{VERSION 4 0 "DEC ALPHA UNIX" "4.0" } {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 Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 1 "#" }}{PARA 0 "" 0 "" {TEXT -1 97 "# 4-stage 4th-order explicit Runge-Kutta formula with step-doub ling -> 11-stage 5th order formula" }}{PARA 0 "" 0 "" {TEXT -1 1 "#" } }{PARA 0 "" 0 "" {TEXT -1 42 "# Use LinearAlgebra and Groebner package s." }}{PARA 0 "" 0 "" {TEXT -1 1 "#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "with(LinearAlgebra): with(Groebner):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 "#" }}{PARA 0 "" 0 "" {TEXT -1 41 "# Set up Groebner basis \+ for (4,4) formula" }}{PARA 0 "" 0 "" {TEXT -1 1 "#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "c0 := < 0, c2, c3, c4 >;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "C0 := DiagonalMatrix(c0);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "A0 := << 0 | 0 | 0 | 0 >,< c2 | 0 | 0 | 0 >,< c3-a32 | a32 | 0 | 0 >,< c4-a42-a43 | a42 | a43 | 0 >>;" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 69 "b0 := < b1, b2, b3, b4 >; b0T := Transpose(b0); e0 \+ := < 1, 1, 1, 1 >;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c0G-%'RTABLEG 6$\"+?Xet`-%'MATRIXG6#7&7#\"\"!7#%#c2G7#%#c3G7#%#c4G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#C0G-%'RTABLEG6$\"+3Wwt`-%'MATRIXG6#7&7&\"\"!F.F.F .7&F.%#c2GF.F.7&F.F.%#c3GF.7&F.F.F.%#c4G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A0G-%'RTABLEG6$\"+#RCRP&-%'MATRIXG6#7&7&\"\"!F.F.F.7&%#c2GF.F .F.7&,&%#c3G\"\"\"%$a32G!\"\"F5F.F.7&,(%#c4GF4%$a42GF6%$a43GF6F:F;F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#b0G-%'RTABLEG6$\"+K9ft`-%'MATRIXG 6#7&7#%#b1G7#%#b2G7#%#b3G7#%#b4G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% $b0TG-%'RTABLEG6$\"+!3#ft`-%'VECTORG6#7&%#b1G%#b2G%#b3G%#b4G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e0G-%'RTABLEG6$\"+3?ft`-%'MATRIXG6#7&7#\" \"\"F-F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "rk1 := b0T.e0 - 1; rk2 := b0T.C0.e0 - 1/2;" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$rk1G,,%#b1G\"\"\"%#b2GF'%#b3GF'%#b4GF'F'!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$rk2G,**&%#b2G\"\"\"%#c2GF(F(*&%#b3G F(%#c3GF(F(*&%#b4GF(%#c4GF(F(#F(\"\"#!\"\"" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 55 "rk31 := b0T.C0.C0.e0 - 1/3; rk32 := b0T.A0.C0.e0 - \+ 1/6;" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%rk31G,**&%#b 2G\"\"\")%#c2G\"\"#F(F(*&%#b3GF()%#c3GF+F(F(*&%#b4GF()%#c4GF+F(F(#F(\" \"$!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%rk32G,(*&,&*&%#b3G\"\" \"%$a32GF*F**&%#b4GF*%$a42GF*F*F*%#c2GF*F**(F-F*%$a43GF*%#c3GF*F*#F*\" \"'!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "rk41 := b0T.ma p(x->x^3,c0) - 1/4; rk42 := b0T.C0.A0.C0.e0 - 1/8; rk43 := b0T.A0.C0.C 0.e0 - 1/12; rk44 := b0T.A0.A0.C0.e0 - 1/24;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%rk41G,**&)%#c2G\"\"$\"\"\"%#b2GF*F**&)%#c3GF)F*%#b3G F*F**&)%#c4GF)F*%#b4GF*F*#F*\"\"%!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%rk42G,(*&,&*(%#b3G\"\"\"%#c3GF*%$a32GF*F**(%#b4GF*%#c4GF*%$a4 2GF*F*F*%#c2GF*F***F.F*F/F*%$a43GF*F+F*F*#F*\"\")!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%rk43G,(*&,&*&%#b3G\"\"\"%$a32GF*F**&%#b4GF*%$a4 2GF*F*F*)%#c2G\"\"#F*F**(F-F*%$a43GF*)%#c3GF1F*F*#F*\"#7!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%rk44G,&**%#b4G\"\"\"%$a43GF(%$a32GF (%#c2GF(F(#F(\"#C!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "F := [rk1,rk2,rk31,rk32,rk41,rk42,rk43,rk44]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "B := gbasis(F,plex(a32,a42,a43,b1,b2,b3,b4,c4)); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"BG7*,&!\"\"\"\"\"%#c4GF(,2%#c2 G\"\"%*(\"\"'F(F+F(%#c3GF(F'**\"#7F(F+F(%#b4GF(F/F(F(*(F1F(F+F(F2F(F' \"\"$F'*&F1F(F2F(F(*&F,F(F/F(F(*(F1F(F/F(F2F(F',.*(F+F(%#b3GF(F/F(!#7* (F1F(F:F()F/\"\"#F(F(**F1F(F+F(F:F(F=F(F(*(F1F()F/F4F(F:F(F'*&F>F(F+F( F(F(F',.*&)F+F4F(%#b2GF(F1**F1F()F+F>F(FFF(F/F(F'*(F1F(FFF(FHF(F'**F1F (F+F(FFF(F/F(F(F(F(*&F>F(F/F(F',,*(F/F(F+F(%#b1GF(F1*(F.F(F+F(F/F(F'F( F'*&F>F(F/F(F(*&F>F(F+F(F(,:*(F/F(%$a43GF(FHF(!\"%**F.F(F=F(FTF(FHF(F( **F4F(F/F(FTF(F+F(F(**F.F(FAF(FTF(F+F(F'*(F4F(F=F(FTF(F'*(F,F(FAF(FTF( F(*&F>F(FHF(F(*&F4F(F+F(F'*(F>F(F/F(FHF(F'*(F4F(F+F(F/F(F(F(F(F/F',<*( F/F(FEF(%$a42GF(F1*(\"\")F(FEF(F[oF(F'**F1F(FHF(F=F(F[oF(F'*$FHF(F'*(F .F(FHF(F[oF(F(*(F,F(F+F(F=F(F(**F]oF(F=F(F+F(F[oF(F(**F.F(F/F(F+F(F[oF (F'*(\"\"&F(F+F(F/F(F'*&F4F(F+F(F(*&F,F(F=F(F'*&FeoF(F/F(F(F>F',**&FHF (%$a32GF(F,*(F>F(F+F(F[pF(F'*&F+F(F/F(F'*$F=F(F(" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 1 "#" }}{PARA 0 "" 0 "" {TEXT -1 77 "# Set up Runge-Ku tta coefficient arrays A, b, c (and e) for 5th order formula" }}{PARA 0 "" 0 "" {TEXT -1 1 "#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "p := 4: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "r := 1/(2^p-1):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "c := < 0, c2/2, c3/2, c4/2, \+ 1/2, 1/2+c2/2, 1/2+c3/2, 1/2+c4/2, c2, c3, c4 >:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "C := DiagonalMatrix(c):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1133 "A := << 0 | 0 | 0 | 0 | 0 \+ | 0 | 0 | 0 | 0 | 0 | 0 >,\n < c2/2 \+ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 >,\n < (c3-a32) /2 | a32/2 | 0 | 0 | 0 \+ | 0 | 0 | 0 | 0 | 0 | 0 >,\n < (c4-a42-a43 ) /2 | a42/2 | a43/2 | 0 | 0 | 0 | 0 | 0 | 0 \+ | 0 | 0 >,\n < b1/2 | b2/2 | b3/2 | b4/2 | 0 \+ | 0 | 0 | 0 | 0 | 0 | 0 >,\n < b1/2 \+ | b2/2 | b3/2 | b4/2 | c2/2 | 0 | 0 | 0 | 0 | 0 | 0 >,\n < b1/2 | b2/2 | b3/2 | b4/2 | (c3-a32) /2 | a32/2 | 0 | 0 | 0 | 0 | 0 >,\n < b1/2 \+ | b2/2 | b3/2 | b4/2 | (c4-a42-a43)/2 | a42/2 | a43/2 | 0 | 0 | 0 | 0 >,\n < c2 | 0 | 0 | 0 | 0 \+ | 0 | 0 | 0 | 0 | 0 | 0 >,\n < c3-a32 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | a32 | 0 \+ | 0 >,\n < c4-a42-a43 | 0 | 0 | 0 | 0 \+ | 0 | 0 | 0 | a42 | a43 | 0 >>:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 188 "b := < (1-r)/2*b1, (1+r)/2*b2, (1+r)/2*b3, (1+r)/2*b 4, (1+r)/2*b1, (1+r)/2*b2, (1+r)/2*b3, (1+r)/2*b4, -r*b2, -r*b3, -r*b4 >: \nbT := Transpose(b): e := < 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 >:" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "# " }}{PARA 0 "" 0 "" {TEXT -1 36 "# Verify the conditions for order 5." }}{PARA 0 "" 0 "" {TEXT -1 1 "#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "rk5 := array(1...17 ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "rk5[1] := bT.e - 1:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "rk5[2] := bT.C.e - 1/2:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "rk5[3] := bT.C.C.e - 1/3:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "rk5[4] := bT.A.C.e - 1/6:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "rk5[5] := bT.map(x->x^3,c) - 1/4:" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 27 "rk5[6] := bT.C.A.C.e - 1/8:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "rk5[7] := bT.A.C.C.e - 1/12:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "rk5[8] := bT.A.A.C.e - 1/24:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "rk5[9] := bT.map(x->x^4,c) - 1/5:" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 31 "rk5[10] := bT.C.C.A.C.e - 1/10:" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 31 "rk5[11] := bT.C.A.C.C.e - 1/15:" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 31 "rk5[12] := bT.A.C.C.C.e - 1/20:" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 39 "rk5[13] := bT.map(x->x^2,A.C.e) - 1/20:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "rk5[14] := bT.C.A.A.C.e - 1/30:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "rk5[15] := bT.A.C.A.C.e - 1/40:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "rk5[16] := bT.A.A.C.C.e - 1/60:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "rk5[17] := bT.A.A.A.C.e - 1/120:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "for i from 1 to 17 do\n nor malf(rk5[i], B, plex(a32,a42,a43,b1,b2,b3,b4,c4));\nod;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "# " }}{PARA 0 "" 0 "" {TEXT -1 40 " # Order-6 truncation error coefficients " }}{PARA 0 "" 0 "" {TEXT -1 1 "#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "rk6 := array(1..20) :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "rk6[1] := bT.map(x->x^5,c) - 1 /6:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "rk6[2] := bT.C.C.C.A.C.e - 1 /12:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "rk6[3] := bT.C.C.A.C.C.e - \+ 1/18:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "rk6[4] := bT.C.map(x->x^2, A.C.e) - 1/24:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "rk6[5] := bT.C.A. C.C.C.e - 1/24:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "rk6[6] := bT.A.C .C.C.C.e - 1/30:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "rk6[7] := bT.C. C.A.A.C.e - 1/36:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "rk6[8] := bT.D iagonalMatrix(A.C.e).(A.C.C.e) - 1/36:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "rk6[9] := bT.C.A.C.A.C.e - 1/48:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "rk6[10] := bT.A.C.C.A.C.e - 1/60:" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 53 "rk6[11] := bT.DiagonalMatrix(A.C.e).(A.A.C.e) - 1/7 2:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "rk6[12] := bT.C.A.A.C.C.e - 1 /72:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "rk6[13] := bT.A.C.A.C.C.e - 1/90:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "rk6[14] := bT.A.map(x->x^ 2,A.C.e) - 1/120:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "rk6[15] := bT. A.A.C.C.C.e - 1/120:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "rk6[16] := \+ bT.C.A.A.A.C.e - 1/144:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "rk6[17] \+ := bT.A.C.A.A.C.e - 1/180:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "rk6[1 8] := bT.A.A.C.A.C.e - 1/240:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "rk 6[19] := bT.A.A.A.C.C.e - 1/360:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "rk6[20] := bT.A.A.A.A.C.e - 1/720:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "redrk6 := array(1..20):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "for j from 1 to 20 do\n redrk6[j] := normalf(rk6[j], B, plex(a32,a42,a43,b1,b2,b3,b4,c4));\nod;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'redrk6G6#\"\"\",2*&%#c3GF')%#c2G\"\"#F'#!\"\"\"$!=* &#F'\"$g$F'F+F'F'*&#F'F0F'*&F,F')F*F-F'F'F/*(#F'\"#!*F'F,F'F*F'F'*&#F' \"$S#F'F,F'F/*&F2F'F7F'F'*&#F'F=F'F*F'F/#F'\"$?(F'" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>&%'redrk6G6#\"\"#,,*&%#c2G\"\"\"%#c3GF+#F+\"$g$*&#F+ \"$?(F+F*F+!\"\"*&#F+F1F+)F,F'F+F+*&#F+\"$![F+F,F+F2#F+\"%S9F+" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'redrk6G6#\"\"$,(*&%#c2G\"\"\"%#c3G F+#F+\"$?(*&#F+F.F+F,F+!\"\"#F+\"%g@F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'redrk6G6#\"\"%,$*&,8*&)%#c2G\"\"#\"\"\")%#c3GF.F/\"#O*(\"#YF /F1F/F,F/!\"\"*&\"#9F/F,F/F/*(\"#7F/)F1\"\"$F/F-F/F/*(\"#]F/F-F/F0F/F5 *(\"#[F/F-F/F1F/F/*&\"#8F/F-F/F5*&\"\")F/F:F/F5*&\"#=F/F0F/F/*&FAF/F1F /F5F;F/F/,.*&F1F/F,F/F9*&FCF/F,F/F5*(F7F/F-F/F1F/F5*&\"#5F/F-F/F/*&F'F /F1F/F/F;F5F5#F5\"%!)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'redrk6G6 #\"\"&,,*$)%#c2G\"\"#\"\"\"#F-\"$?(*(#F-\"$g$F-F+F-%#c3GF-F-*&#F-\"$)G F-F+F-!\"\"*&#F-F/F-F3F-F7F.F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%' redrk6G6#\"\"',2*&%#c3G\"\"\")%#c2G\"\"#F+#F+\"$!=*&#F+\"$g$F+*$F,F+F+ !\"\"*(F/F+F-F+)F*F.F+F+*&#F+\"#!*F+*&F-F+F*F+F+F5*&#F+\"$S#F+F-F+F+*& #F+F3F+*$F7F+F+F5*&F=F+F*F+F+#F+\"$?(F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'redrk6G6#\"\"(,&%#c3G#!\"\"\"%S9#\"\"\"\"%?VF." }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>&%'redrk6G6#\"\"),$*&,8*&%#c3G\"\"\")%#c2G\"\"$ F-\"#=*&\"\"'F-F.F-!\"\"*(F1F-)F/\"\"#F-)F,F7F-F-*(\"#IF-F,F-F6F-F4*& \"\"&F-F6F-F-*(F:F-F/F-F8F-F4*(\"#NF-F/F-F,F-F-*&\"\"(F-F/F-F4*&\"#7F- F8F-F-*&\"#8F-F,F-F4F0F-F-,.*&F,F-F6F-FC*&F'F-F6F-F4*(\"#9F-F/F-F,F-F4 *&\"#5F-F/F-F-*&\"\"%F-F,F-F-F0F4F4#F4\"%?V" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'redrk6G6#\"\"*,(%#c2G#!\"\"\"%S9#\"\"\"F,F.*&#F.F,F .%#c3GF.F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'redrk6G6#\"#5,,*&%#c 2G\"\"\"%#c3GF+#!\"\"\"$g$*&#F+\"$?(F+F*F+F+*&#F+F2F+*$)F,\"\"#F+F+F.* &#F+\"$![F+F,F+F+#F+\"%S9F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'red rk6G6#\"#6,$*&,2*&%#c3G\"\"\")%#c2G\"\"#F-\"\"'*&F0F-F.F-!\"\"*(\"#=F- F/F-)F,F0F-F-*(\"#EF-F/F-F,F-F3*&\"\"(F-F/F-F-*&\"#7F-F6F-F3*&\"#8F-F, F-F-\"\"$F3F-,.F+F<*&\"\")F-F.F-F3*(\"#9F-F/F-F,F-F3*&\"#5F-F/F-F-*&\" \"%F-F,F-F-F?F3F3#F-\"%S')" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'redr k6G6#\"#7,&%#c2G#!\"\"\"%S9#\"\"\"\"%g@F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'redrk6G6#\"#8,(*&%#c2G\"\"\"%#c3GF+#!\"\"\"$?(*&#F+ F/F+F,F+F+#F+\"%g@F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'redrk6G6# \"#9,$*&,8*&)%#c2G\"\"#\"\"\")%#c3GF.F/\"#O*(\"#YF/F1F/F,F/!\"\"*&F'F/ F,F/F/*(\"#7F/)F1\"\"$F/F-F/F/*(\"#]F/F-F/F0F/F5*(\"#[F/F-F/F1F/F/*&\" #8F/F-F/F5*&\"\")F/F9F/F5*&\"#=F/F0F/F/*&F@F/F1F/F5F:F/F/,.*&F1F/F,F/F 8*&FBF/F,F/F5*(F'F/F-F/F1F/F5*&\"#5F/F-F/F/*&\"\"%F/F1F/F/F:F5F5#F/\"% !)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'redrk6G6#\"#:,,*$)%#c2G\"\" #\"\"\"#!\"\"\"$?(*&#F-\"$g$F-*&F+F-%#c3GF-F-F/*&#F-\"$)GF-F+F-F-*&#F- F0F-F5F-F-#F-F0F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'redrk6G6#\"#; #\"\"\"\"%?V" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'redrk6G6#\"#<,&%#c 3G#\"\"\"\"%S9#F+\"%?V!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'red rk6G6#\"#=,(%#c2G#\"\"\"\"%S9#F+F,!\"\"*&F*F+%#c3GF+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'redrk6G6#\"#>,&%#c2G#\"\"\"\"%S9#F+\"%g@!\"\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'redrk6G6#\"#?#!\"\"\"%?V" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 1 "#" }}{PARA 0 "" 0 "" {TEXT -1 38 "# Factor numerators of 6th order terms" }}{PARA 0 "" 0 "" {TEXT -1 1 "# " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "for \+ k from 1 to 20 do\n factor(numer(redrk6[k]));\nod;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,$*(,&!\"\"\"\"\"*&\"\"#F'%#c3GF'F'F',&%#c2GF)F'F&F', (F,F'F'F&F*F'F'F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&!\"\"\"\"\"*& \"\"#F&%#c3GF&F&F&,(F)F&*&F(F&%#c2GF&F&F&F%F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&%#c2G\"\"\"%#c3GF&\"\"$*&F(F&F'F&!\"\"F&F&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&!\"\"\"\"\"*&\"\"#F'%#c3GF'F'F', 2*&%#c2GF')F*F)F'\"\"'*&\"\"%F'F.F'F&*(\"#=F'F*F')F-F)F'F'*(\"#AF'F-F' F*F'F&*&\"\"(F'F*F'F'*&\"#9F'F4F'F&*&\"#8F'F-F'F'\"\"$F&F'F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&%#c2G\"\"#\"\"\"!\"\"F',(F%F'*&F&F'%#c3G F'F'F&F(F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(,&!\"\"\"\"\"*&\"\"#F& %#c3GF&F&F&,&%#c2GF(F&F%F&,(F+F&F&F%F)F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%#c3G!\"$\"\"\"F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# ,8*&%#c3G\"\"\")%#c2G\"\"$F&!#=*&\"\"'F&F'F&F&*(\"#=F&)F(\"\"#F&)F%F0F &!\"\"*(\"#IF&F%F&F/F&F&*&\"\"&F&F/F&F2*(F4F&F(F&F1F&F&*(\"#NF&F(F&F%F &F2*&\"\"(F&F(F&F&*&\"#7F&F1F&F2*&\"#8F&F%F&F&F)F2" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,(%#c2G!\"\"%#c3GF%\"\"\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&!\"\"\"\"\"*&\"\"#F'%#c3GF'F'F',(F*F'*&F)F'%#c2GF 'F'F'F&F'F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,2*&%#c3G\"\"\")%#c2G\" \"#F&\"\"'*&F)F&F'F&!\"\"*(\"#=F&F(F&)F%F)F&F&*(\"#EF&F(F&F%F&F,*&\"\" (F&F(F&F&*&\"#7F&F/F&F,*&\"#8F&F%F&F&\"\"$F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%#c2G!\"$\"\"#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&%#c2G\"\"\"%#c3GF&!\"$*&\"\"$F&F'F&F&F&!\"\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#*&,&!\"\"\"\"\"*&\"\"#F&%#c3GF&F&F&,2*&%#c2GF&)F)F(F& \"\"'*&\"\"%F&F-F&F%*(\"#=F&F)F&)F,F(F&F&*(\"#AF&F,F&F)F&F%*&\"\"(F&F) F&F&*&\"#9F&F3F&F%*&\"#8F&F,F&F&\"\"$F%F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&%#c2G\"\"#\"\"\"!\"\"F(,(F&F(*&F'F(%#c3GF(F(F'F)F (F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%#c3G\"\"$\"\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(%#c2G\"\"\"F%!\"\"%#c3GF%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#, &%#c2G\"\"$\"\"#!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"\"" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 1 "#" }}{PARA 0 "" 0 "" {TEXT -1 40 "# Factor denominators of 6th order terms" }}{PARA 0 "" 0 "" {TEXT -1 1 "#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "for m from 1 to 20 do \n factor(denom(redrk6[m]));\nod;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# \"$?(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"%S9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"%g@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&%#c2G\" \"#\"\"\"!\"\"F(,**&F&F(%#c3GF(\"\"'*&\"\"%F(F&F(F)*&F/F(F,F(F)\"\"$F( F(\"%!)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"%S9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$?(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"%?V" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&%#c2G\"\"#\"\"\"!\"\"F(,**&F&F(% #c3GF(\"\"'*&\"\"%F(F&F(F)*&F/F(F,F(F)\"\"$F(F(\"%?V" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"%S9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"%S9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&%#c2G\"\"#\"\"\"!\"\"F(,**&F&F(% #c3GF(\"\"'*&\"\"%F(F&F(F)*&F/F(F,F(F)\"\"$F(F(\"%S')" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"%?V" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"%g@" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&%#c2G\"\"#\"\"\"!\"\"F(,**&F&F(% #c3GF(\"\"'*&\"\"%F(F&F(F)*&F/F(F,F(F)\"\"$F(F(\"%!)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"%S9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"%?V" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"%?V" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"%S9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"%?V" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"%?V" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 "#" }} {PARA 0 "" 0 "" {TEXT -1 18 "# c2 cannot be 1/2" }}{PARA 0 "" 0 "" {TEXT -1 1 "#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "25 0 0" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 } {RTABLE_HANDLES 5373584520 5373764408 5373924392 5373591432 5373592080 5373592008 }{RTABLE M6R0 I7RTABLE_SAVE/5373584520X*%)anythingG6"6"\[[[[[t%"%""!%#c2G%#c3G%#c4GF& } {RTABLE M6R0 I7RTABLE_SAVE/5373764408X,%)anythingG6#%)diagonalG6"][[[[co%"%"%""!%#c2G%#c3G%# c4GF' } {RTABLE M6R0 I7RTABLE_SAVE/5373924392X,%)anythingG6"6"][[[[[p1"%"%""!%#c2G,&%#c3G"""%$a32G!" 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