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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 257 27 "Newton's Method for Sys
tems" }}{PARA 0 "" 0 "" {TEXT 256 31 "Mth 351 October 29 2002 Maple 6
" }}{PARA 0 "" 0 "" {TEXT 258 16 "Bent E. Petersen" }{TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 259 
12 "Introduction" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 122 "In this worksheet I give an example to i
llustrate Newton's iterative method for solving a system of (nonlinear
) equations." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 11 "Digits:=18:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(l
inalg):" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names n
orm and trace have been redefined and unprotected\n" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 
50 "Warning, the name changecoords has been redefined\n" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "f:=x^2 - 4*x*y + 2*y^2 + 3*x - 2*y:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "g:=x^2 + 4*x*y + 2*y^2 \+
+ 3*x - 2*y:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 41 "We wish to solve the system of equations:" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 " eqn
f:=f=2; eqng:=g=5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqnfG/,,*$)%
\"xG\"\"#\"\"\"F+*(\"\"%F+F)F+%\"yGF+!\"\"*&F*F+)F.F*F+F+*&\"\"$F+F)F+
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G\"\"#\"\"\"F+*(\"\"%F+F)F+%\"yGF+F+*&F*F+)F.F*F+F+*&\"\"$F+F)F+F+*&F*
F+F.F+!\"\"\"\"&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 41 "Let's begin by plotting the level curves." }}
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0 81 "imgf:=contourplot(f,x=-6..6,y=-4..3,contours=[rhs(eqnf)],color=b
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contourplot(g,x=-6..6,y=-4..3,contours=[rhs(eqng)],color=red,thickness
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cx7$F]go7$$\"3.$yM/8Rn]$F1$!3Ocp3Eyf&Q$F17$7$$\"3$\\J\\o]J\\W$F1FdyFag
o7$Fggo7$$\"3+_F\"p`!3sLF1$!3)>xl)zp/FIF17$7$$\"3-AAAAAA)R$F1F[\\lF[ho
7$Faho7$$\"3GZ*y:j_5O$F1$!3G>x3NShSFF17$7$Fb[n$!3xxxxxxxFEF1Feho7$7$$
\"3-b>y7^/=QF1Fjt7$$\"3/KH<pw1(o$F1$!3q5Uot%*yqPF17$7$$\"3i4Q_4Q_tOF1F
gwFbio7$Fhio7$Fb[n$!3gmmmmm;fNF17$F[io7$$\"3#Q_4Q_4Qi$F1$!3$*)))))))))
))Qh#F17$7$Fa\\n$!3WiZ!>w/>l#F1F`jo7$7$Fa\\n$!3)>w/>w/>l#F17$$\"3a_cp3
EyuTF1$!3ujC2b)*GbEF17$7$Fg]n$!3ivvvvvvXFF1F][p7$7$Fg]n$!31wvvvvvXFF17
$$\"3')**********zW\\F1$!3YmmmmmYCGF17$7$Fc^n$!3Sbbbbbb`GF1Fj[p7$7$Fi_
n$!39+++++gyHF17$$\"3'GLLLLLL8&F1F[\\l7$Fg\\pF`\\p7$Fd\\p7$$\"3#z$p&4?
iI!eF1$!3o!)[A<'>^/$F17$7$F_an$!3)*Q[N>un/JF1F\\]p-Fdan6&FfanFhanFganF
ganF[bn-%+AXESLABELSG6%%\"xG%\"yG%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 
1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 112 "From the plot \+
we see there are 4 soultions. Maple can find the solutions if we give \+
some hints of where to look." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "soln[1]:=fsolve(\{f=rhs(eqnf
),g=rhs(eqng)\},\{x,y\},\{x=-5..-3\});" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>&%%solnG6#\"\"\"<$/%\"xG$!3$*p!*QJ`&H&Q!#</%\"yG$!32lV?r')yK(*!
#>" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "soln[2]:=fsolve(\{f=r
hs(eqnf),g=rhs(eqng)\},\{x,y\},\{x=-1..0\});" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%%solnG6#\"\"#<$/%\"yG$!3)o.j;vD)p5!#</%\"xG$!3k;kKAP
C0N!#=" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "soln[3]:=fsolve(
\{f=rhs(eqnf),g=rhs(eqng)\},\{x,y\},\{x=0..1\});" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%%solnG6#\"\"$<$/%\"xG$\"3WjOIq[6&4#!#=/%\"yG$\"3lD#*
R$*z()*y\"!#<" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "soln[4]:=f
solve(\{f=rhs(eqnf),g=rhs(eqng)\},\{x,y\},\{x=0.5..2\});" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>&%%solnG6#\"\"%<$/%\"xG$\"3]_M\"f;#oR**!#=/%\"y
G$\"3\")[&e(\\kvsPF-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 141 "We will use these Maple solutions to test the \+
convergence of Newton's method. We can convert the solutions to points
 in the plane as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "for k from 1 to 4 do pnt[k]:=subs(s
oln[k],[x,y]); od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$pntG6#\"\"\"
7$$!3$*p!*QJ`&H&Q!#<$!32lV?r')yK(*!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>&%$pntG6#\"\"#7$$!3k;kKAPC0N!#=$!3)o.j;vD)p5!#<" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>&%$pntG6#\"\"$7$$\"3WjOIq[6&4#!#=$\"3lD#*R$*z()*y\"!
#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$pntG6#\"\"%7$$\"3]_M\"f;#oR*
*!#=$\"3\")[&e(\\kvsPF+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 108 "We will need the Jacobian matrix (the ma
trix of first order partial derivatives) of our system of equations:" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 25 "J:=jacobian([f,g],[x,y]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%\"JG-%'matrixG6#7$7$,(%\"xG\"\"#*&\"\"%\"\"\"%\"yGF/!\"\"\"\"$F/,(F+!
\"%*&F.F/F0F/F/F,F17$,(F+F,*&F.F/F0F/F/F2F/,(F+F.*&F.F/F0F/F/F,F1" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 7 "Po
int 1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "
Based on the plot let's take for our initial guess  [-4,0]" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "est
1[0]:=[-4.0,0];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%%est1G6#\"\"!7$$
!#S!\"\"F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "J1[0]:=subs(x
=est1[0][1],y=est1[0][2],evalm(J));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>&%#J1G6#\"\"!-%'matrixG6#7$7$$!#]!\"\"$\"$S\"F/7$F-$!$!=F/" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 116 "N
ote above we evaluate the matrix J in order to make subs() work compon
entwise. This simple idea was shown to me by " }{TEXT 264 11 "David Fi
nch" }{TEXT -1 153 ". There are some unpleasant and much more complica
ted ways to achieve the same effect - for example by using loops over \+
indices or by using an index map." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "F1[0]:=subs(x=est1[0][1],y=e
st1[0][2],evalm([f-rhs(eqnf),g-rhs(eqng)]));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%#F1G6#\"\"!-%'vectorG6#7$$\"$+#!\"#$!$+\"F." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Delta1[1]:=linsolve(J1[0],-F
1[0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'Delta1G6#\"\"\"-%'vector
G6#7$$\"3+++++++v8!#=$!3,++++++v$*!#>" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 37 "est1[0]+Delta1[1]: est1[1]:=evalm(%);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>&%%est1G6#\"\"\"-%'vectorG6#7$$!3++++++]iQ!#<$!3,
++++++v$*!#>" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 26 "Let's do a few more steps:" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "for k from 1 to 3 do
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "J1[k]:=subs(x=est1[k][1],y=est1
[k][2],evalm(J));  " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "F1[k]:=subs(
x=est1[k][1],y=est1[k][2],evalm([f-rhs(eqnf),g-rhs(eqng)]));" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 36 "Delta1[k+1]:=linsolve(J1[k],-F1[k]);" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "est1[k+1]:= evalm(est1[k]+Delta1[k+
1]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%#J1G6#\"\"\"-%'matrixG6#7$7$$!3+++++++]V!#<$\"3+++++
+]28!#;7$$!3++++++++^F/$!3++++++]#y\"F2" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>&%#F1G6#\"\"\"-%'vectorG6#7$$\"3+++++vo/))!#>$!3+++++D\"y]\"F.
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'Delta1G6#\"\"#-%'vectorG6#7$$
\"3%41B4]l^^*!#?$!3EU6tf#H$oNF." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%
%est1G6#\"\"#-%'vectorG6#7$$!3%p2*\\M[)H&Q!#<$!3C9J(f#H$=t*!#>" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#J1G6#\"\"#-%'matrixG6#7$7$$!3JH#f>
N'p;V!#<$\"3KQd4-mE-8!#;7$$!3Xyq.')HC&4&F/$!3CBN]l77!y\"F2" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>&%#F1G6#\"\"#-%'vectorG6#7$$\"1_<%z&H<=D!#>
$!0[$3&>l3)>F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'Delta1G6#\"\"$-%
'vectorG6#7$$\"3LHKnnXG]H!#A$!3bf\"pEXstb*!#B" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%%est1G6#\"\"$-%'vectorG6#7$$!3@+MlJ`&H&Q!#<$!3;\"Q=K
m)yK(*!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#J1G6#\"\"$-%'matrixG6
#7$7$$!3<l!yn6*f;V!#<$\"3c'[3!y4D-8!#;7$$!3nNb$)4AA&4&F/$!3gLUJ(G8,y\"
F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#F1G6#\"\"$-%'vectorG6#7$$\",
'=M)4=#!#>$!*92uZ(F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'Delta1G6#
\"\"%-%'vectorG6#7$$\"3v\")H;dGIVE!#F$!3:Fhd[R)f)z!#G" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>&%%est1G6#\"\"%-%'vectorG6#7$$!3#*p!*QJ`&H&Q!#<$!
36lV?r')yK(*!#>" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 28 "Let's compare the residuals:" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "'Maple'; su
bs(x=pnt[1][1], y=pnt[1][2], f); subs(x=pnt[1][1], y=pnt[1][2], g);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&MapleG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"3,+++++++?!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"
3,+++++++]!#<" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "'Newton'; \+
subs(x=est1[4][1], y=est1[4][2], f); subs(x=est1[4][1], y=est1[4][2], \+
g);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%'NewtonG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"3\"***************>!#<" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#$\"3#***************\\!#<" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 39 "Let's also compare the solution poin
ts:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "evalm(pnt[1]-est1[4]);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#-%'vectorG6#7$$!\"\"!#<$\"\"%!#>" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 7 "Point 2" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "Based on the plot let's
 take for our initial guess  [0,-1]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "est2[0]:=[0,-1.0];" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%%est2G6#\"\"!7$F'$!#5!\"\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 20 "for k from 0 to 3 do" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 50 "J2[k]:=subs(x=est2[k][1],y=est2[k][2],evalm(J));  " }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 72 "F2[k]:=subs(x=est2[k][1],y=est2[k][2],evalm
([f-rhs(eqnf),g-rhs(eqng)]));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "De
lta2[k+1]:=linsolve(J2[k],-F2[k]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
39 "est2[k+1]:= evalm(est2[k]+Delta2[k+1]);" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 3 "od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#J2G6#\"\"!-
%'matrixG6#7$7$$\"#q!\"\"$!#gF/7$$!#5F/F0" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%#F2G6#\"\"!-%'vectorG6#7$$\"#?!\"\"$!$+\"!\"#" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'Delta2G6#\"\"\"-%'vectorG6#7$$!3**
************\\P!#=$!3mmmmmmmT5F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&
%%est2G6#\"\"\"-%'vectorG6#7$$!3**************\\P!#=$!3nmmmmm;/6!#<" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#J2G6#\"\"\"-%'matrixG6#7$7$$\"3om
mmmmmmm!#<$!3ommmmmm;\\F/7$$!3ommmmmmm@F/$!3ommmmmm;zF/" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>&%#F2G6#\"\"\"-%'vectorG6#7$$\"0#*)))))))Qwg!#<
$\"2#*)))))))Qw&=$F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'Delta2G6#
\"\"#-%'vectorG6#7$$\"3J*>n$p'HNR#!#>$\"3koa3H:0pLF." }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>&%%est2G6#\"\"#-%'vectorG6#7$$!31!GjI.Z1^$!#=$!3)>
\"eP^hZq5!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#J2G6#\"\"#-%'matri
xG6#7$7$$\"3\">f!*))>v(zl!#<$!3!f$zF#zXw([F/7$$!3$R!f67S.%)>F/$!3%*f&G
(=M;'o(F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#F2G6#\"\"#-%'vectorG6
#7$$!/]c![)pDQ!#<$\"0)y$f)*p&ogF." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
&%'Delta2G6#\"\"$-%'vectorG6#7$$\"3FNx5/*44S&!#@$\"32k*pTP28]'F." }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%%est2G6#\"\"$-%'vectorG6#7$$!3Ks\"f
J7Y_]$!#=$!3)\\RQ1-E)p5!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#J2G6
#\"\"$-%'matrixG6#7$7$$\"3YX<#z&[Dyl!#<$!3*4\"**GLc?x[F/7$$!3Q9a=2LN!)
>F/$!3&)[s\"=`-9o(F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#F2G6#\"\"$
-%'vectorG6#7$$!,5'y![n#!#<$\"-%[Cd:a#F." }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>&%'Delta2G6#\"\"%-%'vectorG6#7$$\"3)3n))4JK3S#!#C$\"3\\M&of4`(*
o#F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%%est2G6#\"\"%-%'vectorG6#7$
$!3LhoKAPC0N!#=$!3Q&3j;vD)p5!#<" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 28 "Let's compare the residuals:" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 78 "'Maple'; subs(x=pnt[2][1], y=pnt[2][2], f); subs(x=pnt[2][1], y=
pnt[2][2], g);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&MapleG" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#$\"3,+++++++?!#<" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"3.+++++++]!#<" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 83 "'Newton'; subs(x=est2[4][1], y=est2[4][2], f); subs(x=est2[4][
1], y=est2[4][2], g);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%'NewtonG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"3-W************>!#<" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#$\"3ig/+++++]!#<" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Let's also compare the solution
 points:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "evalm(pnt[2]-est2[4]);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#-%'vectorG6#7$$\"&pW%!#=$\"%][!#<" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 262 7 "Point 3" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "Based on the plot let's
 take for our initial guess  [0.2, 1.8]" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "est3[0]:=[0.2,1.8];" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%%est3G6#\"\"!7$$\"\"#!\"\"$\"#=F+
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 20 "for k from 0 to 3 do" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 50 "J3[k]:=subs(x=est3[k][1],y=est3[k][2],evalm(J));  " }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "F3[k]:=subs(x=est3[k][1],y=est3[k]
[2],evalm([f-rhs(eqnf),g-rhs(eqng)]));" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 36 "Delta3[k+1]:=linsolve(J3[k],-F3[k]);" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 39 "est3[k+1]:= evalm(est3[k]+Delta3[k+1]);" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 3 "od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#J3G
6#\"\"!-%'matrixG6#7$7$$!#Q!\"\"$\"#WF/7$$\"$1\"F/$\"#gF/" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>&%#F3G6#\"\"!-%'vectorG6#7$$\"\")!\"#$!\"%F." 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'Delta3G6#\"\"\"-%'vectorG6#7$$\"
3*3$\\H3Y+Z%*!#?$!3zPaYZTI-5!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%
%est3G6#\"\"\"-%'vectorG6#7$$\"3J\\H3Y+Z%4#!#=$\"3iX`_ep(**y\"!#<" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#J3G6#\"\"\"-%'matrixG6#7$7$$!3i#z%
)[#Q,TP!#<$\"3w-#oc\")>@K%F/7$$\"3B(zJV=!)y0\"!#;$\"3?iX`_ep(*fF/" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#F3G6#\"\"\"-%'vectorG6#7$$\"/=vG\\
>*o'!#<$!.OE\"fAe))F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'Delta3G6#
\"\"#-%'vectorG6#7$$\"3^CN-btzZk!#A$!3%zlBGldd*)*F." }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>&%%est3G6#\"\"#-%'vectorG6#7$$\"3m^%=e#[6&4#!#=$\"3Q
<)[4+y)*y\"!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#J3G6#\"\"#-%'mat
rixG6#7$7$$!3>z:jQ!*[SP!#<$\"3')))yYtgY@VF/7$$\"3)f*e*o\\`y0\"!#;$\"3=
]E7Mzb(*fF/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#F3G6#\"\"#-%'vector
G6#7$$\"+**[\\E\\!#<$!*0B(z<F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'
Delta3G6#\"\"$-%'vectorG6#7$$\"3b:Fvb6_[W!#E$!3BRuv#p\"f\\vF." }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%%est3G6#\"\"$-%'vectorG6#7$$\"3AjOI
q[6&4#!#=$\"3pD#*R$*z()*y\"!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#
J3G6#\"\"$-%'matrixG6#7$7$$!37qh`***)[SP!#<$\"3ZPaZDgY@VF/7$$\"3ajdw%
\\`y0\"!#;$\"30o$=<#zb(*fF/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#F3G
6#\"\"$-%'vectorG6#7$$\"#D!#<$F'F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>&%'Delta3G6#\"\"%-%'vectorG6#7$$\"3GFcpL*R*4?!#M$!3&pfp`\")\\`/%F." }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%%est3G6#\"\"%-%'vectorG6#7$$\"3UjO
Iq[6&4#!#=$\"3lD#*R$*z()*y\"!#<" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 28 "Let's compare the residuals:" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 78 "'Maple'; subs(x=pnt[3][1], y=pnt[3][2], f); subs(x=pnt[3][1], y=
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