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{SECT 0 {EXCHG {PARA 3 "" 0 "" {TEXT 256 37 "Runge-Kutta Methods, Tree
s, and Maple" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 44 "Version: 9. Feb. 2001, (c) Folkmar Bornemann" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "Author's address: Munic
h University of Technology, 80290 Munich, Germany" }}{PARA 0 "" 0 "" 
{TEXT -1 85 "                           E-mail: bornemann@ma.tum.de, U
RL: http://www.ma.tum.de/m3/" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 43 "Maple source code from: Folkmar Bornemann, " }
{TEXT 258 28 "Runge-Kutta Methods, Trees, " }}{PARA 0 "" 0 "" {TEXT 
260 11 "and Maple, " }{TEXT -1 37 "Selcuk J. Appl. Math. 2, 3--15, 200
1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 31 "`type/tree`:=list(tree): f:=[]:" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 26 "TreeSort:=proc(beta::tree)" }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 27 "  sort(map(TreeSort,beta));" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "end: # TreeSort" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 43 "beta[1]:=[[f,[f]],f]: beta[2]:=[f,[[f],f]]:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "evalb(TreeSort(beta[1])=Tree
Sort(beta[2]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "TreeOrder:=proc(beta::tree)" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "  option remember;" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 33 "  1+`+`(op(map(TreeOrder,beta)));" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 16 "end: # TreeOrder" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 39 "beta:=[[[f],[f],f],f]: TreeOrder(beta);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#\"\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
31 "TreeFactorial:=proc(beta::tree)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
18 "  option remember;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "  TreeOrd
er(beta)*`*`(op(map(TreeFactorial,beta)));" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "end: # TreeFactorial" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "TreeFactorial(beta);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#\"$#>" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "TreeAlpha:=proc
(beta::tree)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "  option remember;
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "  local n;" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 16 "  n:=nops(beta);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
37 "  nops(combinat[permute](beta,n))/n!*" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 35 "      `*`(op(map(TreeAlpha,beta)));" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 16 "end: # TreeAlpha" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "TreeAlpha(beta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##
\"\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "TreeA:=proc(
beta::tree,no::posint)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "  option \+
remember;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "  local vars,val,son;
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "  vars:=[i,j,k,l,m,p,q,r,u,v,w]
;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "  val:=1;" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "  for son in beta do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 42 "     val:= val*Sum(a[vars[no],vars[no+1]]*" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 42 "         TreeA(son,no+1),vars[no+1]=1..s);" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 5 "  od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 
"  val;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "end: # TreeA" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 36 "TreeOrderCondition:=proc(beta::tree)" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 33 "  Sum(b[i]*TreeA(beta,1),i=1..s)=" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 28 "      1/TreeFactorial(beta);" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 25 "end: # TreeOrderCondition" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 25 "TreeOrderCondition(beta);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%$SumG6$*(&%\"bG6#%\"iG\"\"\"-F%6$*(&%\"aG6$F+%\"jGF,
)-F%6$*&&F16$F3%\"kGF,-F%6$&F16$F:%\"lG/F?;F,%\"sGF,/F:FA\"\"#F,-F%6$F
8FCF,/F3FAF,-F%6$F0FGF,/F+FA#F,\"$#>" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "Trees:=proc(order::posint)" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "  option remember;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
34 "  local Replace,AddLeaf,all,trees;" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "  Replace:=proc(new::tre
e,old::posint,beta::tree)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "    so
rt(subsop(old=new,beta)); # order-independent..." }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "  end: # Replace" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 
"" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "  AddLeaf:=proc(beta::tree)" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "    option remember;" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 24 "    local val,child,new;" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 31 "    val:=\{sort([[],op(beta)])\};" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 37 "    for child from 1 to nops(beta) do" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 32 "      new:=AddLeaf(beta[child]);" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "      val:=val union map(Replace,ne
w,child,beta);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "    od;" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 8 "    val;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 16 "  end: # AddLeaf" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 14 "  trees:=\{[]\};" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "  all:=[trees];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 
"  to order-1 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "    trees:=`uni
on`(op(map(AddLeaf,trees)));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "   \+
 all:=[op(all),trees];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "  od:" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "  all;" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "end: # Trees" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 9 "Trees(4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&<#7\"<#7#F%<$7#F'7
$F%F%<&7$F%F'7#F)7#F*7%F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 20 "map(nops,Trees(10));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7,\"\"\"
F$\"\"#\"\"%\"\"*\"#?\"#[\"$:\"\"$'G\"$>(" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 11 "`+`(op(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"%0
7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "OrderConditions:=proc(
order::posint,stages::posint)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "  \+
option remember;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "  local eqs,var
s,auto,explicit;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "  explicit:=\{s
eq(seq(a[i,j]=0,j=i..stages)," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "  \+
    i=1..stages)\};" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "  vars:=eval
(\{seq(b[i],i=1..stages)," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "      \+
seq(seq(a[i,j],j=1..stages),i=1..stages)," }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 49 "      seq(c[i],i=1..stages)\},explicit) minus \{0\};
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "  auto:=eval(\{seq(sum(a[i,j],j
=1..stages)=c[i]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "      i=1..sta
ges)\},explicit);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "  eqs:=value(E
val(map(TreeOrderCondition," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "    \+
  `union`(op(Trees(order)))),s=stages));" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 37 "  eqs:=eval(eval(eqs,explicit),auto);" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 16 "  eqs,auto,vars;" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "end: # OrderConditions" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "OrderConditions(4,4): %[1];" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#<*/,*&%\"bG6#\"\"\"F)&F'6#\"\"#F)&F'6#\"\"$F)&F'6#\"\"%
F)F)/,(*&F*F)&%\"cGF+F)F)*&F-F)&F7F.F)F)*&F0F)&F7F1F)F)#F)F,/,&*(F-F)&
%\"aG6$F/F,F))F6F,F)F)*&F0F),&*&&FA6$F2F,F)FCF)F)*&&FA6$F2F/F))F9F,F)F
)F)F)#F)\"#7/,(*&F*F))F6F/F)F)*&F-F))F9F/F)F)*&F0F))F;F/F)F)#F)F2/,&*(
F-F)F@F)F6F)F)*&F0F),&*&FGF)F6F)F)*&FJF)F9F)F)F)F)#F)\"\"'/**F0F)FJF)F
@F)F6F)#F)\"#C/,(*&F*F)FCF)F)*&F-F)FLF)F)*&F0F))F;F,F)F)#F)F//,&**F-F)
F9F)F@F)F6F)F)*(F0F)F;F)FfnF)F)#F)\"\")" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 52 "RungeKuttaMethod:=proc(p::posint,s::posint,pre::set)
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "  # explicit methods only" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "  local conds,auto,vars,eqs,sols,so
l,val;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "  conds,auto,vars:=OrderC
onditions(p,s);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "  eqs:=conds uni
on auto union pre;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "  sols:=\{sol
ve(eqs,vars)\};" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "  val:=NULL;" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "  for sol in sols do" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 14 "    val:=val,[" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 50 "    a=matrix([seq([seq(eval(a[i,j],sol),j=1..i-1)," }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 39 "              seq(0,j=i..s)],i=1..s)])," }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "    b=vector([seq(eval(b[i],sol),i=
1..s)])," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "    c=vector([seq(eval(
c[i],sol),i=1..s)])];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "  od:" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "  val;" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "end: # RungeKuttaMethod" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 35 "RungeKuttaMethod(2,2,\{b[2]=theta\});" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#7%/%\"aG-%'matrixG6#7$7$\"\"!F+7$,$*&\"\"\"F/%&the
taG!\"\"#F/\"\"#F+/%\"bG-%'vectorG6#7$,&F0F1F/F/F0/%\"cG-F76#7$F+F-" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "pre:=\{c[2]=u,c[3]=1/4,c[4
]=1/2,c[5]=3/4,c[6]=1," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "   b[2]=0
,a[4,3]=v\}:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "RungeKuttaM
ethod(5,6,pre): %[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"aG-%'matr
ixG6#7(7(\"\"!F*F*F*F*F*7(%\"uGF*F*F*F*F*7(,$*&,&!\"\"\"\"\"*&\"\")F2F
,F2F2F2F,F1#F2\"#K,$*&F2F2F,F1F5F*F*F*F*7(,$*&,*%\"vG!\"#F2F2*(F4F2F=F
2F,F2F2*&\"\"%F2F,F2F1F2F,F1#F1F4,$*&,&F=\"\"#F2F1F2F,F1FBF=F*F*F*7(,$
*&,*F2F2F=F1*&\"\"$F2F,F2F1*(FAF2F=F2F,F2F2F2F,F1#FL\"#;,$*&,&F1F2F=F2
F2F,F1FN,&#FLFAF2*&#FLFAF2F=F2F1#\"\"*FOF*F*7(,$*&,*\"\"(F2*&\"\"'F2F=
F2F1*(\"#CF2F=F2F,F2F2*&\"#AF2F,F2F1F2F,F1#F1\"#9,$*&,&!\"(F2*&FinF2F=
F2F2F2F,F1F^o,$F=#\"#7Fgn#!#7Fgn#F4FgnF*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "%%[2],%%[3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%\"b
G-%'vectorG6#7(#\"\"(\"#!*\"\"!#\"#;\"#X#\"\"#\"#:F-F)/%\"cG-F&6#7(F,%
\"uG#\"\"\"\"\"%#F:F1#\"\"$F;F:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}}{MARK "0 0 0" 37 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
{PAGENUMBERS 0 1 2 33 1 1 }