Q9F-7$$\"3;_8N^$ye6)F-$\"3C$\"3aT8Yqv-h6F>7$$\"3oKCVKs3o@F>$\"3c?q**e 5wz?F>7$$\"3$4\"3\"3T.Ds#F>$\"3+#H`Y\")*G6KF>7$$\"3jy$y$y\"=lB$F>$\"3L 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ca,!\\@%QJFho7$$\"3!*['[')ep#**>Fho$\"3Or<<>%)R)=$Fho7$$\"31Yf%faPE0#F ho$\"3!)Qt2wy;NKFho7$$\"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$$\"3oq-F IB#>F#Fho$\"3%>b4%)\\*>0MFho7$$\"3vvcnb(p?K#Fho$\"35nW;n-%*QMFho7$$\"3 eCVKkVazBFho$\"3))))QO[-#QDFho$\"3]n/[Ud8hNFho 7$$\"3AdnvO#pkf#Fho$\"3[Y'[5$oF(e$Fho7$$\"3'>;i@A4pk#Fho$\"3e\"*)HR3Wu g$Fho7$$\"3$ovcn0fTq#Fho$\"3E1Ex&>gui$Fho7$$\"3;Yf%f1Ojv#Fho$\"3)*3G:e W%Hk$Fho7$$\"3[aS09&4M\"GFho$\"3#G5xO+Znl$Fho7$$\"3OnvcZUliGFho$\"3?xe _?\\\"fm$Fho7$$\"3163\"3DQ(=HFho$\"3I;`e!RAJn$Fho7$$\"3k*=*=HE\"H(HFho $\"3IP\"\\xLemn$Fho7$$\"3%pvcnh^q-$Fho$\"3e^O4S8lwOFho7$$\"3-dnvO9*43$ Fho$\"3y,I^j&GHn$Fho7$$\"3m(H(H<2\"G8$Fho$\"3]\\w?2VllOFho7$$\"3'>*=*= tG))=$Fho$\"3#R\\Gi]bMl$Fho7$$\"3-A;ihz@UKFho$\"3q^o+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)$\"3CxC=r RoRDFho-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%%FONTG6$%*HELVETICAG\"\"*-%+A XESLABELSG6$%\"xG%%y(x)G-%%VIEWG6$;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 cons tructs a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " based on e ach of the methods and gives the " }{TEXT 260 22 "root mean square err or" }{TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 757 "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 := [`Huta's scheme A`,`Huta's scheme B`,`a compan ion to Huta's scheme B`,`an efficient scheme with the same nodes and w eights as Huta's two schemes`,`a stage order 3 scheme with small princ ipal error norm`]: errs := []:\nDigits := 30:\nfor ct to 5 do\n Wn_R K6_||ct := RK6_||ct(W(x,y),x,y,x0,y0,hh,numsteps,false);\n sm := 0: \+ numpts := nops(Wn_RK6_||ct):\n for ii to numpts do\n sm := sm+( Wn_RK6_||ct[ii,2]-w(Wn_RK6_||ct[ii,1]))^2;\n end do:\n errs := [op (errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[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\"$+%Q)pprint376\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%0Huta's~ scheme~AG$\"+cq@?7!#97$%0Huta's~scheme~BG$\"+SK&)yR!#:7$%?a~companion~ to~Huta's~scheme~BG$\"++\\YC>F07$%doan~efficient~scheme~with~the~same~ nodes~and~weights~as~Huta's~two~schemesG$\"+KK[vf!#;7$%Wa~stage~order~ 3~scheme~with~small~principal~error~normG$\"+.xRfNF9Q)pprint386\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The follo wing code constructs " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for solutions based on each of the Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each of the \+ methods at the point where " }{XPPEDIT 18 0 "x = 3.499;" "6#/%\"xG-%& FloatG6$\"%*\\$!\"$" }{TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 688 "W := (x,y) -> x*(9-x^2)/(1+y^2): hh := 0.0 1: numsteps := 400: x0 := 0: y0 := 0:\nmatrix([[`slope field: `,W(x, y)],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of ste ps: `,numsteps]]);``;\nmthds := [`Huta's scheme A`,`Huta's scheme B` ,`a companion to Huta's scheme B`,`an efficient scheme with the same n odes and weights as Huta's two schemes`,`a stage order 3 scheme with s mall principal error norm`]: errs := []:\nDigits := 30:\nfor ct to 5 d o\n wn_RK6_||ct := RK6_||ct(W(x,y),x,y,x0,y0,hh,numsteps,true);\nend do:\nxx := 3.499: wxx := evalf(w(xx)):\nfor ct to 5 do\n errs := [o p(errs),abs(wn_RK6_||ct(xx)-wxx)];\nend do:\nDigits := 10:\nlinalg[tra nspose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'ma trixG6#7&7$%0slope~field:~~~G*(%\"xG\"\"\",&\"\"*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)pprint396\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7' 7$%0Huta's~scheme~AG$\"+GV\"e0\"!#97$%0Huta's~scheme~BG$\"+j&[ak$!#:7$ %?a~companion~to~Huta's~scheme~BG$\"+*=&=b=F07$%doan~efficient~scheme~ with~the~same~nodes~and~weights~as~Huta's~two~schemesG$\"+U.yMb!#;7$%W a~stage~order~3~scheme~with~small~principal~error~normG$\"+=4vpMF9Q)pp rint406\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0,4]" "6#7$\"\"!\"\"%" }{TEXT -1 82 " of each Runge-Kutta method is estimated as follows using the specia l procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perform numerical integration by the 7 point Newton-Cotes method over 200 equal subinte rvals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 438 "mthds := [`Huta's scheme A`,`Huta's scheme B`,`a companion to Huta's scheme B`,`an effi cient scheme with the same nodes and weights as Huta's two schemes`,`a stage order 3 scheme with small principal error norm`]: errs := []:\n Digits := 20:\nfor ct to 5 do\n sm := NCint((w(x)-'wn_RK6_||ct'(x))^ 2,x=0..4,adaptive=false,numpoints=7,factor=200);\n errs := [op(errs) ,sqrt(sm/4)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf( errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%0Huta's~s cheme~AG$\"+C:j(>\"!#97$%0Huta's~scheme~BG$\"+>V7 " 0 "" {MPLTEXT 1 0 540 "evalf [30](plot([w(x)-'wn_RK6_1'(x),w(x)-'wn_RK6_2'(x),w(x)-'wn_RK6_3'(x),w( x)-'wn_RK6_4'(x),w(x)-'wn_RK6_5'(x)],\nx=0..4,-1.69e-5..1.69e-5,color= [COLOR(RGB,.95,0,.2),COLOR(RGB,1,.55,0),COLOR(RGB,.5,0,.95),COLOR(RGB, 0,.75,.2),COLOR(RGB,0,.55,.95)],\nlegend=[`Huta's scheme A`,`Huta's sc heme B`,`a companion to Huta's scheme B`,`an efficient scheme with the same nodes and weights as Huta's two schemes`,`a stage order 3 scheme with small principal error norm`],font=[HELVETICA,9],title=`error cur ves for 8 stage order 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 1031 551 551 {PLOTDATA 2 "6+-%'CURVESG6%7go7$$\"\"!F)F(7$$\" ?LLLLLLLLLLL3VfV!#J$\"8OW\"4yd#\\x>Vp%!#K7$$\"?mmmmmmmmmmm;')=()F-$\"9 ,A9#oaI4tA`8*F-7$$\"?MLLLLLLLLeR?ah5!#I$\":a$z`l1,k\"*GFBAF-7$$\"?++++ +++++]7z>^7F9$\":xz(4o2$[vr\"p8YF-7$$\"?mmmmmmmmmT&y`3W\"F9$\":An;:%)= RFHU'f&)F-7$$\"?LLLLLLLLLLe'40j\"F9$\":s]u@Mmgfs.hX\"F97$$\"?+++++++++ 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bl$\"9:]F'p=8*Gg)pi#Ffq-F[cl6&F]clF($\"#vF`clFacl-Fecl6#%doan~efficien t~scheme~with~the~same~nodes~and~weights~as~Huta's~two~schemesG-F$6%7 \\pF'7$F+$!9m/R$f\"G^[*f..$F07$F2$!9#\\@xxK)yL>Q\"Q\"F-7$F=$!9+9)\\i3yU3F9$ !8]#*G?&*=R&Hs9hF97$FL$!8a%=;3rXt%=O-'F97$$\"?nmmmmmmm;z>$HP;#F9$!8XTK 0C-&Rr2!p&F97$$\"?MLLLLLLLL3_KPqAF9$!8QO\\l9'HHaNu]F97$$\"?++++++++]P% =9`\\GF97$FV$\"8W7:TCU\"G)pej&F 97$Fen$\"9Eb/'yu%)4&G]%p\"F97$F_o$\"9W-s1([m-#*f-\"GF97$Fcp$\"95r)HqPR daU**e$F97$$\"?+++++++++](oZiH%F9$\"9\\'e/>b*Gz%[Pm$F97$$\"?mmmmmmmmmm ;EI&R%F9$\"9o_r8$)R)o@u'3PF97$$\"?LLLLLLLLL$eadV\\%F9$\"9gNwqT%[qX9es$ F97$Fhp$\"9t%H?S$*RQ3[mr$F97$$\"?LLLLLLLLLLLB_\"z%F9$\"9hB#oAUk!yVoEOF 97$F]q$\"9`(f]sx_zCh\\X$F97$$\"?NLLLLLLLLLeZ*)*R&F9$\"9?!oJKL$oX+i1HF9 7$Fbq$\"8UY-)>dx&3b9A#Ffq7$Fhq$\"7=g9(G@HL&G7yFfq7$F]r$!7jj9YVLnYs**[F fq7$Fbr$!8]%)p!*[WXzrOR\"Ffq7$Fgr$!8?^C+\"R(zhCx)>Ffq7$F\\s$!8%f=7i&*) H3B)*[#Ffq7$Fas$!8#QJ#[g>$>][(z#Ffq7$Ffs$!8&>vxaM-uUgPIFfq7$F[t$!8X1S8 UpzxmO;$Ffq7$F`t$!85dt*)4hD[GbG$Ffq7$Fet$!8r4A5HP)*ylHO$Ffq7$Fjt$!8xtw w%)))Q0&4FMFfq7$F_u$!8\")H6V@Q57?+Z$Ffq7$Fdu$!8&36lgKGE.9#\\$Ffq7$Fiu$ !8`DVIvvA8t2_$Ffq7$F^v$!8'oZ.&pY#Ro$Ffq7 $Ffx$!8.05S562aT(3PFfq7$F[y$!8a['pj\"eGPe0u$Ffq7$F`y$!8yOR;_c@GS5x$Ffq 7$Fey$!8YJE7[i>bGT!QFfq7$Fjy$!8#*Ql\"ez&)zD_MQFfq7$F_z$!8(\\@+!o/\\8>L 'QFfq7$Fdz$!8InwB!RWd#)4()QFfq7$Fiz$!8Jx&H.c$eYKL!RFfq7$F^[l$!81sr['3L Cj.0RFfq7$Fc[l$!8A^]-CY#[s\")yQFfq7$Fh[l$!8ew@Ka3O*3h?QFfq7$F]\\l$!8$e =U=wy0ki\"p$Ffq7$Fb\\l$!8/z*QWp?)3N9Y$Ffq7$Fg\\l$!88K%[,\\L#Gp*zIFfq7$ F\\]l$!8)[`oM)o`d$=iBFfq7$Fa]l$!81CT&z$\\3&z&['=Ffq7$Ff]l$!8o-ymS0_G(Q /7Ffq7$F[^l$!7$>!*)*G^V0BIk\"Ffq7$F`^l$\"8FVG!H@&p!H+K6Ffq7$Fe^l$\"8A2 1*=*o72:#f>Ffq7$Fj^l$\"8Y[&>p[JM>hWHFfq7$F__l$\"8JIm![@h,wjETFfq7$Fd_l $\"8=;yl7eIs[db&Ffq7$Fi_l$\"8W\"[O%\\G\\b3oT'Ffq7$F^`l$\"8L&G.d#*\\;_H !R(Ffq7$Fc`l$\"8'G&*pI#Ggq0!)\\)Ffq7$Fh`l$\"8P_4`9]/ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "Test 9 of 8 stage order 6 Runge-Kutta methods " }} {PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "dy/dx=-(1+cos(2*x)) *y^3" "6#/*&%#dyG\"\"\"%#dxG!\"\",$*&,&F&F&-%$cosG6#*&\"\"#F&%\"xGF&F& F&*$%\"yG\"\"$F&F(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(0) = sqrt(2); " "6#/-%\"yG6#\"\"!-%%sqrtG6#\"\"#" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = 1/sqrt(sin(2*x)+2*x+1/2)" "6#/%\"yG*&\"\"\"F&-%%sqrtG6#,(-%$ 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 0 770 "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 := [`Huta's scheme A`,`Huta's scheme B`,`a companion to Huta's scheme B`,`an efficient scheme with the same nodes and weig hts as Huta's two schemes`,`a stage order 3 scheme with small principa l error norm`]: errs := []:\nDigits := 20:\nfor ct to 5 do\n Mn_RK6_ ||ct := RK6_||ct(M(x,y),x,y,x0,evalf(y0),hh,numsteps,false);\n sm := 0: numpts := nops(Mn_RK6_||ct):\n for ii to numpts do\n sm := \+ sm+(Mn_RK6_||ct[ii,2]-m(Mn_RK6_||ct[ii,1]))^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\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)pprint426\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6 #7'7$%0Huta's~scheme~AG$\"+x0@S7!#87$%0Huta's~scheme~BG$\"+&\\+\\p%!#9 7$%?a~companion~to~Huta's~scheme~BG$\"+RupyCF07$%doan~efficient~scheme ~with~the~same~nodes~and~weights~as~Huta's~two~schemesG$\"+omD5n!#:7$% Wa~stage~order~3~scheme~with~small~principal~error~normG$\"+Ki2nEF9Q)p print436\"" }}}{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 = 2.99 9;" "6#/%\"xG-%&FloatG6$\"%**H!\"$" }{TEXT -1 16 " is also given." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 701 "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 := [`Huta's scheme A`,`Huta's scheme B`,`a companion to Huta's scheme B`,`an efficient s cheme with the same nodes and weights as Huta's two schemes`,`a stage \+ order 3 scheme with small principal error norm`]: errs := []:\nDigits \+ := 20:\nfor ct to 5 do\n mn_RK6_||ct := RK6_||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_RK6_||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)pprint446\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%0Huta's~scheme~AG$\"+!e.9Z$!#9 7$%0Huta's~scheme~BG$\"+pgZ&H\"F+7$%?a~companion~to~Huta's~scheme~BG$ \"+MN66o!#:7$%doan~efficient~scheme~with~the~same~nodes~and~weights~as ~Huta's~two~schemesG$\"+yq[t=F47$%Wa~stage~order~3~scheme~with~small~p rincipal~error~normG$\"+6bWo#)!#;Q)pprint456\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root \+ mean square error" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 3];" "6#7$\"\"!\"\"$" }{TEXT -1 82 " of each Runge-Kutta method \+ is estimated as follows using the special procedure " }{TEXT 0 5 "NCi nt" }{TEXT -1 98 " to perform numerical integration by the 7 point Ne wton-Cotes method over 150 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 438 "mthds := [`Huta's scheme A`,`Huta's scheme B`,` a companion to Huta's scheme B`,`an efficient scheme with the same nod es and weights as Huta's two schemes`,`a stage order 3 scheme with sma ll principal error norm`]: errs := []:\nDigits := 20:\nfor ct to 5 do \n sm := NCint((m(x)-'mn_RK6_||ct'(x))^2,x=0..3,adaptive=false,numpo ints=7,factor=150);\n errs := [op(errs),sqrt(sm/3)];\nend do:\nDigit s := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%0Huta's~scheme~AG$\"+&*e*GB\"!#87$%0Hu ta's~scheme~BG$\"+Y-!fm%!#97$%?a~companion~to~Huta's~scheme~BG$\"+:>:j CF07$%doan~efficient~scheme~with~the~same~nodes~and~weights~as~Huta's~ two~schemesG$\"+6g;qm!#:7$%Wa~stage~order~3~scheme~with~small~principa l~error~normG$\"+yE'fl#F9Q)pprint466\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following error graphs are const ructed using the numerical procedures for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 522 "evalf[25](plot([m(x)-'mn_RK6_1'(x) ,m(x)-'mn_RK6_2'(x),m(x)-'mn_RK6_3'(x),m(x)-'mn_RK6_4'(x),m(x)-'mn_RK6 _5'(x)],\nx=0..3,color=[COLOR(RGB,.95,0,.2),COLOR(RGB,1,.55,0),COLOR(R GB,.5,0,.95),COLOR(RGB,0,.75,.2),COLOR(RGB,0,.55,.95)],\nlegend=[`Huta 's scheme A`,`Huta's scheme B`,`a companion to Huta's scheme B`,`an ef ficient scheme with the same nodes and weights as Huta's two schemes`, `a stage order 3 scheme with small principal error norm`],font=[HELVET ICA,9],title=`error curves for 8 stage order 6 Runge-Kutta methods`)); " }}{PARA 13 "" 1 "" {GLPLOT2D 1122 583 583 {PLOTDATA 2 "6+-%'CURVESG6 %7ap7$$\"\"!F)$\"%*o\"!#C7$$\":++++++DJ?$[V?!#F$\"4aH1uU)Gv$4\"F,7$$\" :++++++]iSmp3%F0$\"45(F,7$$\":+++++]i:gT<-\"!#E$ \"6\"yCbNU0)HAI\"F,7$$\":++++++v=#**3E7FE$\"6%eV'H$)4O!e%G\"F,7$$\":++ +++](=U#Q/V\"FE$\"6LU\"HO:8i(QK\"F,7$$\":+++++++Dc'yM;FE$\"6#o^di=J&[9 Z\"F,7$$\":+++++]7G)[8R=FE$\"6]9$)*y3xNTzGYeFF,7$$ F5FE$\"6EAkqWK>uI3$F,7$$\":+++++]ils98H%FE$\"6%e8TD!Gv$yQIF,7$$\":++++ ++vo/jc\\%FE$\"6$H%[E#><*=>.$F,7$$\":+++++](=n8,+ZFE$\"6eo+^s>0,z3$F,7 $$\":+++++++vofV!\\FE$\"65O2@Gi=xCB$F,7$$\":++++++](oHv@dFE$\"66@,)*\\ n:V@H$F,7$$\":++++++++DY\"RlFE$\"6q?sLathOzP$F,7$$\":+++++]PMxb.D(FE$ \"6lfxc6w\\ErZ$F,7$$\":++++++voHl:'zFE$\"6$*RAxg*[@&y`$F,7$$\":+++++v$ fe+<<$)FE$\"6O9P\"p\")H4!f[$F,7$$\":+++++]7.#[xs')FE$\"6$eHpMq)=5;Y$F, 7$$\":++++](=<,sd]))FE$\"6'oe@np%z26\\$F,7$$\":+++++DJ?ez$G!*FE$\"6[>0 Sqzc`#QNF,7$$\":++++]i!*G'>=1#*FE$\"6\")R$*RuQHqo\\$F,7$$\":++++++]PM% )RQ*FE$\"6O8dHFF99RY$F,7$$\":+++++]i!R.k!3\"!#D$\"6c*[\"=,6N&e:MF,7$$ \":++++++]PC#)GA\"Fju$\"6V&fOF(ev?UM$F,7$$\":++++++D1/9Ga\"Fju$\"7o2wG P3#4*\\'4$Fju7$$\":+++++++v$eui=Fju$\"7uUcCR$3T:,&GFju7$$\":++++++++N) z%=#Fju$\"7r&G[!z+E95BEFju7$$\":+++++++D'3&o]#Fju$\"7dWMBgS.eU:CFju7$$ 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E#fg'F,7$$\":++++]PMxnn$R\")FE$\"4G%**zB8l_\"e'F,7$Fjs$\"4G*pG'**3J*=l F,7$$\":++++]7`%RC(\\\\)FE$\"487%e2qUx&\\'F,7$F_t$\"4othRA<6U_'F,7$Fdt $\"4wu$GD1:/5mF,7$Fit$\"4\">u;KR'yKr'F,7$F^u$\"4XDp\"e9Y;PmF,7$Fcu$\"4 Oo&4_kgb&e'F,7$$\":+++++voa5*eR(*FE$\"4g)fWn!GSug'F,7$$\":+++++v=nQ>&4 5Fju$\"4G8\"QgE&\\'4nF,7$$\":++++vo/[i*HF5Fju$\"4wvD#Ho7OUmF,7$$\":+++ +]i!*G')z]/\"Fju$\"4,Yb*p8EW,mF,7$$\":++++Dcw45gG1\"Fju$\"4&))zcEQ)exf 'F,7$Fhu$\"4VLgz+=xzj'F,7$$\":++++vV[rd?%)4\"Fju$\"4+Obzcm!pCnF,7$$\": ++++]PM_\"3?;6Fju$\"4%oHKj0ALomF,7$$\":++++DJ?L0\")R8\"Fju$\"4v!>n5*z'[q$fm'F,7$F^v$\"4[(RM8rz()*f'F,7$Fcv$\"5-A7y+s(R 0I'Fju7$Fhv$\"5VwQ6M9e)H%fFju7$F]w$\"5caJ7+c%e;e&Fju7$Fbw$\"50!)\\I=01 'QA&Fju7$Fgw$\"5Z.W%G)>qKFju7$Fey$\"5&HQ(e9(*zO#)HFju7$Fjy$\"5p'G22VkAZu#Fju7$F_z$\"5'HJm4 m#H*Hd#Fju7$Fdz$\"5Lm$Hr9l%H;CFju7$Fiz$\"5j\\c_](4+iG#Fju7$F^[l$\"5mWc .z`\\u&=#Fju7$Fc[l$\"5_)*3qY;%G46#Fju7$Fh[l$\"5OYR[7N-qR?Fju7$F]\\l$\" 5zW>#Qt8->*>Fju7$Fb\\l$\"5]P>*>;HF$[>Fju7$Fg\\l$\"5iN!=n:#4V>>Fju7$F\\ ]l$\"5HLk'*p0'4k*=Fju7$Fa]l$\"57/ykH]*p8)=Fju7$Ff]l$\"5#G!H.1@)z:(=Fju 7$F[^l$\"5//!)G4impm=Fju7$F`^l$\"5N]8\"*zU\\\\k=Fju7$Fe^l$\"5aAr?n8\"o S'=Fju7$Fj^l$\"5:S5va&eYR'=Fju7$F__l$\"5H;4z/NZoi=Fju7$Fd_l$\"5#*Q')** *e`O'e=Fju7$Fi_l$\"5ylH?+!or/&=Fju7$F^`l$\"5rhnlbrEHP=Fju7$Fc`l$\"5lmW %z&fNE:=Fju7$Fh`l$\"5\"))*)*Hm:%yzy\"Fju7$F]al$\"5lwo**\\8Bb]p(Q9Ow9E)Fju-F]fl6&F_flF(FgfmF`fl-Fgfl6#%Wa~stage~order~3~scheme~ with~small~principal~error~normG-%&TITLEG6#%Uerror~curves~for~8~stage~ order~6~Runge-Kutta~methodsG-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6 $Q\"x6\"Q!Fc\\q-%%VIEWG6$;F(Fhel%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Huta's scheme A" "Huta's scheme B" " a companion to Huta's scheme B" "an efficient scheme with the same nod es and weights as Huta's two schemes" "a stage order 3 scheme with sma ll principal error norm" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 46 "Test 10 of 8 stage order 6 Runge-Kutta methods" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = -(2*sin(5*x)+3*cos( 7*x))*sinh(y);" "6#/*&%#dyG\"\"\"%#dxG!\"\",$*&,&*&\"\"#F&-%$sinG6#*& \"\"&F&%\"xGF&F&F&*&\"\"$F&-%$cosG6#*&\"\"(F&F3F&F&F&F&-%%sinhG6#%\"yG F&F(" }{TEXT -1 5 " , " }{XPPEDIT 18 0 "y(0)=sqrt(5)/2" "6#/-%\"yG6# \"\"!*&-%%sqrtG6#\"\"&\"\"\"\"\"#!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 207 "de := diff(y(x),x)=-(2*sin( 5*x)+3*cos(7*x))*sinh(y(x));\nic := y(0)=sqrt(5)/2;\ndsolve(\{de,ic\}, y(x));\nsimplify(convert(%,exp));\np := unapply(rhs(%),x):\nplot(p(x), x=0..5,font=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,$*&,&*&\"\"#\"\"\"-%$s inG6#,$*&\"\"&F2F,F2F2F2F2*&\"\"$F2-%$cosG6#,$*&\"\"(F2F,F2F2F2F2F2-%% sinhG6#F)F2!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\" \"!,$*&\"\"#!\"\"\"\"&#\"\"\"F,F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ -%\"yG6#%\"xG-%#lnG6#-%%tanhG6#,*#\"\"\"\"\"&F0*&#F0\"\"#F0-F)6#,$*&,& -%$expG6#,$*&F4!\"\"F1F3F0F0F0F0F0,&F:F0F0F?F?F?F0F0*&#\"\"$\"#9F0-%$s inG6#,$*&\"\"(F0F'F0F0F0F0*&#F0F1F0-%$cosG6#,$*&F1F0F'F0F0F0F?" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%#lnG6#*&,*-%$expG6#,4# \"\"#\"\"&!\"\"*&#\"$#>\"\"(\"\"\"*&)-%$cosGF&\"\"'F9-%$sinGF&F9F9F4*& #\"$S#F8F9*&)F<\"\"%F9F?F9F9F9*&#\"#sF8F9*&)F$\"33uw,WSt45F,7$$\"3WmmmT&)G\\aF0$\"3yUB%H69F5*!#=7$$\"3m** **\\7G$R<)F0$\"3[G6@7@G;#)F87$$\"3GLLL3x&)*3\"F8$\"3u_\"Hlv:eW(F87$$\" 3))**\\i!R(*Rc\"F8$\"3aT]N\"zi(yjF87$$\"3umm\"H2P\"Q?F8$\"3xQ:-NK+KcF8 7$$\"3YLek.pu/BF8$\"3:Vt%)yf.P`F87$$\"3!***\\PMnNrDF8$\"3\"zkU]7kD7&F8 7$$\"37$eR(\\;m/FF8$\"3#f>&4'\\tL/&F87$$\"3MmT5ll'z$GF8$\"3#3U'>%*z$=) \\F87$$\"37](o/[r7(HF8$\"3oQ2*>TRs$\\F87$$\"3MLL$eRwX5$F8$\"359>\\xg!* 3\\F87$$\"3:L$3F%\\wQKF8$\"3wA2?_M<'*[F87$$\"3_LLe*[`HP$F8$\"3Y^)o(RAn )*[F87$$\"3*QLek.Ur]$F8$\"3K()))eR[\"e\"\\F87$$\"3rLLL$eI8k$F8$\"35_F/ 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4l5FZ$G\"F,7$$\"3-]P4rr=M:F,$\"3i,%*Q\"yBqC\"F,7$$\"3UL$e*[z(yb\"F,$\" 3+\")o!f;2L>\"F,7$$\"3w;/Ev&[ge\"F,$\"3#3#R!QDA@6\"F,7$$\"34+Dc,#>Uh\" F,$\"39\"HI%*[n*=5F,7$$\"3V$eky#)*QU;F,$\"3mms=Lk!y?*F87$$\"3wmm;a/cq; F,$\"3Pe)[!HQiL#)F87$$\"3\"pm;a)))G=F,$\"35Bp0?YWZMF87$$\"3KLe9;0?E>F,$\"34 :XWhm;,LF87$$\"3pTg-gl[Q>F,$\"3-rZpoe$*\\KF87$$\"31]i!Rgs2&>F,$\"35MI# 4&[)H@$F87$$\"3WekyZ'eI'>F,$\"3%ovQ?ez,>$F87$$\"3gmmm\"pW`(>F,$\"3Q%HR /(>[\"=$F87$$\"3_ek.HW#)))>F,$\"3SdFyRyC)=$F87$$\"3?]iSmTI-?F,$\"3]%Ge CR0B@$F87$$\"3*=/wP!Ry:?F,$\"3OG\"o!ej/aKF87$$\"3dLe9TOEH?F,$\"32!yA?( )GSJ$F87$$\"3'pT&)e6Bi0#F,$\"3EaW;d/\"=\\$F87$$\"3K+]i!f#=$3#F,$\"35[n iL?f`PF87$$\"3/++D\"=EX8#F,$\"3)=7zi$ePDXF87$$\"3?+](=xpe=#F,$\"3rT'ox RDBu&F87$$\"3$pTNrfbE@#F,$\"3I&**Rb6vOf'F87$$\"3mLeRA9WRAF,$\"3Y7cI=h \\:wF87$$\"3S]ilZsAmAF,$\"3O*zb%4IS@))F87$$\"37nm\"H28IH#F,$\"3I'yDR:j =-\"F,7$$\"3!oTN@#3hF,7$$\"3WeRseStdCF,$\"3d+$en#[n??F,7$$\"3)oT5l0+5Z#F,$\"3%)[_;#*)R 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0\"F,7$$\"3=vo/Bh$)yMF,$\"3i,C<-Qm\\5F,7$$\"3mm\"H#oZ1\"\\$F,$\"3%43uA U\"*f/\"F,7$$\"36]Pfe?_:NF,$\"3]?:\"y&yYI5F,7$$\"35L$e*[$z*RNF,$\"3WZc wFIo05F,7$$\"3%o;Hd!fX$f$F,$\"3\\R'zK#G91$*F87$$\"3e++]iC$pk$F,$\"3E?I x(pVQY)F87$$\"3ILe*[t\\sp$F,$\"3-#z!\\!\\E.v(F87$$\"3[m;H2qcZPF,$\"3!Q '*ep\"3v-sF87$$\"3s***\\7.lQx$F,$\"3Il\"\\:&*zo*pF87$$\"3UL$3_0j,!QF,$ \"3cWvs*pf#\\oF87$$\"3F+v=n?J8QF,$\"3KU^fg!Quz'F87$$\"36nm;z5YEQF,$\"3 -)RYLJ)=gnF87$$\"3^Le9\"45'RQF,$\"39oisVpNPnF87$$\"3O+]7.\"fF&QF,$\"3V Eh1S\\sGnF87$$\"3i3_vlYhlQF,$\"33cF9)GTPt'F87$$\"3)oT&QG-ZyQF,$\"3aR=Q 3Gt^nF87$$\"39Dc,\"zD8*QF,$\"3!zI/$G#3By'F87$$\"3TLek`8=/RF,$\"33OP7dc +DoF87$$\"3$*\\i!*yC*)HRF,$\"3reG3V'=X%pF87$$\"3Ymm;/OgbRF,$\"33DC(y#> %[5(F87$$\"3*G$e*[$zV4SF,$\"39]/NuwfQvF87$$\"3w**\\ilAFjSF,$\"3D8xFXpB @!)F87$$\"3#G3_]p'>*3%F,$\"3qve-u54K#)F87$$\"3ym\"zW7@^6%F,$\"3%eL`Rp% o0%)F87$$\"3w3F>RL3GTF,$\"3!)Q$yF^SJZ)F87$$\"3t]i!RbX59%F,$\"3i)pQA'3/ D&)F87$$\"3#=z>'ox+aTF,$\"37$4T=0@'f&)F87$$\"3yLLL$)*pp;%F,$\"3/<(p'zR Pv&)F87$$\"3!Q3_+sD-=%F,$\"3Ga;6C(p2d)F87$$\"3#Q$3xc9[$>%F,$\"3X[ZFn2M W&)F87$$\"3'Qe*[$>Pn?%F,$\"3u\\G!HhJc\\)F87$$\"3)QL3-$H**>UF,$\"3[:#3' R6kC%)F87$$\"3#R$ek.W]YUF,$\"3+Xc4Am3=#)F87$$\"3)RL$3xe,tUF,$\"3%p?zi! 3VLzF87$$\"3Cn;HdO=yVF,$\"3;#>X)HduejF87$$\"3a+++D>#[Z%F,$\"3qyAO;Hx\" )\\F87$$\"3TM$3_5,-`%F,$\"3'RSzCT^zW%F87$$\"3SnmT&G!e&e%F,$\"3%=ER;G9D :%F87$$\"3/]i:NK'zf%F,$\"3*f`:i,h67%F87$$\"3fLe*[=Y.h%F,$\"3U'>WD]()H5 %F87$$\"386but%F,$\"3')>+X\\xqwZF87$$\"37+]iSjE!z%F,$\"3<\"z**[W(=J cF87$$\"3y*\\7G))Rb\"[F,$\"3[YJ\"GK]h>'F87$$\"3L+++DM\"3%[F,$\"3!**fZR -9n(oF87$$\"3)3](=np3m[F,$\"3gVt^2I*Ho(F87$$\"3a+]P40O\"*[F,$\"3`9x`tb ,B')F87$$\"3>]7.#Q?&=\\F,$\"3Ik*[g6ply*F87$$\"3s+voa-oX\\F,$\"3)pQ=b([ U56F,7$$\"3O]PMF,%G(\\F,$\"3_;`pzy]b7F,7$$\"\"&F)$\"3ftg')yo>49F,-%'CO LOURG6&%$RGBG$\"#5!\"\"F(F(-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6$ %\"xG%%y(x)G-%%VIEWG6$;F(Ficn%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "The following code constructs a discrete solution based on each of the methods and gives the " } {TEXT 260 22 "root mean square error" }{TEXT -1 18 " of each solution. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 787 "P := (x,y) -> -(2*sin(5 *x)+3*cos(7*x))*sinh(y): hh := 0.01: numsteps := 500: x0 := 0: y0 := s qrt(5)/2:\nmatrix([[`slope field: `,P(x,y)],[`initial point: `,``(x0 ,y0)],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmth ds := [`Huta's scheme A`,`Huta's scheme B`,`a companion to Huta's sche me B`,`an efficient scheme with the same nodes and weights as Huta's t wo schemes`,`a stage order 3 scheme with small principal error norm`]: errs := []:\nDigits := 30:\nfor ct to 5 do\n Pn_RK6_||ct := RK6_||c t(P(x,y),x,y,x0,evalf(y0),hh,numsteps,false);\n sm := 0: numpts := n ops(Pn_RK6_||ct):\n for ii to numpts do\n sm := sm+(Pn_RK6_||ct [ii,2]-p(Pn_RK6_||ct[ii,1]))^2;\n end do:\n errs := [op(errs),sqrt (sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf( errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~fi eld:~~~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~st eps:~~~G\"$+&Q)pprint476\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%0Huta's~scheme~AG$\"+ d7chB!#87$%0Huta's~scheme~BG$\"+3F[5z!#97$%?a~companion~to~Huta's~sche me~BG$\"+H/_;UF07$%doan~efficient~scheme~with~the~same~nodes~and~weigh ts~as~Huta's~two~schemesG$\"+%=\\]T\"F07$%Wa~stage~order~3~scheme~with ~small~principal~error~normG$\"+hw7DLF0Q)pprint486\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code con structs " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for solu tions 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 0 718 "P := (x,y) -> -(2*sin(5*x)+3*cos(7*x))*sinh(y): hh : = 0.01: numsteps := 500: x0 := 0: y0 := sqrt(5)/2:\nmatrix([[`slope fi eld: `,P(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,hh], \n[`no. of steps: `,numsteps]]);``;\nmthds := [`Huta's scheme A`,`Hu ta's scheme B`,`a companion to Huta's scheme B`,`an efficient scheme w ith the same nodes and weights as Huta's two schemes`,`a stage order 3 scheme with small principal error norm`]: errs := []:\nDigits := 30: \nfor ct to 5 do\n pn_RK6_||ct := RK6_||ct(P(x,y),x,y,x0,evalf(y0),h h,numsteps,true);\nend do:\nxx := 4.999: pxx := evalf(p(xx)):\nfor ct \+ to 5 do\n errs := [op(errs),abs(pn_RK6_||ct(xx)-pxx)];\nend do:\nDig its := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,$*&,&*&\"\"#\"\"\" -%$sinG6#,$*&\"\"&F.%\"xGF.F.F.F.*&\"\"$F.-%$cosG6#,$*&\"\"(F.F5F.F.F. F.F.-%%sinhG6#%\"yGF.!\"\"7$%0initial~point:~G-%!G6$\"\"!,$*&F-FBF4#F. F-F.7$%/step~width:~~~G$F.!\"#7$%1no.~of~steps:~~~G\"$+&Q)pprint496\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%0Huta's~scheme~AG$\"+CUJ\"f\"!#87$%0Huta's~sche me~BG$\"+nvYIw!#97$%?a~companion~to~Huta's~scheme~BG$\"+=$e@[)F07$%doa n~efficient~scheme~with~the~same~nodes~and~weights~as~Huta's~two~schem esG$\"+KUmgMF07$%Wa~stage~order~3~scheme~with~small~principal~error~no rmG$\"+)RH>6\"F+Q)pprint506\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square erro r" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "[0, 5];" "6#7$ \"\"!\"\"&" }{TEXT -1 82 " of each Runge-Kutta method is estimated as follows using the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perform numerical integration by the 7 point Newton-Cotes met hod over 200 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 438 "mthds := [`Huta's scheme A`,`Huta's scheme B`,`a companion to Huta's scheme B`,`an efficient scheme with the same nodes and weights as Huta's two schemes`,`a stage order 3 scheme with small principal e rror norm`]: errs := []:\nDigits := 20:\nfor ct to 5 do\n sm := NCin t((p(x)-'pn_RK6_||ct'(x))^2,x=0..5,adaptive=false,numpoints=7,factor=2 00);\n errs := [op(errs),sqrt(sm/5)];\nend do:\nDigits := 10:\nlinal g[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# K%'matrixG6#7'7$%0Huta's~scheme~AG$\"+vWEjB!#87$%0Huta's~scheme~BG$\"+ i0k;z!#97$%?a~companion~to~Huta's~scheme~BG$\"+i=&[@%F07$%doan~efficie nt~scheme~with~the~same~nodes~and~weights~as~Huta's~two~schemesG$\"+FO s79F07$%Wa~stage~order~3~scheme~with~small~principal~error~normG$\"+!= '[5LF0Q)pprint516\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 92 "The following error graphs are constructed using the nu merical procedures for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 522 "evalf[30](plot(['pn_RK6_1'(x)-p(x),'pn_RK6_2'(x)-p(x ),'pn_RK6_3'(x)-p(x),'pn_RK6_4'(x)-p(x),'pn_RK6_5'(x)-p(x)],\nx=0..5,c olor=[COLOR(RGB,.95,0,.2),COLOR(RGB,1,.55,0),COLOR(RGB,.5,0,.95),COLOR (RGB,0,.75,.2),COLOR(RGB,0,.55,.95)],\nlegend=[`Huta's scheme A`,`Huta 's scheme B`,`a companion to Huta's scheme B`,`an efficient scheme wit h the same nodes and weights as Huta's two schemes`,`a stage order 3 s cheme with small principal error norm`],font=[HELVETICA,9],title=`erro r curves for 8 stage order 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 1091 603 603 {PLOTDATA 2 "6+-%'CURVESG6%7ju7$$\"\"!F)F(7$ $\"?MLLLLLLLLL3x&)*3\"!#I$!;G^O\")>'f$R!QFlR)F-7$$\"?++++++++]i!R(*Rc \"F-$!Wr@F-$!<.Qb>]!GA'4(G>n7F-7$$\"?MLLLLLLLek.pu/BF -$!=0QCF-$!<4*QE!)[6D6sKM/8F-7$$\" ?++++++++]PMnNrDF-$!1:8F-7$$\"?nmmmmmmmT5ll'z$GF-$!97L\"F-7$$\"?MLLLLLLLL$eRwX5$F-$!$)fut,q+&>4n\"F-7$$\"?NLLLLLLLLeR-/PiF-$!<8GaI)*4ZTN/x8&=F- 7$$\"?+++++++++vVVX$\\'F-$!F-7$$\"?qmmmmmmmm\"zWo)\\nF -$!<46xosorWW7.E+#F-7$$\"?+++++++++++b2yoF-$!V**=L(47E?F-7$$\"? 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RbX59%Fdt$\";cL_Uh()eSoV-'ox+aTFdt$\";(>fuoe&Rw#3 4CA$F-7$F\\im$\";JAzou(yWK'*yl@$F-7$F`jm$\";%oVp<)[i*)>>%G4$F-7$Fjjm$ \";A![I3\\cxb!*R-'GF-7$F_[n$\";S0u^g!*H^$))obC#F-7$Fi[n$\";c.&=j'\\5Z? un;m3v7e.e4K8F-7$Fh\\n$\";XA%)=hP'f>=_ML\"F- 7$$\"?NLLLLLL$e9T.&\\ZYFdt$\";WN>kZemr?V,T8F-7$F_eu$\";myd!38O$)[C!zb8 F-7$$\"?-++++++]PfL4EsYFdt$\";'*HRrB#)Qs$F-7$F_`n$\";&)*G))HnVzF**zEh%F-7 $$\"?-++++++DJqX/%\\!\\Fdt$\";7z'G&*R@X>8XE8&F-7$Fd`n$\";A;z96Q%\\CxXH x&F-7$$\"?-++++++v$f$=.5K\\Fdt$\":^&*y'3f7x^![W['Fdt7$Fi`n$\":?Ag0W'*f oc!HNsFdt7$Fa`s$\":Au;t5/#f(Gl/9)Fdt7$F^an$\":G.E))eXF(zX*o2*Fdt7$Fi`s $\";-`%eoc?\"*R=+$35Fdt7$Fcan$\";&=_$>]Gst^\"QE7\"Fdt-Fhan6&FjanF(Fehp F[bn-Fbbn6#%Wa~stage~order~3~scheme~with~small~principal~error~normG-% &TITLEG6#%Uerror~curves~for~8~stage~order~6~Runge-Kutta~methodsG-%%FON TG6$%*HELVETICAG\"\"*-%+AXESLABELSG6$Q\"x6\"Q!F]^x-%%VIEWG6$;F(Fcan%(D EFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Huta 's scheme A" "Huta's scheme B" "a companion to Huta's scheme B" "an ef ficient scheme with the same nodes and weights as Huta's two schemes" "a stage order 3 scheme with small principal error norm" }}}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "Test 11 of 8 stage order 6 Runge -Kutta methods " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 143 "This example is similar to one that appears in an articl e by F. G. Lether: Mathematics of Computation, Vol. 20, no. 95, (July 1966) page 382. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=exp(-x)/(x-1)^2" "6#/*&%#dyG\" \"\"%#dxG!\"\"*&-%$expG6#,$%\"xGF(F&*$,&F.F&F&F(\"\"#F(" }{TEXT -1 2 " " }{XPPEDIT 18 0 "cos(1/(x-1))-y" "6#,&-%$cosG6#*&\"\"\"F(,&%\"xGF(F (!\"\"F+F(%\"yGF+" }{TEXT -1 5 ", " }{XPPEDIT 18 0 "y(0)=sin*1" "6# /-%\"yG6#\"\"!*&%$sinG\"\"\"F*F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "y = -exp(-x)*sin(1/(x-1))" "6#/%\"yG,$*&-%$expG6#,$%\"x G!\"\"\"\"\"-%$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 "d e := diff(y(x),x)=exp(-x)/(x-1)^2*cos(1/(x-1))-y(x);\nic := y(0)=sin(1 );\ndsolve(\{de,ic\},y(x));\nq := unapply(rhs(%),x):\nplot(q(x),x=0..1 -1/(6*Pi),font=[HELVETICA,9],labels=[`x`,`y(x)`]);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,&*(-%$expG6#,$F,!\" \"\"\"\",&F,F4F4F3!\"#-%$cosG6#*&F4F4F5F3F4F4F)F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!-%$sinG6#\"\"\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%\"yG6#%\"xG,$*&-%$expG6#,$F'!\"\"\"\"\"-%$sinG6#*& F/F/,&F'F/F/F.F.F/F." }}{PARA 13 "" 1 "" {GLPLOT2D 603 396 396 {PLOTDATA 2 "6&-%'CURVESG6$7[r7$$\"\"!F)$\"30l*y![)4ZT)!#=7$$\"3#>=\"* )>z2k?!#>$\"3m%p=Qu6DN)F,7$$\"3kMQU\"3@+'QF0$\"3m=O<,5$yO#)F,7$$\"38UQ!)p4'G\"zF0$\"3/Z^$zTlU<)F,7$$\" 3BY$*R0>JO**F0$\"36ty1)z*36\")F,7$$\"3wbXC%*4B\"=\"F,$\"3A;o(=P!Q^!)F, 7$$\"3M!)fxC(zaP\"F,$\"3E8^!ydw!))zF,7$$\"3kgswR?Pw:F,$\"3T8>lD8j?zF,7 $$\"3Qnlb!4?mx\"F,$\"3%p4\\s^P4&yF,7$$\"3OsvSC)*f#)>F,$\"3/$H(=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$$\" 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\\RTgHcmRGF,7$$\"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($\"+BN[p%*!#5%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 93 "The following code constructs a discrete \+ solution based on each of the methods and gives the " }{TEXT 260 22 "r oot mean square error" }{TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 810 "Q := (x,y) -> exp(-x)/(x-1)^2*cos( 1/(x-1))-y: hh := 1/500-1/(3000*Pi): numsteps := 500: x0 := 0: y0 := s in(1):\nmatrix([[`slope field: `,Q(x,y)],[`initial point: `,``(x0,y0 )],[`step width: `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds \+ := [`Huta's scheme A`,`Huta's scheme B`,`a companion to Huta's scheme \+ B`,`an efficient scheme with the same nodes and weights as Huta's two \+ schemes`,`a stage order 3 scheme with small principal error norm`]: er rs := []:\nDigits := 30:\nfor ct to 5 do\n Qn_RK6_||ct := RK6_||ct(Q (x,y),x,y,x0,evalf[33](y0),evalf[33](hh),numsteps,false);\n sm := 0: numpts := nops(Qn_RK6_||ct):\n for ii to numpts do\n sm := sm+ (Qn_RK6_||ct[ii,2]-q(Qn_RK6_||ct[ii,1]))^2;\n end do:\n errs := [o p(errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpose]([ mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7& 7$%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$%1no.~of~steps :~~~GFFQ)pprint526\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%0Huta's~scheme~AG$\"+Gomoi! #97$%0Huta's~scheme~BG$\"+:-t_8!#87$%?a~companion~to~Huta's~scheme~BG$ \"+t(\\NI\"F07$%doan~efficient~scheme~with~the~same~nodes~and~weights~ as~Huta's~two~schemesG$\"+QhzSGF+7$%Wa~stage~order~3~scheme~with~small ~principal~error~normG$\"+3Ss_\\F+Q)pprint536\"" }}}{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 = 0;" "6#/%\"xG\"\"!" }{TEXT -1 21 ".9 469 is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 734 "Q := (x,y) -> exp(-x)/(x-1)^2*cos(1/(x-1))-y: hh := 1/500-1/(3000*Pi): num steps := 500: x0 := 0: y0 := sin(1):\nmatrix([[`slope field: `,Q(x,y )],[`initial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of step s: `,numsteps]]);``;\nmthds := [`Huta's scheme A`,`Huta's scheme B`, `a companion to Huta's scheme B`,`an efficient scheme with the same no des and weights as Huta's two schemes`,`a stage order 3 scheme with sm all principal error norm`]: errs := []:\nDigits := 30:\nfor ct to 5 do \n qn_RK6_||ct := RK6_||ct(Q(x,y),x,y,x0,evalf(y0),evalf(hh),numstep s,true);\nend do:\nxx := 0.9469: qxx := evalf(q(xx)):\nfor ct to 5 do \n errs := [op(errs),abs(qn_RK6_||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)pprint546\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$ %0Huta's~scheme~AG$\"+EO#=1*!#87$%0Huta's~scheme~BG$\"+X+Mr?!#77$%?a~c ompanion~to~Huta's~scheme~BG$\"+.B3'*>F07$%doan~efficient~scheme~with~ the~same~nodes~and~weights~as~Huta's~two~schemesG$\"+>,K5UF+7$%Wa~stag e~order~3~scheme~with~small~principal~error~normG$\"+mgDPpF+Q)pprint55 6\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Th e " }{TEXT 260 22 "root mean square error" }{TEXT -1 19 " over the int erval " }{XPPEDIT 18 0 " [0, 1-1/(6*Pi)] " "6#7$\"\"!,&\"\"\"F&*&F&F&* &\"\"'F&%#PiGF&!\"\"F+" }{TEXT -1 82 " of each Runge-Kutta method is \+ estimated as follows using the special procedure " }{TEXT 0 5 "NCint " }{TEXT -1 98 " to perform numerical integration by the 7 point Newt on-Cotes method over 200 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 458 "mthds := [`Huta's scheme A`,`Huta's scheme B`,`a \+ companion to Huta's scheme B`,`an efficient scheme with the same nodes and weights as Huta's two schemes`,`a stage order 3 scheme with small principal error norm`]: errs := []:\nDigits := 20:\nfor ct to 5 do\n \+ sm := NCint((q(x)-'qn_RK6_||ct'(x))^2,x=0..1-1/(6*Pi),adaptive=false ,numpoints=7,factor=200);\n errs := [op(errs),sqrt(sm/(1-1/(6*Pi)))] ;\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%0Huta's~scheme~AG$\"+ aw&en%!#97$%0Huta's~scheme~BG$\"++h " 0 "" {MPLTEXT 1 0 524 "evalf[20](plot(['qn _RK6_1'(x)-q(x),'qn_RK6_2'(x)-q(x),'qn_RK6_3'(x)-q(x),'qn_RK6_4'(x)-q( x),'qn_RK6_5'(x)-q(x)],\nx=0..0.7,color=[COLOR(RGB,.95,0,.2),COLOR(RGB ,1,.55,0),COLOR(RGB,.5,0,.95),COLOR(RGB,0,.75,.2),COLOR(RGB,0,.55,.95) ],\nlegend=[`Huta's scheme A`,`Huta's scheme B`,`a companion to Huta's scheme B`,`an efficient scheme with the same nodes and weights as Hut a's two schemes`,`a stage order 3 scheme with small principal error no rm`],font=[HELVETICA,9],title=`error curves for 8 stage order 6 Runge- Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 1102 541 541 {PLOTDATA 2 "6+-%'CURVESG6%7S7$$\"\"!F)F(7$$\"5mmmmm\"z+e_\"!#@$\",(QG 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9],title=`error curves for 8 stage order 6 Runge-Kutta methods`));" }} {PARA 13 "" 1 "" {GLPLOT2D 996 521 521 {PLOTDATA 2 "6+-%'CURVESG6%7]o7 $$\"\"(!\"\"$\"1OVGD_N.C!#@7$$\"5LLLL3i^0Pq!#?$\"0O)*)HW)fT#F17$$\"5nm m\"z/m'HpqF1$\"0LP0niJU#F17$$\"5LLL$eufbb5(F1$\"0&o)p)))GCCF17$$\"5LLL $3A)[0UrF1$\"0v[\\!R&yT#F17$$\"5nmm\"HAp!QyrF1$\"0sU)3'3HS#F17$$\"5LLL e9o$f?@(F1$\"0SK98LRQ#F17$$\"5+++DJ'oJpC(F1$\"0fnI++`N#F17$$\"5LLLek(o '*HG(F1$\"0\")Gy=MXJ#F17$$\"5+++D1Gg%*=tF1$\"0]sDi+:E#F17$$\"5nmmmTVV# fN(F1$\"0LWD=#R$>#F17$$\"5LLL$3P'[\\)Q(F1$\"0c/Yg5B6#F17$$\"5++++Db:;D uF1$\"0`29(>46?F17$$\"5++++v.)y>Y(F1$\"0SmXL(p!*=F17$$\"5++++vW!fu\\(F 1$\"0Wole;Ox\"F17$$\"5LLLek-&y'HvF1$\"0wF^4JLl\"F17$$\"5nmmmTa0*zc(F1$ \"01i^LLAZ\"F17$$\"5nmmmm()eW+wF1$\"0hL@c;qG\"F17$$\"5+++D\"e:*>QwF1$ \"04Y.xUk0\"F17$$\"5nmmm;&><;n(F1$\"/n4Qgd'*zF17$$\"5+++D\"33#G3xF1$\" /F\"eRH72&F17$$\"5+++vVAd>VxF1$\"/EHtVShCF17$$\"5nmm;zWWizxF1$!.c<3Ms4 )F17$$\"5nmm\"HPQxI\"yF1$!/5$[>zF1$!07i_j5$[fE6z#F17$$\"5+++] 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J8QF2$\"3yKp#[4RfP\"F27$$\"3\"R3xJd'))>QF2$\"3/CVy4j'=U\"F27$$\"36nm;z 5YEQF2$\"3%GUA'*pv(o9F27$$\"3J]i:&eNI$QF2$\"3eJ%\\YU_f^\"F27$$\"3^Le9 \"45'RQF2$\"3][!)RoHki:F27$$\"3s;a8(f%=YQF2$\"3Zk-ND-43;F27$$\"3O+]7. \"fF&QF2$\"3]5UW.;d^;F27$$\"3i3_vlYhlQF2$\"3S)o\\Z'>`GF27$$\"3Ymm;/O gbRF2$\"3e5rkQQq>>F27$$\"3*G$e*[$zV4SF2$\"3^Ar&3889#>F27$$\"3w**\\ilAF jSF2$\"3Q')**40OT@>F27$$\"3#G3_]p'>*3%F2$\"3!o$)[rAz8#>F27$$\"3ym\"zW7 @^6%F2$\"3371C6\\d?>F27$$\"3@Qf$=B-;7%F2$\"39!e*f[J()>>F27$$\"3w3F>RL3 GTF2$\"3A(\\vy/K(=>F27$$\"3Iz%\\lWkX8%F2$\"38-([etep\">F27$$\"3t]i!RbX 59%F2$\"38M:lS7J9>F27$$\"3#=z>'ox+aTF2$\"3&>*GfbB=0>F27$$\"3yLLL$)*pp; %F2$\"3C=Lu(zB&))=F27$$\"3!Q3_+sD-=%F2$\"3ug(ptnA/'=F27$$\"3#Q$3xc9[$> %F2$\"3:,22Ku<==F27$$\"3'Qe*[$>Pn?%F2$\"3i6,,)=G,w\"F27$$\"3)QL3-$H**> UF2$\"33wUWwLu'o\"F27$$\"3X3xc)z?mA%F2$\"3/e\"zYY>^k\"F27$$\"3\"R3Fpm[ KB%F2$\"3x%eAK2J4g\"F27$$\"3OfkGNl()RUF2$\"3/aZW5%e[b\"F27$$\"3#R$ek.W ]YUF2$\"3_pHqN-l2:F27$$\"3]3_+sA8`UF2$\"3?puckU3g9F27$$\"3'Rek.9g(fUF2 $\"30z2N6y#HT\"F27$$\"3SfRs3!)QmUF2$\"3E%zqA:-pO\"F27$$\"3)RL$3xe,tUF2 $\"3Hzu(eD`EK\"F27$$\"3kv$4'\\=;'G%F2$\"32(*p'*eu>U7F27$$\"3G#[Z %F2$\"3S)\\h/++++\"F27$$\"3TM$3_5,-`%F2$\"3y/++\"F27$$\"3SnmT&G!e &e%F2$\"3$eA?a3RF+\"F27$$\"3fLe*[=Y.h%F2$\"3M:+/vy285F27$$\"3m+]P%37^j %F2$\"356M^f6but%F2$\"3%\\.QNQyVc\"F27$$\"3ID19>zl]ZF2$\"3S[]>-K'z\"F27$$\"37+]i SjE!z%F2$\"3&)4L&Q*oyW=F27$$\"3y*\\7G))Rb\"[F2$\"3)fCVDB`!)*=F27$$\"3L +++DM\"3%[F2$\"3$Gxb\\1an\">F27$$\"3i]P4'>]M&[F2$\"3p-qe54u>>F27$$\"3) 3](=np3m[F2$\"3zS9J5F#4#>F27$$\"3G]7GQPsy[F2$\"3j:7^7UI@>F27$$\"3a+]P4 0O\"*[F2$\"3L:$**HR(R@>F27$$\"3s+voa-oX\\F2$\"3%e\"e*Rr89#>F27$$\"\"&F )$\"3n\\kX'z.7#>F2-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%%FONTG6$%*HELVETIC AG\"\"*-%+AXESLABELSG6$%\"xG%%y(x)G-%%VIEWG6$;F(F_[q%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The following c ode constructs a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " ba sed on each of the methods and gives the " }{TEXT 260 22 "root mean sq uare error" }{TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 754 "R := (x,y) -> 5*y*sin(7*x)^7: hh := 0.01: numstep s := 500: x0 := 0: y0 := 1:\nmatrix([[`slope field: `,R(x,y)],[`init ial point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,nu msteps]]);``;\nmthds := [`Huta's scheme A`,`Huta's scheme B`,`a compan ion to Huta's scheme B`,`an efficient scheme with the same nodes and w eights as Huta's two schemes`,`a stage order 3 scheme with small princ ipal error norm`]: errs := []:\nDigits := 30:\nfor ct to 5 do\n Rn_R K6_||ct := RK6_||ct(R(x,y),x,y,x0,y0,hh,numsteps,false);\n sm := 0: \+ numpts := nops(Rn_RK6_||ct):\n for ii to numpts do\n sm := sm+( Rn_RK6_||ct[ii,2]-r(Rn_RK6_||ct[ii,1]))^2;\n end do:\n errs := [op (errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpose]([m thds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7 $%0slope~field:~~~G,$*(\"\"&\"\"\"%\"yGF,)-%$sinG6#,$*&\"\"(F,%\"xGF,F ,F4F,F,7$%0initial~point:~G-%!G6$\"\"!F,7$%/step~width:~~~G$F,!\"#7$%1 no.~of~steps:~~~G\"$+&Q)pprint576\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%0Huta's~schem e~AG$\"+!3)fpE!#87$%0Huta's~scheme~BG$\"+4U@/9F+7$%?a~companion~to~Hut a's~scheme~BG$\"+d!Qf5\"F+7$%doan~efficient~scheme~with~the~same~nodes ~and~weights~as~Huta's~two~schemesG$\"+je)f?$!#97$%Wa~stage~order~3~sc heme~with~small~principal~error~normG$\"+%))*=iiF8Q)pprint586\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The follo wing code constructs " }{TEXT 260 20 "numerical procedures" }{TEXT -1 56 " for solutions based on each of the Runge-Kutta schemes." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each of the \+ methods at the point where " }{XPPEDIT 18 0 "x = 4.999;" "6#/%\"xG-%& FloatG6$\"%**\\!\"$" }{TEXT -1 16 " is also given." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 685 "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 := [`Huta's scheme A`,`Huta's scheme B`,`a companion to Huta's scheme B`,`an efficient scheme with the same node s and weights as Huta's two schemes`,`a stage order 3 scheme with smal l principal error norm`]: errs := []:\nDigits := 25:\nfor ct to 5 do\n rn_RK6_||ct := RK6_||ct(R(x,y),x,y,x0,y0,hh,numsteps,true);\nend do :\nxx := 4.999: rxx := evalf(r(xx)):\nfor ct to 5 do\n errs := [op(e rrs),abs(rn_RK6_||ct(xx)-rxx)];\nend do:\nDigits := 10:\nlinalg[transp ose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matri xG6#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)pprint596\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$ %0Huta's~scheme~AG$\"+:)eoT$!#87$%0Huta's~scheme~BG$\"+Wi^*Q#F+7$%?a~c ompanion~to~Huta's~scheme~BG$\"+u+U*4#F+7$%doan~efficient~scheme~with~ the~same~nodes~and~weights~as~Huta's~two~schemesG$\"+jf&*>g!#97$%Wa~st age~order~3~scheme~with~small~principal~error~normG$\"+u!>aH\"F+Q)ppri nt606\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square error" }{TEXT -1 20 " over th e 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 subinter vals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 438 "mthds := [`Huta's \+ scheme A`,`Huta's scheme B`,`a companion to Huta's scheme B`,`an effic ient scheme with the same nodes and weights as Huta's two schemes`,`a \+ stage order 3 scheme with small principal error norm`]: errs := []:\nD igits := 20:\nfor ct to 5 do\n sm := NCint((r(x)-'rn_RK6_||ct'(x))^2 ,x=0..5,adaptive=false,numpoints=7,factor=200);\n errs := [op(errs), sqrt(sm/5)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(e rrs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%0Huta's~sc heme~AG$\"+-R=qE!#87$%0Huta's~scheme~BG$\"+s(eZS\"F+7$%?a~companion~to ~Huta's~scheme~BG$\"+My(f5\"F+7$%doan~efficient~scheme~with~the~same~n odes~and~weights~as~Huta's~two~schemesG$\"+c[o0K!#97$%Wa~stage~order~3 ~scheme~with~small~principal~error~normG$\"+YeCdiF8Q)pprint616\"" }}} {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 0 522 "evalf[20]( plot(['rn_RK6_1'(x)-r(x),'rn_RK6_2'(x)-r(x),'rn_RK6_3'(x)-r(x),'rn_RK6 _4'(x)-r(x),'rn_RK6_5'(x)-r(x)],\nx=0..5,color=[COLOR(RGB,.95,0,.2),CO LOR(RGB,1,.55,0),COLOR(RGB,.5,0,.95),COLOR(RGB,0,.75,.2),COLOR(RGB,0,. 55,.95)],\nlegend=[`Huta's scheme A`,`Huta's scheme B`,`a companion to Huta's scheme B`,`an efficient scheme with the same nodes and weights as Huta's two schemes`,`a stage order 3 scheme with small principal e rror norm`],font=[HELVETICA,9],title=`error curves for 8 stage order 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 1124 527 527 {PLOTDATA 2 "6+-%'CURVESG6%7jdl7$$\"\"!F)F(7$$\"5NLLL$3FWYs#!#@$!,>.]* =Z!#>7$$\"5qmmmmT&)G\\aF-$!.;\")pi95$F07$$\"50+++]7G$R<)F-$!/2='**4!RE F07$$\"5MLLLL3x&)*3\"!#?$!/-#)R'H$ylF07$$\"5++]i!*GER37F>$!/hH6-MzuF07 $$\"5nmm\"z%\\v#pK\"F>$!/iZ:$\".(zQ>dEVF07$$ \"5+++]i!R(*Rc\"F>$\"0+&3t$oH^\"F07$$\"5MLe9\"4&[EB;F>$\"0=[W%)RJ&HF07 $$\"5nm;z>6B`#o\"F>$\"0Xry%)yL^%F07$$\"5++vV[r(*zT$\"0=W0\"GFAhF07$ $\"5MLL3xJs1,=F>$\"0\"*p\\3rv#)*F07$$\"5nm\"Hd?pM.'=F>$\"1#4(*>]Hj5\"F 07$$\"5++]PM_@g>>F>$\"1pF>$\"1IB!oXP<#=F07 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7$F^cr$\"0PGe*Rq)f(F07$Fhcr$\"0I!3sB^+&)F07$Fbdr$\"0\"oIDuSz!*F07$F\\e r$\"0ZO)yos\"))*F07$Ffer$\"1Qj;y(G<4\"F07$F[fr$\"1`4\"3K]'\\6F07$F`fr$ \"1()G)HM*RF7F07$Fefr$\"12u)o;!**)H\"F07$Fjfr$\"1$4d>mF'Q8F07$Fibw$\"1 %*ydM$Q\"e8F07$F^cw$\"1js:tp8w8F07$Fccw$\"1]1S=(y[Q\"F07$F_gr$\"1xlhQr \\y8F07$F[dw$\"1)fx[1ePR\"F07$F`dw$\"1j@Q'[2IR\"F07$Fedw$\"1:)[6N@EO\" F07$Fdgr$\"1hSASU*[P\"F07$F^hr$\"1Rpk<5&*>8F07$Fhhr$\"1]^\"[6gQ@\"F07$ F]ir$\"1o$3pL*=y6F07$Fbir$\"1q'34z%zd6F07$F\\jr$\"150k<6O`6F07$Ffjr$\" 1E!)eP+wf6F07$F[[s$\"1%H*f*y*Gn6F07$F`[s$\"1sZ$=-'*p=\"F07$Fe[s$\"1<;- A\\iQ7F07$Fj[s$\"1\"*y/=lIy7F07$F_\\s$\"1j(=U%Qj&G\"F07$Fd\\s$\"1$f@$y h(yH\"F0-Fi\\s6&F[]sF(F]hwF\\]s-Fc]s6#%Wa~stage~order~3~scheme~with~sm all~principal~error~normG-%&TITLEG6#%Uerror~curves~for~8~stage~order~6 ~Runge-Kutta~methodsG-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6$Q\"x6 \"Q!Fbjel-%%VIEWG6$;F(Fd\\s%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Huta's scheme A" "Huta's scheme B" " a companion to Huta's scheme B" "an efficient scheme with the same nod es and weights as Huta's two schemes" "a stage order 3 scheme with sma ll principal error norm" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "Test 13 of 8 stage order 6 Runge-Kutta methods " }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "See: \"Mathe matica in Action\" by Stan Wagon, Springer-Verlag, page 302. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 3 " " } {XPPEDIT 18 0 "dy/dx = cos*x+2*y;" "6#/*&%#dyG\"\"\"%#dxG!\"\",&*&%$co sGF&%\"xGF&F&*&\"\"#F&%\"yGF&F&" }{TEXT -1 8 ", " }{XPPEDIT 18 0 "y(0) = -2/5;" "6#/-%\"yG6#\"\"!,$*&\"\"#\"\"\"\"\"&!\"\"F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "y = 1/5;" "6#/%\"yG*&\"\"\"F&\"\"&!\" \"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin*x-2/5" "6#,&*&%$sinG\"\"\"%\"x GF&F&*&\"\"#F&\"\"&!\"\"F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "cos*x" "6# *&%$cosG\"\"\"%\"xGF%" }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 53 "The general s olution of the differential equation " }{XPPEDIT 18 0 "dy/dx = cos* x+2*y;" "6#/*&%#dyG\"\"\"%#dxG!\"\",&*&%$cosGF&%\"xGF&F&*&\"\"#F&%\"yG F&F&" }{TEXT -1 64 " contains an exponential term, but with the initi al condition " }{XPPEDIT 18 0 "y(0) = -2/5" "6#/-%\"yG6#\"\"!,$*&\"\" #\"\"\"\"\"&!\"\"F-" }{TEXT -1 23 " this term disappears." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "de : = diff(y(x),x)=cos(x)+2*y(x);\ndsolve(de,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,,&-%$cosGF+\"\"\"*&\"\" #F0F)F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,(*&#\"\"# \"\"&\"\"\"-%$cosGF&F-!\"\"*&#F-F,F--%$sinGF&F-F-*&-%$expG6#,$*&F+F-F' F-F-F-%$_C1GF-F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 105 "Any slight deviation of a numerical solution from the co rrect solution tends to become rapidly magnified." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 154 "de := diff( y(x),x)=cos(x)+2*y(x);\nic := y(0)=-2/5;\ndsolve(\{de,ic\},y(x));\ne : = unapply(rhs(%),x):\nplot(e(x),x=0..8,font=[HELVETICA,9],labels=[`x`, `y(x)`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6# %\"xGF,,&-%$cosGF+\"\"\"*&\"\"#F0F)F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!#!\"#\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/-%\"yG6#%\"xG,&*&#\"\"#\"\"&\"\"\"-%$cosGF&F-!\"\"*&#F-F,F--%$sinGF &F-F-" }}{PARA 13 "" 1 "" {GLPLOT2D 703 312 312 {PLOTDATA 2 "6&-%'CURV ESG6$7gn7$$\"\"!F)$!3A+++++++S!#=7$$\"3ELLLLBxV5E$F,$!35;%fC]([[JF,7$$\"3MLLLLAKn\\F,$!3C&4%=OwYjDF,7$$\"3 =LLLLc$\\o'F,$!31c1[)*fT**=F,7$$\"3)emmm^&Q%R)F,$!39J7$$\"3))*****\\YJ?;\"!#<$\"3m!=?Y3*>`CFK 7$$\"3?LLL=\"\\g**FK7$$\"3\")*****\\[A4]\"FO$\"3Xgu ?U;&er\"F,7$$\"3wmmm'3Q\\n\"FO$\"3S\\4.g(y\\S#F,7$$\"3OLLLB6@G=FO$\"3e *[f2BGC&HF,7$$\"3&)******f-w+?FO$\"375@EVOJ&[$F,7$$\"3%*********y,u@FO 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mm;\\%H&\\dFO$!3y[ey96=hWF,7$$\"3eLLe%p5Bz&FO$!3=)zg%Q%y/Z%F,7$$\"3')* *****R>4NeFO$!3waa0%)\\frWF,7$$\"3HLL$ed*f:fFO$!3]_J$4k<:X%F,7$$\"3#em m;@2h*fFO$!3V5vHeMg-WF,7$$\"37LLL))3E!3'FO$!3=l`a'y%*4K%F,7$$\"3]***** \\c9W;'FO$!3>=$e-d.)3UF,7$$\"3Lmmmmd'*GjFO$!3Gy*y<4!G/RF,7$$\"3j***** \\iN7]'FO$!3;B6I^7jsMF,7$$\"3aLLLt>:nmFO$!37+2hu:afHF,7$$\"35LLL.a#o$o FO$!3;\"e/Z#4*3N#F,7$$\"3ammm^Q40qFO$!3!4`1I$pa!o\"F,7$$\"3y******z]rf rFO$!3pfL'*)RTA-\"F,7$$\"3gmmmc%GpL(FO$!3?j;%3XMsQ#FK7$$\"3/LLL8-V&\\( FO$\"3qi(R>/(R\"p%FK7$$\"3=+++XhUkwFO$\"3ZX^U-))=F,7$$\"\")F)$\"3s<7[GmrgDF,-%'COLOURG6&%$RGBG$\"#5!\"\"F (F(-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6$%\"xG%%y(x)G-%%VIEWG6$;F (Fg]l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The following code constructs a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " based on each of the methods and gives the \+ " }{TEXT 260 22 "root mean square error" }{TEXT -1 18 " of each soluti on." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 760 "E := (x,y) -> cos(x) +2*y: hh := 0.02: numsteps := 400: x0 := 0: y0 := -2/5:\nmatrix([[`slo pe field: `,E(x,y)],[`initial point: `,``(x0,y0)],[`step width: `, hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Huta's scheme A` ,`Huta's scheme B`,`a companion to Huta's scheme B`,`an efficient sche me with the same nodes and weights as Huta's two schemes`,`a stage ord er 3 scheme with small principal error norm`]: errs := []:\nDigits := \+ 20:\nfor ct to 5 do\n En_RK6_||ct := RK6_||ct(E(x,y),x,y,x0,evalf(y0 ),hh,numsteps,false);\n sm := 0: numpts := nops(En_RK6_||ct):\n fo r ii to numpts do\n sm := sm+(En_RK6_||ct[ii,2]-e(En_RK6_||ct[ii, 1]))^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\nend do:\n Digits := 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$%/s tep~width:~~~G$F0F97$%1no.~of~steps:~~~G\"$+%Q)pprint626\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'ma trixG6#7'7$%0Huta's~scheme~AG$\"+o3r(Q#!\")7$%0Huta's~scheme~BG$\"+8se dz!\"*7$%?a~companion~to~Huta's~scheme~BG$\"+3[y;TF07$%doan~efficient~ scheme~with~the~same~nodes~and~weights~as~Huta's~two~schemesG$\"+c4V88 F07$%Wa~stage~order~3~scheme~with~small~principal~error~normG$\"+.\"Hj A\"F0Q)pprint636\"" }}}{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 = 7.999;" "6#/%\"xG-%&FloatG6$\"%**z!\"$" }{TEXT -1 16 " is also gi ven." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 691 "E := (x,y) -> cos(x )+2*y: hh := 0.02: numsteps := 400: x0 := 0: y0 := -2/5:\nmatrix([[`sl ope field: `,E(x,y)],[`initial point: `,``(x0,y0)],[`step width: ` ,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Huta's scheme A `,`Huta's scheme B`,`a companion to Huta's scheme B`,`an efficient sch eme with the same nodes and weights as Huta's two schemes`,`a stage or der 3 scheme with small principal error norm`]: errs := []:\nDigits := 20:\nfor ct to 5 do\n en_RK6_||ct := RK6_||ct(E(x,y),x,y,x0,evalf(y 0),hh,numsteps,true);\nend do:\nxx := 7.999: exx := evalf(e(xx)):\nfor ct to 5 do\n errs := [op(errs),abs(en_RK6_||ct(xx)-exx)];\nend do: \nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,&-%$cosG6#%\" xG\"\"\"*&\"\"#F.%\"yGF.F.7$%0initial~point:~G-%!G6$\"\"!#!\"#\"\"&7$% /step~width:~~~G$F0F97$%1no.~of~steps:~~~G\"$+%Q)pprint646\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'ma trixG6#7'7$%0Huta's~scheme~AG$\"+#yzKK\"!\"(7$%0Huta's~scheme~BG$\"+=L Y4W!\")7$%?a~companion~to~Huta's~scheme~BG$\"+\\uB\"G#F07$%doan~effici ent~scheme~with~the~same~nodes~and~weights~as~Huta's~two~schemesG$\"+A #4$ys!\"*7$%Wa~stage~order~3~scheme~with~small~principal~error~normG$ \"+f]j&z'F9Q)pprint656\"" }}}{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$\"\"! \"\")" }{TEXT -1 82 " of each Runge-Kutta method is estimated as foll ows using the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " \+ to perform numerical integration by the 7 point Newton-Cotes method ov er 200 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 438 "mthds := [`Huta's scheme A`,`Huta's scheme B`,`a companion to Hut a's scheme B`,`an efficient scheme with the same nodes and weights as \+ Huta's two schemes`,`a stage order 3 scheme with small principal error norm`]: errs := []:\nDigits := 20:\nfor ct to 5 do\n sm := NCint((e (x)-'en_RK6_||ct'(x))^2,x=0..8,adaptive=false,numpoints=7,factor=200); \n errs := [op(errs),sqrt(sm/8)];\nend do:\nDigits := 10:\nlinalg[tr anspose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'m atrixG6#7'7$%0Huta's~scheme~AG$\"+1=lVB!\")7$%0Huta's~scheme~BG$\"+n&= 3\"y!\"*7$%?a~companion~to~Huta's~scheme~BG$\"+.6&3/%F07$%doan~efficie nt~scheme~with~the~same~nodes~and~weights~as~Huta's~two~schemesG$\"+tH ?*G\"F07$%Wa~stage~order~3~scheme~with~small~principal~error~normG$\"+ jyq.7F0Q)pprint666\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following error graphs are constructed using the n umerical procedures for the solutions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 522 "evalf[20](plot(['en_RK6_1'(x)-e(x),'en_RK6_2'(x)-e(x ),'en_RK6_3'(x)-e(x),'en_RK6_4'(x)-e(x),'en_RK6_5'(x)-e(x)],\nx=0..2,c olor=[COLOR(RGB,.95,0,.2),COLOR(RGB,1,.55,0),COLOR(RGB,.5,0,.95),COLOR (RGB,0,.75,.2),COLOR(RGB,0,.55,.95)],\nlegend=[`Huta's scheme A`,`Huta 's scheme B`,`a companion to Huta's scheme B`,`an efficient scheme wit h the same nodes and weights as Huta's two schemes`,`a stage order 3 s cheme with small principal error norm`],font=[HELVETICA,9],title=`erro r curves for 8 stage order 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 999 548 548 {PLOTDATA 2 "6+-%'CURVESG6%7S7$$\"\"!F)F(7$$ \"5MLLLLL3VfV!#@$\"2)=E4Mp#z/#!#A7$$\"5nmmmm\"H[D:)F-$\"1')4()=k4%>%F- 7$$\"5LLLLLe0$=C\"!#?$\"1#4U.kL+`'F-7$$\"5LLLLL3RBr;F9$\"1ow29C-j!*F-7 $$\"5nmmm;zjf)4#F9$\"2qmI4*R_!=\"F-7$$\"5MLLL$e4;[\\#F9$\"2=x]8R=qX\"F -7$$\"5++++]i'y]!HF9$\"2)R7nE[ef+&F9$\"2%yZy*ob/n$F-7$ $\"5+++++]Z/NaF9$\"1rdp\\-&>8%F97$$\"5+++++]$fC&eF9$\"1V?ew889YF97$$\" 5MLLL$ez6:B'F9$\"1aU(\\U=+4&F97$$\"5nmmmm;=C#o'F9$\"1+KE5#=(zcF97$$\"5 nmmmmm#pS1(F9$\"1Gd!HO/gA'F97$$\"5,+++]i`A3vF9$\"1S$p5yU#**oF97$$\"5nm mmmm(y8!zF9$\"1v,B`ywUvF97$$\"5,+++]i.tK$)F9$\"1PZ%pfd&*H)F97$$\"5,+++ ](3zMu)F9$\"1iQ#>0GW2*F97$$\"5nmmmm\"H_?<*F9$\"1!pcK?kc%**F97$$\"5nmmm 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DG6#%0Huta's~scheme~AG-F$6%7SF'7$F+$!17hJs)R&eoF07$F2$!1kjp\\[c/9F-7$F 7$!13NEWV$o=#F-7$F=$!1itd%H/^.$F-7$FB$!1!Q.w-JM&RF-7$FG$!1A93g3Dz[F-7$ FL$!1#>:K%>Q#*eF-7$FQ$!1ao\\HSK8qF-7$FV$!1e_*Gqfz@)F-7$Fen$!1Gz0G-Du&* F-7$Fjn$!2)o..?U9x5F-7$F_o$!2wM-iw')*G7F-7$Fdo$!1Kd]TP[$Q\"F97$Fio$!1* \\%)fk+\\a\"F97$F^p$!1H:re/?/F97$Fhp$!1^F(=ibW3#F9 7$F]q$!176N-e!)4BF97$Fbq$!1>j()=U>DDF97$Fgq$!1(o(F97$Fiu$!1+8#4!HBg$)F97$F^v$!1k;i(y(y?!*F97$Fcv$!1^n(oDm()z*F97$F hv$!2$ywX^FFf5F97$F]w$!2St!>S)e!\\6F97$Fbw$!2=mpL2(HW7F97$Fgw$!2Um6))F97$Fcp$!0:dBy#GK)*F97$Fhp$!1t s$)*G!yx5F97$F]q$!11[?5OH%>\"F97$Fbq$!1)G'4xym08F97$Fgq$!1*yf'ogjO9F97 $F\\r$!1DGnX@tq:F97$Far$!1;***p++:s\"F97$Ffr$!13k9>7`p=F97$F[s$!1Cg>dK pT?F97$F`s$!1SJ*\\MCTB#F97$Ffs$!1IDuDqh9CF97$F[t$!1345tOF97$F du$!1&puL&)**[(RF97$Fiu$!1'z8\"*3KHK%F97$F^v$!1\"**fuY4Xm%F97$Fcv$!1.@ r?/!o1&F97$Fhv$!1!QSjYitZ&F97$F]w$!1!y%pQK)\\[#y8F97$F^v$\"1JM%*f*er[\"F97$Fcv$\"1I%f*QWU:;F97$Fhv$\"1 UTg5+LY+56iV*=F97$Fbw$\"1Hz`_0R^?F97$Fgw$\"1'Q9+[i*HAF97 $F\\x$\"1X$=ck6kT#F97$Fax$\"1+SRl[KBEF97$Ffx$\"1WvmYb+YGF97$F[y$\"1#\\ !**zpHnIF97$F`y$\"11))4jUQULF97$Fey$\"1K(GY4i'4OF97$Fjy$\"1l$R*40==RF9 7$F_z$\"1%QD[**z(QUF97$Fdz$\"1i\\f4O&>h%F9-Fiz6&F[[lF($\"#vF^[lF_[l-Fb [l6#%doan~efficient~scheme~with~the~same~nodes~and~weights~as~Huta's~t wo~schemesG-F$6%7SF'7$F+$\"1))o6g6%[0\"F07$F2$\"0ckg9%oh@F-7$F7$\"0AX; D_YO$F-7$F=$\"0)>:yK3nYF-7$FB$\"0]KA(4;wgF-7$FG$\"0Qm)pK,-vF-7$FL$\"0[ ENYF*f!*F-7$FQ$\"1'4Irg;#y5F-7$FV$\"17Wu#HeNE\"F-7$Fen$\"17x0'Q%)HZ\"F -7$Fjn$\"1-0QZ^%ol\"F-7$F_o$\"1%G'pT4c\"*=F-7$Fdo$\"0DvY([JH@F97$Fio$ \"0QGA0(oxBF97$F^p$\"0XJ_#>(Hi#F97$Fcp$\"0!4*QYCl#HF97$Fhp$\"0/UyRJ\"3 KF97$F]q$\"0\\3b`(oaNF97$Fbq$\"0nPmvL#>=6yF97$F`t$\"0 h!o9i\"p])F97$Fet$\"0f\"[f0wU#*F97$Fjt$\"1!=>;2)3,5F97$F_u$\"1ei4dxX$4 \"F97$Fdu$\"1j2eP/J$=\"F97$Fiu$\"1'zH:kBpG\"F97$F^v$\"1tOg7F97$Fgw$\"1I#H+bU@3#F97$F\\x$\"14'RB.RiD#F97$Fax$\"1V0V?aX\\CF97 $Ffx$\"14(G[.&RdEF97$F[y$\"1C#HQn-S'GF97$F`y$\"1K$=c%=*37$F97$Fey$\"1) y@%y.XqLF97$Fjy$\"1*fN?v:&eOF97$F_z$\"1k;:0v&y&RF97$Fdz$\"1MV7e^=1VF9- Fiz6&F[[lF(F\\elF\\[l-Fb[l6#%Wa~stage~order~3~scheme~with~small~princi pal~error~normG-%&TITLEG6#%Uerror~curves~for~8~stage~order~6~Runge-Kut ta~methodsG-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABELSG6$Q\"x6\"Q!Fjbn-%% VIEWG6$;F(Fdz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Huta's scheme A" "Huta's scheme B" "a companion to Huta 's scheme B" "an efficient scheme with the same nodes and weights as H uta's two schemes" "a stage order 3 scheme with small principal error \+ norm" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 522 "evalf[20](plot(['en_RK6_1'(x)-e(x),'en_RK6_2'(x)-e(x ),'en_RK6_3'(x)-e(x),'en_RK6_4'(x)-e(x),'en_RK6_5'(x)-e(x)],\nx=2..8,c olor=[COLOR(RGB,.95,0,.2),COLOR(RGB,1,.55,0),COLOR(RGB,.5,0,.95),COLOR (RGB,0,.75,.2),COLOR(RGB,0,.55,.95)],\nlegend=[`Huta's scheme A`,`Huta 's scheme B`,`a companion to Huta's scheme B`,`an efficient scheme wit h the same nodes and weights as Huta's two schemes`,`a stage order 3 s cheme with small principal error norm`],font=[HELVETICA,9],title=`erro r curves for 8 stage order 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 1148 550 550 {PLOTDATA 2 "6+-%'CURVESG6%7Z7$$\"\"#\"\"!$ \"2S$Rn>M0i$)!#?7$$\"5+++++DHyI@!#>$\"3e(RA=Z()*z5F-7$$\"5++++v[kdWAF1 $\"39H35t5r\\8F-7$$\"5++++]n\"\\DP#F1$\"3/Mb*o&z@O,%)HPQAF-7$$\"5++++v8*y&HEF1$\"4'Q(fa'f+=&)G!#@7$$\"5++++vG[ W[FF1$\"4])\\YCp)3Al$FH7$$\"5++++v)fB:(GF1$\"4=O6o'e(pTm%FH7$$\"5++++v Q=\"))*HF1$\"4'=$G)R31l3gFH7$$\"5++++vj=pDJF1$\"5'ou4TV]Qvt(!#A7$$\"5+ ++++lN?cKF1$\"5aCtx^6;(Q+\"FH7$$\"5++++]U$e6P$F1$\"5y>4O.'R`GE\"FH7$$ \"5+++++&>q0]$F1$\"5#y=)\\0j+QN;FH7$$\"5+++++DM^IOF1$\"5%=8^[M')f.7#FH 7$$\"5+++++0ytbPF1$\"4+$*GbI]\\Ns#F-7$$\"5++++vQNXpQF1$\"4[i1$f))H%*=M F-7$$\"5+++++XDn/SF1$\"4s\\h\"HPxm![%F-7$$\"5+++++!y?#>TF1$\"4&ep9udtW McF-7$$\"5++++v3wY_UF1$\"497s,.=vaN(F-7$$\"5+++++IOTqVF1$\"5\"e!*>tZTm GJ*FH7$$\"5++++v3\">)*\\%F1$\"5!*)=,@O#=Z17F-7$$\"5++++DEP/BYF1$\"5yDq Z>\"*3wV:F-7$$\"5++++](o:;v%F1$\"51$*o8&GB!f'*>F-7$$\"5++++v$)[op[F1$ \"5ve#)Ra,FcGDF-7$$\"5++++]i%Qq*\\F1$\"5'>c1vgp6BE$F-7$$\"5++++vQIKH^F 1$\"5%Q_b20-*o]UF-7$$\"5++++D^rZW_F1$\"5c`2:!f#y#>N&F-7$$\"5++++]Zn%)o `F1$\"52$*yv6Py\"Q'oF-7$$\"5+++++5FL(\\&F1$\"5eH#=)[!G6c())F-7$$\"5+++ +]d6.BcF1$\"5#*=UoW&=@89\"F17$$\"5++++vo3lWdF1$\"5$[[xma[-dX\"F17$$\"5 ++++]A))ozeF1$\"5k/2su2))>2>F17$$\"5+++++Ik-,gF1$\"5BQ.7-!)y;JCF17$$\" 5+++++D-eIhF1$\"5DBc&)pE`V]JF17$$\"5++++v=_(zC'F1$\"5@#>0wZ14W)RF17$$ \"5+++++b*=jP'F1$\"5<(4W#))oyt]^F17$$\"5++++v3/3(\\'F1$\"5^h^9#GiQ#elF 17$$\"5++++vB4JBmF1$\"5Rl=>2Ek@U%)F17$$\"5+++++DVsYnF1$\"5&e\"QIN%pI13 \"!#=7$$\"5++++v=n#f(oF1$\"5`s%>;\">dN*R\"Fiw7$$\"5+++++!)RO+qF1$\"5pD Ld\"36**[z\"Fiw7$$\"5++++]_!>w7(F1$\"5#3c5Bqh[_J#Fiw7$$\"5++++v)Q?QD(F 1$\"5uxv'4Jvo,)HFiw7$$\"5+++++5jyptF1$\"5!eXd#zaMh)*4AbFiw7$$\"5+++++gEd@wF1$\"5h%[w]Z\\O$>iFiw7$$\"5+++ ]PMh%\\o(F1$\"5-dH'QAcd*fqFiw7$$\"5++++v3'>$[xF1$\"5S%3ukBP-U,)Fiw7$$ \"5+++++5h(*3yF1$\"5#****y/ZY@\"[!*Fiw7$$\"5++++D6EjpyF1$\"5u*RY#RiFa@ 5!#<7$$\"5+++vVeWA-zF1$\"5ia&p'zX&o.4\"F[\\l7$$\"5+++]i0j\"[$zF1$\"5:q $=?f5EQ;\"F[\\l7$$\"5+++D\"G:3u'zF1$\"5hF6R66_BU7F[\\l7$$\"\")F*$\"5!o @gscxIfK\"F[\\l-%&COLORG6&%$RGBG$\"#&*!\"#$F*F*$F)!\"\"-%'LEGENDG6#%0H uta's~scheme~AG-F$6%7Z7$F($!2g`as1-sz#F-7$F/$!2M`v21qBh$F-7$F5$!2l\"e! GU'=9XF-7$F:$!29tkib$Q1eF-7$F?$!2E)*\\#Ge/&[(F-7$FD$!3/B+17_:Z'*FH7$FJ $!4Iy?eej*3@7FH7$FO$!4_&*)yo=bIf:FH7$FT$!4/rU%[o@h3?FH7$FY$!59yc?)f@[j e#Fgn7$Fin$!416B&fQiDbLFH7$F^o$!4-8Se\\L[0A%FH7$Fco$!4oC2LFo!3laFH7$Fh o$!4'Qu1`o!=_3(FH7$F]p$!3<%>)H'o\\\"F-7$F\\q$!4DkAp-el@)=F-7$Faq$!4hN?GdRpoX#F-7$Ffq$!4T`YF>& >W5JF-7$F[r$!4-`1:C&Q>HSF-7$F`r$!4v?v4/sD_:&F-7$Fer$!4Py'*)Qcb(om'F-7$ Fjr$!4Fx:W'\\acU%)F-7$F_s$!5)okE'y(Gc\"*3\"F-7$Fds$!59)e*o'y;>!>9F-7$F is$!5p_Cp14W^'y\"F-7$F^t$!5j#G.Z$Gw,\"H#F-7$Fct$!5*p/B>S6qA'HF-7$Fht$! 5*R>W)Rbk*)3QF-7$F]u$!5Zx43@n**pd[F-7$Fbu$!5k-Q)>XA(yjjF-7$Fgu$!5knIA. f1]6\")F-7$F\\v$!5$>#oS:Q'[50\"F17$Fav$!5\")p9Y.\")*y\"H8F17$Ffv$!5!3a 4;ai=\"=ChxXu=#F17$F`w$!58Rjf%3A-c\"GF17$Few$!5tass0\\R y.OF17$F[x$!5LB#Q?H>7jm%F17$F`x$!5vs#\\8_W-[)fF17$Fex$!5C'f\\V*G)G#>xF 17$Fjx$!5&=o_dp%GLN**F17$F_y$!5d0$)*z<=cGD\"Fiw7$Fdy$!5\"zx2\"pw2$4V\" Fiw7$Fiy$!5n%ykntx:Vj\"Fiw7$F^z$!5a\"on\">F!*fS=Fiw7$Fcz$!5xChjDH$>H2# Fiw7$Fhz$!57E#*=^Y*4IN#Fiw7$F][l$!5)ya!)RPtX4n#Fiw7$Fb[l$!5b)zA\"*=\"3 T:IFiw7$Fg[l$!5;\\y^dU1I/MFiw7$F]\\l$!5n#pG$=/fdLOFiw7$Fb\\l$!5(>\"3SP g$)HyQFiw7$Fg\\l$!50\\8O]$)**\\RTFiw7$F\\]l$!5DKvT,>\"*G=WFiw-Fa]l6&Fc ]l$\"\"\"F*$\"#bFf]lFg]l-F[^l6#%0Huta's~scheme~BG-F$6%7Z7$F($!2J\"*=t7 akW\"F-7$F/$!2kxU$pc+o=F-7$F5$!2Oh4^?mVL#F-7$F:$!2`b7,$*=E+$F-7$F?$!2U JE0)>tqQF-7$FD$!3/\\TBVI()))\\FH7$FJ$!35'))H`@IZJ'FH7$FO$!3#*zgS\"\\cQ 1)FH7$FT$!4/&QN([i_(Q5FH7$FY$!59L.3Mk8aP8Fgn7$Fin$!4E.wwVo._t\"FH7$F^o $!4Ab`:&zqr#=#FH7$Fco$!4=G!>[CYPEGFH7$Fho$!4wGp.I9'HkOFH7$F]p$!3Tf;gN> O1ZF-7$Fbp$!3$G.QZ9Zw!fF-7$Fgp$!3ZUZ/x(>;u(F-7$F\\q$!3j&\\^)[+YM(*F-7$ Faq$!4)Gn[P)H#pq7F-7$Ffq$!4C#)e7rfM(3;F-7$F[r$!4f#4Mz+$RR3#F-7$F`r$!4; \\V\"Rs\"fjm#F-7$Fer$!4QxL]o,S#[MF-7$Fjr$!4xS&=cS#*F-7$F^t$!58&zjQYX;]=\" F-7$Fct$!5U%=D5jxLA`\"F-7$Fht$!55!ptx!H0F-7$F]u$!5nH!o.Aw#p7DF-7$Fb u$!5\")p/K,mlw\"H$F-7$Fgu$!5Z`[\\R4v$e>%F-7$F\\v$!5*Q/*oZGK\"oV&F-7$Fa v$!5ma'*=r;advoF-7$Ffv$!5EoR#o>Prv)))F-7$F[w$!5YG:qcYPaJ6F17$F`w$!5`k& pV2=(\\c9F17$Few$!5&\\$*3pFZbJrA?_*F17$F cz$!5*>/)z>yTRs5Fiw7$Fhz$!5P/U&\\jh+t@\"Fiw7$F][l$!56aztJFty\"Q\"Fiw7$ Fb[l$!5nJ2$GQB++c\"Fiw7$Fg[l$!5w?er]Rx>h8kN$Gk+#Fiw7$Fg\\l$!5XLIw@8ecT@Fiw7$F\\]l$!5.H*\\*puR!eG#F iw-Fa]l6&Fc]l$\"\"&Fi]lFg]lFd]l-F[^l6#%?a~companion~to~Huta's~scheme~B G-F$6%7Z7$F($\"1i\\f4O&>h%F-7$F/$\"14\"p?'G:cfF-7$F5$\"1G5QrHBVuF-7$F: $\"1j3zoJ9u&*F-7$F?$\"2\"*>9a*oBM7F-7$FD$\"3'))z_e)\\z!f\"FH7$FJ$\"3]1 Bx[_f8?FH7$FO$\"3)oF/Z=z8d#FH7$FT$\"3;#R26.\"R7LFH7$FY$\"4'G7\"H#y&H_E %Fgn7$Fin$\"3kh^QbwRLbFH7$F^o$\"33p\"G11e0'pFH7$Fco$\"3i![0'HWF8!*FH7$ Fho$\"4%)e?_O.b&o6FH7$F]p$\"3IXv1w bqI^F-7$F[r$\"3ykO!yzijk'F-7$F`r$\"3!oP1Mf)*R])F-7$Fer$\"4D*HX;1[y*4\" F-7$Fjr$\"41Rc;4RPFR\"F-7$F_s$\"4WEv4^$>y'z\"F-7$Fds$\"4'[6b/wE,TBF-7$ Fis$\"4%f[RF,#[t%HF-7$F^t$\"4jYOU>r\\(zPF-7$Fct$\"4Xm3&y@$3t)[F-7$Fht$ \"4d.#*=mFjUG'F-7$F]u$\"4I#zf?-9&[,)F-7$Fbu$\"5LIKj%*R`+]5F-7$Fgu$\"5l -\\EX(R-%Q8F-7$F\\v$\"5'*HI7ZNcFM#F-7$Ffv$\"5\" z>a#p%)*)4NGF-7$F[w$\"5zSfYU*)[i4OF-7$F`w$\"5,b*G\")zF*GYYF-7$Few$\"57 +[gzmO2ZfF-7$F[x$\"5%oL!QU^In+xF-7$F`x$\"5;u<&)>mFxw)*F-7$Fex$\"5l\"R- $4Z&RRF\"F17$Fjx$\"5I!***R.86rR;F17$F_y$\"5Ps#e=jJPx1#F17$Fdy$\"5`*43%yO)QF17$F][l$\"57p!HaKu&[3WF17$Fb[l$\"5ME;E%*=@ 4x\\F17$Fg[l$\"5W$RvfmLP!>cF17$F]\\l$\"5D^[lH0Q^(*fF17$Fb\\l$\"5N9[Y8% R\"[,kF17$Fg\\l$\"5CD)RT\"*QfE$oF17$F\\]l$\"5Xag_=P4)GH(F1-Fa]l6&Fc]lF g]l$\"#vFf]lFh]l-F[^l6#%doan~efficient~scheme~with~the~same~nodes~and~ weights~as~Huta's~two~schemesG-F$6%7Z7$F($\"1MV7e^=1VF-7$F/$\"1-N)*3HM hbF-7$F5$\"1Kp)\\KX(\\pF-7$F:$\"15:pETZR*)F-7$F?$\"2-'eKxCS_6F-7$FD$\" 3OLPG:kK&[\"FH7$FJ$\"3]0*>)Q,4!)=FH7$FO$\"33F!\\r9%*3S#FH7$FT$\"3OK\\ \"*Row#4$FH7$FY$\"4'Qs4;n0U#)RFgn7$Fin$\"3/\"Q4-A4l;&FH7$F^o$\"3oh=4IS .*\\'FH7$Fco$\"3#=+Y$Q,j:%)FH7$Fho$\"4afA]!e<2\"4\"FH7$F]p$\"3:Lxl&=t8 S\"F-7$Fbp$\"3G!ps&)\\*3fihsil)*GF -7$Faq$\"3LmEU\\V#Qy$F-7$Ffq$\"38H3\\?P\\!z%F-7$F[r$\"3ii0c(\\Zc?'F-7$ F`r$\"3r^B&\\s-,%zF-7$Fer$\"4forT\"y#eo-\"F-7$Fjr$\"4H1`*>0\\Q+8F-7$F_ s$\"4$QcmH0njx;F-7$Fds$\"4)['4q,tyd=#F-7$Fis$\"4;y[xS62>v#F-7$F^t$\"4# R())pAM5\"HNF-7$Fct$\"48Y4e03CKc%F-7$Fht$\"4dK:bYZVv'eF-7$F]u$\"42!*=! om8P$[(F-7$Fbu$\"5c'QS`/trP!)*FH7$Fgu$\"5$QQ*))\\S'\\'\\7F-7$F\\v$\"5W ;F-7$Fav$\"5@YY8#4N4y/#F-7$Ffv$\"5%>Ia'psU4ZEF-7$F[w$\"5KH7U9 \")yDqLF-7$F`w$\"5%=z4X5:w\"QVF-7$Few$\"5$fi'Q&Q+*p_bF-7$F[x$\"59Q#Q<# [y+!>(F-7$F`x$\"5EmQRF17$Fdy$\"5[cU'=i[\\]?#F17$Fiy$\"5\"zi]H_H &\\=DF17$F^z$\"53\\6ufy/TOGF17$Fcz$\"5&*\\6vc&ycW>$F17$Fhz$\"5Mc/txtz7 EOF17$F][l$\"5$)f%Q30`Jh6%F17$Fb[l$\"5H`eTd6%Hqk%F17$Fg[l$\"5Ouh[63CSY _F17$F]\\l$\"5c&H*>&G=z(*f&F17$Fb\\l$\"5))f)R^Gccp(fF17$Fg\\l$\"5H=F!4 \"R)R&zjF17$F\\]l$\"5RC$HP " 0 "" {MPLTEXT 1 0 2 " ;" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "Test 14 of 8 stage order 6 Runge-Kutta me thods " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "dy/dx = 10*x*cos*x-10*y;" "6#/*&%#dyG\"\"\"%#d xG!\"\",&**\"#5F&%\"xGF&%$cosGF&F,F&F&*&F+F&%\"yGF&F(" }{TEXT -1 8 ", \+ " }{XPPEDIT 18 0 "y(0) = sqrt(5);" "6#/-%\"yG6#\"\"!-%%sqrtG6#\" \"&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "y=100/101" "6#/%\"yG *&\"$+\"\"\"\"\"$,\"!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x*cos*x-990 /10201" "6#,&*(%\"xG\"\"\"%$cosGF&F%F&F&*&\"$!**F&\"&,-\"!\"\"F+" } {TEXT -1 1 " " }{XPPEDIT 18 0 "cos*x+10/101" "6#,&*&%$cosG\"\"\"%\"xGF &F&*&\"#5F&\"$,\"!\"\"F&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x*sin*x-200/ 10201" "6#,&*(%\"xG\"\"\"%$sinGF&F%F&F&*&\"$+#F&\"&,-\"!\"\"F+" } {TEXT -1 1 " " }{XPPEDIT 18 0 "sin*x+(990/10201+sqrt(5))*exp(-10*x)" " 6#,&*&%$sinG\"\"\"%\"xGF&F&*&,&*&\"$!**F&\"&,-\"!\"\"F&-%%sqrtG6#\"\"& F&F&-%$expG6#,$*&\"#5F&F'F&F-F&F&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 163 "de := diff( y(x),x)=10*x*cos(x)-10*y(x);\nic := y(0)=sqrt(5);\ndsolve(\{de,ic\},y( x));\nb := unapply(rhs(%),x):\nplot(b(x),x=0..5,font=[HELVETICA,9],lab els=[`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-*&#\"$!**\"&,-\"F-F/F-!\"\"*&#\"#5F,F-*&-%$si nGF&F-F'F-F-F-*&#\"$+#F4F-F:F-F5*&-%$expG6#,$*&F8F-F'F-F5F-,&#F3F4F-*$ \"\"&#F-\"\"#F-F-F-" }}{PARA 13 "" 1 "" {GLPLOT2D 703 312 312 {PLOTDATA 2 "6&-%'CURVESG6$7hn7$$\"\"!F)$\"3\")*y*\\xz1OA!#<7$$\"3ALL$ 3FWYs#!#>$\"3*pc)\\jF;12h\"H\"48F,7$$ \"3m****\\7G$R<)F0$\"3<_u(oLbK,\"F,7$$\"3GLLL3x&)*3\"!#=$\"3(**[ro!GyV zF@7$$\"3))**\\i!R(*Rc\"F@$\"3A'ysO]2xW&F@7$$\"3umm\"H2P\"Q?F@$\"3/$)o qvSKmSF@7$$\"3YLek.pu/BF@$\"3$Qjx*Gs<7OF@7$$\"3!***\\PMnNrDF@$\"3M:4%* 3rt@LF@7$$\"3MmT5ll'z$GF@$\"3/?Np5C\\bJF@7$$\"3MLL$eRwX5$F@$\"3)GTJ!oG 0$3$F@7$$\"3rLLL$eI8k$F@$\"3FyHM$p'GKJF@7$$\"33ML$3x%3yTF@$\"3n$**Q]` \"yRLF@7$$\"3emm\"z%4\\Y_F@$\"3%)*)G8T#p2%RF@7$$\"3`LLeR-/PiF@$\"3PZ.% R2Cm^%F@7$$\"3]***\\il'pisF@$\"31e'*fKlL9]F@7$$\"3>MLe*)>VB$)F@$\"3%3) yy-pAk`F@7$$\"3Y++DJbw!Q*F@$\"3Kg$RQBm7^&F@7$$\"3%ommTIOo/\"F,$\"3xrTB 'zz$GaF@7$$\"3YLL3_>jU6F,$\"3!p\\3Dp!RX^F@7$$\"37++]i^Z]7F,$\"3Kvxv\"* =%>e%F@7$$\"33++](=h(e8F,$\"3C^UQ-(*\\]PF@7$$\"3/++]P[6j9F,$\"3icMTpV \"zp#F@7$$\"3UL$e*[z(yb\"F,$\"3C-jVJM+L:F@7$$\"3wmm;a/cq;F,$!33XIKjA5+ 5F07$$\"3%ommmJF,$!3!\\F1VDv$)p&F@7$$\"3K+]i!f#=$3#F,$!3V$*[LN4F4!)F @7$$\"3?+](=xpe=#F,$!3$[6ral`?.\"F,7$$\"37nm\"H28IH#F,$!3s6ToLL*4G\"F, 7$$\"3um;zpSS\"R#F,$!3/xdj'GyI^\"F,7$$\"3GLL3_?`(\\#F,$!3MGc,i$*4jpxg#F,$!3XXbRF+x>IF,$!3r\"fP(*)y]OGF,7$$\"3F+]i!RU07$F,$!3[Tb]!*H%e)HF,7$$\"3+++v =S2LKF,$!3Tyi#Q\\'[=JF,7$$\"3Jmmm\"p)=MLF,$!3N`)=cl>V?$F,7$$\"3B++](=] @W$F,$!3Y_BiN[odKF,7$$\"3mm\"H#oZ1\"\\$F,$!3)o4&)z%=-oKF,7$$\"35L$e*[$ z*RNF,$!3%)Q61)ek$pKF,7$$\"3%o;Hd!fX$f$F,$!3#*y!45Ut-E$F,7$$\"3e++]iC$ pk$F,$!3LO[nw')*)RKF,7$$\"3ILe*[t\\sp$F,$!3D1>x`HA5KF,7$$\"3[m;H2qcZPF ,$!3/[q%\\V.-<$F,7$$\"3O+]7.\"fF&QF,$!3KL?tX&>E0$F,7$$\"3Ymm;/OgbRF,$! 3KQEMNc$G*GF,7$$\"3w**\\ilAFjSF,$!3/QR)44g!yEF,7$$\"3yLLL$)*pp;%F,$!30 ,GW_`#fU#F,7$$\"3)RL$3xe,tUF,$!3*G#*H@1([B@F,7$$\"3Cn;HdO=yVF,$!35Q!)* 4x]5y\"F,7$$\"3a+++D>#[Z%F,$!3(y*pyl_QJ9F,7$$\"3SnmT&G!e&e%F,$!3]X/0\" RC%G**F@7$$\"3#RLLL)Qk%o%F,$!3u!*)Q\"4WH+dF@7$$\"37+]iSjE!z%F,$!3+r[gf MO'=*F07$$\"3a+]P40O\"*[F,$\"3+2*eSHde(QF@7$$\"\"&F)$\"3&Q8`\">jC3#*F@ -%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%%FONTG6$%*HELVETICAG\"\"*-%+AXESLABE LSG6$%\"xG%%y(x)G-%%VIEWG6$;F(F\\^l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The following code constructs a " }{TEXT 260 17 "discrete solution" }{TEXT -1 44 " based on each of t he methods and gives the " }{TEXT 260 22 "root mean square error" } {TEXT -1 18 " of each solution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 769 "B := (x,y) -> 10*x*cos(x)-10*y: hh := 0.01: numsteps := 500: \+ x0 := 0: y0 := sqrt(5):\nmatrix([[`slope field: `,B(x,y)],[`initial \+ point: `,``(x0,y0)],[`step width: `,hh],\n[`no. of steps: `,numste ps]]);``;\nmthds := [`Huta's scheme A`,`Huta's scheme B`,`a companion \+ to Huta's scheme B`,`an efficient scheme with the same nodes and weigh ts as Huta's two schemes`,`a stage order 3 scheme with small principal error norm`]: errs := []:\nDigits := 20:\nfor ct to 5 do\n Bn_RK6_| |ct := RK6_||ct(B(x,y),x,y,x0,evalf(y0),hh,numsteps,false);\n sm := \+ 0: numpts := nops(Bn_RK6_||ct):\n for ii to numpts do\n sm := s m+(Bn_RK6_||ct[ii,2]-b(Bn_RK6_||ct[ii,1]))^2;\n end do:\n errs := \+ [op(errs),sqrt(sm/numpts)];\nend do:\nDigits := 10:\nlinalg[transpose] ([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6# 7&7$%0slope~field:~~~G,&*(\"#5\"\"\"%\"xGF,-%$cosG6#F-F,F,*&F+F,%\"yGF ,!\"\"7$%0initial~point:~G-%!G6$\"\"!*$\"\"&#F,\"\"#7$%/step~width:~~~ G$F,!\"#7$%1no.~of~steps:~~~G\"$+&Q)pprint676\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$ %0Huta's~scheme~AG$\"+TEr,H!#87$%0Huta's~scheme~BG$\"+(>,6/\"F+7$%?a~c ompanion~to~Huta's~scheme~BG$\"+B#R86&!#97$%doan~efficient~scheme~with ~the~same~nodes~and~weights~as~Huta's~two~schemesG$\"+))**fz8F47$%Wa~s tage~order~3~scheme~with~small~principal~error~normG$\"+6kOE&)!#;Q)ppr int686\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The following code constructs " }{TEXT 260 20 "numerical procedure s" }{TEXT -1 56 " for solutions based on each of the Runge-Kutta schem es." }}{PARA 0 "" 0 "" {TEXT -1 75 "The error in the value obtained by each of the methods at the point where " }{XPPEDIT 18 0 "x = 4.999; " "6#/%\"xG-%&FloatG6$\"%**\\!\"$" }{TEXT -1 16 " is also given." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 700 "B := (x,y) -> 10*x*cos(x)-1 0*y: hh := 0.01: numsteps := 500: x0 := 0: y0 := sqrt(5):\nmatrix([[`s lope field: `,B(x,y)],[`initial point: `,``(x0,y0)],[`step width: \+ `,hh],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Huta's scheme \+ A`,`Huta's scheme B`,`a companion to Huta's scheme B`,`an efficient sc heme with the same nodes and weights as Huta's two schemes`,`a stage o rder 3 scheme with small principal error norm`]: errs := []:\nDigits : = 20:\nfor ct to 5 do\n bn_RK6_||ct := RK6_||ct(B(x,y),x,y,x0,evalf( y0),hh,numsteps,true);\nend do:\nxx := 4.999: bxx := evalf(b(xx)):\nfo r ct to 5 do\n errs := [op(errs),abs(bn_RK6_||ct(xx)-bxx)];\nend do: \nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G,&*(\"#5\"\"\" %\"xGF,-%$cosG6#F-F,F,*&F+F,%\"yGF,!\"\"7$%0initial~point:~G-%!G6$\"\" !*$\"\"&#F,\"\"#7$%/step~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$+&Q) pprint696\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#K%'matrixG6#7'7$%0Huta's~scheme~AG$\"++w%Gb#!#97$%0Hu ta's~scheme~BG$\"+(ow9!*)!#:7$%?a~companion~to~Huta's~scheme~BG$\"+u#e .A%F07$%doan~efficient~scheme~with~the~same~nodes~and~weights~as~Huta' s~two~schemesG$\"+N69`6F07$%Wa~stage~order~3~scheme~with~small~princip al~error~normG$\"+SD2z`!# " 0 "" {MPLTEXT 1 0 438 "mthds := [`Huta's scheme A`,`Huta's scheme B`,`a com panion to Huta's scheme B`,`an efficient scheme with the same nodes an d weights as Huta's two schemes`,`a stage order 3 scheme with small pr incipal error norm`]: errs := []:\nDigits := 20:\nfor ct to 5 do\n s m := NCint((b(x)-'bn_RK6_||ct'(x))^2,x=0..5,adaptive=false,numpoints=7 ,factor=200);\n errs := [op(errs),sqrt(sm/5)];\nend do:\nDigits := 1 0:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$%0Huta's~scheme~AG$\"+&\\R@$G!#87$%0Hut a's~scheme~BG$\"+'4fe,\"F+7$%?a~companion~to~Huta's~scheme~BG$\"+Ny,)) \\!#97$%doan~efficient~scheme~with~the~same~nodes~and~weights~as~Huta' s~two~schemesG$\"+9@9Z8F47$%Wa~stage~order~3~scheme~with~small~princip al~error~normG$\"+#)\\)[x)!#;Q)pprint716\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "The following error graphs are \+ constructed using the numerical procedures for the solutions." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 539 "evalf[20](plot([b(x)-'bn_RK 6_1'(x),b(x)-'bn_RK6_2'(x),b(x)-'bn_RK6_3'(x),b(x)-'bn_RK6_4'(x),b(x)- 'bn_RK6_5'(x)],\nx=0..0.65,numpoints=100,color=[COLOR(RGB,.95,0,.2),CO LOR(RGB,1,.55,0),COLOR(RGB,.5,0,.95),COLOR(RGB,0,.75,.2),\nCOLOR(RGB,0 ,.55,.95)],legend=[`Huta's scheme A`,`Huta's scheme B`,`a companion to Huta's scheme B`,`an efficient scheme with the same nodes and weights as Huta's two schemes`,`a stage order 3 scheme with small principal e rror norm`],font=[HELVETICA,9],title=`error curves for 8 stage order 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 1035 475 475 {PLOTDATA 2 "6+-%'CURVESG6%7`z7$$\"\"!F)F(7$$\"5+-----xv'e)!#B$\".`GEy 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principal error norm" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 525 "evalf[25](plot([b(x)-'bn_RK6_1'(x),b(x)-'bn_RK6_2'(x),b(x)-'bn_RK 6_3'(x),b(x)-'bn_RK6_4'(x),b(x)-'bn_RK6_5'(x)],\nx=0.65..5,color=[COLO R(RGB,.95,0,.2),COLOR(RGB,1,.55,0),COLOR(RGB,.5,0,.95),COLOR(RGB,0,.75 ,.2),\nCOLOR(RGB,0,.55,.95)],legend=[`Huta's scheme A`,`Huta's scheme \+ B`,`a companion to Huta's scheme B`,`an efficient scheme with the same nodes and weights as Huta's two schemes`,`a stage order 3 scheme with small principal error norm`],font=[HELVETICA,9],title=`error curves f or 8 stage order 6 Runge-Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 880 564 564 {PLOTDATA 2 "6+-%'CURVESG6%7js7$$\"#l!\"#$\"8TVo &3$e#[mD/>!#F7$$\":+++++]i:0Wqt'!#D$\"8&['RE(=M[N'*\\6F-7$$\":++++++DJ 5)3upF1$\"7]ru(*f;`D]%p%F-7$$\":+++++](oa@86sF1$!7Mz(H[L1h/\"o:F-7$$\" :++++++]i?w\"[uF1$!7ozFN!e^b/$)z&F-7$$\":+++++vV)HxngyF1$!9`$p$['zuQ8y 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d))Q5)>gUF*)Ff[r7$F`gm$!:'p52T&z%z(**4y^(Ff[r7$Fegm$!:!oZ6&Q=(ye5;mmF f[r7$Fjgm$!3N-9GIuQ7fF1-F`hm6&FbhmFehmF_doFchm-Fjhm6#%Wa~stage~order~3 ~scheme~with~small~principal~error~normG-%&TITLEG6#%Uerror~curves~for~ 8~stage~order~6~Runge-Kutta~methodsG-%%FONTG6$%*HELVETICAG\"\"*-%+AXES LABELSG6$Q\"x6\"Q!Fj^y-%%VIEWG6$;F(Fjgm%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Huta's scheme A" "Huta's scheme \+ B" "a companion to Huta's scheme B" "an efficient scheme with the same nodes and weights as Huta's two schemes" "a stage order 3 scheme with small principal error norm" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 339 "evalf[25](plot([b(x)-'bn_RK 6_4'(x),b(x)-'bn_RK6_5'(x)],x=0.65..5,color=[COLOR(RGB,0,.75,.2),COLOR (RGB,0,.55,.95)],\nlegend=[`an efficient scheme with the same nodes an d weights as Huta's two schemes`,`a stage order 3 scheme with small pr incipal error norm`],font=[HELVETICA,9],title=`error curves for 8 stag e order 6 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d))Q5)>gUF*)F]\\l7$F\\]n$!:'p52T&z%z(**4y^(F]\\l7$Fa]n$!:!oZ6&Q= (ye5;mmF]\\l7$Ff]n$!3N-9GIuQ7fF1-F\\^n6&F^^nF_^n$\"#bF*$\"#&*F*-Ff^n6# %Wa~stage~order~3~scheme~with~small~principal~error~normG-%&TITLEG6#%U error~curves~for~8~stage~order~6~Runge-Kutta~methodsG-%%FONTG6$%*HELVE TICAG\"\"*-%+AXESLABELSG6$Q\"x6\"Q!Fa]t-%%VIEWG6$;F(Ff]n%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "an efficient sc heme with the same nodes and weights as Huta's two schemes" "a stage o rder 3 scheme with small principal error norm" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "Test 15 of 8 stage, order 6 Runge-Kutta methods" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 198 "See: Comparing Numerical Methods for Ordinary Differential Equati ons, Hull, Enright, Fellen and Sedgwick,\n Siam Journal on Nume rical Analysis, Vol. 9, No. 4 (Dec. 1972), page 617, Example A5." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 3 " " } {XPPEDIT 18 0 "dy/dx = (y-x)/(y+x);" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&,&% \"yGF&%\"xGF(F&,&F+F&F,F&F(" }{TEXT -1 8 ", " }{XPPEDIT 18 0 "y( 1) = 1;" "6#/-%\"yG6#\"\"\"F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "Solution: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "2*ln((x^2+y^2)/(x^2))+4*arctan(y/x)+4*ln*x-2*ln*2-Pi = 0;" "6#/, ,*&\"\"#\"\"\"-%#lnG6#*&,&*$%\"xGF&F'*$%\"yGF&F'F'*$F.F&!\"\"F'F'*&\" \"%F'-%'arctanG6#*&F0F'F.F2F'F'*(F4F'F)F'F.F'F'*(F&F'F)F'F&F'F2%#PiGF2 \"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "de := diff(y(x),x)=(y(x)-x)/(y(x)+x );\nic := y(1)=1;\ndsolve(\{de,ic\},y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,*&,&F)\"\"\"F,!\"\"F/,& F)F/F,F/F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"\"F) " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%'RootOfG6#,,*&\"\" #\"\"\"-%#lnG6#*&,&*$)F'F-F.F.*$)%#_ZGF-F.F.F.F'!\"#F.!\"\"*&\"\"%F.-% 'arctanG6#*&F8F.F'F:F.F:*&F " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "The solution can be given more \+ simply as " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " ln(x^2+y^2)+2*arctan(y/x)=ln*2+Pi/2" "6#/,&- %#lnG6#,&*$%\"xG\"\"#\"\"\"*$%\"yGF+F,F,*&F+F,-%'arctanG6#*&F.F,F*!\" \"F,F,,&*&F&F,F+F,F,*&%#PiGF,F+F4F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 18 "The solution (for " }{TEXT 267 1 "x" }{TEXT -1 47 " inc reasing) is the section of the polar curve " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "r=sqrt(2)*exp(Pi/4-theta)" "6#/%\"rG*&- %%sqrtG6#\"\"#\"\"\"-%$expG6#,&*&%#PiGF*\"\"%!\"\"F*%&thetaGF2F*" } {TEXT -1 5 ", " }{XPPEDIT 18 0 "-Pi/4<=theta" "6#1,$*&%#PiG\"\"\"\" \"%!\"\"F)%&thetaG" }{XPPEDIT 18 0 "``<=Pi/4" "6#1%!G*&%#PiG\"\"\"\"\" %!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 " ln((x^2+y^2))+2*arctan(y/x)=ln(2)+Pi/2;\nimplicitdiff(%,y,x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%#lnG6#,&*$)%\"xG\"\"#\"\"\"F-*$)% \"yGF,F-F-F-*&F,F--%'arctanG6#*&F0F-F+!\"\"F-F-,&-F&6#F,F-*&F,F6%#PiGF -F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&%\"xG\"\"\"%\"yG!\"\"F',& F(F'F&F'F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 471 "p1 := plot([sqrt(2)*exp(Pi/4-t),t,t=-Pi/4..Pi/4 ],coords=polar,thickness=2,color=red):\np2 := plot([sqrt(2)*exp(Pi/4-t ),t,t=Pi/4..2*Pi],coords=polar,color=black,linestyle=2):\np3 := plot([ sqrt(2)*exp(Pi/4-t),t,t=-Pi/3..-Pi/4],coords=polar,color=black,linesty le=2):\np4 := plot([[[1,1],[uu,-uu]]$4],style=point,symbol=[circle$2,d iamond,cross],\n symbolsize=[12,10$3],color=[black,gr een$3]):\nplots[display]([p1,p2,p3,p4],font=[HELVETICA,9],labels=[`x`, `y(x)`]);" }}{PARA 13 "" 1 "" {GLPLOT2D 567 520 520 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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 265 1 "y " }{TEXT -1 25 " numerically in terms of " }{TEXT 266 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 0 821 "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 := [`Hut a's scheme A`,`Huta's scheme B`,`a companion to Huta's scheme B`,`an e fficient scheme with the same nodes and weights as Huta's two schemes` ,`a stage order 3 scheme with small principal error norm`]:\nerrs := [ ]: vals := []:\nDigits := 25:\nfor ct to 5 do\n Cn_RK6_||ct := RK6_ ||ct(C(x,y),x,y,x0,y0,hh,numsteps,false);\n sm := 0: numpts := nops( Cn_RK6_||ct):\n for ii to numpts do\n if ct=1 then vals := [op( vals),phi(Cn_RK6_||ct[ii,1])] end if;\n sm := sm+(Cn_RK6_||ct[ii, 2]-vals[ii])^2;\n end do:\n errs := [op(errs),sqrt(sm/numpts)];\ne nd do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G*&,& %\"yG\"\"\"%\"xG!\"\"F,,&F-F,F+F,F.7$%0initial~point:~G-%!G6$F,F,7$%/s tep~width:~~~G$F,!\"#7$%1no.~of~steps:~~~G\"$v$Q)pprint726\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'ma trixG6#7'7$%0Huta's~scheme~AG$\"+h3M96!#87$%0Huta's~scheme~BG$\"+Lk#H \\$!#97$%?a~companion~to~Huta's~scheme~BG$\"+;8N`;F07$%doan~efficient~ scheme~with~the~same~nodes~and~weights~as~Huta's~two~schemesG$\"+]iHHa !#:7$%Wa~stage~order~3~scheme~with~small~principal~error~normG$\"+g1P[ RF9Q)pprint736\"" }}}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{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 = 4.749;" "6#/%\"xG-%&FloatG6$\"%\\Z!\"$" }{TEXT -1 16 " is also gi ven." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 684 "C := (x,y) -> (y-x) /(y+x): hh := 0.01: numsteps := 375: x0 := 1: y0 := 1:\nmatrix([[`slop e field: `,C(x,y)],[`initial point: `,``(x0,y0)],[`step width: `,h h],\n[`no. of steps: `,numsteps]]);``;\nmthds := [`Huta's scheme A`, `Huta's scheme B`,`a companion to Huta's scheme B`,`an efficient schem e with the same nodes and weights as Huta's two schemes`,`a stage orde r 3 scheme with small principal error norm`]: errs := []:\nDigits := 3 0:\nfor ct to 5 do\n cn_RK6_||ct := RK6_||ct(C(x,y),x,y,x0,y0,hh,num steps,true);\nend do:\nxx := 4.749: cxx := evalf(phi(xx)):\nfor ct to \+ 5 do\n errs := [op(errs),abs(cn_RK6_||ct(xx)-cxx)];\nend do:\nDigits := 10:\nlinalg[transpose]([mthds,evalf(errs)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7$%0slope~field:~~~G*&,&%\"yG\"\"\"%\"xG! \"\"F,,&F-F,F+F,F.7$%0initial~point:~G-%!G6$F,F,7$%/step~width:~~~G$F, !\"#7$%1no.~of~steps:~~~G\"$v$Q)pprint746\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7'7$ %0Huta's~scheme~AG$\"+C7a1\"*!#87$%0Huta's~scheme~BG$\"+u!Rvq#F+7$%?a~ companion~to~Huta's~scheme~BG$\"+^$)457F+7$%doan~efficient~scheme~with ~the~same~nodes~and~weights~as~Huta's~two~schemesG$\"+=(4Y?%!#97$%Wa~s tage~order~3~scheme~with~small~principal~error~normG$\"+a*[]2$F8Q)ppri nt756\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "root mean square error" }{TEXT -1 20 " over th e interval " }{XPPEDIT 18 0 "[1, 4.75];" "6#7$\"\"\"-%&FloatG6$\"$v%! \"#" }{TEXT -1 82 " of each Runge-Kutta method is estimated as follow s using the special procedure " }{TEXT 0 5 "NCint" }{TEXT -1 98 " to perform numerical integration by the 7 point Newton-Cotes method over 200 equal subintervals." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 445 "mthds := [`Huta's scheme A`,`Huta's scheme B`,`a companion to Huta's \+ scheme B`,`an efficient scheme with the same nodes and weights as Huta 's two schemes`,`a stage order 3 scheme with small principal error nor m`]: errs := []:\nDigits := 20:\nfor ct to 5 do\n sm := NCint(('phi' (x)-'cn_RK6_||ct'(x))^2,x=1..4.75,adaptive=false,numpoints=7,factor=20 0);\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$%6Butcher's~scheme~..~1G$\"+%yq(>>!#@7$%6Butcher's~sche me~..~2G$\"+bkr#>$F+7$*&%7Chammud's~scheme~with~G\"\"\"6%/&%\"cG6#\"\" ##\"\"%\"\"(/&F76#\"\"$#\"\"&F#F@FB/FD#F;FBF3$\"+!H>l\\\"F+7$*&FLF36%FNFP/FD #\"\")\"#6F3$\"+4uz\"**)!#A7$*&FLF36%/F6#\"#C\"#h/F>#\"#<\"#G/FD#\"#Q \"#`F3$\"+o0\"47)FinQ*pprint1736\"" }}}{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 0 545 "evalf[30](plot(['phi'(x)-'cn_RK6_1'(x),' phi'(x)-'cn_RK6_2'(x),'phi'(x)-'cn_RK6_3'(x),'phi'(x)-'cn_RK6_4'(x),\n 'phi'(x)-'cn_RK6_5'(x)],x=1..3.75,color=[COLOR(RGB,.95,0,.2),COLOR(RGB ,1,.55,0),COLOR(RGB,.5,0,.95),\nCOLOR(RGB,0,.75,.2),COLOR(RGB,0,.55,.9 5)],legend=[`Huta's scheme A`,`Huta's scheme B`,`a companion to Huta's scheme B`,`an efficient scheme with the same nodes and weights as Hut a's two schemes`,`a stage order 3 scheme with small principal error no rm`],font=[HELVETICA,9],title=`error curves for 8 stage order 6 Runge- Kutta methods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 918 655 655 {PLOTDATA 2 "6+-%'CURVESG6%7S7$$\"\"\"\"\"!$F*F*7$$\"?LLLLLLLL$eRnZ%*GF27$$\"?++++++++]7`'=tu\"F/$\":SO#fXMY*=(GG'=$F27$$ \"?++++++++]igJr/=F/$\":<(en-p(G$o&=F/$\":.C %>GDF/$\":^@WBm?6IF/$\":U&*>&>,=:Q5>$R%F27$$\"?+++++++]PM()4QK?F/$\":T\\$4<\"f% =o+T[ZF27$$\"?nmmmmmmmmT!eRk3#F/$\":@M^&yG()o,CPt]F27$$\"?+++++++]P%[U ]d9#F/$\":uVELi\"[f(oK1X&F27$$\"?+++++++]7`u$GA?#F/$\":cOiHwwU6^sB$eF2 7$$\"?nmmmmmmmT5!>d6E#F/$\":zZ\"3t5DZ8hgXiF27$$\"?nmmmmmm;aQQAF:BF/$\" :Gi;%fe$\\HV%yGmF27$$\"?LLLLLLLLekGEktBF/$\":Rf/8YJ626t`2(F27$$\"?LLLL LLL$3F%fIFMCF/$\":[%[z/J:&fhy*evF27$$\"?+++++++]ilF?0([#F/$\":$oW-q\"[ 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