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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "# A Maple Session to
 compute the next approximation" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 28 "with(plots):with(plottools):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 70 "f:=x^3-3*x*y+y^3; # This is the function on problems \+
#7 & #5 of test 2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG,(*$)%\"xG
\"\"$\"\"\"F**(F)F*F(F*%\"yGF*!\"\"*$)F,F)F*F*" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 54 "P0:=[1/2,9/8]; # This is roughly the Q in #5 o
f test 2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#P0G7$#\"\"\"\"\"##\"\"*
\"\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "x0:=P0[1]; y0:=P0[
2]; # P0 is a Maple list, x0 is the first element, y0 the 2nd" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G#\"\"\"\"\"#" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%#y0G#\"\"*\"\")" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 54 "z0:=eval(f,\{x=x0,y=y0\}); #For later use in the 3d p
lot" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z0G#!#r\"$7&" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "f_x:=diff(f,x); f_y:=diff(f,y); # T
his the Maple way to take derivatives" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%$f_xG,&*$)%\"xG\"\"#\"\"\"\"\"$*&F+F*%\"yGF*!\"\"" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%$f_yG,&%\"xG!\"$*&\"\"$\"\"\")%\"yG\"\"#F*F*" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "a:=eval(f_x,\{x=x0,y=y0\});
 # This computes f_x(x0,y0)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG#
!#@\"\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "b:=eval(f_y,\{x
=x0,y=y0\}); # This computes f_y(x0,y0)" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"bG#\"$Z\"\"#k" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "
u:=[-a, -b]; # this is the direction of steepest descent" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%\"uG7$#\"#@\"\")#!$Z\"\"#k" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 64 "PuLine:=[x0-a*t,y0-b*t,t=0..1]; #Something \+
you can put into plot" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'PuLineG7%,
&#\"\"\"\"\"#F(*&#\"#@\"\")F(%\"tGF(F(,&#\"\"*F-F(*&#\"$Z\"\"#kF(F.F(!
\"\"/F.;\"\"!F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 286 "display
(\ncontourplot(f,x=-1..2,y=-1..2,contours=[z0, -1/2,0,1,2,4,8]), #Note
 contours\nplot(PuLine,color=green), # This is the Puline\ntextplot([x
0,y0,\"P0\"],color=green), # The text label P0\narrow([x0,y0],vector(u
),.1,.2,.2,color=red) # An arrow, just to be fancy\n); # end display c
ommand" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "g:=eval(f,\{x=x0-
a*t,y=y0-b*t\}); # g(t) = f(x(t),y(t))" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"gG,(*$),&#\"\"\"\"\"#F**&#\"#@\"\")F*%\"tGF*F*\"\"$F*F**(F1F
*F(F*,&#\"\"*F/F**&#\"$Z\"\"#kF*F0F*!\"\"F*F:*$)F3F1F*F*" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "diff(g,t); # looking for critical p
oints of g" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**$),&#\"\"\"\"\"#F(*&#
\"#@\"\")F(%\"tGF(F(F)F(#\"#jF-#\"$$p\"$G\"!\"\"*&#\"%h#*\"$c#F(F.F(F(
*&#\"$T%\"#kF(*$),&#\"\"*F-F(*&#\"$Z\"F<F(F.F(F4F)F(F(F4" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "expand(diff(g,t)); # force maple to
 expand the powers" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(#!&L)\\\"%'4%
\"\"\"*&#\"(N[^\"\"&%Q;F'%\"tGF'F'*&#\"(F`p%\"'W@EF')F,\"\"#F'F'" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "solve(diff(g,t)=0,t); # find
 the critical point" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&#!%g\"*\"%\\N
\"\"\"*&#\"#KF&F'-%%sqrtG6#\"&%H!*F'F',&F$F'*&#F*F&F'*$F+F'F'!\"\"" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "# There are two critical po
ints, we want the local min which " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "# has to be the first. Why?" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 62 "t1:=solve(diff(g,t)=0,t)[1]; # A way to pull the
 first one off" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#t1G,&#!%g\"*\"%\\
N\"\"\"*&#\"#KF(F)-%%sqrtG6#\"&%H!*F)F)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 80 "# If f was a quadratic instead of a cubic, t0 would b
e a fraction and not a surd" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
54 "x1:=eval(x0-a*t,\{t=t1\}); # new approximation's x value" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#x1G,&#!%@@\"$Q$\"\"\"*&#\"\"%\"$p\"F)-%%s
qrtG6#\"&%H!*F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "y1:=ev
al(y0-b*t,\{t=t1\}); # new approximation's y value" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%#y1G,&#\"%#>\"\"$p\"\"\"\"*&#\"\"(\"$Q$F)*$-%%sqrtG
6#\"&%H!*F)F)!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "P1:=[
x1,y1]; # new approximation" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#P1G7
$,&#!%@@\"$Q$\"\"\"*&#\"\"%\"$p\"F*-%%sqrtG6#\"&%H!*F*F*,&#\"%#>\"F.F*
*&#\"\"(F)F**$F/F*F*!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
63 "z1:=eval(f,\{x=x1,y=y1\}); # value of f(P1) for contourplot below
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z1G,(*$),&#!%@@\"$Q$\"\"\"*&#\"
\"%\"$p\"F,-%%sqrtG6#\"&%H!*F,F,\"\"$F,F,*(F5F,F(F,,&#\"%#>\"F0F,*&#\"
\"(F+F,*$F1F,F,!\"\"F,F>*$)F7F5F,F," }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 283 "display(\ncontourplot(f,x=0..1.5,y=0..1.5,contours=[
z0, z1,-1/2,0,1,2,4,8]), #Notice z0 z1\nline(P0,P1,color=green), # lin
e seqment P0 to P1\ntextplot([x0,y0,\"P0\"],color=green), # starting p
oint\ntextplot([x1,y1,\"P1\"],color=blue), # next approximation\nscali
ng=constrained\n); #end display" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 348 "display(\nplot3d(f,x=0..1.5,y=0..1.5,style=patchcontour,shadi
ng=z,\ncontours=[z0, evalf(z1),-1/2,0,1,2,4,8]), #Note z0, z1 \nspacec
urve([x0-a*t,y0-b*t,g],t=0..t1,color=green, #\nthickness=3), #plot g a
s a spacecurve\ntextplot3d([x0,y0,z0,\"P0\"],color=green), #starting p
oint\ntextplot3d([x1,y1,z1,\"P1\"],color=red) #next approximation poin
t\n); #end display" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
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