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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "read \"multifac_1.2.mpl\":\npath:=
\"examples\\/\":" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "Zeng's sixth \+
example" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "p1:=x^2+x*y-y^2+
1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p1G,**$)%\"xG\"\"#\"\"\"F**&F
(F*%\"yGF*F**$)F,F)F*!\"\"F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 28 "p2:=expand((x^3+1)*(y^2-2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%#p2G,**&)%\"xG\"\"$\"\"\")%\"yG\"\"#F*F**&F-F*F'F*!\"\"*$F+F*F*F-F
/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "p3:=x^3-y^3-3*x*y^2+2;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p3G,**$)%\"xG\"\"$\"\"\"F**$)%
\"yGF)F*!\"\"*(F)F*F(F*)F-\"\"#F*F.F1F*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 57 "pp:=randpoly([x,y],degree=4,coeffs=rand(-5..5),terms=
10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ppG,4\"\"%!\"\"*&\"\"#\"\"
\"%\"yGF*F'*&\"\"$F*)%\"xGF)F*F**(F&F*F/F*F+F*F**(F&F*F.F*F+F*F**&F-F*
)F+F-F*F'*&F-F*)F/F&F*F'*(F&F*)F/F-F*F+F*F**&F/F*F3F*F'" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "cleanF14:=p1^4*p2^3*p3:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "noiseF14:='`*`'(10^(-8),pp):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "save cleanF14, noiseF14, cat
(path,\"exF14\");" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "zengex
6:=evalf(expand(cleanF14+noiseF14)):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 68 "The example has univariate  multiple factors  (x^3+1)^3, (y^2-2
)^3. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "t14:=time():\nres:
=multifac_cont(zengex6,[x,y],8);\nt14:=time()-t14;" }}{PARA 6 "" 1 "" 
{TEXT -1 108 "` the biggest gap, the last r-th singular values and the
 number of factors `, 19325210.41, .6474953137e-9, 5" }}{PARA 6 "" 1 "
" {TEXT -1 58 "` The time for computing the number of factors*****`, .
211" }}{PARA 6 "" 1 "" {TEXT -1 53 "`The time for the entire factoriza
tion******`, 17.986" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$resG7(7$,0$
\"+.h-\\c!\"*\"\"\"*&$\"3[&**GMk#Ghq!#=F+)%\"yG\"\"'F+!\"\"*&$\"3'4X`z
c=]G\"!#EF+)F1\"\"&F+F+*&$\"37!)y!RdpnB%!#<F+)F1\"\"%F+F+*&$\"3$**=%)e
Cyq`#F7F+)F1\"\"$F+F+*&$\"3I'Qs:;RNZ)F=F+)F1\"\"#F+F3*&$\"3w=5@,WJ.>F7
F+F1F+F+F+7$,6*&^$$!32Md.W9@9%)F/$\"\"!FTF+FCF+F+*(^$$!3JBh\\VMECDF=FS
F+%\"xGF+FHF+F+*(^$$\"3)z\")[<M\"4FYF7FSF+)FYFIF+F1F+F+*&^$$\"3vq&epZ6
UT)F/FSF+)FYFDF+F+*&^$$!3mxp!fy2)R<F7FSF+FHF+F+*(^$$\"3<n#Q'eUi,kF7FSF
+FYF+F1F+F+*&^$$\"3c<!*[`vizn!#FFSF+FhnF+F+*&^$$\"3_6@KGBWd')!#GFSF+F1
F+F+*&^$$!334B7.aK[fF7FSF+FYF+F+^$$\"+&HUGo\"F*FSF+F+7$,(*&^$$!3sOoCHD
$RC\"F7$\"3CKr?6ey'G#F7F+F1F+F+*&^$$\"3]y\\-xq+inF/$!3IbhzWFz\\8F=F+FY
F+F+^$$\"+HWa3$)!#5$\"+hL]g7F*F+FD7$,(*&^$$\"3_<5T,C!R;$F_p$\"3;(=(>_J
]-EF7F+F1F+F+*&^$$\"3&)*Q1>A6sK$!#>$!3n57=[7L4:F=F+FYF+F+^$$!+]_vB8F*$
\"+C8^esFhqF+FD7$,(*&^$$!3w$3,GN%yuqF7FSF+F1F+F+*&^$$!3)eKFSV[)yNF=FSF
+FYF+F+^$$!+K%[)yNF*FSF+FD7$,.*&^$$\"3mo+O$fyX2\"F=FSF+FHF+F+*(^$$!3E6
t`s$yX2\"F=FSF+FYF+F1F+F+*&^$$!3CBG13'yX2\"F=FSF+FhnF+F+*&^$$!3Sdl*R_!
yr5!#CFSF+F1F+F+*&^$$\"3J*>s@)f1=?F^uFSF+FYF+F+^$$!+H&yX2\"F*FSF+F?" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$t14G$\"&;N(!\"$" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 84 "result:=expand(res[1][1]*res[2][1]*res[3]
[1]^3*res[4][1]^3*res[5][1]^3*res[6][1]^4):" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 37 "backward_error(result,zengex6,[x,y]);" }}{PARA 6 "
" 1 "" {TEXT -1 50 "Approximate factorization is not a real polynomial
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"cG^$$!+'4C.y%!\")$!+L?'GF%!\"(
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+#)[(z3#!#;" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 14 "h1:=res[6][1];" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%#h1G,.*&^$$\"3mo+O$fyX2\"!#<$\"\"!F,\"\"\")%\"yG\"\"#
F-F-*(^$$!3E6t`s$yX2\"F*F+F-%\"xGF-F/F-F-*&^$$!3CBG13'yX2\"F*F+F-)F5F0
F-F-*&^$$!3Sdl*R_!yr5!#CF+F-F/F-F-*&^$$\"3J*>s@)f1=?F?F+F-F5F-F-^$$!+H
&yX2\"!\"*F+F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "backward_
error(h1,p1,[x,y]);" }}{PARA 6 "" 1 "" {TEXT -1 50 "Approximate factor
ization is not a real polynomial" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%
\"cG$!+E&yX2\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+/L?r8!#;" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "h2:=expand(res[1][1]*res[3]
[1]^3*res[4][1]^3*res[5][1]^3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 38 "backward_error(h2,expand(p2^3),[x,y]);" }}{PARA 6 "" 1 "" 
{TEXT -1 50 "Approximate factorization is not a real polynomial" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"cG^$$!+6;!3E%!\")$!+*G!\\3Q!\"(" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+)3)yqB!#<" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 14 "h3:=res[2][1];" }}{PARA 12 "" 1 "" {XPPMATH 
20 "6#>%#h3G,6*&^$$!32Md.W9@9%)!#=$\"\"!F,\"\"\")%\"yG\"\"$F-F-*(^$$!3
JBh\\VMECD!#<F+F-%\"xGF-)F/\"\"#F-F-*(^$$\"3)z\")[<M\"4FY!#EF+F-)F6F8F
-F/F-F-*&^$$\"3vq&epZ6UT)F*F+F-)F6F0F-F-*&^$$!3mxp!fy2)R<F=F+F-F7F-F-*
(^$$\"3<n#Q'eUi,kF=F+F-F6F-F/F-F-*&^$$\"3c<!*[`vizn!#FF+F-F>F-F-*&^$$
\"3_6@KGBWd')!#GF+F-F/F-F-*&^$$!334B7.aK[fF=F+F-F6F-F-^$$\"+&HUGo\"!\"
*F+F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "backward_error(h3,
p3,[x,y]);" }}{PARA 6 "" 1 "" {TEXT -1 50 "Approximate factorization i
s not a real polynomial" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"cG$\"+t
9@9%)!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+/pgIK!#=" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "facF14:=res:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 44 "save facF14, t14, cat(path,\"exF14-factors\");
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}}{MARK "23 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
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