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{SECT 0 {SECT 0 {PARA 3 "" 0 "" {TEXT -1 46 "Constrained Optimization:
 Lagrange multipliers" }}{EXCHG {PARA 0 "" 0 "" {TEXT 256 12 "Descript
ion:" }{TEXT -1 40 " Two examples for Lagrange multipliers (" }{TEXT 
257 9 "animation" }{TEXT -1 2 ")." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "Code" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 51 "restart():with(plottools):with(plots):with(linalg):" 
}}{PARA 7 "" 1 "" {TEXT -1 62 "Warning, the names arrow and changecoor
ds have been redefined\n" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the \+
protected names norm and trace have been redefined and unprotected\n" 
}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 11 "Example 1: " }{XPPEDIT 18 0 "f
(x,y)=x^2+2y^2" "6#/-%\"fG6$%\"xG%\"yG,&*$F'\"\"#\"\"\"*&F+F,*$F(F+F,F
," }{TEXT -1 26 " subject to the constrain " }{XPPEDIT 18 0 "g(x,y)=x^
2+y^2" "6#/-%\"gG6$%\"xG%\"yG,&*$F'\"\"#\"\"\"*$F(F+F," }{TEXT -1 2 "=
1" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "g:=(x,y)->x^2+y^2:\nf:=
(x,y)->x^2+2*y^2:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "constrain:=im
plicitplot(g(x,y)=1,x=-1.5..1.5,y=-1.5..1.5,\nnumpoints=800,color=blue
,thickness=3,scaling=constrained):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
77 "gradg:=gradplot(g(x,y),x=-2..2,y=-2..2,color=green,arrows=slim,gri
d=[10,10]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "ex1[1]:=display(\{co
nstrain,gradg\},title=`Constrain and its gradient vectors`):" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 9 "npts:=20:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 154 "arrg:=seq(arrow([cos(i*2*Pi/npts),sin(i*2*Pi/npts)],\neval(norm
alize(grad(g(x,y),[x,y])),\{x=cos(i*2*Pi/npts),y=sin(i*2*Pi/npts)\})\n
,color=green),i=0..npts):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "ex1[2]
:=display(\{constrain,arrg\},title=`Constrain g(x,y)=x^2+y^2`):" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "levelsf:=contourplot(f(x,y),x=-2..
2,y=-2..2,\ncontours=[0.2,0.5,1,1.5,2,2.5,3,4,5],numpoints=1000,color=
red):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "ex1[3]:=display(\{constrai
n,arrg,levelsf\},\ntitle=`Level curves of f(x,y)=x^2+2*y^2`):" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 360 "for n from 1 to 5 do \nex1[n+3]:=d
isplay(\{constrain,arrg,\nimplicitplot(f(x,y)=(n/2),x=-2..2,y=-2..2,co
lor=red,thickness=2,\nnumpoints=2000),\nseq(arrow([sqrt(n/2)*cos(i*2*P
i/npts),1/sqrt(2)*sqrt(n/2)*sin(i*2*Pi/npts)],\neval(normalize(grad(f(
x,y),[x,y])),\{x=sqrt(n/2)*cos(i*2*Pi/npts),\ny=1/sqrt(2)*sqrt(n/2)*si
n(i*2*Pi/npts)\}),color=red),i=0..npts)\},\ntitle=``)\nod:" }}}}{SECT 
1 {PARA 5 "" 0 "" {TEXT -1 11 "Example 2: " }{XPPEDIT 18 0 "f(x,y)=x^2
+3*y^2-3*x*y" "6#/-%\"fG6$%\"xG%\"yG,(*$F'\"\"#\"\"\"*&\"\"$F,*$F(F+F,
F,*(F.F,F'F,F(F,!\"\"" }{TEXT -1 26 " subject to the constrain " }
{XPPEDIT 18 0 "g(x,y)=x^2/4+y^2" "6#/-%\"gG6$%\"xG%\"yG,&*&F'\"\"#\"\"
%!\"\"\"\"\"*$F(F+F." }{TEXT -1 2 "=1" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 47 "g:=(x,y)->x^2/4+y^2:\nf:=(x,y)->x^2+3*y^2-3*x*y:" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "constrain:=implicitplot(g(x,y)=1,x
=-2..2,y=-1.5..1.5,\nnumpoints=800,color=blue,thickness=3,scaling=cons
trained):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 185 "gradg:=gradplot(g(x,y
),x=-3..3,y=-2..2,color=green,arrows=slim,grid=[10,10]):\nlevelsf:=con
tourplot(f(x,y),x=-2..2,y=-1.5..1.5,contours=[seq(0.2+i/4,i=0..30)],\n
color=red,numpoints=2000):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "gradf
:=gradplot(f(x,y),x=-2..2,y=-1.5..1.5,arrows=SLIM,color=red,grid=[10,1
0]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "npts:=26:" }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 158 "arrg:=seq(arrow([2*cos(i*2*Pi/npts),sin(i*2*Pi/np
ts)],\neval(normalize(grad(g(x,y),[x,y])),\{x=2*cos(i*2*Pi/npts),y=sin
(i*2*Pi/npts)\})\n,color=green),i=0..npts):" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 156 "arrf:=seq(arro
w([2*cos(i*2*Pi/npts),sin(i*2*Pi/npts)],\neval(normalize(grad(f(x,y),[
x,y])),\{x=2*cos(i*2*Pi/npts),y=sin(i*2*Pi/npts)\})\n,color=red),i=0..
npts):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 266 "arrt:=seq(arrow([2*cos(i
*2*Pi/npts),sin(i*2*Pi/npts)],\neval(evalm(normalize(grad(f(x,y),[x,y]
))-\ndotprod(normalize(grad(f(x,y),[x,y])),normalize(grad(g(x,y),[x,y]
)))*normalize(grad(g(x,y),\n[x,y]))),\{x=2*cos(i*2*Pi/npts),y=sin(i*2*
Pi/npts)\}),color=yellow),i=0..npts):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 561 "ex2[1]:=display(\{constrain,levelsf\},title=`Constrain g(x,y)=1
/4*x^2+y^2 and levels of f(x,y)=x^2+3*y^2-3*x*y`):\nex2[2]:=display(\{
constrain,levelsf,gradf\},title=`Gradients of f(x,y)`):\nex2[3]:=displ
ay(\{constrain,levelsf,arrg,gradf\},title=`Gradient of the constrain`)
:\nex2[4]:=display(\{constrain,levelsf,arrg,arrf\},title=`Gradient of \+
f(x,y) along the constrain curve`):\nex2[5]:=display(\{constrain,level
sf,arrg,arrf,arrt\},title=`Component of grad(f) along the constrain`):
\nex2[6]:=display(\{constrain,levelsf,arrt\},title=`Component of grad(
f) along the constrain`):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 
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\\=&=&=g?FK7$7$Fbe]nFjadlFae]n7$Fge]n7$$\"3U666htEr:Fj]m$\"3y_xxx_@w6F
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i_flFb]pF[^pFbcglF[^p" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 
45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 11 "Example 2: " }{XPPEDIT 18 0 "f(x
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$F(F+F,F,*(F.F,F'F,F(F,!\"\"" }{TEXT -1 26 " subject to the constrain \+
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\"\"%!\"\"\"\"\"*$F(F+F." }{TEXT -1 2 "=1" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 46 "display([seq(ex2[i],i=1..6)],insequence=true);" }}
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