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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "MCS 595 Thursday 4 Septemb
er 2003" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 600 "This very simple \+
worksheet illustrates the main theorem of elimination theory applied t
o justify numerical homotopy continuation methods.  For a simple homot
opy, we set up the discriminant system which defines all singular poin
ts.  From this system we eliminate the orginal independent variables t
o arrive at one polynomial in the continuation parameter t.  This show
s there are only finitely many singular values for the continuation pa
rameter t.  By random choice of complex coefficients in the start syst
em we can ensure that all roots of this discriminant polynomial will m
iss the interval [0,1]." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 60 "Consider the intersection of two simple quadratic equatio
ns:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "f
 := [x^2 + 4*y^2 - 4,2*y^2- x];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
\"fG7$,(*$)%\"xG\"\"#\"\"\"F+*&\"\"%F+)%\"yGF*F+F+F-!\"\",&*$F.F+F*F)F
0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "By the \"method of degenera
tion\", we replace each equation in f by an equation of the same degre
e, but much simpler:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 24 "g := [x^2 - 1, y^2 - 1];" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"gG7$,&*$)%\"xG\"\"#\"\"\"F+F+!\"\",&*$)%\"yGF*F+F+F
+F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "So g is the start system i
n the homotopy:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "h := t*f + (1-t)*g;" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%\"hG,&*&%\"tG\"\"\"7$,(*$)%\"xG\"\"#F(F(*&\"\"%F()%\"yGF.F(F(F0!\"
\",&*$F1F(F.F-F3F(F(*&,&F(F(F'F3F(7$,&F+F(F(F3,&F5F(F(F3F(F(" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 151 "The continuation parameter t will
 move from 0 to 1.  At t=0, we have the start system, at t=1, we have \+
the target system (the system we want to solve)." }{MPLTEXT 1 0 0 "" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "For further manipulations in Map
le, we will expand the homotopy:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 16 "eh := expand(h);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#ehG7$,**$)%\"xG\"\"#\"\"\"F+F+!\"\"*&\"\"$F+%\"tGF+F
,*(\"\"%F+)%\"yGF*F+F/F+F+,,*$F2F+F+*&F2F+F/F+F+F+F,F/F+*&F)F+F/F+F," 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "We are interested in the singu
lar points of the homotopy.  Therefore, we compute the Jacobian matrix
 of the homotopy:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 80 "jh := matrix(2,2,[[diff(eh[1],x),diff(eh[1],y)],[diff
(eh[2],x),diff(eh[2],y)]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#jhG-
%'matrixG6#7$7$,$%\"xG\"\"#,$*&%\"yG\"\"\"%\"tGF0\"\")7$,$F1!\"\",&F/F
,*(F,F0F/F0F1F0F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 88 "Then all sin
gular points of the homotopy are defined by the following polynomial s
ystem:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
37 "sys := [eh[1],eh[2],linalg[det](jh)];" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$sysG7%,**$)%\"xG\"\"#\"\"\"F+F+!\"\"*&\"\"$F+%\"tGF+
F,*(\"\"%F+)%\"yGF*F+F/F+F+,,*$F2F+F+*&F2F+F/F+F+F+F,F/F+*&F)F+F/F+F,,
(*&F)F+F3F+F1**F1F+F)F+F3F+F/F+F+*(\"\")F+F3F+)F/F*F+F+" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 119 "With a pure lexicographical Groebner bas
is we can eliminate the variables x and y, to arrive at a polynomial i
n t only:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 40 "gb := grobner[gbasis](sys,[x,y,t],plex);" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%#gbG7&,2%\"xG\"#e*&\"#g\"\"\")%\"tG\"\"'F+F+*&\"#LF+)
F-\"\"&F+F+*&\"$v\"F+)F-\"\"%F+F+*&\"$;\"F+)F-\"\"$F+F+*&\"$`#F+)F-\"
\"#F+F+*&\"#JF+F-F+F+\"#?F+,2*$)%\"yGF>F+F8*&\"$O$F+F5F+!\"\"*&\"#()F+
F9F+F+*&\"#\\F+F=F+F+*&\"$)>F+F-F+F+\"#zFH*&\"$C$F+F,F+FH*&\"$(QF+F1F+
FH,,FEF+*&FEF+F-F+F+*(\"\"(F+FEF+F=F+F+*(FWF+FEF+F9F+F+*(F6F+FEF+F5F+F
+,2FHF+F-F+*&\"#5F+F9F+F+*&\"#HF+F1F+F+*&\"#8F+F5F+F+*&F2F+F=F+FH*&\"#
7F+)F-FWF+F+*&\"#@F+F,F+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "We \+
find this polynomial in t as the last polynomial of our lexicographica
l Groebner basis:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "gb[nops(gb)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,2!
\"\"\"\"\"%\"tGF%*&\"#5F%)F&\"\"$F%F%*&\"#HF%)F&\"\"&F%F%*&\"#8F%)F&\"
\"%F%F%*&F.F%)F&\"\"#F%F$*&\"#7F%)F&\"\"(F%F%*&\"#@F%)F&\"\"'F%F%" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "As the degree of this \"discrimina
nt polynomial\" is seven, we can expect seven complex roots:" }
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "fsolve(
gb[nops(gb)],t,complex);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6)^$$!+Yw`=)
)!#5$!+wD+x\"*F&^$F$$\"+wD+x\"*F&^$$!+!p*f6?F&$!+t$*Gx))F&^$F-$\"+t$*G
x))F&^$$\"+nXw`o!#7$!+Gt'z#RF&^$F5$\"+Gt'z#RF&$\"+\"Q*>BSF&" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 133 "We should be particularily troubl
ed by the root around 0.4.  This means that as t moves from 0 to 1, we
 will encounter a singularity." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 200 "We will choose a random constant, which we call g
amma.  The instruction which is commented out shows how we should real
ly do it, but in view of Maple's \"grobner\", we just choose a much si
mpler gamma. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "unprotect(
gamma):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "#gamma := exp(st
ats[random,uniform[0,2*Pi]](1)*I);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "gamma := 1+I;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&g
ammaG^$\"\"\"F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "We have a new \+
homotopy and a new discriminant system, we go through all the same mot
ions as above:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "nh := t*f + gamma*(1-t)*g;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#nhG,&*&%\"tG\"\"\"7$,(*$)%\"xG\"\"#F(F(*&\"\"%F()%\"
yGF.F(F(F0!\"\",&*$F1F(F.F-F3F(F(*(^$F(F(F(,&F(F(F'F3F(7$,&F+F(F(F3,&F
5F(F(F3F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "enh := expan
d(nh);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$enhG7$,0*$)%\"xG\"\"#\"\"
\"F+*&^#F+F+F(F+F+*(^#!\"\"F+F(F+%\"tGF+F+^$F0F0F+*&\"\"$F+F1F+F0*&F-F
+F1F+F+*(\"\"%F+)%\"yGF*F+F1F+F+,2*$F8F+F+*&F-F+F8F+F+*&F8F+F1F+F+*(F/
F+F8F+F1F+F+F2F+F1F+F5F+*&F)F+F1F+F0" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 85 "jnh := matrix(2,2,[[diff(enh[1],x),diff(enh[1],y)],[d
iff(enh[2],x),diff(enh[2],y)]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
$jnhG-%'matrixG6#7$7$,(%\"xG\"\"#*&^#F,\"\"\"F+F/F/*(^#!\"#F/F+F/%\"tG
F/F/,$*&%\"yGF/F3F/\"\")7$,$F3!\"\",*F6F,*&F.F/F6F/F/*(F,F/F6F/F3F/F/*
(F1F/F6F/F3F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "nsys := \+
[enh[1],enh[2],linalg[det](jnh)];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>
%%nsysG7%,0*$)%\"xG\"\"#\"\"\"F+*&^#F+F+F(F+F+*(^#!\"\"F+F(F+%\"tGF+F+
^$F0F0F+*&\"\"$F+F1F+F0*&F-F+F1F+F+*(\"\"%F+)%\"yGF*F+F1F+F+,2*$F8F+F+
*&F-F+F8F+F+*&F8F+F1F+F+*(F/F+F8F+F1F+F+F2F+F1F+F5F+*&F)F+F1F+F0,.*(^#
\"\")F+F)F+F9F+F+**\"#7F+F)F+F9F+F1F+F+**^#!\"%F+F)F+F9F+F1F+F+**FGF+F
)F+)F1F*F+F9F+F+**F7F+F)F+FJF+F9F+F0*(FCF+F9F+FJF+F+" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 42 "ngb := grobner[gbasis](nsys,[x,y,t],plex)
;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$ngbG7&,>%\"xG\"$;\"*&^#\"$x#\"
\"\")%\"tG\"\"'F,F,*&\"#'*F,F-F,!\"\"*&^#\"$)yF,)F.\"\"&F,F,*&\"%o<F,F
6F,F2*&^#!%?IF,)F.\"\"%F,F,*&\"%Q<F,F=F,F2*&^#\"\"#F,)F.\"\"$F,F,*&\"$
U)F,FDF,F,*&^#!%w9F,)F.FCF,F,*&\"%$y\"F,FKF,F,*&^#\"%\"[\"F,F.F,F,*&\"
%$4\"F,F.F,F2^$\"#')!$s%F,,>*$)%\"yGFCF,F(*&^#\"$$=F,F-F,F,*&\"$O#F,F-
F,F,*&^#\"%W>F,F6F,F,*&\"$!=F,F6F,F,*&^#\"$c&F,F=F,F,*&\"%5MF,F=F,F2*&
^#!$e)F,FDF,F,*&\"$'eF,FDF,F,*&^#!%q?F,FKF,F,*&\"$T'F,FKF,F2*&^#\"%l;F
,F.F,F,*&\"%V7F,F.F,F,^$!$'e!$!GF,,4*&FYF,F=F,F7*(^#\"#7F,FDF,FYF,F,*&
FYF,FDF,F,*(^#F>F,FKF,FYF,F,*(FEF,FYF,FKF,F2*(FBF,F.F,FYF,F,*(\"\"*F,F
YF,F.F,F2*&^#!\"$F,FYF,F,FYF,,>*$)F.\"\"(F,F7*&^#\"#FF,F-F,F,*&\"#@F,F
-F,F,*&^#\"#]F,F6F,F,*&\"#gF,F6F,F2*&^#!#_F,F=F,F,*&\"#;F,F=F,F2*&^#!#
:F,FDF,F,*&\"#?F,FDF,F2*&^#!\"#F,FKF,F,*&\"#aF,FKF,F,*&^#\"#=F,F.F,F,*
&\"#EF,F.F,F2^$FC!\"'F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "
fsolve(ngb[nops(ngb)],t,complex);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6)^
$$!+6)yf!\\!\"*$!+gI$\\n#F&^$$!+xJ)4G'!#5$\"+d+s_mF,^$$!+#=*)4H#F,$!+t
\"o#>?F&^$$\"+T_qbGF,$!+9`/&y#F,^$$\"+=rE;PF,$!+6I*\\n(F,^$$\"+I\"[R?%
F,$!+RvE?`F,^$$\"+x*Re&[F,$\"+U\")fp?F," }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 109 "Now we see that none of the roots is real.  So as t move
s from 0 to 1, we will not encounter any singularity." }{MPLTEXT 1 0 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "2 0 0" 
575 }{VIEWOPTS 1 1 0 2 1 1805 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }