{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 }{PSTYLE "Norm al" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Maple O utput" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple 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 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 41 "Derivation of order 3 Runge-Kutta schemes" }}{PARA 0 "" 0 "" {TEXT -1 45 "by Peter Stone, Gabriola Isla nd, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 18 "Version: 5.12.2011" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 53 "load procedures for constructing \+ Runge-Kutta schemes " }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file \+ " }{TEXT 262 9 "butcher.m" }{TEXT -1 32 " is required by this workshee t. " }}{PARA 0 "" 0 "" {TEXT -1 121 "It can be read into a Maple sessi on by a command similar to the one that follows, where the file path g ives its location." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "read \+ \"C:\\\\Maple/procdrs/butcher.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 64 "A system of equations for 3 stage , order 3 Runge-Kutta schemes: " }{TEXT 0 8 "RK3_3eqs" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "Abreviated form of order conditions" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Abreviated order condit ions can obtained using the procedure " }{TEXT 0 15 "OrderConditions" }{TEXT -1 17 " from the m-file " }{TEXT 0 9 "butcher.m" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 160 "convert(ListTools[Enumerate](OrderConditions(3)),matrix):\nli nalg[augment](linalg[delcols](%,2..2),matrix([[` `]$(linalg[rowdim](% ))]),linalg[delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'m atrixG6#7&7%\"\"\"%#~~G/*&%\"bGF(%\"eGF(F(7%\"\"#F)/*&F,F(%\"cGF(#F(F/ 7%\"\"$F)/*&F,F(-%!G6#*&%\"aGF(F2F(F(#F(\"\"'7%\"\"%F)/*&F,F()F2F/F(#F (F5Q(pprint86\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 50 "Ord er conditions and their associated rooted trees" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 245 "trs3 := All Trees(3):\np1 := DrawTree(trs3[1],height=2,show_ordercondition=true): \nfor i from 2 to nops(trs3) do \n p||i := DrawTree(trs3[i],height=2, show_ordercondition=true,scale=.7);\nend do:\nplots[display](convert([ [p||(1..nops(trs3))]],array));" }}{PARA 13 "" 1 "" {GLPLOT2D 586 199 199 {PLOTDATA 2 "6G-%%TEXTG6&7$$\"#!*\"\"!$!+0reA))!\"*Q(b~c~~=~6\"-%& COLORG6&%$RGBG$F)F)F3$\"\"\"F)-%%FONTG6$%(COURIERG\"#5-F$6&7$F'$!+81^D zF,Q*~~~~~~~~1F.F/F6-F$6&7$F'$!+gF,7$$\"+++++!)FL$\"+m%)Q!*>F,7$$\"+++++5!\"(Fhn-%'SYM BOLG6#%'CIRCLEG-F06&F2F3F3F3-%&STYLEG6#%&POINTG-FU6&FW-F_o6#%(DIAMONDG FboFdo-FU6&FW-F_o6#%&CROSSGFboFdo-FU6%7$FXFen-F06&F2F4F3F3-%*THICKNESS G6#\"\"#-FU6%7$FXFjnFepFgp-FU6%7#7$F'$F\\oFL-%'COLOURG6&F2$F:!\"\"F3F3 -Feo6#%%LINEG-F$6&7$F3FJQ(b~e~=~1F.F/F6-FU6&7#7$F3$!+++++gF,F^oFboFdo- FU6&FarFjoFboFdo-FU6&FarF_pFboFdo-FU6%7#7$F3FbqFcqFhq-F$6&7$$\"#IF)F*Q 'b~c~=~F.F/F6-F$6&7$F`sF>Q)~~~~~~~1F.F/F6-F$6&7$F`sFDQ)~~~~~~~_F.F/F6- F$6&7$F`sFJQ)~~~~~~~2F.F/F6-FU6&7$7$F`sFY7$F`sFhnF^oFboFdo-FU6&FatFjoF boFdo-FU6&FatF_pFboFdo-FU6%FatFepFgp-FU6%7#7$F`sFbqFcqFhq-FU6#-%'LEGEN DG6#QB__never_display_this_legend_entryF.-F$6&7$$\"#gF)F*Q+b~(a~c)~=~F .F/F6-F$6&7$FguF>Q-~~~~~~~~~~~1F.F/F6-F$6&7$FguFDQ-~~~~~~~~~~~_F.F/F6- F$6&7$FguFJQ-~~~~~~~~~~~6F.F/F6-FU6&7%7$FguFY7$FguFhn7$FguFbqF^oFboFdo -FU6&FhvFjoFboFdo-FU6&FhvF_pFboFdo-FU6%7$FivFjvFepFgp-FU6%7$FjvF[wFepF gp-FU6%7#F[wFcqFhq-%*AXESSTYLEG6#%%NONEG" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17 " "Curve 18" "Curve 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" "C urve 24" "Curve 25" "Curve 26" "Curve 27" "Curve 28" "Curve 29" "Curve 30" "Curve 31" "Curve 32" "Curve 33" "Curve 34" "Curve 35" "Curve 36 " }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 41 "Long (summ ation) form of order conditions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 75 "The order conditions in summation form ca n be obtained using the procedure " }{TEXT 0 15 "OrderConditions" } {TEXT -1 17 " from the m-file " }{TEXT 0 9 "butcher.m" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "convert(ListTools[Enumerate](OrderConditions(3,3)),matrix):\n linalg[augment](linalg[delcols](%,2..2),matrix([[` `]$(linalg[rowdim] (%))]),linalg[delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K% 'matrixG6#7&7%\"\"\"%#~~G/-%$SumG6$&%\"bG6#%\"iG/F1;F(\"\"$F(7%\"\"#F) /-F,6$*&F.F(&%\"cGF0F(F2#F(F67%F4F)/-F,6$*&F.F(-F,6$*&&%\"aG6$F1%\"jGF (&F<6#FIF(/FI;F(,&F1F(F(!\"\"F(F2#F(\"\"'7%\"\"%F)/-F,6$*&F.F()F;F6F(F 2#F(F4Q(pprint96\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 75 " Set up order conditions together with row sum conditions in expanded f orm: " }{TEXT 0 8 "RK4_4eqs" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 90 "The system of equations for the Runge-Kutta coe fficients can be set with Maple as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "RK3_3eqs := [a[2, 1]=c[2],\na[3,1]+a[3,2]=c[3],\nb[1]+b[2]+b[3]=1,\nb[3]*c[3]+b[2]*c[2]= 1/2,\nb[3]*a[3,2]*c[2]=1/6,\nb[2]*c[2]^2+b[3]*c[3]^2=1/3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)RK3_3eqsG7(/&%\"aG6$\"\"#\"\"\"&%\"cG6#F* /,&&F(6$\"\"$F+F+&F(6$F3F*F+&F-6#F3/,(&%\"bG6#F+F+&F;F.F+&F;F7F+F+/,&* &F>F+F6F+F+*&F=F+F,F+F+#F+F*/*(F>F+F4F+F,F+#F+\"\"'/,&*&F=F+)F,F*F+F+* &F>F+)F6F*F+F+#F+F3" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 29 "Alternatively the procedures " }{TEXT 0 16 "RowSumCondi tions" }{TEXT -1 5 " and " }{TEXT 0 15 "OrderConditions" }{TEXT -1 13 " can be used." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 174 "RK3_3eqs := [op(RowSumConditions(3,'expanded')) ,op(OrderConditions(3,3,'expanded'))]:\nfor ct to nops(RK3_3eqs) do\n \+ print(`equation `||ct,` ... `,RK3_3eqs[ct]); \nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%+equation~1G%(~~...~~G/&%\"aG6$\"\"#\"\"\"& %\"cG6#F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%+equation~2G%(~~...~~G/, &&%\"aG6$\"\"$\"\"\"F+&F(6$F*\"\"#F+&%\"cG6#F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%+equation~3G%(~~...~~G/,(&%\"bG6#\"\"\"F*&F(6#\"\"#F*& F(6#\"\"$F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%+equation~4G%(~~...~ ~G/,&*&&%\"bG6#\"\"$\"\"\"&%\"cGF*F,F,*&&F)6#\"\"#F,&F.F1F,F,#F,F2" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%%+equation~5G%(~~...~~G/*(&%\"bG6#\"\" $\"\"\"&%\"aG6$F*\"\"#F+&%\"cG6#F/F+#F+\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%+equation~6G%(~~...~~G/,&*&&%\"bG6#\"\"#\"\"\")&%\"cGF *F+F,F,*&&F)6#\"\"$F,)&F/F2F+F,F,#F,F3" }}}{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 78 "Pr incipal error terms and associated conditions for a 3 stage order 3 sc heme: " }{TEXT 0 14 "RK3_3err_terms" }{TEXT -1 2 ", " }{TEXT 0 12 "RK3 _3err_eqs" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 116 "The principal error terms are those terms which correspond to the order 4 terms in the Taylor series of a solution. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 31 "A breviated form of error terms " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 167 "PrincipalErrorTerms(3):\nco nvert(ListTools[Enumerate](%),matrix):\nlinalg[augment](linalg[delcols ](%,2..2),matrix([[` `]$(linalg[rowdim](%))]),linalg[delcols](%,1..1) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7%\"\"\"%#~~G,&*&% \"bGF(-%!G6#*&%\"aGF(-F.6#*&F1F(%\"cGF(F(F(F(#F(\"#C!\"\"7%\"\"#F),&*( F,F(F5F(F2F(F(#F(\"\")F87%\"\"$F),$*&#F(F:F(-F.6#,&*&F,F(-F.6#*&)F5F:F (F1F(F(F(#F(\"#7F8F(F(7%\"\"%F),$*&#F(\"\"'F(-F.6#,&*&F,F()F5F@F(F(#F( FOF8F(F(Q(pprint06\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 37 "Long (summation) form of error terms " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 169 "PrincipalErrorTerms( 3,3):\nconvert(ListTools[Enumerate](%),matrix):\nlinalg[augment](linal g[delcols](%,2..2),matrix([[` `]$(linalg[rowdim](%))]),linalg[delcols ](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7&7%\"\"\"% #~~G,&-%$SumG6$*&&%\"bG6#%\"iGF(-F,6$*&&%\"aG6$F2%\"jGF(-F,6$*&&F76$F9 %\"kGF(&%\"cG6#F?F(/F?;F(,&F9F(F(!\"\"F(/F9;F(,&F2F(F(FFF(/F2;F(\"\"$F (#F(\"#CFF7%\"\"#F),&-F,6$*(F/F(&FAF1F(-F,6$*&F6F(&FA6#F9F(FGF(FJF(#F( \"\")FF7%FLF),&*&#F(FPF(-F,6$*&F/F(-F,6$*&F6F()FYFPF(FGF(FJF(F(#F(FNFF 7%\"\"%F),&*&#F(\"\"'F(-F,6$*&F/F()FUFLF(FJF(F(#F(FNFFQ(pprint16\"" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 37 "Set up error terms in expanded form: " }{TEXT 0 14 "RK3_3err_terms" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 167 "RK3_3err_te rms := PrincipalErrorTerms(3,3,'expanded'):\nfor ct to nops(RK3_3err_t erms) do\n print(`error term `||ct,` ... `,RK3_3err_terms[ct]);pri nt(``); \nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%-error~term~1G %(~~...~~G#!\"\"\"#C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%-error~term~2G%(~~...~~G,&#\"\"\"\"\")!\"\" **&%\"bG6#\"\"$F'&%\"cGF-F'&%\"aG6$F.\"\"#F'&F06#F4F'F'" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%-error~t erm~3G%(~~...~~G,&#\"\"\"\"#C!\"\"*&#F'\"\"#F'*(&%\"bG6#\"\"$F'&%\"aG6 $F1F,F')&%\"cG6#F,F,F'F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%%-error~term~4G%(~~...~~G,(#\"\"\"\"#C !\"\"*&#F'\"\"'F'*&&%\"bG6#\"\"#F')&%\"cGF0\"\"$F'F'F'*&F+F'*&&F/6#F5F ')&F4F9F5F'F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 42 "Set up error conditions in expand ed form: " }{TEXT 0 12 "RK3_3err_eqs" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "Note that the first error condition is \+ impossible to satisfy with a 3 stage scheme." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 164 "RK3_3err_eqs := P rincipalErrorConditions(3,3,'expanded'):\nfor ct to nops(RK3_3err_eqs) do\n print(`equation `||ct,` ... `,RK3_3err_eqs[ct]);print(``); \+ \nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%+equation~1G%(~~...~~G/ \"\"!#\"\"\"\"#C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6%%+equation~2G%(~~...~~G/**&%\"bG6#\"\"$\"\"\"&% \"cGF)F+&%\"aG6$F*\"\"#F+&F-6#F1F+#F+\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%+equation~3G%(~ ~...~~G/*(&%\"bG6#\"\"$\"\"\"&%\"aG6$F*\"\"#F+)&%\"cG6#F/F/F+#F+\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%+equation~4G%(~~...~~G/,&*&&%\"bG6#\"\"#\"\"\")&%\"cGF*\"\"$F,F,*& &F)6#F0F,)&F/F3F0F,F,#F,\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}}{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 48 "Examples of 3 stage, order 3 Runge-Kutta \+ schemes" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "Classical order 3 " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[1/2, 1/2, ``, ``], [1, -1, 2, \+ ``], [``, 1/6, 2/3, 1/6]])" "6#-%'matrixG6#7%7&*&\"\"\"F)\"\"#!\"\"*&F )F)F*F+%!GF-7&F),$F)F+F*F-7&F-*&F)F)\"\"'F+*&F*F)\"\"$F+*&F)F)F2F+" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 43 "We specify two of the co efficients, namely " }{XPPEDIT 18 0 "c[2] = 1/2,c[3] = 1;" "6$/&%\"cG6 #\"\"#*&\"\"\"F)F'!\"\"/&F%6#\"\"$F)" }{TEXT -1 59 ", so as to obtain \+ a system of 6 equations with 6 unknowns. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "RK3_3eqs := [op(RowSu mConditions(3,'expanded')),op(OrderConditions(3,3,'expanded'))]:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 169 "e1 := \{c[2]=1/2,c[3]=1\}: \ne2 := solve(subs(e1,\{op(RK3_3eqs)\})):\ne3 := `union`(e1,e2);\nsubs (e3,matrix([[c[2],a[2,1],``,``],\n[c[3],a[3,1],a[3,2],``],[``,b[1],b[2 ],b[3]]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e3G<*/&%\"cG6#\"\"## \"\"\"F*/&F(6#\"\"$F,/&%\"bGF/#F,\"\"'/&%\"aG6$F0F,!\"\"/&F86$F0F*F*/& F36#F,F4/&F86$F*F,F+/&F3F)#F*F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%' matrixG6#7%7&#\"\"\"\"\"#F(%!GF+7&F)!\"\"F*F+7&F+#F)\"\"'#F*\"\"$F/Q(p print36\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "subs(e3,RK3 _3eqs);\nmap(u->lhs(u)-rhs(u),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7 (/#\"\"\"\"\"#F%/F&F&F(F$/#F&\"\"'F*/#F&\"\"$F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(\"\"!F$F$F$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 94 "We can calculate the principal error norm , that is, the 2-norm of the principal error terms. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 149 "errterms := PrincipalErrorTerms(3,3,'exp anded'):\nsm := 0:\nfor ct to nops(errterms) do\n sm := sm+(evalf(su bs(e3,errterms[ct])))^2;\nend do:\nsqrt(sm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+5lb#*e!#6" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 53 "Two of the principal error conditions are satisfied. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "RK3_3err_eq s := PrincipalErrorConditions(3,3,'expanded'):\nsubs(e3,RK3_3err_eqs): \nmap(u->lhs(u)-rhs(u),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&#!\"\" \"#C#\"\"\"F&\"\"!F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "PrincipalErrorConditions(3,3,'expanded'); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&/\"\"!#\"\"\"\"#C/**&%\"bG6#\"\" $F'&%\"cGF-F'&%\"aG6$F.\"\"#F'&F06#F4F'#F'\"\")/*(F+F'F1F')F5F4F'#F'\" #7/,&*&&F,F6F')F5F.F'F'*&F+F')F/F.F'F'#F'\"\"%" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([b[3]*a[3,2]*c[2] ^2 = 1/12,``],[ b[2]*c[2]^3+b[3]*c[3]^3 = 1/4,``])" "6#-%*PIECEWISEG6$ 7$/*(&%\"bG6#\"\"$\"\"\"&%\"aG6$F,\"\"#F-&%\"cG6#F1F1*&F-F-\"#7!\"\"%! G7$/,&*&&F*6#F1F-*$&F36#F1F,F-F-*&&F*6#F,F-*$&F36#F,F,F-F-*&F-F-\"\"%F 7F8" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 39 "The weights of the system are those of " }{TEXT 260 14 "Simpson's Rule" }{TEXT -1 283 " for numerical integration. It is well -known that Simpson's Rule, derived to give exact results when integra ting a degree 2 polynomial, actually integrates a degree 3 polynomial \+ exactly. Obtaining the extra quadrature condition as a bonus for this \+ scheme is therefore not a surprise." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 145 "The classical scheme is the unique sche me which satisfies the row sum conditions, order conditions and the ex tra two principal error conditions. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 193 "err_eqns := [b[3]*a[3,2]*c[2]^2=1/12,b[2]*c[2]^3+b[3 ]*c[3]^3=1/4]:\ne1 := solve(\{op(RK3_3eqs),op(err_eqns)\});\nsubs(e3,m atrix([[c[2],a[2,1],``,``],\n[c[3],a[3,1],a[3,2],``],[``,b[1],b[2],b[3 ]]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e1G<*/&%\"aG6$\"\"#\"\"\" #F+F*/&%\"bG6#F+#F+\"\"'/&%\"cG6#\"\"$F+/&F56#F*F,/&F/F6F1/&F(6$F7F*F* /&F/F:#F*F7/&F(6$F7F+!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matri xG6#7%7&#\"\"\"\"\"#F(%!GF+7&F)!\"\"F*F+7&F+#F)\"\"'#F*\"\"$F/Q(pprint 16\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 18 "Variant 1 \+ order 3 " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[ 1/3, 1/3, ``, ``], [2/3, 0, 2/3, ``], [``, 1/4, 0, 3/4]])" "6#-%'matri xG6#7%7&*&\"\"\"F)\"\"$!\"\"*&F)F)F*F+%!GF-7&*&\"\"#F)F*F+\"\"!*&F0F)F *F+F-7&F-*&F)F)\"\"%F+F1*&F*F)F5F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "We specify two of the coe fficients, namely " }{XPPEDIT 18 0 "c[2] = 1/3,c[3] = 2/3;" "6$/&%\"cG 6#\"\"#*&\"\"\"F)\"\"$!\"\"/&F%6#F**&F'F)F*F+" }{TEXT -1 59 ", so as t o obtain a system of 6 equations with 6 unknowns. " }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 85 "RK3_3eqs := [op(RowSumConditions(3,'expanded ')),op(OrderConditions(3,3,'expanded'))]:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 171 "e1 := \{c[2]=1/3,c[3]=2/3\}:\ne2 := solve(subs(e1, \{op(RK3_3eqs)\})):\ne3 := `union`(e1,e2);\nsubs(e3,matrix([[c[2],a[2, 1],``,``],\n[c[3],a[3,1],a[3,2],``],[``,b[1],b[2],b[3]]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e3G<*/&%\"bG6#\"\"#\"\"!/&F(6#\"\"$#F/\" \"%/&%\"cGF)#\"\"\"F//&F4F.#F*F//&%\"aG6$F*F6F5/&F(6#F6#F6F1/&F<6$F/F6 F+/&F<6$F/F*F9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7%7&#\" \"\"\"\"$F(%!GF+7&#\"\"#F*\"\"!F-F+7&F+#F)\"\"%F/#F*F2Q(pprint46\"" }} }{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "subs(e3,RK3_3eqs);\nmap( u->lhs(u)-rhs(u),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(/#\"\"\"\"\" $F%/#\"\"#F'F)/F&F&/#F&F*F-/#F&\"\"'F/F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(\"\"!F$F$F$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 78 "The principal error norm is less than that of the \+ classical order 3 scheme. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 149 "errterms := PrincipalErrorTerms(3,3,'expanded'):\nsm := 0:\nfor c t to nops(errterms) do\n sm := sm+(evalf(subs(e3,errterms[ct])))^2; \nend do:\nsqrt(sm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+J'H'HY!#6 " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "None of the principal error conditions are satisfied. " }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 105 "RK3_3err_eqs := PrincipalErrorConditions(3, 3,'expanded'):\nsubs(e3,RK3_3err_eqs):\nmap(u->lhs(u)-rhs(u),%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7&#!\"\"\"#C#F%\"#s#F%\"#OF)" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 18 "Variant 2 order 3 " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[3/7, 3/7, `` , ``], [16/21, -64/945, 112/135, ``], [``, 19/96, 1/3, 15/32]])" "6#-% 'matrixG6#7%7&*&\"\"$\"\"\"\"\"(!\"\"*&F)F*F+F,%!GF.7&*&\"#;F*\"#@F,,$ *&\"#kF*\"$X*F,F,*&\"$7\"F*\"$N\"F,F.7&F.*&\"#>F*\"#'*F,*&F*F*F)F,*&\" #:F*\"#KF," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "We specify two of the coefficients, namely " } {XPPEDIT 18 0 "c[2] = 3/7,b[2] = 1/3;" "6$/&%\"cG6#\"\"#*&\"\"$\"\"\" \"\"(!\"\"/&%\"bG6#F'*&F*F*F)F," }{TEXT -1 59 ", so as to obtain a sys tem of 6 equations with 6 unknowns. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "RK3_3eqs := [op(RowSumCond itions(3,'expanded')),op(OrderConditions(3,3,'expanded'))]:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 171 "e1 := \{c[2]=3/7,b[2]=1/3\} :\ne2 := solve(subs(e1,\{op(RK3_3eqs)\})):\ne3 := `union`(e1,e2);\nsub s(e3,matrix([[c[2],a[2,1],``,``],\n[c[3],a[3,1],a[3,2],``],[``,b[1],b[ 2],b[3]]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e3G<*/&%\"cG6#\"\"# #\"\"$\"\"(/&%\"bGF)#\"\"\"F,/&%\"aG6$F*F2F+/&F06#F,#\"#:\"#K/&F(F9#\" #;\"#@/&F56$F,F*#\"$7\"\"$N\"/&F56$F,F2#!#k\"$X*/&F06#F2#\"#>\"#'*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7%7&#\"\"$\"\"(F(%!GF+7&# \"#;\"#@#!#k\"$X*#\"$7\"\"$N\"F+7&F+#\"#>\"#'*#\"\"\"F)#\"#:\"#KQ(ppri nt56\"" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "subs(e3,RK3_3e qs);\nmap(u->lhs(u)-rhs(u),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(/# \"\"$\"\"(F%/#\"#;\"#@F)/\"\"\"F-/#F-\"\"#F//#F-\"\"'F2/#F-F&F5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7(\"\"!F$F$F$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "The principal error norm \+ is less than that of the previous two schemes. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 149 "errterms := PrincipalErrorTerms(3,3,'expande d'):\nsm := 0:\nfor ct to nops(errterms) do\n sm := sm+(evalf(subs(e 3,errterms[ct])))^2;\nend do:\nsqrt(sm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+d@aAU!#6" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 54 "None of the principal error conditions are satisfied. \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "RK3_3err_eqs := Princip alErrorConditions(3,3,'expanded'):\nsubs(e3,RK3_3err_eqs):\nmap(u->lhs (u)-rhs(u),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&#!\"\"\"#C#\"\"\" \"$/&#F%\"#%)#!#H\"%k<" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 18 "Variant 3 order 3 " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "matrix([[1/2, 1/2, ``, ``], [1/4+sqrt(33)/12, -5/12+sqr t(33)/12, 2/3, ``], [``, -1/4+sqrt(33)/12, 3/4-sqrt(33)/12, 1/2]])" "6 #-%'matrixG6#7%7&*&\"\"\"F)\"\"#!\"\"*&F)F)F*F+%!GF-7&,&*&F)F)\"\"%F+F )*&-%%sqrtG6#\"#LF)\"#7F+F),&*&\"\"&F)F7F+F+*&-F46#F6F)F7F+F)*&F*F)\" \"$F+F-7&F-,&*&F)F)F1F+F+*&-F46#F6F)F7F+F),&*&F?F)F1F+F)*&-F46#F6F)F7F +F+*&F)F)F*F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 43 "We specify two of the coefficients, namel y " }{XPPEDIT 18 0 "c[2] = 1/2,b[3] = 1/2;" "6$/&%\"cG6#\"\"#*&\"\"\"F )F'!\"\"/&%\"bG6#\"\"$*&F)F)F'F*" }{TEXT -1 59 ", so as to obtain a sy stem of 6 equations with 6 unknowns. " }}{PARA 0 "" 0 "" {TEXT 260 4 " Note" }{TEXT -1 72 ": If you re-execute these commands, you may have t o change the index in " }{TEXT 261 19 "[allvalues(sol)][2]" }{TEXT -1 32 " to obtain the desired solution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "RK3_3eqs := [op(RowSumCond itions(3,'expanded')),op(OrderConditions(3,3,'expanded'))]:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 214 "e1 := \{c[2]=1/2,b[3]=1/2\} :\nsol := solve(subs(e1,\{op(RK3_3eqs)\})):\ne2 := [allvalues(sol)][2] ;\ne3 := `union`(e1,e2);\nsubs(e3,matrix([[c[2],a[2,1],``,``],\n[c[3], a[3,1],a[3,2],``],[``,b[1],b[2],b[3]]]));\n``;\nevalf(%%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e2G<(/&%\"aG6$\"\"#\"\"\"#F+F*/&F(6$\"\"$ F+,&#\"\"&\"#7!\"\"*&F4F5\"#LF,F+/&F(6$F0F*#F*F0/&%\"bG6#F*,&#F0\"\"%F +*&F4F5F7F,F5/&%\"cG6#F0,&#F+FBF+*&F4F5F7F,F+/&F>6#F+,&#F+FBF5*&F4F5F7 F,F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e3G<*/&%\"aG6$\"\"#\"\"\"#F +F*/&%\"cG6#F*F,/&F(6$\"\"$F+,&#\"\"&\"#7!\"\"*&F8F9\"#LF,F+/&F(6$F4F* #F*F4/&%\"bG6#F4F,/&FBF0,&#F4\"\"%F+*&F8F9F;F,F9/&F/FC,&#F+FHF+*&F8F9F ;F,F+/&FB6#F+,&#F+FHF9*&F8F9F;F,F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# K%'matrixG6#7%7&#\"\"\"\"\"#F(%!GF+7&,&#F)\"\"%F)*&\"#7!\"\"\"#LF(F),& #\"\"&F1F2*&F1F2F3F(F)#F*\"\"$F+7&F+,&#F)F/F2*&F1F2F3F(F),&#F9F/F)*&F1 F2F3F(F2F(Q)pprint196\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7%7&$\"+++++]!#5F(%!GF+7&$ \"+Rb8(G(F*$\"*s)o/iF*$\"+nmmmmF*F+7&F+$\"+Rb8(G#F*$\"+hW'Gr#F*F(Q)ppr int206\"" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "simplify(su bs(e3,RK3_3eqs));\nmap(u->lhs(u)-rhs(u),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(/#\"\"\"\"\"#F%/,&#F&\"\"%F&*&\"#7!\"\"\"#LF%F&F)/F&F &F$/#F&\"\"'F2/#F&\"\"$F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(\"\"!F$ F$F$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "The principal error norm is less than that of the previous three s chemes. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 149 "errterms := P rincipalErrorTerms(3,3,'expanded'):\nsm := 0:\nfor ct to nops(errterms ) do\n sm := sm+(evalf(subs(e3,errterms[ct])))^2;\nend do:\nsqrt(sm) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+/Oo)>%!#6" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "One of the principal er ror conditions are satisfied. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "RK3_3err_eqs := PrincipalErrorConditions(3,3,'expanded'):\nsimp lify(subs(e3,RK3_3err_eqs)):\nmap(u->lhs(u)-rhs(u),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&#!\"\"\"#C,&#\"\"\"\"#7F%*&\"#sF%\"#L#F)\"\"#F) \"\"!,&#F)\"#;F%*&\"$W\"F%F-F.F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "By incorporating the extra error condition " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[3]*a[3,2]*c[2]^2=1/12" "6#/*( &%\"bG6#\"\"$\"\"\"&%\"aG6$F(\"\"#F)&%\"cG6#F-F-*&F)F)\"#7!\"\"" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 100 "in the system of equati ons to be solved, this scheme can be derived by specifying the single \+ weight " }{XPPEDIT 18 0 "b[3]=1/2" "6#/&%\"bG6#\"\"$*&\"\"\"F)\"\"#!\" \"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 214 "e1 := \{b[3]=1/2\}:\nsol := solve(subs(e1, \{op(RK3_3eqs),b[3]*a[3,2]*c[2]^2=1/12\})):\ne2 := [allvalues(sol)][2] ;\ne3 := `union`(e1,e2);\nsubs(e3,matrix([[c[2],a[2,1],``,``],\n[c[3], a[3,1],a[3,2],``],[``,b[1],b[2],b[3]]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e2G<)/&%\"aG6$\"\"#\"\"\"#F+F*/&%\"cG6#F*F,/&%\"bG6# F+,&#F+\"\"%!\"\"*&\"#7F8\"#LF,F+/&F(6$\"\"$F*#F*F?/&F3F0,&#F?F7F+*&F: F8F;F,F8/&F/6#F?,&#F+F7F+*&F:F8F;F,F+/&F(6$F?F+,&#\"\"&F:F8*&F:F8F;F,F +" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e3G<*/&%\"aG6$\"\"#\"\"\"#F+F* /&%\"cG6#F*F,/&%\"bG6#F+,&#F+\"\"%!\"\"*&\"#7F8\"#LF,F+/&F36#\"\"$F,/& F(6$F?F*#F*F?/&F3F0,&#F?F7F+*&F:F8F;F,F8/&F/F>,&#F+F7F+*&F:F8F;F,F+/&F (6$F?F+,&#\"\"&F:F8*&F:F8F;F,F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%' matrixG6#7%7&#\"\"\"\"\"#F(%!GF+7&,&#F)\"\"%F)*&\"#7!\"\"\"#LF(F),&#\" \"&F1F2*&F1F2F3F(F)#F*\"\"$F+7&F+,&#F)F/F2*&F1F2F3F(F),&#F9F/F)*&F1F2F 3F(F2F(Q)pprint326\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "Variant 4 order 3 " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "matrix([[1/2, 1/2, ``, ``], [109/145, -109/21025, 15914 /21025, ``], [``, 145/654, 74/219, 21025/47742]])" "6#-%'matrixG6#7%7& *&\"\"\"F)\"\"#!\"\"*&F)F)F*F+%!GF-7&*&\"$4\"F)\"$X\"F+,$*&F0F)\"&D5#F +F+*&\"&9f\"F)F4F+F-7&F-*&F1F)\"$a'F+*&\"#uF)\"$>#F+*&F4F)\"&Ux%F+" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "By incorporating the extra error condition " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "b[3]*a[3,2]*c[2]^2=1/12" "6#/*( &%\"bG6#\"\"$\"\"\"&%\"aG6$F(\"\"#F)&%\"cG6#F-F-*&F)F)\"#7!\"\"" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 94 "in the system of equatio ns to be solved, a system of 7 equations with 8 unknowns is obtained. \+ " }}{PARA 0 "" 0 "" {TEXT -1 41 "We can obtain a solution with the wei ght " }{XPPEDIT 18 0 "b[2]" "6#&%\"bG6#\"\"#" }{TEXT -1 17 " as a para meter. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "RK3_3eqs := [op(RowSumConditions(3,'expanded')),op(Or derConditions(3,3,'expanded'))]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "e1 := solve(\{op(RK3_3eqs),b[3]*a[3,2]*c[2]^2=1/12\} ):\nsubs(e1,matrix([[c[2],a[2,1],``,``],\n[c[3],a[3,1],a[3,2],``],[``, b[1],b[2],b[3]]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7%7 &#\"\"\"\"\"#F(%!GF+7&,$*&#F)\"\"'F)*&,&*&\"\"$F)&%\"bG6#F*F)F)\"\"%! \"\"F),&F5F)F)F9F9F)F),$*&#F)\"#=F)*(,&*&F4F)F5F)F)F)F9F)F2F)F:!\"#F)F ),$*&#F)\"\"*F)*&,(*$)F5F*F)F)*&F*F)F5F)F9F)F)F9F2F)F)F9F+7&F+*&F:F)F2 F9F5,$*(F4F)FHF)F2F9F9Q)pprint336\"" }}}{PARA 0 "" 0 "" {TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "simplify(subs(e1,RK3_3eqs));\nmap(u->lhs(u)-rhs(u),%) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(/#\"\"\"\"\"#F%/,$*&#F&\"\"'F&* &,&*&\"\"$F&&%\"bG6#F'F&F&\"\"%!\"\"F&,&F1F&F&F5F5F&F&F)/F&F&F$/F+F+/# F&F0F:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(\"\"!F$F$F$F$F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "The sum of the \+ sqauares of the principal errors can be expressed in terms of " } {XPPEDIT 18 0 "b[2]" "6#&%\"bG6#\"\"#" }{TEXT -1 5 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 134 "errterms := PrincipalErrorTerms(3, 3,'expanded'):\nsm := 0:\nfor ct to nops(errterms) do\n sm := sm+sub s(e1,errterms[ct])^2;\nend do:\nsm;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #,(#\"\"\"\"$w&F%*$),&#F%\"\")!\"\"*&#F%\"#OF%*&,&*&\"\"$F%&%\"bG6#\" \"#F%F%\"\"%F,F%,&F4F%F%F,F,F%F%F7F%F%*$),(#F%\"#CF,*&#F%\"#[F%F4F%F%* &#F%\"$K%F%*(,(*$)F4F7F%F%*&F7F%F4F%F,F%F%F%F1F7F9!\"$F%F,F7F%F%" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "plot(sm,b[2]=0..1,0..0.008,color=blue);" }}{PARA 13 "" 1 "" {GLPLOT2D 303 239 239 {PLOTDATA 2 "6&-%'CURVESG6#7\\q7$$\"\"!F)$\"3-dl =;eW]>!#?7$$\"3')*****fT:(z@!#>$\"3w^u'pVgJ$>F,7$$\"33+++XTFwSF0$\"3.. !QNu[\"=>F,7$$\"3&******4z_\"4iF0$\"3iaZS1-N,>F,7$$\"3o******R&phN)F0$ \"3)yZ75[&f%)=F,7$$\"3++++*=)H\\5!#=$\"3#=*pPD^:o=F,7$$\"3%******z/3uC \"FE$\"3'=v\\RO/K&=F,7$$\"35+++J$RDX\"FE$\"3L`g\")fi6Q=F,7$$\"37+++)R' ok;FE$\"32(4-e@aI#=F,7$$\"3-+++1J:w=FE$\"3dZtIDyt3=F,7$$\"33+++3En$4#F E$\"31kq>Xa#\\z\"F,7$$\"3-+++/RE&G#FE$\"3aob**=uq$y\"F,7$$\"3\")***** \\K]4]#FE$\"33.fZ\"[#QsxWv\"F,7$$\"31+++$Q*o]RFE$\"3o6P5EQ \"Rw\"F,7$$\"3?+++\"=lj;%FE$\"3EcBI*er.y\"F,7$$\"3++++V&RF,7$$\"3A+++(RQb@&FE$ \"3g'o3W`L#4?F,7$$\"3`+++=>Y2aFE$\"3'=]c&f2&Q4#F,7$$\"3\\+++yXu9cFE$\" 3-IxH;Yx3AF,7$$\"3d******[y))GeFE$\"3#GZfp04&fBF,7$$\"3.+++i_QQgFE$\"3 'RjmnGjna#F,7$$\"3A+++!y%3TiFE$\"3ybE/%>8cx#F,7$$\"35+++O![hY'FE$\"3yN ezRJp+JF,7$$\"3I+++#Qx$omFE$\"3MSo@'))*yvMF,7$$\"3s*****RP+V)oFE$\"3u \\3SpVT\"*RF,7$$\"3Y*****ppe*zqFE$\"3Wcnc'f%>&f%F,7$$\"3u*****R#\\'QH( FE$\"3%H(H=LONdaF,7$$\"3#******H,M^\\(FE$\"3*>!>Yv`)R`'F,7$$\"3S+++0#= bq(FE$\"3#=hq(=*Q)\\!)F,7$$\"3Y*****p?27\"zFE$\"3AG&e[+C#45F07$$\"3a++ +IXaE\")FE$\"31T$)))3E(HJ\"F07$$\"37+++l*RRL)FE$\"3&[sHz;eCu\"F07$$\"3 i*****HvJga)FE$\"3w9Z#*=x98CF07$$\"3s*****HJnjv)FE$\"3[pY$>et%)[$F07$$ \"3k******[Qk\\*)FE$\"3#[lT7x0,;&F07$$\"3M+++5ASg!*FE$\"3A&H%4g3*[l'F0 7$$\"3w******o0;r\"*FE$\"3H[I@E5MI))F07$$\"3Y*****\\;C-F*FE$\"3%4]&y;. \\s6FE7$$\"3[*****\\w(Gp$*FE$\"3IkV9?=D;;FE7$$\"3;+++?-\"\\Z*FE$\"3hj] _n0>1CFE7$$\"37+++!oK0e*FE$\"3kz&R]IT9*QFE7$$\"3^+++?i!eg*FE$\"3hw:\"f >Z*RWFE7$$\"3=+++l(z5j*FE$\"3jOE%p*FE$\"3 /(*)Q%)>v&zvFE7$$\"3\\*******Q+pq*FE$\"3o.1?P?my#)FE7$$\"3B+++gr`>(*FE $\"3gF#[d(*\\f2*FE7$$\"3G+++NRZu`luk/6!#<7$$\"3a*****\\ZZuv*FE$\"3**R5`4HTF7F[^l7$$\"3!*** ****fewj(*FE$\"3uyYZ.[T'H\"F[^l7$$\"3G+++XU3q(*FE$\"3go=)zMd7P\"F[^l7$ $\"3a******HESw(*FE$\"3_]$Q%4Uh_9F[^l7$$\"33+++<5s#y*FE$\"3wG%>R\")e7a \"F[^l7$$\"3]*****\\)4^*y*FE$\"3#H\"[fuYmX;F[^l7$$\"3?+++b4I'z*FE$\"3/ ^osoh!3w\"F[^l7$$\"3\\+++?44.)*FE$\"3ipRAax>))=F[^l7$$\"33+++!*3))4)*F E$\"3\\JYK\\.jH?F[^l7$$\"3n******f3n;)*FE$\"3U\"e;-OLs=#F[^l7$$\"3P+++ I3YB)*FE$\"3SD&3]'[bjBF[^l7$$\"3'********z]-$)*FE$\"3!>V(eiTmhDF[^l7$$ \"3C+++l2/P)*FE$\"3O,xQA[G&y#F[^l7$$\"3`+++I2$Q%)*FE$\"3+7pRZ)p*QIF[^l 7$$\"37++++2i])*FE$\"3<(4!pGILGLF[^l7$$\"3s******p1Td)*FE$\"3Y@894\"f. m$F[^l7$$\"3U+++S1?k)*FE$\"3?tP#R[EQ/%F[^l7$$\"3++++51*4()*FE$\"39R*\\ ^#))))*[%F[^l7$$\"3G+++v0yx)*FE$\"3Y/J\"=/*)G,&F[^l7$$\"3Y******R0d%)) *FE$\"3$H'[7Z*>:j&F[^l7$$\"3;+++50O\"*)*FE$\"3gsF,$\\10P'F[^l7$$\"3u** ****z/:)*)*FE$\"3u*>EtD)4jsF[^l7$$\"3/+++X/%\\!**FE$\"3%>!GeFU&[N)F[^l 7$$\"3K+++5/t6**FE$\"3BR'*\\SIa4(*F[^l7$$\"3\"*******z._=**FE$\"3(y%4J W?'=9\"!#;7$$\"3^******\\.JD**FE$\"3!px'4s**ph8Fiel7$$\"3@+++?.5K**FE$ \"3mNQgW'R5l\"Fiel7$$\"3!)*******G!*)Q**FE$\"3CW$\\AT0D/#Fiel7$$\"33++ +b-oX**FE$\"3)[SnP%HN!f#Fiel7$$\"3Q+++?-Z_**FE$\"3')*fU@.k-R$Fiel7$$\" 3'********=g#f**FE$\"3QCEJ#o+Si%Fiel7$$\"3a******f,0m**FE$\"3OTGu9GBsm Fiel7$$\"3C+++I,%G(**FE$\"3/'GaL0xY/\"!#:7$$\"3%)*******4I'z**FE$\"3yl ky.t,h=Fbhl7$$\"37+++l+U')**FE$\"3'Qb6R>&)e>%Fbhl7$$\"3S+++I+@$***FE$ \"3Ao%*G&z)z\"o\"!#97$%*undefinedGFdil-%'COLOURG6&%$RGBGF(F($\"*++++\" !\")-%+AXESLABELSG6$Q%b[2]6\"Q!F`jl-%%VIEWG6$;F($\"\"\"F);F($\"\")!\"$ " 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }} }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "We can \+ find the value of " }{XPPEDIT 18 0 "b[2]" "6#&%\"bG6#\"\"#" }{TEXT -1 37 " that minimizes the principal error. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "minimize(sm,location);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$#\"#t\"&g<%<#7$<#/&%\"bG6#\"\"##\"#u\"$>#F#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "The asociated scheme can \+ the be obtained as follows." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 204 "e1 := \{b[2]=74/219\}:\ne2 := solve(subs(e1,\{op(RK3_3eqs),b[3]*a [3,2]*c[2]^2=1/12\})):\ne3 := `union`(e1,e2);\nsubs(e3,matrix([[c[2],a [2,1],``,``],\n[c[3],a[3,1],a[3,2],``],[``,b[1],b[2],b[3]]]));\n``;\ne valf(%%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e3G<*/&%\"bG6#\"\"\"# \"$X\"\"$a'/&%\"aG6$\"\"#F*#F*F2/&%\"cG6#F2F3/&F06$\"\"$F*#!$4\"\"&D5# /&F66#F;#\"$4\"F,/&F06$F;F2#\"&9f\"F>/&F(FA#F>\"&Ux%/&F(F7#\"#u\"$>#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7%7&#\"\"\"\"\"#F(%!GF+7 &#\"$4\"\"$X\"#!$4\"\"&D5##\"&9f\"F2F+7&F+#F/\"$a'#\"#u\"$>##F2\"&Ux%Q )pprint366\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#K%'matrixG6#7%7&$\"+++++]!#5F(%!GF+7&$\"+z8C " 0 "" {MPLTEXT 1 0 149 "errterms := Princip alErrorTerms(3,3,'expanded'):\nsm := 0:\nfor ct to nops(errterms) do\n sm := sm+(evalf(subs(e3,errterms[ct])))^2;\nend do:\nsqrt(sm);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+&z45=%!#6" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 18 "Variant 5 order 3 " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[p, p, ``, ``], [q, q*(q+3*p* (p-1))/(3*p-2)/p, p, ``], [``, (3*p*(2*q-1)-3*q+2)/(6*p*q), (3*q-2)/(6 *(q-p)*p), p]]);" "6#-%'matrixG6#7%7&%\"pGF(%!GF)7&%\"qG**F+\"\"\",&F+ F-*(\"\"$F-F(F-,&F(F-F-!\"\"F-F-F-,&*&F0F-F(F-F-\"\"#F2F2F(F2F(F)7&F)* &,(*(F0F-F(F-,&*&F5F-F+F-F-F-F2F-F-*&F0F-F+F-F2F5F-F-*(\"\"'F-F(F-F+F- F2*&,&*&F0F-F+F-F-F5F2F-*(F>F-,&F+F-F(F2F-F(F-F2F(" }{TEXT -1 3 " , " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "p=6^(-1/3)" "6#/ %\"pG)\"\"',$*&\"\"\"F)\"\"$!\"\"F+" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "q = (1+sqrt(9-12*p))*p/2" "6#/%\"qG*(,&\"\"\"F'-%%sqrtG6#,&\"\"*F'* &\"#7F'%\"pGF'!\"\"F'F'F/F'\"\"#F0" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "We specify that three o f the coefficients " }{XPPEDIT 18 0 "b[3]" "6#&%\"bG6#\"\"$" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a[2,1]" "6#&%\"aG6$\"\"#\"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "a[3,2]" "6#&%\"aG6$\"\"$\"\"#" }{TEXT -1 137 " are equal so as to obtain a system of 9 equations with 9 unknowns inc luding the 8 Runge-Kutta coefficients and the unknown common value " } {TEXT 265 1 "p" }{TEXT -1 31 " of three of the coefficients. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "RK3_3eqs := [op(RowSumConditions(3,'expanded')),op(OrderConditions (3,3,'expanded'))]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 247 "e1 \+ := \{b[3]=p,a[3,2]=p,a[2,1]=p\}:\nsubs(e1,\{op(RK3_3eqs)\}):\nsolve(%) :\ne2 := remove(has,[allvalues(%)],Complex(1))[2]:\ne3 := `union`(subs (e2,e1),e2):\nsubs(e3,matrix([[c[2],a[2,1],``,``],\n[c[3],a[3,1],a[3,2 ],``],[``,b[1],b[2],b[3]]]));\n``:\nevalf(%%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7%7&,$*&\"\"'!\"\"F*#\"\"#\"\"$\"\"\"F(%!GF 07&,&*&\"#7F+F*F,F/*&\"\"%F+,&\"\")F+*&F*F/)F*#F/F.F/F/#F/F-F/,&*&F4F+ F*F,F+*&F6F+F7F " 0 "" {MPLTEXT 1 0 53 "simplify(subs(e3,RK3_3eqs));\nmap(u ->lhs(u)-rhs(u),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(/,$*&\"\"'!\" \"F'#\"\"#\"\"$\"\"\"F%/,&*&\"#7F(F'F)F,*&\"\"%F(,&\"\")F(*&F'F,)F'#F, F+F,F,#F,F*F,F./F,F,/F8F8/#F,F'F " 0 "" {MPLTEXT 1 0 149 "errterms := PrincipalErrorTerms(3,3,'expanded'):\nsm := 0:\nfor ct to nops(errterms) do\n sm := sm+(evalf(subs(e3,errterms[ct])))^2;\nend do:\nsqrt(sm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+eM(yG%!#6" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "None of t he principal error conditions are satisfied. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "RK3_3err_eqs := PrincipalErrorConditions(3,3,'e xpanded'):\nsimplify(subs(e3,RK3_3err_eqs)):\nmap(u->lhs(u)-rhs(u),%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&#!\"\"\"#C,(#\"\"\"\"\")F%*&\"#s F%\"\"'#\"\"#\"\"$F)*&F&F%,&F*F%*&F-F))F-#F)F0F)F)#F)F/F),&*&\"#OF%F-F .F)#F)\"#7F%,,*&F&F%F-F5F%*&F;F%F-F.F)*&F;F%F2F6F)*(\"#[F%F-F.F2F6F%#F )\"\"%F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "Description" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 274 "p := 'p' :\nq := 'q':\np = 6^(-1/3);\na21 := c2: a32 := c2: b3 := c2:\nc3 := q: \nq = (1+sqrt(9-12*c2))*c2/2;\nt := 3*c3-2:\na31 := (c3*(c3+3*c2*(c2-1 )))/((3*c2-2)*c2):\nb2 := t/(6*(c3-c2)*c2):\nb1 := (3*c2*(2*c3-1)-t)/( 6*c2*c3):\nmatrix([[c2,a21,``,``],[c3,a31,a32,``],[``,b1,b2,b3]]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"pG,$*&\"\"'!\"\"F'#\"\"#\"\"$\"\" \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"qG,$*(\"\"#!\"\",&\"\"\"F**$ ,&\"\"*F**&\"#7F*%\"pGF*F(#F*F'F*F*F0F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7%7&%\"pGF(%!GF)7&%\"qG**F+\"\"\",&F+F-*(\" \"$F-F(F-,&F(F-F-!\"\"F-F-F-,&*&F0F-F(F-F-\"\"#F2F2F(F2F(F)7&F),$**\" \"'F2,(*(F0F-F(F-,&*&F5F-F+F-F-F-F2F-F-*&F0F-F+F-F2F5F-F-F(F2F+F2F-,$* *F9F2,&*&F0F-F+F-F-F5F2F-,&F+F-F(F2F2F(F2F-F(Q)pprint726\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 312 "D igits := 14:\nc2 := evalf((1/6)^(1/3)): a21 := c2:\na32 := c2: b3 := c 2:\nc3 := (1+sqrt(9-12*c2))*c2/2:\nt := 3*c3-2:\na31 := (c3*(c3+3*c2*( c2-1)))/((3*c2-2)*c2):\nb2 := t/(6*(c3-c2)*c2):\nb1 := (3*c2*(2*c3-1)- t)/(6*c2*c3):\nmm := matrix([[c2,a21,``,``],\n[c3,a31,a32,``],[``,b1,b 2,b3]]):\nDigits := 10:\nevalf(evalm(mm));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7%7&$\"+\"37K]&!#5F(%!GF+7&$\"+B8&4,(F*$\"+ T#Rx]\"F*F(F+7&F+$\"+xi8ACF*$\"+T;lu?F*F(Q)pprint466\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "Variant 6 order 3 " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "matrix([[1-sqrt(3)/3, 1-sqrt(3 )/3, ``, ``], [2/3, 1/3-sqrt(3)/9, 1/3+sqrt(3)/9, ``], [``, 1/4, 0, 3/ 4]])" "6#-%'matrixG6#7%7&,&\"\"\"F)*&-%%sqrtG6#\"\"$F)F.!\"\"F/,&F)F)* &-F,6#F.F)F.F/F/%!GF47&*&\"\"#F)F.F/,&*&F)F)F.F/F)*&-F,6#F.F)\"\"*F/F/ ,&*&F)F)F.F/F)*&-F,6#F.F)F=F/F)F47&F4*&F)F)\"\"%F/\"\"!*&F.F)FEF/" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "We specify two of the coefficients, namely " }{XPPEDIT 18 0 "c[2] = 1-1/sqrt(3),b[2] = 0;" "6$/&%\"cG6#\"\"#,&\"\"\"F)*&F)F)- %%sqrtG6#\"\"$!\"\"F//&%\"bG6#F'\"\"!" }{TEXT -1 59 ", so as to obtain a system of 6 equations with 6 unknowns. " }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 85 "RK3_3eqs := [op(RowSumConditions(3,'expanded')),op( OrderConditions(3,3,'expanded'))]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 192 "e1 := \{c[2]=1-1/sqrt(3),b[2]=0\}:\ne2 := solve(subs (e1,\{op(RK3_3eqs)\})):\ne3 := `union`(e1,e2);\nsubs(e3,matrix([[c[2], a[2,1],``,``],\n[c[3],a[3,1],a[3,2],``],[``,b[1],b[2],b[3]]]));\n``:\n evalf(%%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e3G<*/&%\"bG6#\"\"\"# F*\"\"%/&F(6#\"\"#\"\"!/&%\"cGF/,&F*F**&\"\"$!\"\"F7#F*F0F8/&%\"aG6$F7 F0,&#F*F7F**&\"\"*F8F7F9F*/&F<6$F7F*,&F?F**&FAF8F7F9F8/&F<6$F0F*F5/&F( 6#F7#F7F,/&F4FL#F0F7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7% 7&,&\"\"\"F)*&\"\"$!\"\"F+#F)\"\"#F,F(%!GF/7&#F.F+,&#F)F+F)*&\"\"*F,F+ F-F,,&F3F)*&F5F,F+F-F)F/7&F/#F)\"\"%\"\"!#F+F:Q(pprint26\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7%7&$\"+2t\\EU!#5F(%!GF+7&$\"+nm mmmF*$\"+NC$)39F*$\"+JU$yD&F*F+7&F+$\"+++++DF*$\"\"!F7$\"+++++vF*Q(ppr int36\"" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 7 "Check: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "simplify(subs( e3,RK3_3eqs));\nmap(u->lhs(u)-rhs(u),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(/,&\"\"\"F&*&\"\"$!\"\"F(#F&\"\"#F)F%/#F+F(F-/F&F&/F*F*/#F&\" \"'F1/#F&F(F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(\"\"!F$F$F$F$F$" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "The prin cipal error norm is less than that of the previous two schemes. " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 149 "errterms := PrincipalErrorT erms(3,3,'expanded'):\nsm := 0:\nfor ct to nops(errterms) do\n sm := sm+(evalf(subs(e3,errterms[ct])))^2;\nend do:\nsqrt(sm);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+yp " 0 "" {MPLTEXT 1 0 115 "RK3_3err _eqs := PrincipalErrorConditions(3,3,'expanded'):\nsimplify(subs(e3,RK 3_3err_eqs)):\nmap(u->lhs(u)-rhs(u),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&#!\"\"\"#C#F%\"#s,&*(\"$3\"F%,&\"\"$\"\"\"*$F-#F.\"\"#F.F.,&F- F%F/F.F1F.#F.\"#7F%#F%\"#O" }}}{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 44 "Test-bed p rocedures for some of the examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 7 "RK3step" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1875 "rk3step := proc(x_rk3step::realcons)\n local c2,c3,c4,a21,a31, a32,f1,f2,f3,\n b1,b2,b3,xk,yk,t,jF,jM,jS,n,h,data,fn,xx,ys,saveDigi ts;\n options `Copyright 2004 by Peter Stone`;\n \n data := SOLN _;\n\n saveDigits := Digits;\n Digits := max(trunc(evalhf(Digits)) ,Digits+5);\n\n # procedure to evaluate the gradient field\n fn := proc(X_,Y_)\n local val; \n val := traperror(evalf(FXY_)); \n if val=lasterror or not type(val,numeric) then\n error \"evaluation of gradient field failed at %1\",evalf([X_,Y_],saveDigit s);\n end if;\n val;\n end proc;\n\n xx := evalf(x_rk 3step);\n n := nops(data);\n\n if (data[1,1]dat a[n,1] or xxdata[1,1])) then\n error \"independent variable is outs ide the interpolation interval: %1\",evalf(data[1,1])..evalf(data[n,1] );\n end if;\n\n c2 := c2_; c3 := c3_;\n a21 := c2; a31 := a31_; a32 := a32_;\n b1 := b1_; b2 := b2_; b3 := b3_;\n # Perform a bin ary search for the interval containing x.\n n := nops(data);\n jF \+ := 0;\n jS := n+1;\n\n if data[1,1]1 do\n jM := trunc((jF+jS)/2);\n if xx>=data[jM,1] t hen jF := jM else jS := jM end if;\n end do;\n if jM = n the n jF := n-1; jS := n end if;\n else\n while jS-jF> 1 do\n \+ jM := trunc((jF+jS)/2);\n if xx<=data[jM,1] then jF := jM els e jS := jM end if;\n end do;\n if jM = n then jF := n-1; jS \+ := n end if;\n end if;\n \n # Get the data needed from the list. \n xk := data[jF,1];\n yk := data[jF,2];\n\n # Do one step with \+ step-size ..\n h := xx-xk;\n f1 := fn(xk,yk);\n t := a21*f1;\n \+ f2 := fn(xk + c2*h,yk + t*h);\n t := a31*f1 + a32*f2;\n f3 := fn( xk + c3*h,yk + t*h);\n \n t := b1*f1 + b2*f2 + b3*f3; \n ys : = yk + t*h;\n evalf[saveDigits](ys);\nend proc: # of rk3step" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 16 "RK3_1 Classical " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1050 "RK3_1 := proc(fxy,x,y,xx,yy,h,stps,bb)\n local c2,c3,a21,a31,a 32,f1,f2,f3,\n b1,b2,b3,t,k,fn,xk,yk,soln,eqns,A,saveDigits;\n\n s aveDigits := Digits;\n Digits := max(trunc(evalhf(Digits)),Digits+5) ;\n\n fn := unapply(fxy,x,y);\n\n A := matrix([[1/2,1/2,0,0],[1,-1 ,2,0],[0,1/6,2/3,1/6]]);\n \n c2 := evalf(A[1,1]);\n c3 := evalf(A [2,1]);\n a21 := c2;\n a31 := evalf(A[2,2]);\n a32 := evalf(A[2, 3]);\n b1 := evalf(A[3,2]);\n b2 := evalf(A[3,3]);\n b3 := evalf (A[3,4]);\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\n yk := yk + (b1*f1 + b2 *f2 + b3*f3)*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 a31_=a31,a32_=a32,b1_=b1,b2_=b2,b3_=b3 \};\n return subs(eqns,eval(rk3step)); \n else\n return ev alf[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 16 "RK3_2 Variant 1 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1052 "RK3_2 := p roc(fxy,x,y,xx,yy,h,stps,bb)\n local c2,c3,a21,a31,a32,f1,f2,f3,\n \+ b1,b2,b3,t,k,fn,xk,yk,soln,eqns,A,saveDigits;\n\n saveDigits := Dig its;\n Digits := max(trunc(evalhf(Digits)),Digits+5);\n\n fn := un apply(fxy,x,y);\n\n A := matrix([[1/3,1/3,0,0],[2/3,0,2/3,0], [0,1/4 ,0,3/4]]);\n \n c2 := evalf(A[1,1]);\n c3 := evalf(A[2,1]);\n a2 1 := c2;\n a31 := evalf(A[2,2]);\n a32 := evalf(A[2,3]);\n b1 := evalf(A[3,2]);\n b2 := evalf(A[3,3]);\n b3 := evalf(A[3,4]);\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 f 2 := fn(xk + c2*h,yk + t*h);\n t := a31*f1 + a32*f2;\n f3 := fn(xk + c3*h,yk + t*h);\n\n yk := yk + (b1*f1 + b2*f2 + b3*f3)*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 a31_=a31,a32_=a32,b1_=b1,b2_=b2,b3_=b3\};\n ret urn subs(eqns,eval(rk3step)); \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 16 "RK3_3 Variant 2 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1069 "RK3_3 := proc(fxy,x,y,xx,y y,h,stps,bb)\n local c2,c3,a21,a31,a32,f1,f2,f3,\n b1,b2,b3,t,k,fn ,xk,yk,soln,eqns,A,saveDigits;\n\n saveDigits := Digits;\n Digits \+ := max(trunc(evalhf(Digits)),Digits+5);\n\n fn := unapply(fxy,x,y); \n\n A := matrix([[3/7,3/7,0,0],[16/21,-64/945,112/135,0],[0,19/96,1 /3,15/32]]);\n \n c2 := evalf(A[1,1]);\n c3 := evalf(A[2,1]);\n \+ a21 := c2;\n a31 := evalf(A[2,2]);\n a32 := evalf(A[2,3]);\n b1 \+ := evalf(A[3,2]);\n b2 := evalf(A[3,3]);\n b3 := evalf(A[3,4]);\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\n yk := yk + (b1*f1 + b2*f2 + b3*f3) *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_=c 2,c3_=c3,\n a31_=a31,a32_=a32,b1_=b1,b2_=b2,b3_=b3\};\n r eturn subs(eqns,eval(rk3step)); \n else\n return evalf[saveDigi ts]([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 16 "RK3_4 Variant 3 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1133 "RK3_4 := proc(fxy,x,y,xx ,yy,h,stps,bb)\n local c2,c3,a21,a31,a32,f1,f2,f3,\n b1,b2,b3,t,k, fn,xk,yk,soln,eqns,A,saveDigits,SQRT;\n\n saveDigits := Digits;\n \+ Digits := max(trunc(evalhf(Digits)),Digits+5);\n SQRT := evalf(sqrt( 33));\n\n fn := unapply(fxy,x,y);\n\n A := matrix([[1/2,1/2,0,0],[ 1/4+SQRT/12,-5/12+SQRT/12,2/3,0],\n [0,-1/4+SQRT/12,3/4-SQRT/1 2,1/2]]);\n \n c2 := evalf(A[1,1]);\n c3 := evalf(A[2,1]);\n a21 := c2;\n a31 := evalf(A[2,2]);\n a32 := evalf(A[2,3]);\n b1 := \+ evalf(A[3,2]);\n b2 := evalf(A[3,3]);\n b3 := evalf(A[3,4]);\n x k := evalf(xx);\n yk := evalf(yy);\n soln := [xk,yk]; \n for k f rom 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\n yk := yk + (b1*f1 + b2*f2 + b3*f3)*h; \n xk := xk + h:\n soln := soln,[xk,yk];\n end do;\n if \+ bb=true then\n eqns := \{SOLN_=[soln],FXY_=fxy,X_=x,Y_=y,c2_=c2,c 3_=c3,\n a31_=a31,a32_=a32,b1_=b1,b2_=b2,b3_=b3\};\n retu rn subs(eqns,eval(rk3step)); \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 16 "RK3_5 Variant 4 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1098 "RK3_5 := proc(fxy,x,y,xx,y y,h,stps,bb)\n local c2,c3,a21,a31,a32,f1,f2,f3,\n b1,b2,b3,t,k,fn ,xk,yk,soln,eqns,A,saveDigits;\n\n saveDigits := Digits;\n Digits \+ := max(trunc(evalhf(Digits)),Digits+5);\n\n fn := unapply(fxy,x,y); \n\n A := matrix([[1/2,1/2,0,0],[109/145,-109/21025,15914/21025,0], \n [0,145/654,74/219,21025/47742]]);\n \n c2 := evalf(A[1,1]) ;\n c3 := evalf(A[2,1]);\n a21 := c2;\n a31 := evalf(A[2,2]);\n \+ a32 := evalf(A[2,3]);\n b1 := evalf(A[3,2]);\n b2 := evalf(A[3,3 ]);\n b3 := evalf(A[3,4]);\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\n yk \+ := yk + (b1*f1 + b2*f2 + b3*f3)*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 a31_=a31,a32_=a32,b 1_=b1,b2_=b2,b3_=b3\};\n return subs(eqns,eval(rk3step)); \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 16 "RK3_6 Variant 5 " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1185 "RK3_6 := proc(fxy,x,y,xx,yy,h,stps,bb)\n local c2,c3,a21,a31,a 32,f1,f2,f3,\n b1,b2,b3,t,k,fn,xk,yk,soln,eqns,A,saveDigits,p,q;\n\n saveDigits := Digits;\n Digits := max(trunc(evalhf(Digits)),Digit s+5);\n p := evalf((1/6)^(1/3));\n q := evalf((1+sqrt(9-12*p))*p/2 );\n\n fn := unapply(fxy,x,y);\n\n A := matrix([[p,p,0,0],[q,q*(q+ 3*p*(p-1))/(3*p-2)/p,p,0],\n [0,(3*p*(2*q-1)-3*q+2)/(6*p*q),(3* q-2)/(6*(q-p)*p),p]]);\n \n c2 := evalf(A[1,1]);\n c3 := evalf(A[2 ,1]);\n a21 := c2;\n a31 := evalf(A[2,2]);\n a32 := evalf(A[2,3] );\n b1 := evalf(A[3,2]);\n b2 := evalf(A[3,3]);\n b3 := evalf(A [3,4]);\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\n yk := yk + (b1*f1 + b2* f2 + b3*f3)*h;\n xk := xk + h:\n soln := soln,[xk,yk];\n e nd do;\n if bb=true then\n eqns := \{SOLN_=[soln],FXY_=fxy,X_=x ,Y_=y,c2_=c2,c3_=c3,\n a31_=a31,a32_=a32,b1_=b1,b2_=b2,b3_=b3 \};\n return subs(eqns,eval(rk3step)); \n else\n return ev alf[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 16 "RK3_7 Variant 6 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1116 "RK3_7 := p roc(fxy,x,y,xx,yy,h,stps,bb)\n local c2,c3,a21,a31,a32,f1,f2,f3,\n \+ b1,b2,b3,t,k,fn,xk,yk,soln,eqns,A,saveDigits,SQRT;\n\n saveDigits : = Digits;\n Digits := max(trunc(evalhf(Digits)),Digits+5);\n SQRT \+ := evalf(sqrt(3));\n\n fn := unapply(fxy,x,y);\n\n A := matrix([[1 -SQRT/3,1-SQRT/3,0,0],[2/3,1/3-SQRT/9,1/3+SQRT/9,0],\n [0,1/4,0,3 /4]]);\n \n c2 := evalf(A[1,1]);\n c3 := evalf(A[2,1]);\n a21 := c2;\n a31 := evalf(A[2,2]);\n a32 := evalf(A[2,3]);\n b1 := eva lf(A[3,2]);\n b2 := evalf(A[3,3]);\n b3 := evalf(A[3,4]);\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\n yk := yk + (b1*f1 + b2*f2 + b3*f3)*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 a31_=a31,a32_=a32,b1_=b1,b2_=b2,b3_=b3\};\n return \+ subs(eqns,eval(rk3step)); \n else\n return evalf[saveDigits]([s oln]);\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 21 "Testing the examples " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "Test 1 of order 3 Runge-Kutta met hods " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=12*x*cos(4*x)*exp(-x)*y" "6#/*&%#dyG\"\"\"%# dxG!\"\"*,\"#7F&%\"xGF&-%$cosG6#*&\"\"%F&F+F&F&-%$expG6#,$F+F(F&%\"yGF &" }{TEXT -1 3 ", " }{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)*ex p(-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-%$expG6#,,*,\"#7\"\"\"\"# " 0 "" {MPLTEXT 1 0 190 "de := diff(y(x),x)=12*x*cos(4*x)*exp(-x)*y(x);\nic : = y(0)=1;\ndsolve(\{de,ic\},y(x)):\ny(x)=simplify(numer(rhs(%))/conver t(denom(rhs(%)),exp));\nf := unapply(rhs(%),x):\nplot(f(x),x=0..5,0..1 .45);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"x GF,,$*,\"#7\"\"\"F,F0-%$cosG6#,$*&\"\"%F0F,F0F0F0-%$expG6#,$F,!\"\"F0F )F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!\"\"\"" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG-%$expG6#,,*&#\"#7\"#< \"\"\"*(F'F0-%$cosG6#,$*&\"\"%F0F'F0F0F0-F)6#,$F'!\"\"F0F0F;*&#\"$!=\" 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}}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "F := (x,y) -> 12*x*cos(4*x)*exp(-x)*y;\nDigits := 20 :\nfor ct to 7 do\n fn_RK3_||ct := RK3_||ct(F(x,y),x,y,0,1,.1,50,tru e);\nend do;\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FGf *6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),$*,\"#7\"\"\"9$F0-%$cosG6#,$*& \"\"%F0F1F0F0F0-%$expG6#,$F1!\"\"F09%F0F0F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "Digits := 2 0:\nxx := 4.9;\nfxx := evalf(f(xx));\nfor ct to 7 do\n fnxx := fn_RK 3_||ct(xx);\n abs(fnxx-fxx);\nend do;\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"#\\!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$fxxG$\"5Z'**Rd&**3,wc!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%f nxxG$\"53xJD**Q**R6d!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"3h!=8N%R !*QN!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%fnxxG$\"59'H'Qlo&)[dc!#? 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "h := x -> exp(-x^2/2);\nDigits := 20:\nfor ct to 7 do\n hn_RK3_||ct := RK3 _||ct(-x*y,x,y,0,1,.1,100,true);\nend do;\nDigits := 10:" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"hGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%$expG 6#,$*&#\"\"\"\"\"#F2*$)9$F3F2F2!\"\"F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "xx := 9.9;\nDigit s := 18:\nhxx := evalf(h(xx));\nfor ct to 7 do\n hnxx := hn_RK3_||ct (xx);\n abs(hnxx-hxx);\nend do;\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"#**!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% $hxxG$\"32^q-imt;_!#R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%hnxxG$\"3c X\"oF2)ze5!#R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"3^0*e#*eQz:%!#R" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%hnxxG$\"3Fn&oW0^F7\"!#R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"3!Q[evg&)R4%!#R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%hnxxG$\"3D@@!\\/dy4\"!#R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"3#)H\\7<'z)=T!#R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%%hnxxG$\"34GCSV(HF4\"!#R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"3)HiC '=p+CT!#R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%hnxxG$\"3*\\_eH(3\")*3 \"!#R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"33E&o!*yDp7%!#R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%hnxxG$\"3)z#4PLNn*3\"!#R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"34BhlGJ1FT!#R" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%%hnxxG$\"3D[)>cZZ36\"!#R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"3#G ?2k=*)e5%!#R" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 402 "evalf[20](plot([1-'hn_RK3_1'(x)/h(x),1-'hn_RK3_ 2'(x)/h(x),1-'hn_RK3_3'(x)/h(x),\n1-'hn_RK3_4'(x)/h(x),1-'hn_RK3_5'(x) /h(x),1-'hn_RK3_6'(x)/h(x),\n1-'hn_RK3_7'(x)/h(x)],x=0..10,\ncolor=[bl ue,magenta,aquamarine,coral,COLOR(RGB,.4,0,1),brown,green],\nlegend=[` Classical`,`variant 1`,`variant 2`,`variant 3`,`variant 4`,`variant 5` ,\n`variant 6`],title=`relative error curves for order 3 Runge-Kutta m ethods`));" }}{PARA 13 "" 1 "" {GLPLOT2D 674 339 339 {PLOTDATA 2 "6,-% 'CURVESG6%7S7$$\"\"!F)F(7$$\"5mmmmm;arz@!#?$!/.p)>+Q])!#>7$$\"5LLLL$e9 ui2%F-$!0A%eh8QE$yLF07$$\"5++++DJ$RDX\"F0$!0p85$*zrq\"F07$$\"5mmmm\"zR'ok;F0 $\"18WfuU1VBF-7$$\"5++++D1J:w=F0$\"2EAH>)\\5M5F-7$$\"5LLLLL3En$4#F0$\" 2[l37C!\\MEF-7$$\"5mmmm;/RE&G#F0$\"2')>u?12BW%F-7$$\"5+++++D.&4]#F0$\" 2kKh_J@!p#)F-7$$\"5+++++vB_F-7$$\"5mmmm\"z*ev:JF0$\"3*zqL-Y,ev#F-7$$\"5LLLLL347TLF0$\"3 5*RG;^x^)QF-7$$\"5LLLLLLY.KNF0$\"3PgY%\\;W3K&F-7$$\"5*******\\7o7Tv$F0 $\"3-:LMu/=RsF-7$$\"5LLLLL$Q*o]RF0$\"3)4')=W--@c*F-7$$\"5*******\\7=lj ;%F0$\"4em9$4FWdn7F-7$$\"5*******\\PaRY2aF0$\"4)pQ:*H7k*H`F-7$$\"5mmmm;zXu9cF0$\"4!=poQoUt]kF -7$$\"5**********\\y))GeF0$\"4(*G%et'Gp>v(F-7$$\"5********\\i_QQgF0$\" 4(**eomu4)H*Gyj:)F--%'COLOURG6&%$RGB GF(F($\"*++++\"!\")-%'LEGENDG6#%*ClassicalG-F$6%7SF'7$F+$\"0_Ur(z*Hf#F -7$F2$\"0]nr[X,#RF-7$F7$\"0gca)fOTHF-7$F<$!.3Wmz'p!)F07$FA$!/Llxx!f#oF 07$FF$!0UT)Q(QEG\"F07$FK$!0\">dBw20:F07$FP$!/7+fn*\\A'F07$FU$\"1D#F-7$F^o$\"2@WcelRwu%F-7 $Fco$\"2&o'**y,RI-)F-7$Fho$\"3V'35DU;hF\"F-7$F]p$\"3Nfi)*>RzN>F-7$Fbp$ \"3-t%GF-7$Fgp$\"3CD[LD:RNSF-7$F\\q$\"31$p&HNPTdcF-7$Faq$\"3XDwU h&Ril(F-7$Ffq$\"4+6**QI/&yP5F-7$F[r$\"42[M;beZ(*) f$=F-7$Fer$\"4*QL#>iMHbJ#F-7$Fjr$\"4%)e0E=dlc.$F-7$F_s$\"4eAqJ%yX&f&QF -7$Fds$\"4X%yoX'y>yu%F-7$Fis$\"4]C!Raw$)=&z&F-7$F^t$\"4I`/t\")yr#=qF-7 $Fct$\"4-sO4_%)3zV)F-7$Fht$\"5QV)45s`Cj+\"F-7$F]u$\"5OW^Sm\"oD]?\"F-7$ Fbu$\"5\"zrg3Q^=%=9F-7$Fgu$\"5U\\zO')QSY!p\"F-7$F\\v$\"5_)\\gF -7$Fav$\"5<:x`wXcD(H#F-7$Ffv$\"5A`6x/$)p5KEF-7$F[w$\"586$)yg$>wY-$F-7$ F`w$\"5yKD*G:e0(3MF-7$Few$\"5k=q^0wy]>)>l&F07$F2$!/&Q\"zb8#*>F07$F7$!/1uD6quZF07$F<$!/`M, !*3!y)F07$FA$!00O*3aP\"G\"F07$FF$!0'f&R`w`S\"F07$FK$!/;5<#3;&yF07$FP$ \"12I0[v9K9F-7$FU$\"1TBU&z1;g'F-7$FZ$\"2\">:/j(>m!=F-7$Fin$\"2LcUw'GE& =$F-7$F^o$\"2)yT\"4\"=aQiF-7$Fco$\"3x?MRE`$>+\"F-7$Fho$\"3A3n:tRJO:F-7 $F]p$\"3gFDLlSHnAF-7$Fbp$\"3$Rht._%GjKF-7$Fgp$\"3%3[-w?jra%F-7$F\\q$\" 3gtoU\\ll$G'F-7$Faq$\"3u@m>A@_2%)F-7$Ffq$\"48i1hL&>-G6F-7$F[r$\"4j\\K! 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G. L ether: Mathematics of Computation, Vol. 20, no. 95, (July 1966) page 3 81. " }}{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!\"\" ,$**\"#KF&%\"xGF&%\"yGF&-%#lnG6#\"\"#F&F(" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "y(-1) = 1/8;" "6#/-%\"yG6#,$\"\"\"!\"\"*&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 ". 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