{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 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 1 "#" }}{PARA 0 "" 0 "" {TEXT -1 85 "# Derivation of a parametric family of 4-stage order-4 explicit Runge-Kutta formulas." }}{PARA 0 "" 0 "" {TEXT -1 38 "# Here c3=1/2 b ut c2 is unconstrained." }}{PARA 0 "" 0 "" {TEXT -1 1 "#" }}{PARA 0 " " 0 "" {TEXT -1 42 "# Use LinearAlgebra and Groebner packages." }} {PARA 0 "" 0 "" {TEXT -1 1 "#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "with(Linea rAlgebra): with(Groebner):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "with( Ore_algebra):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 "#" }}{PARA 0 "" 0 "" {TEXT -1 56 "# Set up Runge-Kutta coefficient arrays A, b, c (and e)." }}{PARA 0 "" 0 "" {TEXT -1 1 "#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "R := poly_algebra(c2,c4,b1,b2,b3,b4,a32,a42,a43);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RG%,Ore_algebraG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "c := < 0, c2, 1/2, c4 >:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "C := DiagonalMatrix(c):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "A := << 0 | 0 | 0 | 0 >,< c2 | 0 | 0 | 0 >,<1/2-a32 \+ | a32 | 0 | 0 >,< c4-a42-a43 | a42 | a43 | 0 >>:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "b := < b1, b2, b3, b4 >: bT := Transpose(b): e := < 1 , 1, 1, 1 >:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "# " }}{PARA 0 "" 0 "" {TEXT -1 24 "# Conditions for order 4" }}{PARA 0 "" 0 "" {TEXT -1 1 "#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "rk1 := bT.e - 1: rk2 := bT.C.e - 1/2:" }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "rk31 := bT.C.C.e - 1/3: rk32 := bT.A.C.e - 1/6:" }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "rk41 := bT.map(x->x^3,c) - 1/4: \+ rk42 := bT.C.A.C.e - 1/8: rk43 := bT.A.C.C.e - 1/12: rk44 := bT.A.A.C. e - 1/24:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "# " }}{PARA 0 "" 0 " " {TEXT -1 52 "# F is the list of polynomials generating the ideal." } }{PARA 0 "" 0 "" {TEXT -1 83 "# T is the CAREFULLY chosen term order! \+ Unwise choice leads to lengthy computation." }}{PARA 0 "" 0 "" {TEXT -1 27 "# B2 is the Groebner basis." }}{PARA 0 "" 0 "" {TEXT -1 1 "#" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "F := [rk1,rk2,rk31,rk32,rk 41,rk42,rk43,rk44]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "T := termorder(R,plex(b1,b2,b3,b4,a32,a42,a43,c4,c2));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"TG%+term_orderG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "B2 := gbasis(F,T):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "for i from 1 to nops(B2) do factor(B2[i]) od;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&!\"\"\"\"\"%#c4GF%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(,&%#c2G\"\"#\"\"\"!\"\"F',&F%F'F'F(F',&%$a43GF'F&F (F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*&%#c2G\"\"\"%$a42GF&\"\"%*( \"\"#F&%$a43GF&F%F&!\"\"*&\"\"$F&F+F&F&*&F(F&F%F&F&F(F," }}{PARA 11 " " 1 "" {XPPMATH 20 "6#*&,&%$a43G\"\"\"\"\"#!\"\"F&,,*&F%F&%#c2GF&!\"'* &\"\"(F&F%F&F&*&\"\"%F&%$a42GF&F&*&\"\")F&F+F&F&F3F(F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,0*&%#c2G\"\"\"%$a32GF&\"#K*&\"#;F&F'F&!\"\"*&\"\" %F&%$a42GF&F+*(\"\"'F&%$a43GF&F%F&F&*&\"\"(F&F1F&F+*&\"#7F&F%F&F+\"#5F &" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*&%$a43G\"\"\"%$a32GF&\"\"#%$a4 2GF&*(F(F&F%F&%#c2GF&!\"\"*&F(F&F%F&F&*&\"\"%F&F+F&F&F/F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,6*&%$a32G\"\"\"%$a42GF&\"#k*&F(F&F%F&!\"\"*&\" #;F&)F'\"\"#F&F&*&F,F&F'F&F**(\"#IF&)%$a43GF.F&%#c2GF&F&*&\"#JF&F2F&F* *(\"#OF&F3F&F4F&F**&\"#MF&F3F&F&*&\"#[F&F4F&F*\"#cF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&%#c2G\"\"\"F&!\"\"F&,&%#b4G\"\"'F&F'F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,%#c2G!\"#*(\"\"$\"\"\"%#b4GF(%$a43GF(F(F(F (*&F*F(F$F(F(F*!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*&%#b4G\"\" \"%$a42GF&\"#C*&\"\"%F&F'F&!\"\"*(\"\"#F&%$a43GF&%#c2GF&F+F.F&*&F*F&F/ F&F&F-F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&%$a32G\"\")\"\"\"!\"\" F',&%#b4G\"\"'F'F(F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,%#b3G\"\"$*( \"\"#\"\"\"%$a43GF(%#c2GF(F(*&F'F(F)F(!\"\"*&\"\"%F(F*F(F,F'F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,.%#b2G\"\"'*&F%\"\"\"%#b4GF'F'*(\"\"% F'%$a43GF'%#c2GF'!\"\"*&F*F'F+F'F'*&\"\")F'F,F'F'\"\"*F-" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,&%#b1G\"\"'\"\"\"!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "for i from 1 to nops(B2) do subs(\{b1=1/6, b3=2/ 3, c2=1, c4=1, a32=1/8\}, factor(B2[i])) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&%$a42G\"\"%%$a43G\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&%$a43G\"\"\"\"\"#!\"\"F&,&%$a42G\"\"%F%F&F& " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%$a42G!\"%%$a43G!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%$a43G#\"\"\"\"\"%%$a42GF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*%$a42G!\")*&\"#;\"\"\")F$\"\"#F(F(*$)%$a43GF*F(! \"\"*&F*F(F-F(F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&!\"\"\"\"\"*(\"\"$F%%#b4GF%%$a43GF%F%" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,**&%#b4G\"\"\"%$a42GF&\"#C*&\"\"%F&F' F&!\"\"%$a43GF+\"\"#F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(%#b2G\"\"'*&F%\"\"\"%#b4GF'F'F'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "fo r i from 1 to nops(B2) do is(subs(\{b1=1/6,b3=2/3,c2=1,c4=1,a32=1/8,a4 2=-a43/4,b2=(a43-2)/(6*a43),b4=1/(3*a43)\}, B2[i]=0)) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%tr ueG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%%trueG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%tru eG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 "#" }}{PARA 0 "" 0 "" {TEXT -1 103 "# We find that c2=1/2, \+ c2=1, or a43=2. The former gives the confluent c2=c3=1/2 solution alre ady known." }}{PARA 0 "" 0 "" {TEXT -1 135 "# Choosing c2=1 leads to a parametrization of the solution variety by a43 below. The situation \+ where a43=2 will be investigated later." }}{PARA 0 "" 0 "" {TEXT -1 1 "#" }}{PARA 0 "" 0 "" {TEXT -1 18 "# c2 = 1, c4 = 1" }}{PARA 0 "" 0 "" {TEXT -1 25 "# a32=1/8, a42 = -a43/4" }}{PARA 0 "" 0 "" {TEXT -1 59 "# b1 = 1/6, b2 = (a43-2)/(6*a43), b3=2/3, b4 = 1/(3*a43)" }} {PARA 0 "" 0 "" {TEXT -1 1 "#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 "# " }}{PARA 0 "" 0 "" {TEXT -1 48 "# Now get the 5th-order truncation er ror terms, " }}{PARA 0 "" 0 "" {TEXT -1 64 "# Reduce the equations mod ulo the ideal of 5th-order conditions." }}{PARA 0 "" 0 "" {TEXT -1 101 "# Observe that on the solution variety all terms except the \"for ked tree\" rk5[5] reduce to constants." }}{PARA 0 "" 0 "" {TEXT -1 97 "# The coefficient rk5[5] can be made 0 by choosing a43 = 1.9. This mi nimizes the sum of squares " }}{PARA 0 "" 0 "" {TEXT -1 100 "# of tru ncation error coefficients on the solution variety. The minimum, ~.00 05, is not that small." }}{PARA 0 "" 0 "" {TEXT -1 1 "#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "rk5 := array(1..9);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$rk5G-%&arrayG6$;\"\"\"\"\"*7\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "rk5[1] := bT.C.C.C.C.e - 1/5:" } {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "rk5[2] := bT.C.C.A. C.e - 1/10:" }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "rk5[3 ] := bT.C.A.C.C.e - 1/15:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "rk5[4] := bT.A.C.C.C.e - 1/20:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "rk5[5] \+ := bT.map(x->x^2,A.C.e) - 1/20:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 " rk5[6] := bT.C.A.A.C.e - 1/30:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "r k5[7] := bT.A.C.A.C.e - 1/40:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "rk 5[8] := bT.A.A.C.C.e - 1/60:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "rk5 [9] := bT.A.A.A.C.e - 1/120:" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "redrk5 := array(1..9);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'redrk5G-%&arrayG6$;\"\"\"\"\"*7\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "alpha_tau:= [1/24,1/2,1/2,1/6,1/2,1,1,1/2 ,1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*alpha_tauG7+#\"\"\"\"#C#F' \"\"#F)#F'\"\"'F)F'F'F)F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "for i from 1 to 9 do\n redrk5[i] := normalf(rk5[i],B2,T) od;" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'re drk5G6#\"\"\"#F'\"$?\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'redrk5G6 #\"\"##\"\"\"\"$S#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'redrk5G6#\" \"$,&%#c2G#!\"\"\"#C#\"\"\"\"#gF." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> &%'redrk5G6#\"\"%#!\"\"\"$?\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'r edrk5G6#\"\"&,.%$a32G#\"\"\"\"#C*&#F+\"#'*F+%$a42GF+F+*(#F'\"$#>F+%$a4 3GF+%#c2GF+F+*&#F+\"$%QF+F4F+!\"\"*&#F'F/F+F5F+F9#\"\"(\"$g*F+" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'redrk5G6#\"\"'#\"\"\"\"$?\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'redrk5G6#\"\"(#!\"\"\"$S#" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'redrk5G6#\"\"),&#!\"\"\"#g\"\"\"*& #F,\"#CF,%#c2GF,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'redrk5G6#\" \"*#!\"\"\"$?\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "t5sq := evalf(simplify(subs(\{c2=1,a32=1/8,a42=-19/40,a43=19/10\},sum('(alpha _tau[i]*redrk5[i])^2','i'=1..9))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%%t5sqG$\"+M&)R^Z!#8" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "# Now f or the case where a43=2." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "for i from 1 to nops(B2) do subs(\{b1=1/6, b2=0, b 3=2/3, b4=1/6, c4=1, a43=2\}, factor(B2[i])) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%#c2G\"\"\"%$a42GF&\"\"%\"\"#F&" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&%#c2G\"\"\"%$a32GF&\"#K*&\"#;F&F'F&!\"\"*&\"\"%F&%$a42GF&F+F -F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%$a32G\"\"%%$a42G\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,**&%$a32G\"\"\"%$a42GF&\"#k*&F(F&F%F& !\"\"*&\"#;F&)F'\"\"#F&F&*&F,F&F'F&F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{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 "" {MPLTEXT 1 0 108 "for i from 1 to nops(B2) do subs(\{b1=1/6,b 2=0,b3=2/3,b4=1/6,c2=-1/(2*a42),c4=1,a32=-a42/4,a43=2\}, B2[i]) od;" } }{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 1 "#" }} {PARA 0 "" 0 "" {TEXT -1 36 "# This is the situation where a43=2." }} {PARA 0 "" 0 "" {TEXT -1 1 "#" }}{PARA 0 "" 0 "" {TEXT -1 27 "# c2 = -1/(2*a42), c4 = 1" }}{PARA 0 "" 0 "" {TEXT -1 21 "# a32=-a42/4, a4 3=2" }}{PARA 0 "" 0 "" {TEXT -1 39 "# b1 = 1/6, b2 = 0, b3=2/3, b4 \+ = 1/6" }}{PARA 0 "" 0 "" {TEXT -1 1 "#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 "#" }}{PARA 0 "" 0 "" {TEXT -1 109 "# The coefficients rk5[ 3] and rk5[8] can be made 0 by choosing a42 = 0.8. This minimizes the sum of squares " }}{PARA 0 "" 0 "" {TEXT -1 100 "# of truncation erro r coefficients on the solution variety. The minimum, ~.0011, is not t hat small." }}{PARA 0 "" 0 "" {TEXT -1 1 "#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "for i from 1 to 9 do\n coef[i] = subs(\{a32=-a42/ 4,a43=2,c2=-1/(2*a42)\}, redrk5[i]) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%%coefG6#\"\"\"#F'\"$?\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/& %%coefG6#\"\"##\"\"\"\"$S#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%%coef G6#\"\"$,&*&\"\"\"F*%$a42G!\"\"#F*\"#[#F*\"#gF*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%%coefG6#\"\"%#!\"\"\"$?\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%%coefG6#\"\"&#\"\"\"\"$![" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%%coefG6#\"\"'#\"\"\"\"$?\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%%coefG6#\"\"(#!\"\"\"$S#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%%coefG6#\"\"),&#!\"\"\"#g\"\"\"*&#F,\"#[F,*&F,F,%$a4 2GF*F,F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%%coefG6#\"\"*#!\"\"\"$? \"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "solve(subs(c2=-1/(2*a 42)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "t5sq_2 := evalf(sim plify(subs(\{a32=-1/5,a42=4/5,a43=2,c2=-5/8\},sum('(alpha_tau[i]*redrk 5[i])^2','i'=1..9))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'t5sq_2G$ \"+\"zDd2\"!#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {MARK "0 1 0" 12 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }