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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "L-27 MCS 563 Monday 25 Oct
ober 2004" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "In this workshee
t we illustrate Example 8.23 in the textbook." }{MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "p1 := x^2 + 4*y^2 - 4;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "p2 := 9*x^2 + y^2 - 2*y - 8;
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "P := \{p1,p2\};" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 176 "In this example we will show the \+
infeasibility of \{1,y,x,y^2\} as a normal set for this system.  We fi
rst compute a Groebner basis, to have an exact representation of the z
eros." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 
"with(Groebner);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "gb := g
basis(P,plex(x,y));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "yval
s := solve(gb[1],y);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "g2a
 := subs(y=yvals[1],gb[2]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
30 "g2b := subs(y=yvals[2],gb[2]);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "xval1 := solve(g2a,x);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "xval2 := solve(g2b,x);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 36 "The four solutions are listed below:" }{MPLTEXT 1 0 0 "" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "sols := [[x=xval1[1],y=y
vals[1]],[x=xval1[2],y=yvals[1]],[x=xval2[1],y=yvals[2]],[x=xval2[2],y
=yvals[2]]];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "As complicated as
 these zeros seem, it is a good idea to verify what the residuals are:
" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "valu
es := [seq(subs(op(sols[k]),P),k=1..4)];" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 55 "map(t->simplify(t[1]),values);  # verify first equati
on" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "map(t->simplify(t[2])
,values);  # verify second equation" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 409 "To show that the set \{1,y,x,y^2\} is cannot be used as a norm
al set for this system, it suffices to find an example, for which no u
nique normal form exists.  Let us take x^2 and apply interpolation -- \+
we forget about the Groebner basis for now -- at the zeros to write x^
2 as a linear combination of the monomials in the set, interpolating a
 the roots.  The general equation in the four unknown coefficients is
" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "equ \+
:= a[0]*1 + a[1]*y + a[2]*x + a[3]*y^2 = x^2;" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 39 "By substitution we generate the system:" }{MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "sys := seq(subs(op
(sols[k]),equ),k=1..4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "
M := LinearAlgebra[GenerateMatrix]([sys],[a[0],a[1],a[2],a[3]]);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "rr := LinearAlgebra[ReducedR
owEchelonForm](M[1]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 120 "It take
s some simplification to see that this matrix is actually singular, so
 no unique solution if there is a solution." }{MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "simplify(rr);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 220 "Recall that the Groebner basis was used \+
to compute the multiplication matrices, which can be arranged into a b
order basis.  Once a Groebner basis has been computed, the algorithms \+
are fairly simple, and are listed below:" }{MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "interface(verboseproc=3);" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "print(SetBasis);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "print(MulMatrix);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "33" 0 }
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