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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Runge-Kutta Methods, Trees
, 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 42 "modified: 11. Nov. 2005 for lecture at TUM" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "Author's address: \+
Munich University of Technology, 80290 Munich, Germany" }}{PARA 0 "" 
0 "" {TEXT -1 29 "E-mail: bornemann@ma.tum.de, " }}{PARA 0 "" 0 "" 
{TEXT -1 29 "URL: http://www.ma.tum.de/m3/" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "Maple source code from: " }}
{PARA 0 "" 0 "" {TEXT -1 95 "Folkmar Bornemann, Runge-Kutta Methods, T
rees, and Maple, Selcuk J. Appl. Math. 2, 3--15, 2001." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "`type/tree`:=l
ist(tree): f:=[]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "TreeSo
rt:=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 27 "TreeOrder:=proc(beta::t
ree)" }}{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 31 "TreeFactorial:=proc(beta::tree)" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "  option remember;" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 51 "  TreeOrder(beta)*`*`(op(map(TreeFactorial,beta)
));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "end: # TreeFactorial" }}}
{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(co
mbinat[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 34 "TreeA:
=proc(beta::tree,no::posint)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "  o
ption 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: # Tr
eeA" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "TreeOrderCondition:=
proc(beta::tree)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "  Sum(b[i]*Tree
A(beta,1),i=1..s)=" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "      1/TreeF
actorial(beta);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "end: # TreeOrder
Condition" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Trees:=proc(or
der::posint)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "  option remember;
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "  local Replace,AddLeaf,all,tre
es;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 49 "  Replace:=proc(new::tree,old::posint,beta::tree)" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "    sort(subsop(old=new,beta)); # o
rder-independent..." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "  end: # Rep
lace" }}{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,new,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:=`union`(op(map(Ad
dLeaf,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 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,vars,auto,explicit;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "  expli
cit:=\{seq(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 "  va
rs:=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..
stages)\},explicit);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "  eqs:=valu
e(Eval(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 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 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
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