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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "L-21 MCS 563 Monday 11 Oct
ober 2004" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "As running examp
le in the first half of the lecture we looked at the following polynom
ial:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "
f := x^3 - x^2 + y^2;" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 30 "1. Abou
t good parametrizations" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "For plo
tting the " }{TEXT 256 4 "real" }{TEXT -1 77 " curve defined by the po
lynomial f, we may need another representation for f." }{MPLTEXT 1 0 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "plots[implicitplot](
f,x=-2..2,y=-2..2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 159 "The plot \+
does not look so good, because of the singularity at the origin we enc
ounter in rectangular coordinates.  Therefore, let us move to polar co
ordinates:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 36 "rf := subs(x=r*cos(t),y=r*sin(t),f);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 22 "rt := solve(rf/r^2,r);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 29 "plots[polarplot](rt,t=-1..1);" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 103 "To sample in complex space, we just cut the hyper
surface defined by the polynomial  with a random line." }{MPLTEXT 1 0 
0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 30 "2. Nearest points to a s
urface" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 269 "The technique of Lagran
ge multipliers is very useful to find the point on the hypersurface cl
osest to a given point.  Here we illustrate this technique with the gi
ven point (1,1) and look for the point on the hypersurface defined by \+
f, closest to the given point (1,1)." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 37 "g := (x-1)^2 + (y-1)^2; # to minimize" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "sys := \{diff(g,x) - lambda*diff(f,
x), diff(g,y) - lambda*diff(f,y),f\};" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 73 "The command fsolve computes one solution using floating-p
oint arithmetic:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "fsolve(sys,\{x,y,lambda\});" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 55 "Let us verify by computing all solutions to the system.
" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "gb :
= Groebner[gbasis]([op(sys)],plex(x,y,lambda));" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 29 "sols := fsolve(gb[1],lambda);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 80 "We know there are four solutions, but let
 us not bother about the complex roots:" }{MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "sys2 := subs(lambda=sols[2],gb);" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "solve(\{sys2[2],sys2[3]\},
\{x,y\});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "We can already see t
hat this solution is farther away from (1,1) than the other solution.
" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
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