{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 "Warning " -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Ti mes" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "T imes" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } } {SECT 0 {PARA 3 "" 0 "" {TEXT -1 50 "Order conditions for an order 7 R unge-Kutta scheme" }}{PARA 0 "" 0 "" {TEXT -1 46 "by Peter Stone, Gabr iola Island, B.C., Canada " }}{PARA 0 "" 0 "" {TEXT -1 17 "Version: 7. 1.2011" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{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 constru cting 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 wo rksheet. " }}{PARA 0 "" 0 "" {TEXT -1 121 "It can be read into a Maple session by a command similar to the one that follows, where the file \+ path gives 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 ";" }}} {PARA 0 "" 0 "" {TEXT -1 57 "#======================================== ================" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 59 "Order conditi ons for a 9 stage, order 7 Runge-Kutta scheme " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matri x([[c[1] = 0, ``, ``, ``, ``, ``, ``, ``, ``,``], [c[2], a[2,1], ``, ` `, ``, ``, ``, ``, ``,``], [c[3], a[3,1], a[3,2], ``, ``, ``, ``, ``, \+ ``,``], [c[4], a[4,1], a[4,2], a[4,3], ``, ``, ``, ``, ``,``], [c[5], \+ a[5,1], a[5,2], a[5,3], a[5,4], ``, ``, ``, ``,``], [c[6], a[6,1], a[6 ,2], a[6,3], a[6,4], a[6,5], ``, ``, ``,``], [c[7], a[7,1], a[7,2], a[ 7,3], a[7,4], a[7,5], a[7,6], ``, ``,``], [c[8], a[8,1], a[8,2], a[8,3 ], a[8,4], a[8,5], a[8,6], a[8,7], ``,``],[c[9], a[9,1], a[9,2], a[9,3 ], a[9,4], a[9,5], a[9,6], a[9,7], a[9,8],``], [``, b[1], b[2], b[3], \+ b[4], b[5], b[6], b[7], b[8],b[9]]])" "6#-%'matrixG6#7,7,/&%\"cG6#\"\" \"\"\"!%!GF.F.F.F.F.F.F.F.7,&F*6#\"\"#&%\"aG6$F2F,F.F.F.F.F.F.F.F.7,&F *6#\"\"$&F46$F9F,&F46$F9F2F.F.F.F.F.F.F.7,&F*6#\"\"%&F46$FAF,&F46$FAF2 &F46$FAF9F.F.F.F.F.F.7,&F*6#\"\"&&F46$FKF,&F46$FKF2&F46$FKF9&F46$FKFAF .F.F.F.F.7,&F*6#\"\"'&F46$FWF,&F46$FWF2&F46$FWF9&F46$FWFA&F46$FWFKF.F. F.F.7,&F*6#\"\"(&F46$F_oF,&F46$F_oF2&F46$F_oF9&F46$F_oFA&F46$F_oFK&F46 $F_oFWF.F.F.7,&F*6#\"\")&F46$F_pF,&F46$F_pF2&F46$F_pF9&F46$F_pFA&F46$F _pFK&F46$F_pFW&F46$F_pF_oF.F.7,&F*6#\"\"*&F46$FaqF,&F46$FaqF2&F46$FaqF 9&F46$FaqFA&F46$FaqFK&F46$FaqFW&F46$FaqF_o&F46$FaqF_pF.7,F.&%\"bG6#F,& Fdr6#F2&Fdr6#F9&Fdr6#FA&Fdr6#FK&Fdr6#FW&Fdr6#F_o&Fdr6#F_p&Fdr6#Faq" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(a [i,j],j=1..i-1)=c[i]" "6#/-%$SumG6$&%\"aG6$%\"iG%\"jG/F+;\"\"\",&F*F.F .!\"\"&%\"cG6#F*" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "i = 2,3,4,5,6,7,8, 9;" "6*/%\"iG\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(b[i]*c[i]^(q-1),i = 1 .. 9) = 1/q;" "6#/-%$SumG6$*& &%\"bG6#%\"iG\"\"\")&%\"cG6#F+,&%\"qGF,F,!\"\"F,/F+;F,\"\"**&F,F,F2F3 " }{TEXT -1 4 ", " }{XPPEDIT 18 0 "q = 1,2,3,4,5,6,7;" "6)/%\"qG\"\" \"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b [i]*Sum(a[i,j]*c[j]^(q-2),j = 2 .. i-1),i = 3 .. 9) = 1/((q-1)*q);" "6 #/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"-F%6$*&&%\"aG6$F+%\"jGF,)&%\"cG6#F3,&% \"qGF,\"\"#!\"\"F,/F3;F:,&F+F,F,F;F,/F+;\"\"$\"\"**&F,F,*&,&F9F,F,F;F, F9F,F;" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "q = 3,4,5,6,7;" "6'/%\"qG\" \"$\"\"%\"\"&\"\"'\"\"(" }{TEXT -1 2 ". " }}{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]^(q-3),j = 2 .. i-1),i = 3 .. 9) = 1/((q-2)*q);" "6#/-% $SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cG6#F+F,-F%6$*&&%\"aG6$F+%\"jGF,)&F.6# F6,&%\"qGF,\"\"$!\"\"F,/F6;\"\"#,&F+F,F,F=F,/F+;F<\"\"**&F,F,*&,&F;F,F @F=F,F;F,F=" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "q = 4,5,6,7;" "6&/%\"q G\"\"%\"\"&\"\"'\"\"(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*c[i]^2*S um(a[i,j]*c[j]^(q-4),j = 2 .. i-1),i = 3 .. 9) = 1/((q-3)*q);" "6#/-%$ SumG6$*(&%\"bG6#%\"iG\"\"\"*$&%\"cG6#F+\"\"#F,-F%6$*&&%\"aG6$F+%\"jGF, )&F/6#F8,&%\"qGF,\"\"%!\"\"F,/F8;F1,&F+F,F,F?F,/F+;\"\"$\"\"**&F,F,*&, &F=F,FEF?F,F=F,F?" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "q = 5,6,7;" "6%/ %\"qG\"\"&\"\"'\"\"(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*c[i]^3*S um(a[i,j]*c[j]^(q-5),j = 2 .. i-1),i = 3 .. 9) = 1/((q-4)*q);" "6#/-%$ SumG6$*(&%\"bG6#%\"iG\"\"\"*$&%\"cG6#F+\"\"$F,-F%6$*&&%\"aG6$F+%\"jGF, )&F/6#F8,&%\"qGF,\"\"&!\"\"F,/F8;\"\"#,&F+F,F,F?F,/F+;F1\"\"**&F,F,*&, &F=F,\"\"%F?F,F=F,F?" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "q = 6,7;" "6$ /%\"qG\"\"'\"\"(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*c[i]^4*Sum(a [i,j]*c[j]^(q-6),j = 2 .. i-1),i = 3 .. 9) = 1/((q-5)*q);" "6#/-%$SumG 6$*(&%\"bG6#%\"iG\"\"\"*$&%\"cG6#F+\"\"%F,-F%6$*&&%\"aG6$F+%\"jGF,)&F/ 6#F8,&%\"qGF,\"\"'!\"\"F,/F8;\"\"#,&F+F,F,F?F,/F+;\"\"$\"\"**&F,F,*&,& F=F,\"\"&F?F,F=F,F?" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "q = 7;" "6#/% \"qG\"\"(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Sum(b[i]*Sum(a[i,j]*Sum(a[ j,k]*c[k]^(q-3),k = 2 .. j-1),j = 3 .. i-1),i = 4 .. 9) = 1/((q-2)*(q- 1)*q);" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"-F%6$*&&%\"aG6$F+%\"jGF,-F%6 $*&&F16$F3%\"kGF,)&%\"cG6#F9,&%\"qGF,\"\"$!\"\"F,/F9;\"\"#,&F3F,F,FAF, /F3;F@,&F+F,F,FAF,/F+;\"\"%\"\"**&F,F,*(,&F?F,FDFAF,,&F?F,F,FAF,F?F,FA " }{TEXT -1 4 ", " }{XPPEDIT 18 0 "q = 4,5,6,7;" "6&/%\"qG\"\"%\"\"& \"\"'\"\"(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Sum(b[i]*c[i]*Sum(a[i,j]*S um(a[j,k]*c[k]^(q-4),k = 2 .. j-1),j = 3 .. i-1),i = 4 .. 9) = 1/((q-3 )*(q-2)*q);" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cG6#F+F,-F%6$*&&%\" aG6$F+%\"jGF,-F%6$*&&F46$F6%\"kGF,)&F.6#F<,&%\"qGF,\"\"%!\"\"F,/F<;\" \"#,&F6F,F,FCF,/F6;\"\"$,&F+F,F,FCF,/F+;FB\"\"**&F,F,*(,&FAF,FJFCF,,&F AF,FFFCF,FAF,FC" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "q = 5,6,7;" "6%/% \"qG\"\"&\"\"'\"\"(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Sum(b[i]*c[i]^2*Su m(a[i,j]*Sum(a[j,k]*c[k]^(q-5),k = 2 .. j-1),j = 3 .. i-1),i = 4 .. 9) = 1/((q-4)*(q-3)*q);" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"*$&%\"cG6#F+ \"\"#F,-F%6$*&&%\"aG6$F+%\"jGF,-F%6$*&&F66$F8%\"kGF,)&F/6#F>,&%\"qGF, \"\"&!\"\"F,/F>;F1,&F8F,F,FEF,/F8;\"\"$,&F+F,F,FEF,/F+;\"\"%\"\"**&F,F ,*(,&FCF,FOFEF,,&FCF,FKFEF,FCF,FE" }{TEXT -1 4 ", " }{XPPEDIT 18 0 " q = 6,7;" "6$/%\"qG\"\"'\"\"(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Sum( b[i]*c[i]^3*Sum(a[i,j]*Sum(a[j,k]*c[k]^(q-6),k = 2 .. j-1),j = 3 .. i- 1),i = 4 .. 9) = 1/((q-5)*(q-4)*q);" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\" \"*$&%\"cG6#F+\"\"$F,-F%6$*&&%\"aG6$F+%\"jGF,-F%6$*&&F66$F8%\"kGF,)&F/ 6#F>,&%\"qGF,\"\"'!\"\"F,/F>;\"\"#,&F8F,F,FEF,/F8;F1,&F+F,F,FEF,/F+;\" \"%\"\"**&F,F,*(,&FCF,\"\"&FEF,,&FCF,FOFEF,FCF,FE" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "q = 7;" "6#/%\"qG\"\"(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Sum(b[i]*Sum(a[i,j]*c[j]*Sum(a[j,k]*c[k]^(q-4),k = 2 .. j-1),j = 3 .. i-1),i = 4 .. 9) = 1/((q-3)*(q-1)*q);" "6#/-%$SumG6$*&&%\"bG6#%\"iG\" \"\"-F%6$*(&%\"aG6$F+%\"jGF,&%\"cG6#F3F,-F%6$*&&F16$F3%\"kGF,)&F56#F<, &%\"qGF,\"\"%!\"\"F,/F<;\"\"#,&F3F,F,FCF,/F3;\"\"$,&F+F,F,FCF,/F+;FB\" \"**&F,F,*(,&FAF,FJFCF,,&FAF,F,FCF,FAF,FC" }{TEXT -1 4 ", " } {XPPEDIT 18 0 "q = 5,6,7;" "6%/%\"qG\"\"&\"\"'\"\"(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "Sum(b[i]*Sum(a[i,j]*c[j]^2*Sum(a[j,k]*c[k]^(q-5),k = 2 \+ .. j-1),j = 3 .. i-1),i = 4 .. 9) = 1/((q-4)*(q-1)*q);" "6#/-%$SumG6$* &&%\"bG6#%\"iG\"\"\"-F%6$*(&%\"aG6$F+%\"jGF,*$&%\"cG6#F3\"\"#F,-F%6$*& &F16$F3%\"kGF,)&F66#F>,&%\"qGF,\"\"&!\"\"F,/F>;F8,&F3F,F,FEF,/F3;\"\"$ ,&F+F,F,FEF,/F+;\"\"%\"\"**&F,F,*(,&FCF,FOFEF,,&FCF,F,FEF,FCF,FE" } {TEXT -1 4 ", " }{XPPEDIT 18 0 "q = 6,7;" "6$/%\"qG\"\"'\"\"(" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Sum(b[i]*Sum(a[i,j]*c[j]^3*Sum(a[j,k]* c[k]^(q-6),k = 2 .. j-1),j = 3 .. i-1),i = 4 .. 9) = 1/((q-5)*(q-1)*q) ;" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"-F%6$*(&%\"aG6$F+%\"jGF,*$&%\"cG6 #F3\"\"$F,-F%6$*&&F16$F3%\"kGF,)&F66#F>,&%\"qGF,\"\"'!\"\"F,/F>;\"\"#, &F3F,F,FEF,/F3;F8,&F+F,F,FEF,/F+;\"\"%\"\"**&F,F,*(,&FCF,\"\"&FEF,,&FC F,F,FEF,FCF,FE" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "q = 7;" "6#/%\"qG\" \"(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*c[i]*Sum(a[i,j]*c[j]*Sum( a[j,k]*c[k]^(q-5),k=2..j-1),\n j=3..i-1),i=4..9)=1/(q-4)/(q -2)/q" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cG6#F+F,-F%6$*(&%\"aG6$F+ %\"jGF,&F.6#F6F,-F%6$*&&F46$F6%\"kGF,)&F.6#F>,&%\"qGF,\"\"&!\"\"F,/F>; \"\"#,&F6F,F,FEF,/F6;\"\"$,&F+F,F,FEF,/F+;\"\"%\"\"***F,F,,&FCF,FPFEFE ,&FCF,FHFEFEFCFE" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "q=6,7" "6$/%\"qG\" \"'\"\"(" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*c[i]*Sum(a[i,j]*c[ j]^2*Sum(a[j,k]*c[k]^(q-6),k = 2 .. j-1),j = 3 .. i-1),i = 4 .. 9) = 1 /(q-5)/(q-2)/q;" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cG6#F+F,-F%6$*( &%\"aG6$F+%\"jGF,*$&F.6#F6\"\"#F,-F%6$*&&F46$F6%\"kGF,)&F.6#F@,&%\"qGF ,\"\"'!\"\"F,/F@;F:,&F6F,F,FGF,/F6;\"\"$,&F+F,F,FGF,/F+;\"\"%\"\"***F, F,,&FEF,\"\"&FGFG,&FEF,F:FGFGFEFG" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "q = 7;" "6#/%\"qG\"\"(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*c[i]^2*S um(a[i,j]*c[j]*Sum(a[j,k]*c[k]^(q-6),k = 2 .. j-1),j = 3 .. i-1),i = 4 .. 9) = 1/(q-5)/(q-3)/q;" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"*$&%\"cG6 #F+\"\"#F,-F%6$*(&%\"aG6$F+%\"jGF,&F/6#F8F,-F%6$*&&F66$F8%\"kGF,)&F/6# F@,&%\"qGF,\"\"'!\"\"F,/F@;F1,&F8F,F,FGF,/F8;\"\"$,&F+F,F,FGF,/F+;\"\" %\"\"***F,F,,&FEF,\"\"&FGFG,&FEF,FMFGFGFEFG" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "q = 7;" "6#/%\"qG\"\"(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " Sum(b[i]*Sum(a[i,j]*Sum(a[j,k]*Sum(a[k,m]*c[m]^(q-4),m = 2 .. k-1),k = 3 .. j-1),j = 4 .. i-1),i = 5 .. 9) = 1/((q-3)*(q-2)*(q-1)*q);" "6#/- %$SumG6$*&&%\"bG6#%\"iG\"\"\"-F%6$*&&%\"aG6$F+%\"jGF,-F%6$*&&F16$F3%\" kGF,-F%6$*&&F16$F9%\"mGF,)&%\"cG6#F?,&%\"qGF,\"\"%!\"\"F,/F?;\"\"#,&F9 F,F,FGF,/F9;\"\"$,&F3F,F,FGF,/F3;FF,&F+F,F,FGF,/F+;\"\"&\"\"**&F,F,**, &FEF,FNFGF,,&FEF,FJFGF,,&FEF,F,FGF,FEF,FG" }{TEXT -1 4 ", " } {XPPEDIT 18 0 "q = 5,6,7;" "6%/%\"qG\"\"&\"\"'\"\"(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(b[i]*c[i]*Sum(a[i,j]*Sum(a[j,k]*Sum(a[k,m]*c[m]^(q- 5),m = 2 .. k-1),k = 3 .. j-1),j = 4 .. i-1),i = 5 .. 9) = 1/((q-4)*(q -3)*(q-2)*q);" "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"&%\"cG6#F+F,-F%6$*&&% \"aG6$F+%\"jGF,-F%6$*&&F46$F6%\"kGF,-F%6$*&&F46$F<%\"mGF,)&F.6#FB,&%\" qGF,\"\"&!\"\"F,/FB;\"\"#,&F%\"mGF,)&F/6#FD,&%\"qGF,\"\"'!\"\"F,/FD;F1,&F>F,F,FKF,/F>;\"\"$,& F8F,F,FKF,/F8;\"\"%,&F+F,F,FKF,/F+;\"\"&\"\"**&F,F,**,&FIF,FYFKF,,&FIF ,FUFKF,,&FIF,FQFKF,FIF,FK" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "q = 7;" "6#/%\"qG\"\"(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*Sum(a[i,j]*c [j]*Sum(a[j,k]*Sum(a[k,m]*c[m]^(q-5),m = 2 .. k-1),k = 3 .. j-1),j = 4 .. i-1),i = 5 .. 9) = 1/((q-4)*(q-3)*(q-1)*q);" "6#/-%$SumG6$*&&%\"bG 6#%\"iG\"\"\"-F%6$*(&%\"aG6$F+%\"jGF,&%\"cG6#F3F,-F%6$*&&F16$F3%\"kGF, -F%6$*&&F16$F<%\"mGF,)&F56#FB,&%\"qGF,\"\"&!\"\"F,/FB;\"\"#,&F%\"mGF,)&F66#FD,&%\"qGF,\"\"'! \"\"F,/FD;F8,&F>F,F,FKF,/F>;\"\"$,&F3F,F,FKF,/F3;\"\"%,&F+F,F,FKF,/F+; \"\"&\"\"**&F,F,**,&FIF,FYFKF,,&FIF,FUFKF,,&FIF,F,FKF,FIF,FK" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "q = 7;" "6#/%\"qG\"\"(" }{TEXT -1 2 ". " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "Sum(b[i]*Sum(a[i,j]*Sum(a[j,k]*c[k]*Sum(a[k,m]*c[m]^(q- 5),m = 2 .. k-1),k = 3 .. j-1),j = 4 .. i-1),i = 5 .. 9) = 1/((q-4)*(q -2)*(q-1)*q);" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"-F%6$*&&%\"aG6$F+%\"j GF,-F%6$*(&F16$F3%\"kGF,&%\"cG6#F9F,-F%6$*&&F16$F9%\"mGF,)&F;6#FB,&%\" qGF,\"\"&!\"\"F,/FB;\"\"#,&F9F,F,FIF,/F9;\"\"$,&F3F,F,FIF,/F3;\"\"%,&F +F,F,FIF,/F+;FH\"\"**&F,F,**,&FGF,FTFIF,,&FGF,FLFIF,,&FGF,F,FIF,FGF,FI " }{TEXT -1 4 ", " }{XPPEDIT 18 0 "q = 6,7;" "6$/%\"qG\"\"'\"\"(" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Sum(b[i]*Sum(a[i,j]*Sum(a[j,k]*c[k]^2* Sum(a[k,m]*c[m]^(q-6),m = 2 .. k-1),k = 3 .. j-1),j = 4 .. i-1),i = 5 \+ .. 9) = 1/((q-5)*(q-2)*(q-1)*q);" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"-F %6$*&&%\"aG6$F+%\"jGF,-F%6$*(&F16$F3%\"kGF,*$&%\"cG6#F9\"\"#F,-F%6$*&& F16$F9%\"mGF,)&F<6#FD,&%\"qGF,\"\"'!\"\"F,/FD;F>,&F9F,F,FKF,/F9;\"\"$, &F3F,F,FKF,/F3;\"\"%,&F+F,F,FKF,/F+;\"\"&\"\"**&F,F,**,&FIF,FYFKF,,&FI F,F>FKF,,&FIF,F,FKF,FIF,FK" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "q = 7; " "6#/%\"qG\"\"(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*Sum(a[i,j]*c [j]*Sum(a[j,k]*c[k]*Sum(a[k,m]*c[m]^(q-6),m = 2 .. k-1),k = 3 .. j-1), j = 4 .. i-1),i = 5 .. 9) = 1/((q-5)*(q-3)*(q-1)*q);" "6#/-%$SumG6$*&& %\"bG6#%\"iG\"\"\"-F%6$*(&%\"aG6$F+%\"jGF,&%\"cG6#F3F,-F%6$*(&F16$F3% \"kGF,&F56#F%\"mGF,)&F.6# FD,&%\"qGF,\"\"'!\"\"F,/FD;\"\"#,&F>F,F,FKF,/F>;\"\"$,&F6F,F,FKF,/F6; \"\"%,&F+F,F,FKF,/F+;\"\"&\"\"**,F,F,,&FIF,FZFKFK,&FIF,FVFKFK,&FIF,FNF KFKFIFK" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "q=7" "6#/%\"qG\"\"(" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "Sum(b[i]*Sum(a[i,j]*Sum(a[j,k]*Sum(a[k,m]*Sum(a[m,n]*c[ n]^(q-5),n = 2 .. m-1),m = 3 .. k-1),k = 4 .. j-1),j = 5 .. i-1),i = 6 .. 9) = 1/((q-4)*(q-3)*(q-2)*(q-1)*q);" "6#/-%$SumG6$*&&%\"bG6#%\"iG \"\"\"-F%6$*&&%\"aG6$F+%\"jGF,-F%6$*&&F16$F3%\"kGF,-F%6$*&&F16$F9%\"mG F,-F%6$*&&F16$F?%\"nGF,)&%\"cG6#FE,&%\"qGF,\"\"&!\"\"F,/FE;\"\"#,&F?F, F,FMF,/F?;\"\"$,&F9F,F,FMF,/F9;\"\"%,&F3F,F,FMF,/F3;FL,&F+F,F,FMF,/F+; \"\"'\"\"**&F,F,*,,&FKF,FXFMF,,&FKF,FTFMF,,&FKF,FPFMF,,&FKF,F,FMF,FKF, FM" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "q = 6,7;" "6$/%\"qG\"\"'\"\"(" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "Sum(b[i]*c[i]*Sum(a[i,j]*Sum(a[j,k]*Sum(a[k,m]*Sum(a[m, n]*c[n]^(q-6),n = 2 .. m-1),m = 3 .. k-1),k = 4 .. j-1),j = 5 .. i-1), i = 6 .. 9) = 1/((q-5)*(q-4)*(q-3)*(q-2)*q);" "6#/-%$SumG6$*(&%\"bG6#% \"iG\"\"\"&%\"cG6#F+F,-F%6$*&&%\"aG6$F+%\"jGF,-F%6$*&&F46$F6%\"kGF,-F% 6$*&&F46$F<%\"mGF,-F%6$*&&F46$FB%\"nGF,)&F.6#FH,&%\"qGF,\"\"'!\"\"F,/F H;\"\"#,&FBF,F,FOF,/FB;\"\"$,&F" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(b[i]*Sum(a[i,j]*Sum(a[j,k]*c[k],k = 2 .. j-1)^2,j = 3 .. i-1),i = 4 .. 9) = 1/120;" "6#/-%$SumG6$*&&%\"bG6#%\"iG\"\"\"-F% 6$*&&%\"aG6$F+%\"jGF,*$-F%6$*&&F16$F3%\"kGF,&%\"cG6#F:F,/F:;\"\"#,&F3F ,F,!\"\"F@F,/F3;\"\"$,&F+F,F,FBF,/F+;\"\"%\"\"**&F,F,\"$?\"FB" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(b[i]*Sum(a[i,j]*c[j]*Sum(a[j,k]*c[k],k = \+ 2 .. j-1)^2,j = 3 .. i-1),i = 4 .. 9) = 1/168;" "6#/-%$SumG6$*&&%\"bG6 #%\"iG\"\"\"-F%6$*(&%\"aG6$F+%\"jGF,&%\"cG6#F3F,-F%6$*&&F16$F3%\"kGF,& F56#F " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "#========================================================" }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 54 "Trees of order 7 and their associ ated order conditions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 407 "TT := Trees(7):\nm := 3: ## number of tr ees per row\nnn := nops(TT): q := iquo(nn,m,'r'):\nfor i to nn do \n \+ p||i := DrawTree(TT[i],color=COLOR(RGB,.75,.2,.2),show_ordercondition= true,\n font_color=black,height=5,width=2,scale=2.5);\nend d o:\npp := plot([[1,1]],style=line,axes=none):\nplots[display](convert( [seq([p||((k-1)*m+1..m*k)],k=1..q),\n `if`(r>0,[p||(m*q+1..nn),pp$(m* (q+1)-nn)],NULL)],array));" }}{PARA 13 "" 1 "" {GLPLOT2D 841 3897 3897 {PLOTDATA 2 "6b]n-%%TEXTG6&7$$\"+tH(H(f!\")$!+X=]%y#!\"(Q6b~c~(a~ c~(a~c~))~=~~~6\"-%'COLOURG6&%$RGBG\"\"!F3F3-%%FONTG6$%(COURIERG\"#5-F $6&7$F'$!+FPpsFF,Q8~~~~~~~~~~~~~~~~~~~~~~1F.F/F4-F$6&7$F'$!+kPQwFF,Q:~ ~~~~~~~~~~~~~~~~~~~~~___F.F/F4-F$6&7$F'$!+++++GF,Q:~~~~~~~~~~~~~~~~~~~ ~~~105F.F/F4-F$6&7$F'$!+7x@rFF,Q8~~~~~~~~~~~~~~2~~~~~~~~F.F/F4-%'CURVE SG6&7)7$F'$!+w9gZFF,7$$\"+(['['[&F)$!+\"=\"3=FF,7$$\"+f%f%fkF)Fen7$$\" +N^8NhF)$!+')3c)o#F,7$$\"+%y$y$y'F)F]o7$$\"+ovcnlF)$!+!fS!fEF,7$$\"+++ ++qF)Feo-%'SYMBOLG6#%'CIRCLEG-%&COLORG6&F2$F3F3FapFap-%&STYLEG6#%&POIN TG-FR6&FT-F[p6#%(DIAMONDGF^pFbp-FR6&FT-F[p6#%&CROSSGF^pFbp-FR6%7$FUFX- F_p6&F2$\"#v!\"#$\"\"#!\"\"Fhq-%*THICKNESSG6#Fiq-FR6%7$FUFgnFcqF[r-FR6 %7$FgnFjnFcqF[r-FR6%7$FgnF_oFcqF[r-FR6%7$F_oFboFcqF[r-FR6%7$F_oFgoFcqF [r-FR6%7#7$F'$!+++++EF,-F06&F2$F8FjqFapFap-Fcp6#%%LINEG-FR6%7#7$$\"+++ ++]F)FVFcsFfs-FR6%7#7$$\"+Yf%f%pF)FVFcsFfs-F$6&7$$\"#gF3$!+X=]%G%F,Q/b ~c~(a~c~)~=~~F.F/F4-F$6&7$Fht$!+FPpsUF,Q1~~~~~~~~~~~~~~~1F.F/F4-F$6&7$ Fht$!+kPQwUF,Q2~~~~~~~~~~~~~~~__F.F/F4-F$6&7$Fht$!+++++VF,Q2~~~~~~~~~~ ~~~~~35F.F/F4-F$6&7$Fht$!+7x@rUF,Q1~~~~~~~~~4~~~~~~F.F/F4-FR6&7)7$Fht$ !+w9gZUF,7$$\"+++++bF)$!+\"=\"3=UF,7$$\"+++++lF)F^w7$$\"+nmmmhF)$!+')3 c)=%F,7$$\"+*))))))Q'F)Ffw7$$\"+6666mF)Ffw7$$\"+LLLLoF)FfwFjoF^pFbp-FR 6&FgvFhpF^pFbp-FR6&FgvF]qF^pFbp-FR6%7$FhvF[wFcqF[r-FR6%7$FhvF`wFcqF[r- FR6%7$F`wFcwFcqF[r-FR6%7$F`wFhwFcqF[r-FR6%7$F`wF[xFcqF[r-FR6%7$F`wF^xF cqF[r-FR6%7#7$Fht$!+++++TF,FcsFfs-FR6%7#7$F]tFivFcsFfs-FR6%7#7$FhoFivF csFfs-F$6&7$Fap$!+X=]%G\"F,Q9b~c~(a~c~(a~(a~c)))~=~~~F.F/F4-F$6&7$Fap$ !+FPps7F,Q;~~~~~~~~~~~~~~~~~~~~~~~~~1F.F/F4-F$6&7$Fap$!+kPQw7F,Q=~~~~~ ~~~~~~~~~~~~~~~~~~~~___F.F/F4-F$6&7$Fap$!+++++8F,Q=~~~~~~~~~~~~~~~~~~~ ~~~~~~210F.F/F4-FR6&7)7$Fap$!+w9gZ7F,7$$!+++++]!\"*$!+\"=\"3=7F,7$$F^t Ff\\lFg\\l7$$\"+nmmm;Ff\\l$!+')3c)=\"F,7$$\"+LLLL$)Ff\\lF^]l7$Fa]l$!+! 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%7$FbawFcawFcqF[r-FR6%7$FbawFdawFcqF[r-FR6%7$FbawFeawFcqF[r-FR6%7$Feaw FfawFcqF[r-FR6%7$FbawFgawFcqF[r-FR6%7$FgawFhawFcqF[r-FR6%7#7$FhtF]`mFc sFfs-FR6%7#7$F]tF]]mFcsFfs-FR6%7#7$FhoF]]mFcsFfs-F$6&7$Fd`lFbenQ4b~(a~ c~(a~c)~)~=~~~F.F/F4-F$6&7$Fd`lFhenF]\\nF/F4-F$6&7$Fd`lF]fnF_fnF/F4-F$ 6&7$Fd`lFcfnFh_rF/F4-F$6&7$Fd`lFifnF[gnF/F4-FR6&7)7$Fd`lF`gn7$Fd`lFcgn 7$FabnFggn7$Fd`lFggn7$Fd`lFjgn7$FdbnFggn7$FdbnFjgnFjoF^pFbp-FR6&F]ewFh pF^pFbp-FR6&F]ewF]qF^pFbp-FR6%7$F^ewF_ewFcqF[r-FR6%7$F_ewF`ewFcqF[r-FR 6%7$F_ewFaewFcqF[r-FR6%7$FaewFbewFcqF[r-FR6%7$F_ewFcewFcqF[r-FR6%7$Fce wFdewFcqF[r-FR6%7#7$Fd`lFhinFcsFfs-FR6%7#7$FcelF`gnFcsFfs-FR6%7#7$Fiel F`gnFcsFfs-F$6&7$Fd`lFhiqQ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 73 "Simple and non-simple order conditions and their associat ed rooted trees " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 138 ": The \"simple\" order conditions ar e those that correspond to rooted trees with no more than one branch p oint at or above the second level." }}{PARA 0 "" 0 "" {TEXT -1 106 "It is often sufficient to consider only the simple order cnditions when \+ constructing a Runge-Kutta scheme." }}{PARA 0 "" 0 "" {TEXT -1 11 "The re are " }{XPPEDIT 18 0 "2^(n-1)" "6#)\"\"#,&%\"nG\"\"\"F'!\"\"" } {TEXT -1 59 " simple order conditions for a Runge-Kutta scheme of orde r " }{TEXT 265 1 "n" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 233 "As the order of the method increases th e proportion of non-simple order conditions also increases so that eve ntually (when the order reaches 10) the number of non-simple order con ditions exceeds the number of simple order conditions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " matrix([[`order of Runge-Kutta method `, ` no. of order conditions `, \+ ` no. of simple order conditions`], [4, 8, 8], [5, 17, 16], [6, 37, 32 ], [7, 85, 64], [8, 200, 128], [9, 486, 256], [10, 1205, 512], [11, 30 47, 1024], [12, 7813, 2048]])" "6#-%'matrixG6#7,7%%=order~of~Runge-Kut ta~method~G%:~no.~of~order~conditions~G%@~no.~of~simple~order~conditio nsG7%\"\"%\"\")F-7%\"\"&\"#<\"#;7%\"\"'\"#P\"#K7%\"\"(\"#&)\"#k7%F-\"$ +#\"$G\"7%\"\"*\"$'[\"$c#7%\"#5\"%07\"$7&7%\"#6\"%ZI\"%C57%\"#7\"%8y\" %[?" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 121 "There are 21 non-simple order conditions for order 6 R unge-Kutta schemes as indicated in the following table and diagram." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 367 "ind := [13,22,24,29,32,43,44,46,47,51,52,57,59,61,63,64,65,71,7 5,78,79]:\nconvert(ListTools[Enumerate](OrderConditions(7)),matrix):\n linalg[delcols](%,2..2),matrix([[` `]$(linalg[rowdim](%))]),linalg[de lcols](%,1..1),matrix([[` `]$(linalg[rowdim](%))]),linalg[transpose]( convert([[seq(`if`(member(i,ind),`non-simple`,`simple`),i=1..85)]],mat rix)):\nlinalg[augment](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matr ixG6#7ap7'\"\"\"%#~~G/*&%\"bGF(%\"eGF(F(F)%'simpleG7'\"\"#F)/*&F,F(%\" cGF(#F(F0F)F.7'\"\"$F)/*&F,F(-%!G6#*&%\"aGF(F3F(F(#F(\"\"'F)F.7'\"\"%F )/*&F,F()F3F0F(#F(F6F)F.7'\"\"&F)/*&F,F(-F:6#*&F=F(F9F(F(#F(\"#CF)F.7' F?F)/*(F,F(F3F(F9F(#F(\"\")F)F.7'\"\"(F)/*&F,F(-F:6#*&FDF(F=F(F(#F(\"# 7F)F.7'FSF)/*&F,F()F3F6F(#F(FAF)F.7'\"\"*F)/*&F,F(-F:6#*&F=F(FJF(F(#F( \"$?\"F)F.7'\"#5F)/*&F,F(-F:6#*(F=F(F3F(F9F(F(#F(\"#SF)F.7'\"#6F)/*(F, F(F3F(FJF(#F(\"#IF)F.7'FfnF)/*&F,F(-F:6#*&F=F(FXF(F(#F(\"#gF)F.7'\"#8F )/*&F,F()F9F0F(#F(\"#?F)%+non-simpleG7'\"#9F)/*(F,F(F3F(FXF(#F(\"#:F)F .7'FiqF)/*(F,F(FDF(F9F(#F(FfoF)F.7'\"#;F)/*&F,F(-F:6#*&FjnF(F=F(F(FaqF )F.7'\"#F)/*&F,F(-F:6#*&F=F(FioF(F(#F(\"$S#F)F.7'FbqF)/*&F,F(-F :6#*(F=F(F3F(FJF(F(#F(\"$!=F)F.7'\"#@F)/*(F,F(F3F(F`oF(#F(\"$W\"F)F.7' \"#AF)/*(F,F(FJF(F9F(#F(\"#sF)Fcq7'\"#BF)/*&F,F(-F:6#*&F=F(FgpF(F(#F( \"$g$F)F.7'FNF)/*&F,F(-F:6#*&F=F(F`qF(F(FcoF)Fcq7'\"#DF)/*(F,F(F3F(Fio F(#F(\"#[F)F.7'\"#EF)/*&F,F(-F:6#*(F=F(F3F(FXF(F(#F(\"#!*F)F.7'\"#FF)/ *(F,F(F3F(FgpF(F_uF)F.7'\"#GF)/*&F,F(-F:6#*(F=F(FDF(F9F(F(FjpF)F.7'\"# HF)/*(F,F(F9F(FXF(#F(\"#OF)Fcq7'FcpF)/*(F,F(FDF(FJF(F^xF)F.7'\"#JF)/*& F,F(-F:6#*&F=F(FbrF(F(FcoF)F.7'\"#KF)/*(F,F(F3F(F`qF(FMF)Fcq7'\"#LF)/* (F,F(FDF(FXF(#F(F\\sF)F.7'\"#MF)/*(F,F(F3F(FbrF(FMF)F.7'\"#NF)/*(F,F(F jnF(F9F(FenF)F.7'F_xF)/*&F,F(-F:6#*&FirF(F=F(F(FbpF)F.7'\"#PF)/*&F,F() F3FGF(F>F)F.7'\"#QF)/*&F,F(-F:6#*&F=F(F_sF(F(#F(\"%S]F)F.7'\"#RF)/*&F, F(-F:6#*&F=F(FhsF(F(#F(\"%!o\"F)F.7'F]pF)/*&F,F(-F:6#*&F=F(F`tF(F(#F( \"%g7F)F.7'\"#TF)/*&F,F(-F:6#*(F=F(F3F(F`oF(F(#F(\"%35F)F.7'\"#UF)/*(F ,F(F3F(F_sF(#F(\"$S)F)F.7'\"#VF)/*&F,F(-F:6#*(F=F(FJF(F9F(F(#F(\"$/&F) Fcq7'\"#WF)/*(F,F(F`oF(F9F(#F(\"$O$F)Fcq7'\"#XF)/*&F,F(-F:6#*&F=F(FeuF (F(#F(\"%?DF)F.7'\"#YF)/*&F,F(-F:6#*&F=F(F]vF(F(F]]lF)Fcq7'\"#ZF)/*&F, F()FJF0F(#F(\"$_#F)Fcq7'FevF)/*&F,F(-F:6#*(F=F(F3F(FioF(F(F\\^lF)F.7' \"#\\F)/*(F,F(F3F(FhsF(#F(\"$!GF)F.7'\"#]F)/*(F,F(F3F(F`tF(#F(\"$5#F)F .7'\"#^F)/*(F,F(FioF(F9F(#F(\"$7\"F)Fcq7'\"#_F)/**F,F(F3F(F9F(FJF(#F( \"#%)F)Fcq7'\"#`F)/*&F,F(-F:6#*&F=F(FjvF(F(#F(\"$I'F)F.7'\"#aF)/*&F,F( -F:6#*(F=F(F3F(FgpF(F(Ff]lF)F.7'\"#bF)/*&F,F(-F:6#*&F=F(FgwF(F(#F(\"$? %F)F.7'\"#cF)/*(F,F(F3F(FeuF(FjblF)F.7'\"#dF)/*&F,F(-F:6#*(F=F(F9F(FXF (F(Fc_lF)Fcq7'\"#eF)/*&F,F(-F:6#*(F=F(FDF(FJF(F(Fc_lF)F.7'\"#fF)/*(F,F (F9F(FgpF(#F(\"$o\"F)Fcq7'F[qF)/*(F,F(FDF(F`oF(FbdlF)F.7'\"#hF)/*(F,F( FJF(FXF(#F(\"$E\"F)Fcq7'\"#iF)/*&F,F(-F:6#*&F=F(FgxF(F(F]]lF)F.7'\"#jF )/*&F,F(-F:6#*(F=F(F3F(F`qF(F(FbdlF)Fcq7'\"#kF)/*(F,F(F3F(F]vF(#F(\"$S \"F)Fcq7'\"#lF)/*&F,F()F9F6F(#F(F]clF)Fcq7'\"#mF)/*&F,F(-F:6#*(F=F(FDF (FXF(F(F[elF)F.7'\"#nF)/*(F,F(F3F(FjvF(#F(\"$0\"F)F.7'\"#oF)/*(F,F(FDF (FgpF(FaalF)F.7'\"#pF)/*(F,F(F3F(FgwF(#F(\"#qF)F.7'F]hlF)/*(F,F(FDF(Fi oF(FfflF)F.7'\"#rF)/**F,F(F3F(F9F(FXF(#F(Fj\\lF)Fcq7'F`uF)/*&F,F(-F:6# *(F=F(F3F(FbrF(F(FbdlF)F.7'\"#tF)/*(F,F(F3F(FgxF(F_flF)F.7'\"#uF)/*&F, F(-F:6#*(F=F(FjnF(F9F(F(FaalF)F.7'\"#vF)/*(F,F(F9F(FbrF(FfflF)Fcq7'\"# wF)/*(F,F(FjnF(FJF(FehlF)F.7'\"#xF)/*&F,F(-F:6#*&F=F(F^zF(F(Fe`lF)F.7' \"#yF)/*&F,F()FXF0F(#F(FeelF)Fcq7'\"#zF)/*(F,F(FDF(F`qF(#F(FdwF)Fcq7' \"#!)F)/*(F,F(FDF(FbrF(F`[mF)F.7'\"#\")F)/*(F,F(FjnF(FXF(#F(FftF)F.7' \"##)F)/*(F,F(F3F(F^zF(#F(FhyF)F.7'\"#$)F)/*(F,F(FirF(F9F(#F(FeqF)F.7' FbalF)/*&F,F(-F:6#*&FezF(F=F(F(FehlF)F.7'\"#&)F)/*&F,F()F3F?F(#F(FUF)F .Q*pprint1536\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 150 "The following picture shows the rooted trees correspondi ng to the order conditions. The trees corresponding to \"simple\" orde r conditions are coloured " }{TEXT 264 6 "orange" }{TEXT -1 78 " while those that correspond to the remaining \"non-simple\" trees are colou red " }{TEXT 260 6 "purple" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 518 "TT := AllTrees(7) :\nind := [13,22,24,29,32,43,44,46,47,51,52,57,59,61,63,64,65,71,75,78 ,79]:\nm := 3: ## number of trees per row\nnn := nops(TT): q := iquo(n n,m,'r'):\nfor i to nn do \n p||i := DrawTree(TT[i],height=5,width=2, show_ordercondition=true,\n color=`if`(member(i,ind),COLOR(RGB,.5,0 ,1),COLOR(RGB,.85,.5,0)),\n scale=2.2,font_color=black);\nend do:\n pp := plot([[1,1]],style=line,axes=none):\nplots[display](convert([seq ([p||((k-1)*m+1..m*k)],k=1..q),\n `if`(r>0,[p||(m*q+1..nn),pp$(m*(q+1 )-nn)],NULL)],array));" }}{PARA 13 "" 1 "" {GLPLOT2D 1006 6418 6418 {PLOTDATA 2 "6c_p-%%TEXTG6&7$$\"#g\"\"!$!+F*=f=)!\"(Q-b~(a~c~)~=~~6\"- %'COLOURG6&%$RGBGF)F)F)-%%FONTG6$%(COURIERG\"#5-F$6&7$F'$!+!\\!>v\")F, Q/~~~~~~~~~~~~~1F.F/F3-F$6&7$F'$!+FJay\")F,Q0~~~~~~~~~~~~~__F.F/F3-F$6 &7$F'$!+++++#)F,Q0~~~~~~~~~~~~~42F.F/F3-F$6&7$F'$!+O%\\Q<)F,Q/~~~~~~~5 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"Curve 987" "Curve 9 88" "Curve 989" "Curve 990" "Curve 991" "Curve 992" "Curve 993" "Curve 994" "Curve 995" "Curve 996" "Curve 997" "Curve 998" "Curve 999" "Cur ve 1000" "Curve 1001" "Curve 1002" "Curve 1003" "Curve 1004" "Curve 10 05" "Curve 1006" "Curve 1007" "Curve 1008" "Curve 1009" "Curve 1010" " Curve 1011" "Curve 1012" "Curve 1013" "Curve 1014" "Curve 1015" "Curve 1016" "Curve 1017" "Curve 1018" "Curve 1019" "Curve 1020" "Curve 1021 " "Curve 1022" "Curve 1023" "Curve 1024" "Curve 1025" "Curve 1026" "Cu rve 1027" "Curve 1028" "Curve 1029" "Curve 1030" "Curve 1031" "Curve 1 032" "Curve 1033" "Curve 1034" "Curve 1035" "Curve 1036" "Curve 1037" "Curve 1038" "Curve 1039" "Curve 1040" "Curve 1041" "Curve 1042" "Curv e 1043" "Curve 1044" "Curve 1045" "Curve 1046" "Curve 1047" "Curve 104 8" "Curve 1049" "Curve 1050" "Curve 1051" "Curve 1052" "Curve 1053" "C urve 1054" "Curve 1055" "Curve 1056" "Curve 1057" "Curve 1058" "Curve \+ 1059" "Curve 1060" "Curve 1061" "Curve 1062" "Curve 1063" "Curve 1064 " "Curve 1065" "Curve 1066" "Curve 1067" "Curve 1068" "Curve 1069" "Cu rve 1070" "Curve 1071" "Curve 1072" "Curve 1073" "Curve 1074" "Curve 1 075" "Curve 1076" "Curve 1077" "Curve 1078" "Curve 1079" "Curve 1080" "Curve 1081" "Curve 1082" "Curve 1083" "Curve 1084" "Curve 1085" "Curv e 1086" "Curve 1087" "Curve 1088" "Curve 1089" "Curve 1090" "Curve 109 1" "Curve 1092" "Curve 1093" "Curve 1094" "Curve 1095" "Curve 1096" "C urve 1097" "Curve 1098" "Curve 1099" "Curve 1100" "Curve 1101" "Curve \+ 1102" "Curve 1103" "Curve 1104" "Curve 1105" "Curve 1106" "Curve 1107 " "Curve 1108" "Curve 1109" "Curve 1110" "Curve 1111" "Curve 1112" "Cu rve 1113" "Curve 1114" "Curve 1115" "Curve 1116" "Curve 1117" "Curve 1 118" "Curve 1119" "Curve 1120" "Curve 1121" "Curve 1122" "Curve 1123" "Curve 1124" "Curve 1125" "Curve 1126" "Curve 1127" "Curve 1128" "Curv e 1129" "Curve 1130" "Curve 1131" "Curve 1132" "Curve 1133" "Curve 113 4" "Curve 1135" "Curve 1136" "Curve 1137" "Curve 1138" "Curve 1139" "C urve 1140" "Curve 1141" "Curve 1142" "Curve 1143" "Curve 1144" "Curve \+ 1145" "Curve 1146" "Curve 1147" "Curve 1148" "Curve 1149" "Curve 1150 " "Curve 1151" "Curve 1152" "Curve 1153" "Curve 1154" "Curve 1155" "Cu rve 1156" "Curve 1157" "Curve 1158" "Curve 1159" "Curve 1160" "Curve 1 161" "Curve 1162" "Curve 1163" "Curve 1164" "Curve 1165" "Curve 1166" "Curve 1167" "Curve 1168" "Curve 1169" "Curve 1170" "Curve 1171" "Curv e 1172" "Curve 1173" "Curve 1174" "Curve 1175" "Curve 1176" "Curve 117 7" "Curve 1178" "Curve 1179" "Curve 1180" "Curve 1181" "Curve 1182" "C urve 1183" "Curve 1184" "Curve 1185" "Curve 1186" "Curve 1187" "Curve \+ 1188" "Curve 1189" "Curve 1190" "Curve 1191" "Curve 1192" "Curve 1193 " "Curve 1194" "Curve 1195" "Curve 1196" "Curve 1197" "Curve 1198" "Cu rve 1199" "Curve 1200" "Curve 1201" "Curve 1202" "Curve 1203" "Curve 1 204" "Curve 1205" "Curve 1206" "Curve 1207" "Curve 1208" "Curve 1209" "Curve 1210" "Curve 1211" "Curve 1212" "Curve 1213" "Curve 1214" "Curv e 1215" "Curve 1216" "Curve 1217" "Curve 1218" "Curve 1219" "Curve 122 0" "Curve 1221" "Curve 1222" "Curve 1223" "Curve 1224" "Curve 1225" "C urve 1226" "Curve 1227" "Curve 1228" "Curve 1229" "Curve 1230" "Curve \+ 1231" "Curve 1232" "Curve 1233" "Curve 1234" "Curve 1235" "Curve 1236 " "Curve 1237" "Curve 1238" "Curve 1239" "Curve 1240" "Curve 1241" "Cu rve 1242" "Curve 1243" "Curve 1244" "Curve 1245" "Curve 1246" "Curve 1 247" "Curve 1248" "Curve 1249" "Curve 1250" "Curve 1251" "Curve 1252" "Curve 1253" "Curve 1254" "Curve 1255" "Curve 1256" "Curve 1257" "Curv e 1258" "Curve 1259" "Curve 1260" "Curve 1261" "Curve 1262" "Curve 126 3" "Curve 1264" "Curve 1265" "Curve 1266" "Curve 1267" "Curve 1268" "C urve 1269" "Curve 1270" "Curve 1271" "Curve 1272" "Curve 1273" "Curve \+ 1274" "Curve 1275" "Curve 1276" "Curve 1277" "Curve 1278" "Curve 1279 " "Curve 1280" "Curve 1281" "Curve 1282" "Curve 1283" "Curve 1284" "Cu rve 1285" "Curve 1286" "Curve 1287" "Curve 1288" "Curve 1289" "Curve 1 290" "Curve 1291" "Curve 1292" "Curve 1293" "Curve 1294" "Curve 1295" "Curve 1296" "Curve 1297" "Curve 1298" "Curve 1299" "Curve 1300" "Curv e 1301" "Curve 1302" "Curve 1303" "Curve 1304" "Curve 1305" "Curve 130 6" "Curve 1307" "Curve 1308" "Curve 1309" "Curve 1310" "Curve 1311" "C urve 1312" "Curve 1313" "Curve 1314" "Curve 1315" "Curve 1316" "Curve \+ 1317" "Curve 1318" "Curve 1319" "Curve 1320" "Curve 1321" "Curve 1322 " "Curve 1323" "Curve 1324" "Curve 1325" "Curve 1326" "Curve 1327" "Cu rve 1328" "Curve 1329" "Curve 1330" "Curve 1331" "Curve 1332" "Curve 1 333" "Curve 1334" "Curve 1335" "Curve 1336" "Curve 1337" "Curve 1338" "Curve 1339" "Curve 1340" "Curve 1341" "Curve 1342" "Curve 1343" "Curv e 1344" "Curve 1345" "Curve 1346" "Curve 1347" "Curve 1348" "Curve 134 9" "Curve 1350" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 70 "Abr eviated form of order conditions for an order 7 Runge-Kutta scheme " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 160 "convert(ListTools[Enumerate](OrderConditions(7)),matrix):\nlina lg[augment](linalg[delcols](%,2..2),matrix([[` `]$(linalg[rowdim](%)) ]),linalg[delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'mat rixG6#7ap7%\"\"\"%#~~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( F57%\"\"&F)/*&F,F(-F96#*&FF)/*(F,F(F2F(F8F(#F(\"\" )7%\"\"(F)/*&F,F(-F96#*&FCF(FF)/* &F,F(-F96#*&FF(#F(FTQ)pprint276\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 61 "Non-simple order conditions and t heir associated rooted trees" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 139 ": The \"non-simple\" order conditions are those that correspond to rooted trees with more than o ne branch point at or above the second level." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 397 "EOC7 := Lis tTools[Enumerate](OrderConditions(7)):\nSOC7 := SimpleOrderConditions( 7):\nNSOC7 := []: ind := []:\nfor ct to nops(EOC7) do \n if not memb er(EOC7[ct,2],SOC7) then\n ind := [op(ind),ct];\n NSOC7 := [ op(NSOC7),EOC7[ct]];\n end if;\nend do:\nind;\nnops(%);``;\nconvert( NSOC7,matrix):\nlinalg[augment](linalg[delcols](%,2..2),matrix([[` `] $(linalg[rowdim](%))]),linalg[delcols](%,1..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#77\"#8\"#A\"#C\"#H\"#K\"#V\"#W\"#Y\"#Z\"#^\"#_\"#d\"#f \"#h\"#j\"#k\"#l\"#r\"#v\"#y\"#z" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\" #@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#777%\"#8%#~~G/*&%\"bG\"\"\")-%!G6#*&%\"aGF- %\"cGF-\"\"#F-#F-\"#?7%\"#AF)/*(F,F--F06#*&F3F-F/F-F-F/F-#F-\"#s7%\"#C F)/*&F,F--F06#*&F3F-F.F-F-#F-\"$?\"7%\"#HF)/*(F,F-F/F--F06#*&)F4F5F-F3 F-F-#F-\"#O7%\"#KF)/*(F,F-F4F-F.F-#F-FB7%\"#VF)/*&F,F--F06#*(F3F-F " 0 "" {MPLTEXT 1 0 490 "TT := AllTrees(7):\nTTN := 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s(7,9,'expanded'))]:\nfor ct to nops(RK7_9eqs) do\n print(`equation \+ `||ct);print(RK7_9eqs[ct]);print(``); \nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%+equation~1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"a G6$\"\"#\"\"\"&%\"cG6#F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%+equation~2G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&&%\"aG6$\"\"$\"\"\"F)&F&6$F(\"\"#F)&%\"cG6#F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%+equation~3G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(&%\"aG6$\"\"%\"\" \"F)&F&6$F(\"\"#F)&F&6$F(\"\"$F)&%\"cG6#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%+equation~4G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/,*&%\"aG6$\"\"&\"\"\"F)&F&6$F(\"\"#F )&F&6$F(\"\"$F)&F&6$F(\"\"%F)&%\"cG6#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%+equation~5G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,,&%\"aG6$\"\"'\"\"\"F)&F&6$F(\"\"#F)&F&6$F 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F*F**&&F,6$F_rF8F*FUF*F**&&F,6$F_rFJF*FgoF*F**&&F,6$F_rFinF*FcqF*F**&& F,6$F_rFbpF*,.*&&F,6$FbpF1F*F2F*F**&FepF*FEF*F**&FhpF*FenF*F**&F[qF*F^ pF*F**&F^qF*F[rF*F**&FaqF*&F3FhnF*F*F*F*F*F*#F*\"#C" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,equation~14G " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,0**&%\"bG6#\"\"$\"\"\"&%\"cGF(F* &%\"aG6$F)\"\"#F*&F,6#F0F*F**(&F'6#\"\"%F*&F,F5F*,&*&&F.6$F6F0F*F1F*F* *&&F.6$F6F)F*F+F*F*F*F**(&F'6#\"\"&F*&F,FAF*,(*&&F.6$FBF0F*F1F*F**&&F. 6$FBF)F*F+F*F**&&F.6$FBF6F*F7F*F*F*F**(&F'6#\"\"'F*&F,FPF*,**&&F.6$FQF 0F*F1F*F**&&F.6$FQF)F*F+F*F**&&F.6$FQF6F*F7F*F**&&F.6$FQFBF*FCF*F*F*F* *(&F'6#\"\"(F*&F,F\\oF*,,*&&F.6$F]oF0F*F1F*F**&&F.6$F]oF)F*F+F*F**&&F. 6$F]oF6F*F7F*F**&&F.6$F]oFBF*FCF*F**&&F.6$F]oFQF*FRF*F*F*F**(&F'6#\"\" )F*&F,FapF*,.*&&F.6$FbpF0F*F1F*F**&&F.6$FbpF)F*F+F*F**&&F.6$FbpF6F*F7F *F**&&F.6$FbpFBF*FCF*F**&&F.6$FbpFQF*FRF*F**&&F.6$FbpF]oF*F^oF*F*F*F** (&F'6#\"\"*F*&F,FiqF*,0*&&F.6$FjqF0F*F1F*F**&&F.6$FjqF)F*F+F*F**&&F.6$ FjqF6F*F7F*F**&&F.6$FjqFBF*FCF*F**&&F.6$FjqFQF*FRF*F**&&F.6$FjqF]oF*F^ oF*F**&&F.6$FjqFbpF*FcpF*F*F*F*#F*Fbp" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,equation~15G" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#/,0*(&%\"bG6#\"\"$\"\"\"&%\"aG6$F)\"\"#F*)&%\"cG 6#F.F.F*F**&&F'6#\"\"%F*,&*&&F,6$F6F.F*F/F*F**&&F,6$F6F)F*)&F1F(F.F*F* F*F**&&F'6#\"\"&F*,(*&&F,6$FCF.F*F/F*F**&&F,6$FCF)F*F>F*F**&&F,6$FCF6F *)&F1F5F.F*F*F*F**&&F'6#\"\"'F*,**&&F,6$FSF.F*F/F*F**&&F,6$FSF)F*F>F*F **&&F,6$FSF6F*FNF*F**&&F,6$FSFCF*)&F1FBF.F*F*F*F**&&F'6#\"\"(F*,,*&&F, 6$F`oF.F*F/F*F**&&F,6$F`oF)F*F>F*F**&&F,6$F`oF6F*FNF*F**&&F,6$F`oFCF*F [oF*F**&&F,6$F`oFSF*)&F1FRF.F*F*F*F**&&F'6#\"\")F*,.*&&F,6$FfpF.F*F/F* F**&&F,6$FfpF)F*F>F*F**&&F,6$FfpF6F*FNF*F**&&F,6$FfpFCF*F[oF*F**&&F,6$ FfpFSF*FapF*F**&&F,6$FfpF`oF*)&F1F_oF.F*F*F*F**&&F'6#\"\"*F*,0*&&F,6$F _rF.F*F/F*F**&&F,6$F_rF)F*F>F*F**&&F,6$F_rF6F*FNF*F**&&F,6$F_rFCF*F[oF *F**&&F,6$F_rFSF*FapF*F**&&F,6$F_rF`oF*FjqF*F**&&F,6$F_rFfpF*)&F1FepF. F*F*F*F*#F*\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,equation~16G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /,2*&&%\"bG6#\"\"#\"\"\")&%\"cGF(\"\"$F*F**&&F'6#F.F*)&F-F1F.F*F**&&F' 6#\"\"%F*)&F-F6F.F*F**&&F'6#\"\"&F*)&F-FF*FHF*F**&FAF*,(*&&F,6$F )F4F*F5F*F**&FEF*FMF*F**&F+F*&F66#F.F*F*F*F*F*F*F*F**&&F'6#\"\")F*,*** &F,6$FhoF.F*F/F*F2F*F5F*F**&&F,6$FhoF)F*FCF*F**&&F,6$FhoF;F*FgnF*F**&& F,6$FhoFRF*,**(&F,6$FRF1F*F2F*F5F*F**&FUF*FHF*F**&FXF*F]oF*F**&FenF*,* *&&F,6$F;F4F*F5F*F**&FinF*FMF*F**&F>F*FcoF*F**&FAF*&F6F(F*F*F*F*F*F*F* F**&&F'6#\"\"*F*,,**&F,6$FhqF.F*F/F*F2F*F5F*F**&&F,6$FhqF)F*FCF*F**&&F ,6$FhqF;F*FgnF*F**&&F,6$FhqFRF*FfpF*F**&&F,6$FhqFhoF*,,*(&F,6$FhoF1F*F 2F*F5F*F**&F[pF*FHF*F**&F^pF*F]oF*F**&FapF*F]qF*F**&FdpF*,,*&&F,6$FRF4 F*F5F*F**&FhpF*FMF*F**&FUF*FcoF*F**&FXF*FdqF*F**&FenF*&F6F:F*F*F*F*F*F *F*F*#F*\"$?\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,equation~18G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# /,.*,&%\"bG6#\"\"%\"\"\"&%\"aG6$F)\"\"$F*&%\"cG6#F.F*&F,6$F.\"\"#F*&F0 6#F4F*F**&&F'6#\"\"&F*,&**&F,6$F:F.F*F/F*F2F*F5F*F**(&F,6$F:F)F*&F0F(F *,&*&&F,6$F)F4F*F5F*F**&F+F*F/F*F*F*F*F*F**&&F'6#\"\"'F*,(**&F,6$FKF.F *F/F*F2F*F5F*F**(&F,6$FKF)F*FBF*FCF*F**(&F,6$FKF:F*&F0F9F*,(*&&F,6$F:F 4F*F5F*F**&F=F*F/F*F**&F@F*FBF*F*F*F*F*F**&&F'6#\"\"(F*,***&F,6$FjnF.F *F/F*F2F*F5F*F**(&F,6$FjnF)F*FBF*FCF*F**(&F,6$FjnF:F*FVF*FWF*F**(&F,6$ FjnFKF*&F0FJF*,**&&F,6$FKF4F*F5F*F**&FNF*F/F*F**&FQF*FBF*F**&FTF*FVF*F *F*F*F*F**&&F'6#\"\")F*,,**&F,6$FcpF.F*F/F*F2F*F5F*F**(&F,6$FcpF)F*FBF *FCF*F**(&F,6$FcpF:F*FVF*FWF*F**(&F,6$FcpFKF*FhoF*FioF*F**(&F,6$FcpFjn F*&F0FinF*,,*&&F,6$FjnF4F*F5F*F**&F]oF*F/F*F**&F`oF*FBF*F**&FcoF*FVF*F **&FfoF*FhoF*F*F*F*F*F**&&F'6#\"\"*F*,.**&F,6$F`rF.F*F/F*F2F*F5F*F**(& F,6$F`rF)F*FBF*FCF*F**(&F,6$F`rF:F*FVF*FWF*F**(&F,6$F`rFKF*FhoF*FioF*F **(&F,6$F`rFjnF*FdqF*FeqF*F**(&F,6$F`rFcpF*&F0FbpF*,.*&&F,6$FcpF4F*F5F *F**&FfpF*F/F*F**&FipF*FBF*F**&F\\qF*FVF*F**&F_qF*FhoF*F**&FbqF*FdqF*F *F*F*F*F*#F*\"#S" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%,equation~19G" }}{PARA 12 "" 1 "" {XPPMATH 20 " 6#/,.*,&%\"bG6#\"\"%\"\"\"&%\"cGF(F*&%\"aG6$F)\"\"$F*&F.6$F0\"\"#F*&F, 6#F3F*F**(&F'6#\"\"&F*&F,F8F*,&*(&F.6$F9F0F*F1F*F4F*F**&&F.6$F9F)F*,&* &&F.6$F)F3F*F4F*F**&F-F*&F,6#F0F*F*F*F*F*F**(&F'6#\"\"'F*&F,FKF*,(*(&F .6$FLF0F*F1F*F4F*F**&&F.6$FLF)F*FBF*F**&&F.6$FLF9F*,(*&&F.6$F9F3F*F4F* F**&F=F*FGF*F**&F@F*F+F*F*F*F*F*F**(&F'6#\"\"(F*&F,FjnF*,**(&F.6$F[oF0 F*F1F*F4F*F**&&F.6$F[oF)F*FBF*F**&&F.6$F[oF9F*FXF*F**&&F.6$F[oFLF*,**& &F.6$FLF3F*F4F*F**&FPF*FGF*F**&FSF*F+F*F**&FVF*F:F*F*F*F*F*F**(&F'6#\" \")F*&F,FcpF*,,*(&F.6$FdpF0F*F1F*F4F*F**&&F.6$FdpF)F*FBF*F**&&F.6$FdpF 9F*FXF*F**&&F.6$FdpFLF*FjoF*F**&&F.6$FdpF[oF*,,*&&F.6$F[oF3F*F4F*F**&F _oF*FGF*F**&FboF*F+F*F**&FeoF*F:F*F**&FhoF*FMF*F*F*F*F*F**(&F'6#\"\"*F *&F,F`rF*,.*(&F.6$FarF0F*F1F*F4F*F**&&F.6$FarF)F*FBF*F**&&F.6$FarF9F*F XF*F**&&F.6$FarFLF*FjoF*F**&&F.6$FarF[oF*FfqF*F**&&F.6$FarFdpF*,.*&&F. 6$FdpF3F*F4F*F**&FhpF*FGF*F**&F[qF*F+F*F**&F^qF*F:F*F**&FaqF*FMF*F**&F dqF*F\\oF*F*F*F*F*F*#F*\"#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%,equation~20G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,.**&%\"bG6#\"\"%\"\"\"&%\"aG6$F)\"\"$F*&F,6$F.\"\"#F* )&%\"cG6#F1F1F*F**&&F'6#\"\"&F*,&*(&F,6$F9F.F*F/F*F2F*F**&&F,6$F9F)F*, &*&&F,6$F)F1F*F2F*F**&F+F*)&F46#F.F1F*F*F*F*F*F**&&F'6#\"\"'F*,(*(&F,6 $FLF.F*F/F*F2F*F**&&F,6$FLF)F*FAF*F**&&F,6$FLF9F*,(*&&F,6$F9F1F*F2F*F* *&FF*F**&&F,6$FCF6F*)&F1F5F)F*F*F*F**&&F'6#\"\"'F*,**&&F,6$FSF.F*F/F* F**&&F,6$FSF)F*F>F*F**&&F,6$FSF6F*FNF*F**&&F,6$FSFCF*)&F1FBF)F*F*F*F** &&F'6#\"\"(F*,,*&&F,6$F`oF.F*F/F*F**&&F,6$F`oF)F*F>F*F**&&F,6$F`oF6F*F NF*F**&&F,6$F`oFCF*F[oF*F**&&F,6$F`oFSF*)&F1FRF)F*F*F*F**&&F'6#\"\")F* ,.*&&F,6$FfpF.F*F/F*F**&&F,6$FfpF)F*F>F*F**&&F,6$FfpF6F*FNF*F**&&F,6$F fpFCF*F[oF*F**&&F,6$FfpFSF*FapF*F**&&F,6$FfpF`oF*)&F1F_oF)F*F*F*F**&&F '6#\"\"*F*,0*&&F,6$F_rF.F*F/F*F**&&F,6$F_rF)F*F>F*F**&&F,6$F_rF6F*FNF* F**&&F,6$F_rFCF*F[oF*F**&&F,6$F_rFSF*FapF*F**&&F,6$F_rF`oF*FjqF*F**&&F ,6$F_rFfpF*)&F1FepF)F*F*F*F*#F*\"#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,equation~25G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,2*&&%\"bG6#\"\"#\"\"\")&%\"cGF(\"\"%F*F**&&F'6# \"\"$F*)&F-F1F.F*F**&&F'6#F.F*)&F-F7F.F*F**&&F'6#\"\"&F*)&F-FF.F*F/F*F2F*F5F*F8F*F**&&F,6$F>F)F*,&**&F,6$F)F1F*F2F*F5F*F8F*F**& F+F*,&*(&F,6$F.F4F*F5F*F8F*F**&F/F*,&*&&F,6$F1F7F*F8F*F**&F2F*&F96#F4F *F*F*F*F*F*F*F*F*F**&&F'6#\"\")F*,(*,&F,6$FZF.F*F/F*F2F*F5F*F8F*F**&&F ,6$FZF)F*FFF*F**&&F,6$FZF>F*,(**&F,6$F>F1F*F2F*F5F*F8F*F**&FAF*FKF*F** &FDF*,(*(&F,6$F)F4F*F5F*F8F*F**&FHF*FPF*F**&F+F*,(*&&F,6$F.F7F*F8F*F** &FMF*FUF*F**&F/F*&F96#F1F*F*F*F*F*F*F*F*F*F**&&F'6#\"\"*F*,**,&F,6$Ffp F.F*F/F*F2F*F5F*F8F*F**&&F,6$FfpF)F*FFF*F**&&F,6$FfpF>F*F_oF*F**&&F,6$ FfpFZF*,***&F,6$FZF1F*F2F*F5F*F8F*F**&FgnF*FKF*F**&FjnF*FeoF*F**&F]oF* ,**(&F,6$F>F4F*F5F*F8F*F**&FaoF*FPF*F**&FAF*F[pF*F**&FDF*,**&&F,6$F)F7 F*F8F*F**&FgoF*FUF*F**&FHF*FapF*F**&F+F*&F96#F.F*F*F*F*F*F*F*F*F*F*#F* \"$?(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,equation~27G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,,*. &%\"bG6#\"\"&\"\"\"&%\"aG6$F)\"\"%F*&F,6$F.\"\"$F*&%\"cG6#F1F*&F,6$F1 \"\"#F*&F36#F7F*F**&&F'6#\"\"'F*,&*,&F,6$F=F.F*F/F*F2F*F5F*F8F*F**&&F, 6$F=F)F*,&**&F,6$F)F1F*F2F*F5F*F8F*F**(F+F*&F36#F.F*,&*&&F,6$F.F7F*F8F *F**&F/F*F2F*F*F*F*F*F*F*F**&&F'6#\"\"(F*,(*,&F,6$FTF.F*F/F*F2F*F5F*F8 F*F**&&F,6$FTF)F*FEF*F**&&F,6$FTF=F*,(**&F,6$F=F1F*F2F*F5F*F8F*F**(F@F *FJF*FLF*F**(FCF*&F3F(F*,(*&&F,6$F)F7F*F8F*F**&FGF*F2F*F**&F+F*FJF*F*F *F*F*F*F*F**&&F'6#\"\")F*,**,&F,6$FioF.F*F/F*F2F*F5F*F8F*F**&&F,6$FioF )F*FEF*F**&&F,6$FioF=F*FinF*F**&&F,6$FioFTF*,***&F,6$FTF1F*F2F*F5F*F8F *F**(FWF*FJF*FLF*F**(FZF*F_oF*F`oF*F**(FgnF*&F3FF*F*F*F**(&F'6#\"\"(F*,**(&F,6 $FaoF.F*F/F*F2F*F**&&F,6$FaoF)F*F5F*F**&&F,6$FaoF?F*FGF*F**&&F,6$FaoFQ F*FfnF*F*F*,,*&&F,6$FaoF1F*F2F*F**&FdoF*F:F*F**&FgoF*FMF*F**&FjoF*F]oF *F**&F]pF*&F3FPF*F*F*F**(&F'6#\"\")F*,,*(&F,6$F[qF.F*F/F*F2F*F**&&F,6$ F[qF)F*F5F*F**&&F,6$F[qF?F*FGF*F**&&F,6$F[qFQF*FfnF*F**&&F,6$F[qFaoF*F _pF*F*F*,.*&&F,6$F[qF1F*F2F*F**&F^qF*F:F*F**&FaqF*FMF*F**&FdqF*F]oF*F* *&FgqF*FgpF*F**&FjqF*&F3F`oF*F*F*F**(&F'6#\"\"*F*,.*(&F,6$FirF.F*F/F*F 2F*F**&&F,6$FirF)F*F5F*F**&&F,6$FirF?F*FGF*F**&&F,6$FirFQF*FfnF*F**&&F ,6$FirFaoF*F_pF*F**&&F,6$FirF[qF*F\\rF*F*F*,0*&&F,6$FirF1F*F2F*F**&F\\ sF*F:F*F**&F_sF*FMF*F**&FbsF*F]oF*F**&FesF*FgpF*F**&FhsF*FerF*F**&F[tF *&F3FjpF*F*F*F*#F*\"#s" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%,equation~31G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,,*,&%\"bG6#\"\"&\"\"\"&%\"aG6$F)\"\"%F*&F,6$F.\"\"$F* &F,6$F1\"\"#F*)&%\"cG6#F4F4F*F**&&F'6#\"\"'F*,&**&F,6$FF*FIF*F**&FAF*)FCF4F*F*F*F*F*F**&&F'6#\"\"(F*,***&F ,6$F]oF.F*F/F*F2F*F5F*F**(&F,6$F]oF)F*FCF*FDF*F**(&F,6$F]oF;F*FXF*FYF* F**(&F,6$F]oFMF*&F0FLF*,**&&F,6$FMF4F*F5F*F**&FPF*FIF*F**&FSF*FinF*F** &FVF*)FXF4F*F*F*F*F*F**&&F'6#\"\")F*,,**&F,6$FgpF.F*F/F*F2F*F5F*F**(&F ,6$FgpF)F*FCF*FDF*F**(&F,6$FgpF;F*FXF*FYF*F**(&F,6$FgpFMF*F[pF*F\\pF*F **(&F,6$FgpF]oF*&F0F\\oF*,,*&&F,6$F]oF4F*F5F*F**&F`oF*FIF*F**&FcoF*Fin F*F**&FfoF*FcpF*F**&FioF*)F[pF4F*F*F*F*F*F**&&F'6#\"\"*F*,.**&F,6$FerF .F*F/F*F2F*F5F*F**(&F,6$FerF)F*FCF*FDF*F**(&F,6$FerF;F*FXF*FYF*F**(&F, 6$FerFMF*F[pF*F\\pF*F**(&F,6$FerF]oF*FhqF*FiqF*F**(&F,6$FerFgpF*&F0Ffp F*,.*&&F,6$FgpF4F*F5F*F**&FjpF*FIF*F**&F]qF*FinF*F**&F`qF*FcpF*F**&Fcq F*FarF*F**&FfqF*)FhqF4F*F*F*F*F*F*#F*\"#!*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,equation~35G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,.*,&%\"bG6#\"\"%\"\"\"&%\"cGF(F*&% \"aG6$F)\"\"$F*&F.6$F0\"\"#F*)&F,6#F3F3F*F**(&F'6#\"\"&F*&F,F9F*,&*(&F .6$F:F0F*F1F*F4F*F**&&F.6$F:F)F*,&*&&F.6$F)F3F*F4F*F**&F-F*)&F,6#F0F3F *F*F*F*F*F**(&F'6#\"\"'F*&F,FMF*,(*(&F.6$FNF0F*F1F*F4F*F**&&F.6$FNF)F* FCF*F**&&F.6$FNF:F*,(*&&F.6$F:F3F*F4F*F**&F>F*FHF*F**&FAF*)F+F3F*F*F*F *F*F**(&F'6#\"\"(F*&F,F]oF*,**(&F.6$F^oF0F*F1F*F4F*F**&&F.6$F^oF)F*FCF *F**&&F.6$F^oF:F*FZF*F**&&F.6$F^oFNF*,**&&F.6$FNF3F*F4F*F**&FRF*FHF*F* *&FUF*FjnF*F**&FXF*)F;F3F*F*F*F*F*F**(&F'6#\"\")F*&F,FgpF*,,*(&F.6$Fhp F0F*F1F*F4F*F**&&F.6$FhpF)F*FCF*F**&&F.6$FhpF:F*FZF*F**&&F.6$FhpFNF*F] pF*F**&&F.6$FhpF^oF*,,*&&F.6$F^oF3F*F4F*F**&FboF*FHF*F**&FeoF*FjnF*F** &FhoF*FdpF*F**&F[pF*)FOF3F*F*F*F*F*F**(&F'6#\"\"*F*&F,FerF*,.*(&F.6$Ff rF0F*F1F*F4F*F**&&F.6$FfrF)F*FCF*F**&&F.6$FfrF:F*FZF*F**&&F.6$FfrFNF*F ]pF*F**&&F.6$FfrF^oF*FjqF*F**&&F.6$FfrFhpF*,.*&&F.6$FhpF3F*F4F*F**&F\\ qF*FHF*F**&F_qF*FjnF*F**&FbqF*FdpF*F**&FeqF*FbrF*F**&FhqF*)F_oF3F*F*F* F*F*F*#F*\"#s" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,equation~36G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/, .*,&%\"bG6#\"\"%\"\"\"&%\"aG6$F)\"\"$F*)&%\"cG6#F.\"\"#F*&F,6$F.F3F*&F 16#F3F*F**&&F'6#\"\"&F*,&**&F,6$F;F.F*F/F*F4F*F6F*F**(&F,6$F;F)F*)&F1F (F3F*,&*&&F,6$F)F3F*F6F*F**&F+F*F0F*F*F*F*F*F**&&F'6#\"\"'F*,(**&F,6$F MF.F*F/F*F4F*F6F*F**(&F,6$FMF)F*FCF*FEF*F**(&F,6$FMF;F*)&F1F:F3F*,(*&& F,6$F;F3F*F6F*F**&F>F*F0F*F**&FAF*FDF*F*F*F*F*F**&&F'6#\"\"(F*,***&F,6 $F]oF.F*F/F*F4F*F6F*F**(&F,6$F]oF)F*FCF*FEF*F**(&F,6$F]oF;F*FXF*FZF*F* *(&F,6$F]oFMF*)&F1FLF3F*,**&&F,6$FMF3F*F6F*F**&FPF*F0F*F**&FSF*FDF*F** &FVF*FYF*F*F*F*F*F**&&F'6#\"\")F*,,**&F,6$FgpF.F*F/F*F4F*F6F*F**(&F,6$ FgpF)F*FCF*FEF*F**(&F,6$FgpF;F*FXF*FZF*F**(&F,6$FgpFMF*F[pF*F]pF*F**(& F,6$FgpF]oF*)&F1F\\oF3F*,,*&&F,6$F]oF3F*F6F*F**&F`oF*F0F*F**&FcoF*FDF* F**&FfoF*FYF*F**&FioF*F\\pF*F*F*F*F*F**&&F'6#\"\"*F*,.**&F,6$FerF.F*F/ F*F4F*F6F*F**(&F,6$FerF)F*FCF*FEF*F**(&F,6$FerF;F*FXF*FZF*F**(&F,6$Fer FMF*F[pF*F]pF*F**(&F,6$FerF]oF*FhqF*FjqF*F**(&F,6$FerFgpF*)&F1FfpF3F*, .*&&F,6$FgpF3F*F6F*F**&FjpF*F0F*F**&F]qF*FDF*F**&F`qF*FYF*F**&FcqF*F\\ pF*F**&FfqF*FiqF*F*F*F*F*F*#F*\"#g" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,equation~37G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,0*(&%\"bG6#\"\"$\"\"\")&%\"aG6$F)\"\"#F/F*)&%\"c G6#F/F)F*F**(&F'6#\"\"%F*,&*&&F-6$F7F/F*F1F*F**&&F-6$F7F)F*&F2F(F*F*F* ,&*&F:F*)F1F/F*F**&F=F*)F?F/F*F*F*F**(&F'6#\"\"&F*,(*&&F-6$FHF/F*F1F*F **&&F-6$FHF)F*F?F*F**&&F-6$FHF7F*&F2F6F*F*F*,(*&FKF*FBF*F**&FNF*FDF*F* *&FQF*)FSF/F*F*F*F**(&F'6#\"\"'F*,**&&F-6$FfnF/F*F1F*F**&&F-6$FfnF)F*F ?F*F**&&F-6$FfnF7F*FSF*F**&&F-6$FfnFHF*&F2FGF*F*F*,**&FinF*FBF*F**&F\\ oF*FDF*F**&F_oF*FXF*F**&FboF*)FdoF/F*F*F*F**(&F'6#\"\"(F*,,*&&F-6$F^pF /F*F1F*F**&&F-6$F^pF)F*F?F*F**&&F-6$F^pF7F*FSF*F**&&F-6$F^pFHF*FdoF*F* *&&F-6$F^pFfnF*&F2FenF*F*F*,,*&FapF*FBF*F**&FdpF*FDF*F**&FgpF*FXF*F**& FjpF*FjoF*F**&F]qF*)F_qF/F*F*F*F**(&F'6#\"\")F*,.*&&F-6$FjqF/F*F1F*F** &&F-6$FjqF)F*F?F*F**&&F-6$FjqF7F*FSF*F**&&F-6$FjqFHF*FdoF*F**&&F-6$Fjq FfnF*F_qF*F**&&F-6$FjqF^pF*&F2F]pF*F*F*,.*&F]rF*FBF*F**&F`rF*FDF*F**&F crF*FXF*F**&FfrF*FjoF*F**&FirF*FfqF*F**&F\\sF*)F^sF/F*F*F*F**(&F'6#\" \"*F*,0*&&F-6$FjsF/F*F1F*F**&&F-6$FjsF)F*F?F*F**&&F-6$FjsF7F*FSF*F**&& F-6$FjsFHF*FdoF*F**&&F-6$FjsFfnF*F_qF*F**&&F-6$FjsF^pF*F^sF*F**&&F-6$F 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FSF*)&F1FRF3F*F*F*F**&&F'6#\"\")F*,.*&&F,6$FfpF.F*F/F*F**&&F,6$FfpF)F* F>F*F**&&F,6$FfpF3F*FNF*F**&&F,6$FfpFCF*F[oF*F**&&F,6$FfpFSF*FapF*F**& &F,6$FfpF`oF*)&F1F_oF3F*F*F*F**&&F'6#\"\"*F*,0*&&F,6$F_rF.F*F/F*F**&&F ,6$F_rF)F*F>F*F**&&F,6$F_rF3F*FNF*F**&&F,6$F_rFCF*F[oF*F**&&F,6$F_rFSF *FapF*F**&&F,6$F_rF`oF*FjqF*F**&&F,6$F_rFfpF*)&F1FepF3F*F*F*F*#F*\"#I " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,equation~45G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,2*&&%\"bG6# \"\"#\"\"\")&%\"cGF(\"\"&F*F**&&F'6#\"\"$F*)&F-F1F.F*F**&&F'6#\"\"%F*) &F-F7F.F*F**&&F'6#F.F*)&F-F=F.F*F**&&F'6#\"\"'F*)&F-FBF.F*F**&&F'6#\" \"(F*)&F-FHF.F*F**&&F'6#\"\")F*)&F-FNF.F*F**&&F'6#\"\"*F*)&F-FTF.F*F*# F*FC" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,equation~46G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,(*0 &%\"bG6#\"\"(\"\"\"&%\"aG6$F)\"\"'F*&F,6$F.\"\"&F*&F,6$F1\"\"%F*&F,6$F 4\"\"$F*&F,6$F7\"\"#F*&%\"cG6#F:F*F**&&F'6#\"\")F*,&*.&F,6$FAF.F*F/F*F 2F*F5F*F8F*F;F*F**&&F,6$FAF)F*,&*,&F,6$F)F1F*F2F*F5F*F8F*F;F*F**&F+F*, &**&F,6$F.F4F*F5F*F8F*F;F*F**&F/F*,&*(&F,6$F1F7F*F8F*F;F*F**&F2F*,&*&& F,6$F4F:F*F;F*F**&F5F*&F<6#F7F*F*F*F*F*F*F*F*F*F*F*F**&&F'6#\"\"*F*,(* .&F,6$F\\oF.F*F/F*F2F*F5F*F8F*F;F*F**&&F,6$F\\oF)F*FIF*F**&&F,6$F\\oFA F*,(*,&F,6$FAF1F*F2F*F5F*F8F*F;F*F**&FDF*FNF*F**&FGF*,(**&F,6$F)F4F*F5 F*F8F*F;F*F**&FKF*FSF*F**&F+F*,(*(&F,6$F.F7F*F8F*F;F*F**&FPF*FXF*F**&F /F*,(*&&F,6$F1F:F*F;F*F**&FUF*FgnF*F**&F2F*&F<6#F4F*F*F*F*F*F*F*F*F*F* F*F*#F*\"%S]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%,equation~47G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,* *0&%\"bG6#\"\"'\"\"\"&%\"aG6$F)\"\"&F*&F,6$F.\"\"%F*&F,6$F1\"\"$F*&%\" cG6#F4F*&F,6$F4\"\"#F*&F66#F:F*F**&&F'6#\"\"(F*,&*.&F,6$F@F.F*F/F*F2F* F5F*F8F*F;F*F**&&F,6$F@F)F*,&*,&F,6$F)F1F*F2F*F5F*F8F*F;F*F**&F+F*,&** &F,6$F.F4F*F5F*F8F*F;F*F**(F/F*&F66#F1F*,&*&&F,6$F1F:F*F;F*F**&F2F*F5F *F*F*F*F*F*F*F*F*F**&&F'6#\"\")F*,(*.&F,6$FfnF.F*F/F*F2F*F5F*F8F*F;F*F **&&F,6$FfnF)F*FHF*F**&&F,6$FfnF@F*,(*,&F,6$F@F1F*F2F*F5F*F8F*F;F*F**& FCF*FMF*F**&FFF*,(**&F,6$F)F4F*F5F*F8F*F;F*F**(FJF*FRF*FTF*F**(F+F*&F6 6#F.F*,(*&&F,6$F.F:F*F;F*F**&FOF*F5F*F**&F/F*FRF*F*F*F*F*F*F*F*F*F**&& F'6#\"\"*F*,**.&F,6$FhpF.F*F/F*F2F*F5F*F8F*F;F*F**&&F,6$FhpF)F*FHF*F** &&F,6$FhpF@F*FaoF*F**&&F,6$FhpFfnF*,**,&F,6$FfnF1F*F2F*F5F*F8F*F;F*F** &FinF*FMF*F**&F\\oF*FgoF*F**&F_oF*,***&F,6$F@F4F*F5F*F8F*F;F*F**(FcoF* FRF*FTF*F**(FCF*F]pF*F_pF*F**(FFF*&F6F(F*,**&&F,6$F)F:F*F;F*F**&FioF*F 5F*F**&FJF*FRF*F**&F+F*F]pF*F*F*F*F*F*F*F*F*F*#F*\"%!o\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,equatio n~48G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,**0&%\"bG6#\"\"'\"\"\"&%\"a G6$F)\"\"&F*&F,6$F.\"\"%F*&%\"cG6#F1F*&F,6$F1\"\"$F*&F,6$F7\"\"#F*&F36 #F:F*F**&&F'6#\"\"(F*,&*.&F,6$F@F.F*F/F*F2F*F5F*F8F*F;F*F**&&F,6$F@F)F *,&*,&F,6$F)F1F*F2F*F5F*F8F*F;F*F**(F+F*&F36#F.F*,&*(&F,6$F.F7F*F8F*F; F*F**&F/F*,&*&&F,6$F1F:F*F;F*F**&F5F*&F36#F7F*F*F*F*F*F*F*F*F*F**&&F'6 #\"\")F*,(*.&F,6$FhnF.F*F/F*F2F*F5F*F8F*F;F*F**&&F,6$FhnF)F*FHF*F**&&F ,6$FhnF@F*,(*,&F,6$F@F1F*F2F*F5F*F8F*F;F*F**(FCF*FMF*FOF*F**(FFF*&F3F( F*,(*(&F,6$F)F7F*F8F*F;F*F**&FJF*FTF*F**&F+F*,(*&&F,6$F.F:F*F;F*F**&FQ F*FYF*F**&F/F*F2F*F*F*F*F*F*F*F*F*F**&&F'6#\"\"*F*,**.&F,6$FipF.F*F/F* F2F*F5F*F8F*F;F*F**&&F,6$FipF)F*FHF*F**&&F,6$FipF@F*FcoF*F**&&F,6$FipF hnF*,**,&F,6$FhnF1F*F2F*F5F*F8F*F;F*F**(F[oF*FMF*FOF*F**(F^oF*FioF*Fjo F*F**(FaoF*&F3F?F*,**(&F,6$F@F7F*F8F*F;F*F**&FeoF*FTF*F**&FCF*F`pF*F** &FFF*,**&&F,6$F)F:F*F;F*F**&F\\pF*FYF*F**&FJF*F2F*F**&F+F*FMF*F*F*F*F* F*F*F*F*F*#F*\"%g7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,equation~49G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,**0&%\"bG6#\"\"'\"\"\"&%\"aG6$F)\"\"&F*&%\"cG6#F.F*&F,6$F.\"\"%F* &F,6$F4\"\"$F*&F,6$F7\"\"#F*&F06#F:F*F**&&F'6#\"\"(F*,&*.&F,6$F@F.F*F/ F*F2F*F5F*F8F*F;F*F**(&F,6$F@F)F*&F0F(F*,&**&F,6$F)F4F*F5F*F8F*F;F*F** &F+F*,&*(&F,6$F.F7F*F8F*F;F*F**&F2F*,&*&&F,6$F4F:F*F;F*F**&F5F*&F06#F7 F*F*F*F*F*F*F*F*F*F**&&F'6#\"\")F*,(*.&F,6$FgnF.F*F/F*F2F*F5F*F8F*F;F* F**(&F,6$FgnF)F*FHF*FIF*F**(&F,6$FgnF@F*&F0F?F*,(**&F,6$F@F4F*F5F*F8F* F;F*F**&FCF*FNF*F**&FFF*,(*(&F,6$F)F7F*F8F*F;F*F**&FKF*FSF*F**&F+F*,(* &&F,6$F.F:F*F;F*F**&FPF*FXF*F**&F2F*&F06#F4F*F*F*F*F*F*F*F*F*F**&&F'6# \"\"*F*,**.&F,6$FjpF.F*F/F*F2F*F5F*F8F*F;F*F**(&F,6$FjpF)F*FHF*FIF*F** (&F,6$FjpF@F*FboF*FcoF*F**(&F,6$FjpFgnF*&F0FfnF*,***&F,6$FgnF4F*F5F*F8 F*F;F*F**&FjnF*FNF*F**&F]oF*FioF*F**&F`oF*,**(&F,6$F@F7F*F8F*F;F*F**&F eoF*FSF*F**&FCF*F_pF*F**&FFF*,**&&F,6$F)F:F*F;F*F**&F[pF*FXF*F**&FKF*F epF*F**&F+F*F/F*F*F*F*F*F*F*F*F*F*#F*\"%35" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,equation~50G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,**0&%\"bG6#\"\"'\"\"\"&%\"cGF(F*&% \"aG6$F)\"\"&F*&F.6$F0\"\"%F*&F.6$F3\"\"$F*&F.6$F6\"\"#F*&F,6#F9F*F**( &F'6#\"\"(F*&F,F>F*,&*,&F.6$F?F0F*F1F*F4F*F7F*F:F*F**&&F.6$F?F)F*,&**& F.6$F)F3F*F4F*F7F*F:F*F**&F-F*,&*(&F.6$F0F6F*F7F*F:F*F**&F1F*,&*&&F.6$ 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\"\"\",(*&&%\"aG6$F)\"\"#F*&%\"cG6#F0F*F**&&F.6$F)\"\"$F*&F26#F7F*F**& &F.6$F)\"\"%F*&F26#F=F*F*F*F;F*&F.6$F=F7F*&F.6$F7F0F*F1F*F**(&F'6#\"\" 'F*,**&&F.6$FGF0F*F1F*F**&&F.6$FGF7F*F8F*F**&&F.6$FGF=F*F>F*F**&&F.6$F GF)F*&F2F(F*F*F*,&**FPF*F@F*FBF*F1F*F**&FSF*,&*(F5F*FBF*F1F*F**&F;F*,& *&&F.6$F=F0F*F1F*F**&F@F*F8F*F*F*F*F*F*F*F**(&F'6#\"\"(F*,,*&&F.6$F^oF 0F*F1F*F**&&F.6$F^oF7F*F8F*F**&&F.6$F^oF=F*F>F*F**&&F.6$F^oF)F*FUF*F** &&F.6$F^oFGF*&F2FFF*F*F*,(**FgoF*F@F*FBF*F1F*F**&FjoF*FYF*F**&F]pF*,(* (FMF*FBF*F1F*F**&FPF*FfnF*F**&FSF*F+F*F*F*F*F*F**(&F'6#\"\")F*,.*&&F.6 $F[qF0F*F1F*F**&&F.6$F[qF7F*F8F*F**&&F.6$F[qF=F*F>F*F**&&F.6$F[qF)F*FU F*F**&&F.6$F[qFGF*F_pF*F**&&F.6$F[qF^oF*&F2F]oF*F*F*,***FdqF*F@F*FBF*F 1F*F**&FgqF*FYF*F**&FjqF*FdpF*F**&F]rF*,**(FdoF*FBF*F1F*F**&FgoF*FfnF* F**&FjoF*F+F*F**&F]pF*FHF*F*F*F*F*F**(&F'6#\"\"*F*,0*&&F.6$F]sF0F*F1F* F**&&F.6$F]sF7F*F8F*F**&&F.6$F]sF=F*F>F*F**&&F.6$F]sF)F*FUF*F**&&F.6$F ]sFGF*F_pF*F**&&F.6$F]sF^oF*F_rF*F**&&F.6$F]sF[qF*&F2FjpF*F*F*,,**FfsF 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F**&&F,6$F]rF=F*FjnF*F**&&F,6$F]rFUF*FjpF*F**&&F,6$F]rF\\pF*,,*(&F,6$F \\pF1F*F2F*F6F*F**&F_pF*FJF*F**&FbpF*F`oF*F**&FepF*FaqF*F**&FhpF*),,*& &F,6$FUF5F*F7F*F**&F\\qF*FPF*F**&FXF*FgoF*F**&FenF*FiqF*F**&FhnF*&F8F< F*F*F5F*F*F*F*F*F*#F*\"$S)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%,equation~55G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,.**&%\"bG6#\"\"%\"\"\")&%\"aG6$F)\"\"$\"\"#F*)&F-6$F/ F0F0F*)&%\"cG6#F0F0F*F**&&F'6#\"\"&F*),&*(&F-6$F;F/F*F2F*F5F*F**&&F-6$ F;F)F*,&*&&F-6$F)F0F*F5F*F**&F,F*&F66#F/F*F*F*F*F0F*F**&&F'6#\"\"'F*), (*(&F-6$FNF/F*F2F*F5F*F**&&F-6$FNF)F*FDF*F**&&F-6$FNF;F*,(*&&F-6$F;F0F *F5F*F**&F?F*FIF*F**&FBF*&F6F(F*F*F*F*F0F*F**&&F'6#\"\"(F*),**(&F-6$F^ oF/F*F2F*F5F*F**&&F-6$F^oF)F*FDF*F**&&F-6$F^oF;F*FZF*F**&&F-6$F^oFNF*, **&&F-6$FNF0F*F5F*F**&FRF*FIF*F**&FUF*FjnF*F**&FXF*&F6F:F*F*F*F*F0F*F* *&&F'6#\"\")F*),,*(&F-6$FhpF/F*F2F*F5F*F**&&F-6$FhpF)F*FDF*F**&&F-6$Fh pF;F*FZF*F**&&F-6$FhpFNF*F]pF*F**&&F-6$FhpF^oF*,,*&&F-6$F^oF0F*F5F*F** 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HF*F**(&F,6$FjoF?F*FjnF*F[oF*F**(&F,6$FjoFUF*&F0FTF*,***&F,6$FUF4F*F5F *F7F*F:F*F**(FXF*F/F*FMF*F**(FenF*FGF*FaoF*F**(FhnF*FjnF*,**&&F,6$F?F9 F*F:F*F**&F]oF*F5F*F**&FBF*F/F*F**&FEF*FGF*F*F*F*F*F*F*F**&&F'6#\"\"*F *,,*.&F,6$FjqF.F*F/F*F2F*F5F*F7F*F:F*F**(&F,6$FjqF)F*FGF*FHF*F**(&F,6$ FjqF?F*FjnF*F[oF*F**(&F,6$FjqFUF*FhpF*FipF*F**(&F,6$FjqFjoF*&F0FioF*,, **&F,6$FjoF4F*F5F*F7F*F:F*F**(F]pF*F/F*FMF*F**(F`pF*FGF*FaoF*F**(FcpF* FjnF*F`qF*F**(FfpF*FhpF*,,*&&F,6$FUF9F*F:F*F**&F[qF*F5F*F**&FXF*F/F*F* *&FenF*FGF*F**&FhnF*FjnF*F*F*F*F*F*F*F*#F*\"$O$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,equation~57G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,,*0&%\"bG6#\"\"&\"\"\"&%\"cGF(F*&% \"aG6$F)\"\"%F*&F.6$F0\"\"$F*&F,6#F3F*&F.6$F3\"\"#F*&F,6#F8F*F**(&F'6# \"\"'F*&F,F=F*,&*,&F.6$F>F0F*F1F*F4F*F6F*F9F*F**&&F.6$F>F)F*,&**&F.6$F )F3F*F4F*F6F*F9F*F**(F-F*&F,6#F0F*,&*&&F.6$F0F8F*F9F*F**&F1F*F4F*F*F*F *F*F*F*F**(&F'6#\"\"(F*&F,FUF*,(*,&F.6$FVF0F*F1F*F4F*F6F*F9F*F**&&F.6$ FVF)F*FGF*F**&&F.6$FVF>F*,(**&F.6$F>F3F*F4F*F6F*F9F*F**(FBF*FLF*FNF*F* *(FEF*F+F*,(*&&F.6$F)F8F*F9F*F**&FIF*F4F*F**&F-F*FLF*F*F*F*F*F*F*F**(& F'6#\"\")F*&F,FjoF*,**,&F.6$F[pF0F*F1F*F4F*F6F*F9F*F**&&F.6$F[pF)F*FGF *F**&&F.6$F[pF>F*F\\oF*F**&&F.6$F[pFVF*,***&F.6$FVF3F*F4F*F6F*F9F*F**( FZF*FLF*FNF*F**(FgnF*F+F*FboF*F**(FjnF*F?F*,**&&F.6$F>F8F*F9F*F**&F^oF *F4F*F**&FBF*FLF*F**&FEF*F+F*F*F*F*F*F*F*F**(&F'6#\"\"*F*&F,FjqF*,,*,& F.6$F[rF0F*F1F*F4F*F6F*F9F*F**&&F.6$F[rF)F*FGF*F**&&F.6$F[rF>F*F\\oF*F **&&F.6$F[rFVF*FjpF*F**&&F.6$F[rF[pF*,,**&F.6$F[pF3F*F4F*F6F*F9F*F**(F _pF*FLF*FNF*F**(FbpF*F+F*FboF*F**(FepF*F?F*FaqF*F**(FhpF*FWF*,,*&&F.6$ FVF8F*F9F*F**&F\\qF*F4F*F**&FZF*FLF*F**&FgnF*F+F*F**&FjnF*F?F*F*F*F*F* F*F*F*#F*\"$!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,equation~58G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# /,,*0&%\"bG6#\"\"&\"\"\"&%\"cGF(F*&%\"aG6$F)\"\"%F*&F,6#F0F*&F.6$F0\" \"$F*&F.6$F5\"\"#F*&F,6#F8F*F**(&F'6#\"\"'F*&F,F=F*,&*,&F.6$F>F0F*F1F* 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{XPPMATH 20 "6#/,.*.&%\"bG6#\"\"%\"\"\"&%\"aG6$F)\"\"$F*&%\"cG6 #F.F*&F,6$F.\"\"#F*&F06#F4F*,&*&&F,6$F)F4F*F5F*F**&F+F*F/F*F*F*F**(&F' 6#\"\"&F*,&**&F,6$F?F.F*F/F*F2F*F5F*F**(&F,6$F?F)F*&F0F(F*F7F*F*F*,(*& &F,6$F?F4F*F5F*F**&FBF*F/F*F**&FEF*FGF*F*F*F**(&F'6#\"\"'F*,(**&F,6$FQ F.F*F/F*F2F*F5F*F**(&F,6$FQF)F*FGF*F7F*F**(&F,6$FQF?F*&F0F>F*FHF*F*F*, **&&F,6$FQF4F*F5F*F**&FTF*F/F*F**&FWF*FGF*F**&FZF*FfnF*F*F*F**(&F'6#\" \"(F*,***&F,6$FaoF.F*F/F*F2F*F5F*F**(&F,6$FaoF)F*FGF*F7F*F**(&F,6$FaoF ?F*FfnF*FHF*F**(&F,6$FaoFQF*&F0FPF*FgnF*F*F*,,*&&F,6$FaoF4F*F5F*F**&Fd oF*F/F*F**&FgoF*FGF*F**&FjoF*FfnF*F**&F]pF*F_pF*F*F*F**(&F'6#\"\")F*,, **&F,6$F[qF.F*F/F*F2F*F5F*F**(&F,6$F[qF)F*FGF*F7F*F**(&F,6$F[qF?F*FfnF *FHF*F**(&F,6$F[qFQF*F_pF*FgnF*F**(&F,6$F[qFaoF*&F0F`oF*F`pF*F*F*,.*&& F,6$F[qF4F*F5F*F**&F^qF*F/F*F**&FaqF*FGF*F**&FdqF*FfnF*F**&FgqF*F_pF*F **&FjqF*F\\rF*F*F*F**(&F'6#\"\"*F*,.**&F,6$FirF.F*F/F*F2F*F5F*F**(&F,6 $FirF)F*FGF*F7F*F**(&F,6$FirF?F*FfnF*FHF*F**(&F,6$FirFQF*F_pF*FgnF*F** 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*F5F*F8F*F**(&F,6$FWF)F*FFF*FGF*F**(&F,6$FWF>F*&F0F=F*,(*(&F,6$F>F4F*F 5F*F8F*F**&FAF*FLF*F**&FDF*,(*&&F,6$F)F7F*F8F*F**&FIF*FQF*F**&F+F*)F/F 7F*F*F*F*F*F*F*F**&&F'6#\"\")F*,**,&F,6$F]pF.F*F/F*F2F*F5F*F8F*F**(&F, 6$F]pF)F*FFF*FGF*F**(&F,6$F]pF>F*F\\oF*F]oF*F**(&F,6$F]pFWF*&F0FVF*,** (&F,6$FWF4F*F5F*F8F*F**&FZF*FLF*F**&FgnF*FcoF*F**&FjnF*,**&&F,6$F>F7F* F8F*F**&F_oF*FQF*F**&FAF*FioF*F**&FDF*)FFF7F*F*F*F*F*F*F*F**&&F'6#\"\" *F*,,*,&F,6$F^rF.F*F/F*F2F*F5F*F8F*F**(&F,6$F^rF)F*FFF*FGF*F**(&F,6$F^ rF>F*F\\oF*F]oF*F**(&F,6$F^rFWF*F[qF*F\\qF*F**(&F,6$F^rF]pF*&F0F\\pF*, ,*(&F,6$F]pF4F*F5F*F8F*F**&F`pF*FLF*F**&FcpF*FcoF*F**&FfpF*FcqF*F**&Fi pF*,,*&&F,6$FWF7F*F8F*F**&F^qF*FQF*F**&FZF*FioF*F**&FgnF*FjqF*F**&FjnF *)F\\oF7F*F*F*F*F*F*F*F*#F*\"$/&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%! 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F*F/F*F2F*F7F*F9F*F**&&F,6$F\\pF)F*FFF*F**&&F,6$F\\pF>F*F[oF*F**&&F,6$ F\\pFVF*,***&F,6$FVF1F*F2F*F7F*F9F*F**(FYF*FKF*FNF*F**(FfnF*FaoF*FcoF* F**(FinF*)&F4F=F6F*,**&&F,6$F>F6F*F9F*F**&F]oF*F3F*F**&FAF*FLF*F**&FDF *FboF*F*F*F*F*F*F*F**&&F'6#\"\"*F*,,*,&F,6$F]rF.F*F/F*F2F*F7F*F9F*F**& &F,6$F]rF)F*FFF*F**&&F,6$F]rF>F*F[oF*F**&&F,6$F]rFVF*FjpF*F**&&F,6$F]r F\\pF*,,**&F,6$F\\pF1F*F2F*F7F*F9F*F**(F_pF*FKF*FNF*F**(FbpF*FaoF*FcoF *F**(FepF*FaqF*FcqF*F**(FhpF*)&F4FUF6F*,,*&&F,6$FVF6F*F9F*F**&F\\qF*F3 F*F**&FYF*FLF*F**&FfnF*FboF*F**&FinF*FbqF*F*F*F*F*F*F*F*#F*\"$?%" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%,equation~64G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,,*.&%\"bG6#\"\"& \"\"\"&%\"cGF(F*&%\"aG6$F)\"\"%F*&F.6$F0\"\"$F*&F.6$F3\"\"#F*)&F,6#F6F 6F*F**(&F'6#\"\"'F*&F,FF6F*F*F*F*F*F*F*F*#F*\"$?%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,equation~65G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,.**&%\"bG6#\"\"%\"\"\"&%\"aG6$F)\" \"$F*)&F,6$F.\"\"#F2F*)&%\"cG6#F2F.F*F**&&F'6#\"\"&F*,&*(&F,6$F:F.F*F/ F*F3F*F**(&F,6$F:F)F*,&*&&F,6$F)F2F*F4F*F**&F+F*&F56#F.F*F*F*,&*&FDF*) F4F2F*F**&F+F*)FGF2F*F*F*F*F*F**&&F'6#\"\"'F*,(*(&F,6$FQF.F*F/F*F3F*F* *(&F,6$FQF)F*FBF*FIF*F**(&F,6$FQF:F*,(*&&F,6$F:F2F*F4F*F**&F=F*FGF*F** &F@F*&F5F(F*F*F*,(*&FhnF*FKF*F**&F=F*FMF*F**&F@F*)F\\oF2F*F*F*F*F*F**& &F'6#\"\"(F*,**(&F,6$FeoF.F*F/F*F3F*F**(&F,6$FeoF)F*FBF*FIF*F**(&F,6$F eoF:F*FfnF*F]oF*F**(&F,6$FeoFQF*,**&&F,6$FQF2F*F4F*F**&FTF*FGF*F**&FWF *F\\oF*F**&FZF*&F5F9F*F*F*,**&FepF*FKF*F**&FTF*FMF*F**&FWF*FaoF*F**&FZ F*)FjpF2F*F*F*F*F*F**&&F'6#\"\")F*,,*(&F,6$FdqF.F*F/F*F3F*F**(&F,6$Fdq F)F*FBF*FIF*F**(&F,6$FdqF:F*FfnF*F]oF*F**(&F,6$FdqFQF*FcpF*F[qF*F**(&F ,6$FdqFeoF*,,*&&F,6$FeoF2F*F4F*F**&FhoF*FGF*F**&F[pF*F\\oF*F**&F^pF*Fj pF*F**&FapF*&F5FPF*F*F*,,*&FgrF*FKF*F**&FhoF*FMF*F**&F[pF*FaoF*F**&F^p F*F`qF*F**&FapF*)F]sF2F*F*F*F*F*F**&&F'6#\"\"*F*,.*(&F,6$FhsF.F*F/F*F3 F*F**(&F,6$FhsF)F*FBF*FIF*F**(&F,6$FhsF:F*FfnF*F]oF*F**(&F,6$FhsFQF*Fc pF*F[qF*F**(&F,6$FhsFeoF*FerF*F^sF*F**(&F,6$FhsFdqF*,.*&&F,6$FdqF2F*F4 F*F**&FgqF*FGF*F**&FjqF*F\\oF*F**&F]rF*FjpF*F**&F`rF*F]sF*F**&FcrF*&F5 FdoF*F*F*,.*&F^uF*FKF*F**&FgqF*FMF*F**&FjqF*FaoF*F**&F]rF*F`qF*F**&F`r F*FdsF*F**&FcrF*)FeuF2F*F*F*F*F*F*#F*\"$_#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,equation~66G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,,*.&%\"bG6#\"\"&\"\"\"&%\"aG6$F)\" \"%F*)&%\"cG6#F.\"\"#F*&F,6$F.\"\"$F*&F,6$F6F3F*&F16#F3F*F**&&F'6#\"\" 'F*,&*,&F,6$F>F.F*F/F*F4F*F7F*F9F*F**(&F,6$F>F)F*)&F1F(F3F*,&*(&F,6$F) F6F*F7F*F9F*F**&F+F*,&*&&F,6$F.F3F*F9F*F**&F4F*&F16#F6F*F*F*F*F*F*F*F* *&&F'6#\"\"(F*,(*,&F,6$FWF.F*F/F*F4F*F7F*F9F*F**(&F,6$FWF)F*FFF*FHF*F* *(&F,6$FWF>F*)&F1F=F3F*,(*(&F,6$F>F6F*F7F*F9F*F**&FAF*FMF*F**&FDF*,(*& &F,6$F)F3F*F9F*F**&FJF*FRF*F**&F+F*F0F*F*F*F*F*F*F*F**&&F'6#\"\")F*,** ,&F,6$F]pF.F*F/F*F4F*F7F*F9F*F**(&F,6$F]pF)F*FFF*FHF*F**(&F,6$F]pF>F*F \\oF*F^oF*F**(&F,6$F]pFWF*)&F1FVF3F*,**(&F,6$FWF6F*F7F*F9F*F**&FZF*FMF *F**&FgnF*FdoF*F**&FjnF*,**&&F,6$F>F3F*F9F*F**&F`oF*FRF*F**&FAF*F0F*F* *&FDF*FGF*F*F*F*F*F*F*F**&&F'6#\"\"*F*,,*,&F,6$F^rF.F*F/F*F4F*F7F*F9F* F**(&F,6$F^rF)F*FFF*FHF*F**(&F,6$F^rF>F*F\\oF*F^oF*F**(&F,6$F^rFWF*F[q F*F]qF*F**(&F,6$F^rF]pF*)&F1F\\pF3F*,,*(&F,6$F]pF6F*F7F*F9F*F**&F`pF*F MF*F**&FcpF*FdoF*F**&FfpF*FdqF*F**&FipF*,,*&&F,6$FWF3F*F9F*F**&F_qF*FR F*F**&FZF*F0F*F**&FgnF*FGF*F**&FjnF*F]oF*F*F*F*F*F*F*F*#F*\"$_#" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%,equation~67G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,.*,&%\"bG6#\"\"% \"\"\",&*&&%\"aG6$F)\"\"#F*&%\"cG6#F0F*F**&&F.6$F)\"\"$F*&F26#F7F*F*F* F5F*&F.6$F7F0F*)F1F0F*F**(&F'6#\"\"&F*,(*&&F.6$F@F0F*F1F*F**&&F.6$F@F7 F*F8F*F**&&F.6$F@F)F*&F2F(F*F*F*,&*(FFF*F:F*FF*, (*&&F06$FF*FenF*F**(&F06$F^oFNF*FPF*,**&&F06$FNF.F*F7F*F**&FSF*F3F*F**&F VF*F,F*F**&FYF*F>F*F*F*F*F*F**(&F'6#\"\")F*)&F-FgpF.F*,,**&F06$FhpF2F* F3F*F5F*F7F*F**(&F06$FhpF)F*F,F*FFF*F**(&F06$FhpFF*FenF*F**(&F06$ FhpFNF*FPF*F^pF*F**(&F06$FhpF^oF*F`oF*,,*&&F06$F^oF.F*F7F*F**&FcoF*F3F *F**&FfoF*F,F*F**&FioF*F>F*F**&F\\pF*FPF*F*F*F*F*F**(&F'6#\"\"*F*)&F-F erF.F*,.**&F06$FfrF2F*F3F*F5F*F7F*F**(&F06$FfrF)F*F,F*FFF*F**(&F06$Ffr FF*FenF*F**(&F06$FfrFNF*FPF*F^pF*F**(&F06$FfrF^oF*F`oF*F[rF*F**(& F06$FfrFhpF*FjpF*,.*&&F06$FhpF.F*F7F*F**&F]qF*F3F*F**&F`qF*F,F*F**&Fcq F*F>F*F**&FfqF*FPF*F**&FiqF*F`oF*F*F*F*F*F*#F*\"#c" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,equation~79G " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,0**&%\"bG6#\"\"$\"\"\"&%\"cGF(F* )&%\"aG6$F)\"\"#F1F*)&F,6#F1F)F*F***&F'6#\"\"%F*&F,F7F*,&*&&F/6$F8F1F* F3F*F**&&F/6$F8F)F*F+F*F*F*,&*&FF*FIF*F**&FAF*)FCF.F*F*F*F*F* F**&&F'6#\"\"(F*,***&F,6$F]oF.F*F/F*F2F*F5F*F**(&F,6$F]oF)F*FCF*FDF*F* *(&F,6$F]oF;F*FXF*FYF*F**(&F,6$F]oFMF*&F0FLF*,**&&F,6$FMF4F*F5F*F**&FP F*FIF*F**&FSF*FinF*F**&FVF*)FXF.F*F*F*F*F*F**&&F'6#\"\")F*,,**&F,6$Fgp F.F*F/F*F2F*F5F*F**(&F,6$FgpF)F*FCF*FDF*F**(&F,6$FgpF;F*FXF*FYF*F**(&F ,6$FgpFMF*F[pF*F\\pF*F**(&F,6$FgpF]oF*&F0F\\oF*,,*&&F,6$F]oF4F*F5F*F** &F`oF*FIF*F**&FcoF*FinF*F**&FfoF*FcpF*F**&FioF*)F[pF.F*F*F*F*F*F**&&F' 6#\"\"*F*,.**&F,6$FerF.F*F/F*F2F*F5F*F**(&F,6$FerF)F*FCF*FDF*F**(&F,6$ FerF;F*FXF*FYF*F**(&F,6$FerFMF*F[pF*F\\pF*F**(&F,6$FerF]oF*FhqF*FiqF*F **(&F,6$FerFgpF*&F0FfpF*,.*&&F,6$FgpF4F*F5F*F**&FjpF*FIF*F**&F]qF*FinF *F**&F`qF*FcpF*F**&FcqF*FarF*F**&FfqF*)FhqF.F*F*F*F*F*F*#F*\"$o\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%,equation~81G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,.*,&%\"bG6#\"\"% \"\"\"&%\"cGF(F*&%\"aG6$F)\"\"$F*&F.6$F0\"\"#F*)&F,6#F3F0F*F**(&F'6#\" \"&F*&F,F9F*,&*(&F.6$F:F0F*F1F*F4F*F**&&F.6$F:F)F*,&*&&F.6$F)F3F*F4F*F **&F-F*)&F,6#F0F0F*F*F*F*F*F**(&F'6#\"\"'F*&F,FMF*,(*(&F.6$FNF0F*F1F*F 4F*F**&&F.6$FNF)F*FCF*F**&&F.6$FNF:F*,(*&&F.6$F:F3F*F4F*F**&F>F*FHF*F* *&FAF*)F+F0F*F*F*F*F*F**(&F'6#\"\"(F*&F,F]oF*,**(&F.6$F^oF0F*F1F*F4F*F **&&F.6$F^oF)F*FCF*F**&&F.6$F^oF:F*FZF*F**&&F.6$F^oFNF*,**&&F.6$FNF3F* F4F*F**&FRF*FHF*F**&FUF*FjnF*F**&FXF*)F;F0F*F*F*F*F*F**(&F'6#\"\")F*&F ,FgpF*,,*(&F.6$FhpF0F*F1F*F4F*F**&&F.6$FhpF)F*FCF*F**&&F.6$FhpF:F*FZF* F**&&F.6$FhpFNF*F]pF*F**&&F.6$FhpF^oF*,,*&&F.6$F^oF3F*F4F*F**&FboF*FHF *F**&FeoF*FjnF*F**&FhoF*FdpF*F**&F[pF*)FOF0F*F*F*F*F*F**(&F'6#\"\"*F*& F,FerF*,.*(&F.6$FfrF0F*F1F*F4F*F**&&F.6$FfrF)F*FCF*F**&&F.6$FfrF:F*FZF *F**&&F.6$FfrFNF*F]pF*F**&&F.6$FfrF^oF*FjqF*F**&&F.6$FfrFhpF*,.*&&F.6$ FhpF3F*F4F*F**&F\\qF*FHF*F**&F_qF*FjnF*F**&FbqF*FdpF*F**&FeqF*FbrF*F** &FhqF*)F_oF0F*F*F*F*F*F*#F*\"$S\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% !G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,equation~82G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,.*,&%\"bG6#\"\"%\"\"\"&%\"aG6$F)\"\"$F*)&%\"cG6#F. F.F*&F,6$F.\"\"#F*&F16#F5F*F**&&F'6#\"\"&F*,&**&F,6$F;F.F*F/F*F3F*F6F* F**(&F,6$F;F)F*)&F1F(F.F*,&*&&F,6$F)F5F*F6F*F**&F+F*F0F*F*F*F*F*F**&&F '6#\"\"'F*,(**&F,6$FMF.F*F/F*F3F*F6F*F**(&F,6$FMF)F*FCF*FEF*F**(&F,6$F MF;F*)&F1F:F.F*,(*&&F,6$F;F5F*F6F*F**&F>F*F0F*F**&FAF*FDF*F*F*F*F*F**& &F'6#\"\"(F*,***&F,6$F]oF.F*F/F*F3F*F6F*F**(&F,6$F]oF)F*FCF*FEF*F**(&F ,6$F]oF;F*FXF*FZF*F**(&F,6$F]oFMF*)&F1FLF.F*,**&&F,6$FMF5F*F6F*F**&FPF *F0F*F**&FSF*FDF*F**&FVF*FYF*F*F*F*F*F**&&F'6#\"\")F*,,**&F,6$FgpF.F*F /F*F3F*F6F*F**(&F,6$FgpF)F*FCF*FEF*F**(&F,6$FgpF;F*FXF*FZF*F**(&F,6$Fg pFMF*F[pF*F]pF*F**(&F,6$FgpF]oF*)&F1F\\oF.F*,,*&&F,6$F]oF5F*F6F*F**&F` oF*F0F*F**&FcoF*FDF*F**&FfoF*FYF*F**&FioF*F\\pF*F*F*F*F*F**&&F'6#\"\"* F*,.**&F,6$FerF.F*F/F*F3F*F6F*F**(&F,6$FerF)F*FCF*FEF*F**(&F,6$FerF;F* FXF*FZF*F**(&F,6$FerFMF*F[pF*F]pF*F**(&F,6$FerF]oF*FhqF*FjqF*F**(&F,6$ FerFgpF*)&F1FfpF.F*,.*&&F,6$FgpF5F*F6F*F**&FjpF*F0F*F**&F]qF*FDF*F**&F `qF*FYF*F**&FcqF*F\\pF*F**&FfqF*FiqF*F*F*F*F*F*#F*\"#%)" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,equatio n~83G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,0*(&%\"bG6#\"\"$\"\"\")&%\" aG6$F)\"\"#F/F*)&%\"cG6#F/\"\"%F*F**(&F'6#F4F*,&*&&F-6$F4F/F*)F1F)F*F* *&&F-6$F4F)F*)&F2F(F)F*F*F*,&*&F:F*F1F*F**&F>F*FAF*F*F*F**(&F'6#\"\"&F *,(*&&F-6$FHF/F*F " 0 "" {MPLTEXT 1 0 8490 "## si mplifying row sum conditions\nfor i to 8 do\n eq||i := add(a[i+1,j], j=1..i)=c[i+1];\nend do: ## eq1 to eq8\n\n## order 7 quadrature condit ions\neq9 := add(b[j],j=1..9)=1:\n\nfor q from 2 to 7 do\n eq||(8+q) := add(b[j]*c[j]^(q-1),j=2..9)=1/q;\nend do: ## eq10 to eq15\n\n## or der 7 sub-quadrature conditions\n## b.a\n## b.a.c to b.a.c^5\nfor q fr om 3 to 7 do\n eq||(13+q) := add(b[i]*add(a[i,j]*c[j]^(q-2),j=2..i-1 ),i=3..9)=1/(q-1)/q;\nend do: ## eq16 to eq20\n\n## b.c.a.c, b.c.a.c^2 , b.c.a.c^3, b.c.a.c^4\nfor q from 4 to 7 do\n eq||(17+q) := add(b[i ]*c[i]*add(a[i,j]*c[j]^(q-3),j=2..i-1),i=3..9)=1/(q-2)/q;\nend do: ## \+ eq21 to eq24\n\n## b.c^2.a.c, b.c^2.a.c^2, b.c^2.a.c^3 \nfor q from 5 \+ to 7 do\n eq||(20+q) := add(b[i]*c[i]^2*add(a[i,j]*c[j]^(q-4),j=2..i -1),i=3..9)=1/(q-3)/q;\nend do: ## eq25 to eq27\n\n## b.c^3.a.c, b.c^3 .a.c^2\nfor q from 6 to 7 do\n eq||(22+q) := add(b[i]*c[i]^3*add(a[i ,j]*c[j]^(q-5),j=2..i-1),i=3..9)=1/(q-4)/q;\nend do: ## eq28,eq29\n\n# # b.c^4.a.c\nq := 7:\neq30 := add(b[i]*c[i]^4*add(a[i,j]*c[j]^(q-6),j= 2..i-1),i=3..9)=1/(q-5)/q:\n\n## b.a.a\n## b.a.a.c to b.a.a.c^4 \nfor \+ q from 4 to 7 do\n eq||(27+q) := add(b[i]*add(a[i,j]*add(a[j,k]*c[k] ^(q-3),k=2..j-1),j=3..i-1),i=4..9)\n \+ =1/(q-2)/(q-1)/q;\nend do: ## eq31 to eq34\n\n## b.c.a.a.c, b.c.a.a.c^ 2, b.c.a.a.c^3 \nfor q from 5 to 7 do\n eq||(30+q) := add(b[i]*c[i]* add(a[i,j]*add(a[j,k]*c[k]^(q-4),k=2..j-1),j=3..i-1),i=4..9)\n \+ =1/(q-3)/(q-2)/q;\nend do: ## eq35 to eq37\n\n## b.c^2.a.a.c, b.c^2 .a.a.c^2\nfor q from 6 to 7 do\n eq||(32+q) :=add(b[i]*c[i]^2*add(a[ i,j]*add(a[j,k]*c[k]^(q-5),k=2..j-1),j=3..i-1),\n i=4..9)=1/( q-4)/(q-3)/q;\nend do: ## eq38,eq39\n\n## b.c^3.a.a.c\nq := 7:\neq40 : = add(b[i]*c[i]^3*add(a[i,j]*add(a[j,k]*c[k]^(q-6),k=2..j-1),j=3..i-1) ,i=4..9)\n =1/(q-5)/(q-4)/q:\n\n## b.a.c.a.c, b.a.c.a.c^2, b. a.c.a.c^3\nfor q from 5 to 7 do\n eq||(36+q) := add(b[i]*add(a[i,j]* c[j]*add(a[j,k]*c[k]^(q-4),k=2..j-1),j=3..i-1),i=4..9) \n =1 /(q-3)/(q-1)/q;\nend do: ## eq41 to eq43\n\n## b.a.c^2.a.c, b.a.c^2.a. c^2\nfor q from 6 to 7 do\n eq||(38+q) := add(b[i]*add(a[i,j]*c[j]^2 *add(a[j,k]*c[k]^(q-5),k=2..j-1),j=3..i-1),\n i=4..9)=1/(q-4)/(q-1) /q;\nend do: ## eq44,eq45\n\n## b.a.c^3.a.c\nq := 7:\neq46 := add(b[i] *add(a[i,j]*c[j]^3*add(a[j,k]*c[k]^(q-6),k=2..j-1),j=3..i-1),i=4..9)\n =1/(q-5)/(q-1)/q:\n\n## b.c.a.c.a.c, b.c.a.c.a.c^2 NEW 1 8/11\nfor q from 6 to 7 do\n eq||(41+q) := add(b[i]*c[i]*add(a[i,j]* c[j]*add(a[j,k]*c[k]^(q-5),k=2..j-1),\n j=3..i-1),i=4..9)=1 /(q-4)/(q-2)/q:\nend do: ## eq47,eq48\n\n## b.c.a.c^2.a.c NEW 18/11\nq := 7:\neq49 := add(b[i]*c[i]*add(a[i,j]*c[j]^2*add(a[j,k]*c[k]^(q-6), k=2..j-1),\n j=3..i-1),i=4..9)=1/(q-5)/(q-2)/q:\n\n## b.c^2 .a.c.a.c NEW 18/11\neq50 := add(b[i]*c[i]^2*add(a[i,j]*c[j]*add(a[j,k] *c[k]^(q-6),k=2..j-1),\n j=3..i-1),i=4..9)=1/(q-5)/(q-3)/q: \n\n## b.a.a.a\n## b.a.a.a.c to b.a.a.a.c^3\nfor q from 5 to 7 do\n \+ eq||(46+q) := add(b[i]*add(a[i,j]*add(a[j,k]*add(a[k,m]*c[m]^(q-4),m=2 ..k-1),k=3..j-1),\n j=4..i-1),i=5..9)=1/(q-3)/(q-2)/(q -1)/q;\nend do: ## eq51 to eq53\n\n## b.c.a.a.a.c, b.c.a.a.a.c^2 \nfor q from 6 to 7 do\n eq||(48+q) := add(b[i]*c[i]*add(a[i,j]*add(a[j,k ]*add(a[k,m]*c[m]^(q-5),m=2..k-1),\n k=3..j-1),j=4..i-1),i=5..9)=1 /(q-4)/(q-3)/(q-2)/q;\nend do: ## eq54,eq55\n\n## b.c^2.a.a.a.c\nq := \+ 7:\neq56 := add(b[i]*c[i]^2*add(a[i,j]*add(a[j,k]*add(a[k,m]*c[m]^(q-6 ),m=2..k-1),k=3..j-1),\n j=4..i-1),i=5..9)=1/(q-5)/(q-4)/(q-3)/ q:\n\n## b.a.c.a.a.c, b.a.c.a.a.c^2\nfor q from 6 to 7 do\n eq||(51+ q) := add(b[i]*add(a[i,j]*c[j]*add(a[j,k]*add(a[k,m]*c[m]^(q-5),m=2..k -1),\n k=3..j-1),j=4..i-1),i= 5..9)=1/(q-4)/(q-3)/(q-1)/q;\n end do: ## eq57,eq58\n\n## b.a.c^2.a.a.c\nq := 7:\neq59 := add(b[i]*ad d(a[i,j]*c[j]^2*add(a[j,k]*add(a[k,m]*c[m]^(q-6),m=2..k-1),k=3..j-1), \n j=4..i-1),i=5..9)=1/(q-5)/(q-4)/(q-1)/q:\n\n## b.a.a.c.a.c, b .a.a.c.a.c^2\nfor q from 6 to 7 do\n eq||(54+q) := add(b[i]*add(a[i, j]*add(a[j,k]*c[k]*add(a[k,m]*c[m]^(q-5),m=2..k-1),\n k=3..j -1),j=4..i-1),i= 5..9)=1/(q-4)/(q-2)/(q-1)/q;\nend do: ## eq60,eq61\n \n## b.a.a.c^2.a.c\nq := 7:\neq62 := add(b[i]*add(a[i,j]*add(a[j,k]*c[ k]^2*add(a[k,m]*c[m]^(q-6),m=2..k-1),k=3..j-1),\n j=4..i-1),i =5..9)=1/(q-5)/(q-2)/(q-1)/q:\n\n## b.a.c.a.c.a.c NEW\neq63 := add(b[i ]*add(a[i,j]*c[j]*add(a[j,k]*c[k]*add(a[k,m]*c[m]^(q-6),m=2..k-1),\n \+ k=3..j-1),j=4..i-1),i= 5..9)=1/(q-5)/(q-3)/(q-1)/q:\n\n## b.c .a.a.c.a.c NEW 18/11\neq64 := add(b[i]*c[i]*add(a[i,j]*add(a[j,k]*c[k] *add(a[k,m]*c[m]^(q-6),m=2..k-1),\n k=3..j-1),j=4..i-1),i= 5 ..9)=1/(q-5)/(q-3)/(q-2)/q:\n\n## b.c.a.c.a.a.c NEW 18/11\neq65 := add (b[i]*c[i]*add(a[i,j]*c[j]*add(a[j,k]*add(a[k,m]*c[m]^(q-6),m=2..k-1), \n k=3..j-1),j=4..i-1),i= 5..9)=1/(q-5)/(q-4)/(q-2)/q:\n\n## b.a.a.a.a\n## b.a.a.a.a.c, b.a.a.a.a.c^2\nfor q from 6 to 7 do\n eq ||(60+q) := add(b[i]*add(a[i,j]*add(a[j,k]*add(a[k,m]*add(a[m,n]*c[n]^ (q-5),n=2..m-1),\n m=3..k-1),k=4..j-1),j=5..i-1),i=6..9) =1/(q-4)/(q-3)/(q-2)/(q-1)/q;\nend do: ## eq66,eq67\n\n## b.c.a.a.a.a. c\nq := 7:\neq68 := add(b[i]*c[i]*add(a[i,j]*add(a[j,k]*add(a[k,m]*add (a[m,n]*c[n]^(q-6),n=2..m-1),\n m=3..k-1),k=4..j-1),j=5..i-1),i=6 ..9)=1/(q-5)/(q-4)/(q-3)/(q-2)/q:\n\n## b.a.c.a.a.a.c\neq69 := add(b[i ]*add(a[i,j]*c[j]*add(a[j,k]*add(a[k,m]*add(a[m,n]*c[n]^(q-6),n=2..m-1 ),\n m=3..k-1),k=4..j-1),j=5..i-1),i=6..9)=1/(q-5)/(q-4)/(q-3)/(q -1)/q:\n\n## b.a.a.c.a.a.c\neq70 := add(b[i]*add(a[i,j]*add(a[j,k]*c[k ]*add(a[k,m]*add(a[m,n]*c[n]^(q-6),n=2..m-1),\n m=3..k-1),k=4..j- 1),j=5..i-1),i=6..9)=1/(q-5)/(q-4)/(q-2)/(q-1)/q:\n\n## b.a.a.a.c.a.c \neq71 := add(b[i]*add(a[i,j]*add(a[j,k]*add(a[k,m]*c[m]*add(a[m,n]*c[ n]^(q-6),n=2..m-1),\n m=3..k-1),k=4..j-1),j=5..i-1),i=6..9)=1/(q-5 )/(q-3)/(q-2)/(q-1)/q:\n\n## b.a.a.a.a.a\n## b.a.a.a.a.a.c\neq72 := ad d(b[i]*add(a[i,j]*add(a[j,k]*add(a[k,m]*add(a[m,n]*add(a[n,p]*c[p]^(q- 6),\n p=2..n-1),n=3..m-1),m=4..k-1),k=5..j-1),j=6..i-1),i=7..9 )\n =1/(q-5)/(q-4)/(q-3)/(q-2)/(q-1)/q:\n\n## b.(a.c). a.c^2 \neq73 := add(b[i]*add(a[i,j]*c[j],j=2..i-1)*add(a[i,k]*c[k]^2,k =2..i-1),i=3..9)=1/36:\n\n## b.(a.c).a.c^3\neq74 := add(b[i]*add(a[i,j ]*c[j],j=2..i-1)*add(a[i,k]*c[k]^3,k=2..i-1),i=3..9)=1/56:\n\n## b.c.( a.c).a.c^2 \neq75 := add(b[i]*c[i]*add(a[i,j]*c[j],j=2..i-1)*add(a[i,k ]*c[k]^2,k=2..i-1),i=3..9)=1/42:\n\n## b.a.(a.c).a.c^2\neq76 := add(b[ i]*add(a[i,j]*add(a[j,k]*c[k],k=2..j-1)*add(a[j,m]*c[m]^2,\n \+ m=2..j-1),j=3..i-1),i=4..9)=1/252:\n\n## b.(a.c).a.a.c \ne q77 := add(b[i]*add(a[i,j]*c[j],j=2..i-1)*add(a[i,k]*add(a[k,m]*c[m],m =2..k-1),\n k=3..i-1),i=4..9)=1/72:\n\n## b.(a.c).a.a.c^2\ne q78 := add(b[i]*add(a[i,j]*c[j],j=2..i-1)*add(a[i,k]*add(a[k,m]*c[m]^2 ,m=2..k-1),\n k=3..i-1),i=4..9)=1/168:\n\n## b.(a.c^2).a. a.c\neq79 := add(b[i]*add(a[i,j]*c[j]^2,j=2..i-1)*add(a[i,k]*add(a[k,m ]*c[m],m=2..k-1),\n k=3..i-1),i=4..9)=1/126:\n \n## b.(a.c).a.c.a.c\neq80 := add(b[i]*add(a[i,j]*c[j],j=2..i-1)*add(a [i,k]*c[k]*add(a[k,m]*c[m],m=2..k-1),\n k=3..i-1),i=4..9) =1/112:\n\n## b.c.(a.c).a.a.c\neq81 := add(b[i]*c[i]*add(a[i,j]*c[j],j =2..i-1)*add(a[i,k]*add(a[k,m]*c[m],m=2..k-1),\n \+ k=3..i-1),i=4..9)=1/84:\n\n## b.(a.c).a.a.a.c\neq82 := add(b[i]*ad d(a[i,j]*c[j],j=2..i-1)*add(a[i,k]*add(a[k,m]*add(a[m,n]*c[n],n=2..m-1 ),\n m=3..k-1),k=4..i-1),i=5..9)=1/336:\n\n## b.a.(a.c).a.a. c\neq83 := add(b[i]*add(a[i,j]*add(a[j,k]*c[k],k=2..j-1)*add(a[j,m]*ad d(a[m,n]*c[n],n=2..m-1),\n m=3..j-1),j=4..i-1),i= 5..9)=1/504:\n\n## non-linear eqs\n## b.(a.c)^2\neq84 := add(b[i]*add( a[i,j]*c[j],j=2..i-1)^2,i=3..9)=1/20:\n\n## b.c.(a.c)^2\neq85 := add(b [i]*c[i]*add(a[i,j]*c[j],j=2..i-1)^2,i=3..9)=1/24:\n\n## b.c^2.(a.c)^2 \neq86 := add(b[i]*c[i]^2*add(a[i,j]*c[j],j=2..i-1)^2,i=3..9)=1/28:\n \n## b.a.(a.c)^2\neq87 := add(b[i]*add(a[i,j]*add(a[j,k]*c[k],k=2..j-1 )^2,j=3..i-1),i=4..9)=1/120:\n\n## b.a.c.(a.c)^2 \neq88 := add(b[i]*ad d(a[i,j]*c[j]*add(a[j,k]*c[k],k=2..j-1)^2,j=3..i-1),i=4..9)=1/168:\n\n ## b.c.a.(a.c)^2\neq89 := add(b[i]*c[i]*add(a[i,j]*add(a[j,k]*c[k],k=2 ..j-1)^2,j=3..i-1),i=4..9)=1/140:\n\n## b.a.a.(a.c)^2\neq90 := add(b[i ]*add(a[i,j]*add(a[j,k]*add(a[k,m]*c[m],m=2..k-1)^2,k=3..j-1),\n \+ j=4..i-1),i=5..9)=1/840:\n\n## b.(a.a.c)^2 \neq91 := add(b[i]*add(a[i,j]*add(a[j,k]*c[k],k=2..j-1),j=3..i-1)^2,i= 4..9)=1/252:\n\n## b.(a.c^2)^2\neq92 := add(b[i]*add(a[i,j]*c[j]^2,j=2 ..i-1)^2,i=3..9)=1/63:\n\n## b.(a.c)^3\neq93 := add(b[i]*add(a[i,j]*c[ j],j=2..i-1)^3,i=3..9)=1/56:\n\nRK7_9eqsB := [eq||(1..93)]:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "93 equations an d 53 unknowns " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 113 "We can check that this collection of equations is the sa me as that obtained using the Butcher package procedure. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "RK7_ 9eqsA := [op(RowSumConditions(9,'expanded')),op(OrderConditions(7,9,'e xpanded'))]:\nnops(%);\nevalb(\{op(RK7_9eqsA)\}=\{op(RK7_9eqsB)\});" } }{PARA 7 "" 1 "" {TEXT -1 79 "Warning, the variables \{i, q\} have bee n unassigned for possible use as indices\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#$*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 57 "#============================== ==========================" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }