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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "L-9 MCS 563 Monday 13 Sept
ember 2004" }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 51 "Symbolic-Numeric Co
mputations using Newton's method" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
81 "In this worksheet we illustrate the use of Newton's method to cons
truct formulas." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "We co
nsider a very simple polynomial system:" }{MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "p := [x[1]*x[2] - 1, x[1]^2 + t*x[2
]^2 - 1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG7$,&*&&%\"xG6#\"\"
\"F+&F)6#\"\"#F+F+F+!\"\",(*$)F(F.F+F+*&%\"tGF+)F,F.F+F+F+F/" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "with Jacobian matrix J:" }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "J := Ma
trix(2,2,(i,j) -> diff(p[i],x[j]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%\"JG-%'RTABLEG6$\"*G@bO\"-%'MATRIXG6#7$7$&%\"xG6#\"\"#&F/6#\"\"\"7$
,$F2F1,$*&%\"tGF4F.F4F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "We nee
d to evaluate both the system and its Jacobian matrix:" }{MPLTEXT 1 0 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "f := unapply(p,x[1],
x[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6$%$x_1G%$x_2G6\"6$%
)operatorG%&arrowGF)7$,&*&9$\"\"\"9%F1F1F1!\"\",(*$)F0\"\"#F1F1*&%\"tG
F1)F2F7F1F1F1F3F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "Jf
 := (a,b) -> subs(x[1]=a,x[2]=b,J);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%#JfGf*6$%\"aG%\"bG6\"6$%)operatorG%&arrowGF)-%%subsG6%/&%\"xG6#\"\"
\"9$/&F26#\"\"#9%%\"JGF)F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "T
he following calculations show that for t = 0, the point (1,1) is a re
gular solution." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 7 "f(1,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$\"\"!%\"t
G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "Jf(1,1);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\"*+2wP\"-%'MATRIXG6#7$7$\"\"\"F,7$
\"\"#,$%\"tGF." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 216 "Our symbolic N
ewton method takes on input two procedures f and Jf which respectively
 evaluate the system and its Jacobian matrix at the third argument, wh
ich is a list.  This list X can consist of numbers or symbols." }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "Newton \+
:= proc(f::procedure,Jf::procedure,X::list)" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 50 "  description `performs one symbolic Newton step`:" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "  local A,b,dx,sys,sol:" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 17 "  b := -f(op(X));" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "  A := Jf(op(X));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
22 "  dx := [dx[1],dx[2]]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "  sys
 := LinearAlgebra[GenerateEquations](A,dx,Vector(b));" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 35 "  sol := solve(\{op(sys)\},\{op(dx)\});" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "  assign(sol);" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 33 "  return [X[1]+dx[1],X[2]+dx[2]];" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 9 "end proc;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'N
ewtonGf*6%'%\"fG%*procedureG'%#JfGF)'%\"XG%%listG6'%\"AG%\"bG%#dxG%$sy
sG%$solG6\"F5C)>8%,$-9$6#-%#opG6#9&!\"\">8$-9%F<>8&7$&FG6#\"\"\"&FG6#
\"\"#>8'-&%.LinearAlgebraG6#%2GenerateEquationsG6%FCFG-%'VectorG6#F8>8
(-%&solveG6$<#-F>6#FP<#-F>6#FG-%'assignG6#FenO7$,&&F@FJFKFIFK,&&F@FMFK
FLFK6#%Bperforms~one~symbolic~Newton~stepGF5F5" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 26 "psA := Newton(f,Jf,[1,1]);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%$psAG7$,&\"\"\"F'*(#F'\"\"#F'%\"tGF',&!\"\"F'F+F'F-
F',&F'F'*&#F'F*F'*&F+F'F,F-F'F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
176 "Note that instead of [1,1], we can only fill in [a,b].  However, \+
using [a,b] starts to get too expensive after a while, so we stick wit
h [1,1], which is the solution for t = 0." }{MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "collect(psA[1],t);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#,&\"\"\"F$*(#F$\"\"#F$%\"tGF$,&!\"\"F$F(F$F*
F$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "We develop the expression f
ound for the first coordinate x[1] as series around t = 0:" }{MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "sA := series(psA
[1],t=0,9);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#sAG+7%\"tG\"\"\"\"\"
!#!\"\"\"\"#F'F)F+F)\"\"$F)\"\"%F)\"\"&F)\"\"'F)\"\"(F)\"\")-%\"OG6#F'
\"\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "and we continue with thr
ee more Newton steps, observe the quadratic convergence:" }{MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "psB := Newton(f,Jf
,psA):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "sB := series(psB[
1],t=0,9);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#sBG+7%\"tG\"\"\"\"\"!
#!\"\"\"\"#F'#!\"&\"\")F+#!#@\"#;\"\"$#!$2\"\"#K\"\"%#!$$f\"#k\"\"&#!%
&R$\"$G\"\"\"'#!&l'>\"$c#\"\"(#!'rV6\"$7&F.-%\"OG6#F'\"\"*" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "psC := Newton(f,Jf,psB):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "sC := series(psC[1],t=0,9);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#sCG+7%\"tG\"\"\"\"\"!#!\"\"\"\"
#F'#!\"&\"\")F+#!#@\"#;\"\"$#!$H%\"$G\"\"\"%#!%JC\"$c#\"\"&#!&$RH\"%C5
\"\"'#!'Dd=\"%[?\"\"(#!(6PU#\"%#>)F.-%\"OG6#F'\"\"*" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 24 "psD := Newton(f,Jf,psC):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "sD := series(psD[1],t=0,9);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#sDG+7%\"tG\"\"\"\"\"!#!\"\"\"\"#F'#
!\"&\"\")F+#!#@\"#;\"\"$#!$H%\"$G\"\"\"%#!%JC\"$c#\"\"&#!&$RH\"%C5\"\"
'#!'Dd=\"%[?\"\"(#!(X[p*\"&oF$F.-%\"OG6#F'\"\"*" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 24 "psE := Newton(f,Jf,psD):" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 27 "sE := series(psE[1],t=0,9);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%#sEG+7%\"tG\"\"\"\"\"!#!\"\"\"\"#F'#!\"&\"\")F+#!#
@\"#;\"\"$#!$H%\"$G\"\"\"%#!%JC\"$c#\"\"&#!&$RH\"%C5\"\"'#!'Dd=\"%[?\"
\"(#!(X[p*\"&oF$F.-%\"OG6#F'\"\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
67 "The following calculations make the conversions even more explicit
:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "pD \+
:= convert(sD,polynom): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 
"pA := convert(sA,polynom):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
26 "pB := convert(sB,polynom):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 26 "pC := convert(sC,polynom):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 6 "pA-pD;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,0*$)%\"tG\"
\"#\"\"\"#F(\"\")*&#\"#8\"#;F()F&\"\"$F(F(*&#\"$l$\"$G\"F()F&\"\"%F(F(
*&#\"%.B\"$c#F()F&\"\"&F(F(*&#\"&\"))G\"%C5F()F&\"\"'F(F(*&#\"',Z=\"%[
?F()F&\"\"(F(F(*&#\"(h%y'*\"&oF$F()F&F*F(F(" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 6 "pB-pD;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*$)%\"t
G\"\"%\"\"\"#F(\"$G\"*&#\"#f\"$c#F()F&\"\"&F(F(*&#\"%LA\"%C5F()F&\"\"'
F(F(*&#\"&0%G\"%[?F()F&\"\"(F(F(*&#\"(,^P#\"&oF$F()F&\"\")F(F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "pC-pD;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,$*$)%\"tG\"\")\"\"\"#F(\"&oF$" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 138 "After one Newton iteration, the error is O(t^2), then
 the error becomes O(t^4), and after the third Newton iteration, the e
rror is O(t^8)." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 128 "Similar to the quadratic convergence of Newton's method as we \+
know it, we can construct more and more accurate series solutions." }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 205 "Notice howev
er, that the solutions we obtained are purely formal.  The difference \+
between the result of the first and fourth Newton iteration when evalu
ate at a small value for t is not really significant:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "fA := unapply(pA,t);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%#fAGf*6#%\"tG6\"6$%)operatorG%&arrowGF(,4\"\"
\"F-*&#F-\"\"#F-9$F-!\"\"*&#F-F0F-*$)F1F0F-F-F2*&#F-F0F-*$)F1\"\"$F-F-
F2*&#F-F0F-*$)F1\"\"%F-F-F2*&#F-F0F-*$)F1\"\"&F-F-F2*&#F-F0F-*$)F1\"\"
'F-F-F2*&#F-F0F-*$)F1\"\"(F-F-F2*&#F-F0F-*$)F1\"\")F-F-F2F(F(F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "fD := unapply(pD,t);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fDGf*6#%\"tG6\"6$%)operatorG%&arrow
GF(,4\"\"\"F-*&#F-\"\"#F-9$F-!\"\"*&#\"\"&\"\")F-*$)F1F0F-F-F2*&#\"#@
\"#;F-*$)F1\"\"$F-F-F2*&#\"$H%\"$G\"F-*$)F1\"\"%F-F-F2*&#\"%JC\"$c#F-*
$)F1F5F-F-F2*&#\"&$RH\"%C5F-*$)F1\"\"'F-F-F2*&#\"'Dd=\"%[?F-*$)F1\"\"(
F-F-F2*&#\"(X[p*\"&oF$F-*$)F1F6F-F-F2F(F(F(" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 9 "T := 0.1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 6 "fD(T);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+;lm>%*!#5" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "fA(T);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"+]WWW%*!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 18 "pT := subs(t=T,p);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#pTG7$,&
*&&%\"xG6#\"\"\"F+&F)6#\"\"#F+F+F+!\"\",(*$)F(F.F+F+*&$F+F/F+)F,F.F+F+
F+F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "sols := fsolve(\{op
(pT)\},\{x[1]=1.0,x[2]=1.0\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%s
olsG<$/&%\"xG6#\"\"\"$\"+^9l>%*!#5/&F(6#\"\"#$\"+1/hh5!\"*" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "assign(sols);" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 13 "fA(T) - x[1];" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#$\")**HzC!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "fD(T)
 - x[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"&l]\"!#5" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 96 "Even for a large value of t, the last pow
er series give us five correct decimal places for x[1]." }{MPLTEXT 1 
0 0 "" }}}}{MARK "4 0 1" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
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