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{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 c2=1/2 b
ut c3 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;" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name change
coords has been redefined\n" }}{PARA 7 "" 1 "" {TEXT -1 69 "Warning, `
ODE` is implicitly declared local to procedure `readshare`\n" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "with(LinearAlgebra): with(Gr
oebner):" }}{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(c3,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, 1/2, c3, c4 >:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "C := DiagonalMatrix(c):" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 101 "A := << 0 | 0 | 0 | 0 >,< 1/2 | 0 | 0 | 0 >,< c3-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,c3));" }}{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 49 "for i from 1 to nops(B2) do [i,indets(B2[i])] od;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#7$\"\"\"<#%#c3G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7$\"\"#<#%#c4G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$\"
\"$<%%#c3G%$a42G%$a43G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$\"\"%<$%$a
32G%$a43G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$\"\"&<#%#b4G" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#7$\"\"'<%%#c3G%#b3G%$a43G" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#7$\"\"(<%%#c3G%#b2G%$a43G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7$\"\")<%%#b1G%#c3G%$a43G" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 1 "#" }}{PARA 0 "" 0 "" {TEXT -1 95 "# We find that c3=0 or c
3=1/2. The latter gives the confluent c2=c3=1/2 solution already known
." }}{PARA 0 "" 0 "" {TEXT -1 73 "# Choosing c3=0 leads to a parametri
zation of the solution variety by b3:" }}{PARA 0 "" 0 "" {TEXT -1 1 "#
" }}{PARA 0 "" 0 "" {TEXT -1 19 "#    c3 = 0, c4 = 1" }}{PARA 0 "" 0 "
" {TEXT -1 19 "#   a32 = 1/(12*b3)" }}{PARA 0 "" 0 "" {TEXT -1 26 "#  \+
 a42 = 3/2,  a43 = 6*b3" }}{PARA 0 "" 0 "" {TEXT -1 39 "#   b1 = 1/6 -
 b3,  b2 = 2/3,  b4 = 1/6" }}{PARA 0 "" 0 "" {TEXT -1 1 "#" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "for i from 1 to nops(B2) do factor(
B2[i]) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%#c3G\"\"\",&!\"\"F%*&
\"\"#F%F$F%F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&!\"\"\"\"\"%#c4GF
%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*%$a42G\"\"#*(\"\"%\"\"\"%$a43GF
(%#c3GF(F(*&F%F(F*F(F(\"\"$!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&
!\"\"\"\"\"*(\"\"#F%%$a43GF%%$a32GF%F%" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,&!\"\"\"\"\"*&\"\"'F%%#b4GF%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#,(%#b3G\"\"'*(\"\"#\"\"\"%$a43GF(%#c3GF(!\"\"F)F+" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#,(%#b2G\"\"$*(\"\"#\"\"\"%$a43GF(%#c3GF(F(F'!\"\"" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*%#b1G\"\"'*(\"\"#\"\"\"%$a43GF(%#c3
GF(!\"\"F)F(F(F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "for i f
rom 1 to nops(B2) do " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "   simplif
y(subs(c3=0,c4=1,a42=3/2,a43=6*b3,a32=1/12/b3,b1=1/6-b3,b2=2/3,b4=1/6,
B2[i]))" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 " od;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 1 "#" }}{PARA 0 "" 0 "" {TEXT -1 44 "# Get the 5th-order trun
cation error terms, " }}{PARA 0 "" 0 "" {TEXT -1 64 "# Reduce the equa
tions modulo the ideal of 5th-order conditions." }}{PARA 0 "" 0 "" 
{TEXT -1 101 "# Observe that on the solution variety all terms except \+
the \"forked tree\" rk5[5] reduce to constants." }}{PARA 0 "" 0 "" 
{TEXT -1 99 "# The coefficient rk5[5] can be made 0 by choosing b3 = -
5/126. This minimizes  the sum of squares " }}{PARA 0 "" 0 "" {TEXT 
-1 100 "# of truncation error coefficients on the solution variety.  T
he minimum, ~.0009, is not that small." }}{PARA 0 "" 0 "" {TEXT -1 1 "
#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "rk5 := array(1..9);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$rk5G-%&arrayG6$;\"\"\"\"\"*7\"" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "rk5[1] := bT.C.C.C.C.e - 1/
5:" }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "rk5[2] := bT.C
.C.A.C.e - 1/10:" }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "
rk5[3] := bT.C.A.C.C.e - 1/15:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "r
k5[4] := bT.A.C.C.C.e - 1/20:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "rk
5[5] := bT.map(x->x^2,A.C.e) - 1/20:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 30 "rk5[6] := bT.C.A.A.C.e - 1/30:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 30 "rk5[7] := bT.A.C.A.C.e - 1/40:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 30 "rk5[8] := bT.A.A.C.C.e - 1/60:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 31 "rk5[9] := bT.A.A.A.C.e - 1/120:" }{TEXT -1 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 22 "redrk5 := array(1..9);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%'redrk5G-%&arrayG6$;\"\"\"\"\"*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#>&
%'redrk5G6#\"\"\"#F'\"$?\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'redr
k5G6#\"\"#,&#\"\"\"\"#SF**&#F*\"#CF*%#c3GF*!\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%'redrk5G6#\"\"$#!\"\"\"$S#" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%'redrk5G6#\"\"%#!\"\"\"$?\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%'redrk5G6#\"\"&,*#\"\"(\"$g\"\"\"\"*&#F'\"#[F,%#c3GF
,!\"\"*(#F,\"#CF,F0F,%$a32GF,F,*&#F,F/F,F5F,F," }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%'redrk5G6#\"\"'#\"\"\"\"$?\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%'redrk5G6#\"\"(,&#!\"\"\"#S\"\"\"*&#F,\"#CF,%#c3GF,F
," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'redrk5G6#\"\")#\"\"\"\"$S#" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'redrk5G6#\"\"*#!\"\"\"$?\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "t5sq := simplify(subs(c3=0,a
32=1/12/b3,sum('(alpha_tau[i]*redrk5[i])^2','i'=1..9)));" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "evalf(subs(b3=-5/126,t5sq));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+,-p3$*!#8" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "24 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 
1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }