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{SECT 0 {EXCHG {PARA 261 "> " 0 "" {MPLTEXT 1 268 8 "restart;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "path:=\"\":" }}}{EXCHG {PARA 
261 "> " 0 "" {MPLTEXT 1 268 34 "read cat(path,\"multifac_1.3.mpl\"):
" }}}{EXCHG {PARA 261 "> " 0 "" {MPLTEXT 1 268 25 "interface(rtablesiz
e=80);" }}}{EXCHG {PARA 261 "" 0 "" {TEXT 270 183 "The purpose of this
 worksheet is to generate instances of multivariate polynomials with c
omplex coefficients.  Executing this worksheet generates the files exf
1, exf2, exf3, and exf4." }}}{SECT 0 {PARA 263 "" 0 "" {TEXT 271 41 "T
he Jacobian for a Stewart-Gough Platform" }}{PARA 261 "" 0 "" {TEXT 
272 58 "We derive the Jacobian for a platform, first symbolically." }}
{SECT 0 {PARA 264 "" 0 "" {TEXT 273 22 "2. Numerical Instances" }}
{PARA 261 "" 0 "" {TEXT 272 109 "We have three vectors to specify: the
 position vector and the locations at endplate and the baseplate joint
s." }}{EXCHG {PARA 261 "" 0 "" {TEXT 272 154 "For the purposes of appl
ying homotopy continuation to factor the polynomials, we generate rand
om complex number of modulus one, using 16 digits precision." }}}
{SECT 0 {PARA 266 "" 0 "" {TEXT 275 9 "Example 1" }}{PARA 261 "" 0 "" 
{TEXT 272 76 "In this first instance, we pick the locations and positi
on vector at random:" }}{EXCHG {PARA 261 "> " 0 "" {MPLTEXT 1 268 37 "
read cat(path,\"verschelde/svw_exf1\"):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 11 "Digits:=20;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'Dig
itsG\"#?" }}}{EXCHG {PARA 261 "> " 0 "" {MPLTEXT 1 268 22 "ord:=[op(in
dets(f1))];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ordG7&&%\"qG6#\"\"!&
F'6#\"\"#&F'6#\"\"$&F'6#\"\"\"" }}}{EXCHG {PARA 261 "> " 0 "" 
{MPLTEXT 1 268 43 "F1:=subs(\{q[1]=x,q[2]=y,q[0]=z,q[3]=1\},f1):" }}}
{EXCHG {PARA 261 "> " 0 "" {MPLTEXT 1 268 25 "F1:=evalf(F1/norm(F1,2))
:" }}}{EXCHG {PARA 261 "> " 0 "" {MPLTEXT 1 268 11 "st:=time();" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#stG$\"&eQ\"!\"$" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 37 "fac_Ver_f1:=multifac1(F1,[x,y,z],10);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%hnthe~gap~is~too~small,~we~think~the~
polynomial~is~irreducible.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%hnthe~
gap~is~too~small,~we~think~the~polynomial~is~irreducible.G" }}{PARA 
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5P/@iVBc!eZ%F/F0F1F0F7F0F0*(^$$\"50Ni8(p*4H>VF/$!5+%\\J5M$)4f*QF/F0F\\
oF0F7F0F0*(^$$\"5T3XI8kH4:xF/$\"5xFr[8kx$4H#F/F0F\\oF0F2F0F0*(^$$\"5D&
[)e%y\"4n\"G#F/$!5<x#G&=-PF()eF/F0FUF0F>F0F0*(^$$\"5b\"Hm7wR@'=8F,$!5@
@1wuIB([L'F/F0F4F0F1F0F0*(^$$!5$)G6dkr&*43>F/$\"5_6uZY*fC+9\"F,F0FcuF0
F5F0F0*(^$$!55BNK<WIO8lF/$\"5o\\X*4%4T#Q.&F/F0FEF0F>F0F0*(^$$!5C+4bVv!
fw+(F/$\"5(eT)3jQJe<=FisF0FcuF0FFF0F0*&^$$!5U%Ry=++&y@UF/$!5V\"G'G#R%H
g*y\"F/F0FFF0F0*(^$$!50j^ea=e&e0)F/$\"5CoffU>p$y1%F/F0F4F0F7F0F0*&^$$
\"5YNi.'*o7#=T$F/$\"54!zh*zx'RW4\"F/F0FEF0F0F07$,6*&$\"5$)z***********
***\\F,F0FUF0F0*($\"5y\"><`HO*>9!)!#dF0F7F0F2F0!\"\"*&$\"5a-%*********
****\\F,F0FFF0F0*($\"56h-I2Q')QMO!#QF0F2F0F5F0F^]m*($\"5MA_6[z<OXAFe]m
F0F7F0F5F0F0*&$\"5Od/++++++]F,F0F4F0F0*&$\"5MBHg%G,&oJ?!#fF0F2F0F0*&$
\"5**[[$ovw%=HoF_^mF0F7F0F0*&$\"53%[Z\\P%*p,f&!#TF0F5F0F^]m$\"5Gg,++++
++]F,F0F3" }}}{EXCHG {PARA 261 "" 0 "" {TEXT 272 37 "The time for the \+
entire factorization" }}}{EXCHG {PARA 261 "> " 0 "" {MPLTEXT 1 268 17 
"TIME1:=time()-st;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&TIME1G$\"'HT>
!\"$" }}}{EXCHG {PARA 261 "" 0 "" {TEXT 272 25 "Check the backward err
or:" }}}{EXCHG {PARA 261 "> " 0 "" {MPLTEXT 1 268 53 "result1:=expand(
fac_Ver_f1[1][1]*fac_Ver_f1[2][1]^3):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 51 "berror_Ver_F1:=backward_error(result1,F1, [x,y,z]);" 
}}{PARA 6 "" 1 "" {TEXT -1 50 "Approximate factorization is not a real
 polynomial" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"cG^$$\"59t54%)pG!)p
B!#>$!5pw5G1]&Q%)p%!#R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%.berror_Ve
r_F1G$\"5GYfC!))fVNv$!#N" }}}{EXCHG {PARA 261 "> " 0 "" {MPLTEXT 1 
268 15 "ceil(log10(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!#:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "save berror_Ver_F1,TIME1,fac
_Ver_f1, cat(path,\"verschelde/svw_F1-factors\");" }}}}{SECT 0 {PARA 
266 "" 0 "" {TEXT 275 9 "Example 2" }}{PARA 261 "" 0 "" {TEXT 272 61 "
Here we set the third element of all locations equal to zero:" }}
{EXCHG {PARA 261 "> " 0 "" {MPLTEXT 1 268 11 "Digits:=20;" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%'DigitsG\"#?" }}}{EXCHG {PARA 261 "> " 0 "" 
{MPLTEXT 1 268 37 "read cat(path,\"verschelde/svw_exf2\"):" }}{PARA 
268 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 261 "> " 0 "" {MPLTEXT 1 
268 21 "bb:=[op(indets(f2))];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#bb
G7&&%\"qG6#\"\"!&F'6#\"\"#&F'6#\"\"$&F'6#\"\"\"" }}}{EXCHG {PARA 261 "
> " 0 "" {MPLTEXT 1 268 43 "F2:=subs(\{q[1]=x,q[2]=y,q[0]=z,q[3]=1\},f
2):" }}}{EXCHG {PARA 261 "> " 0 "" {MPLTEXT 1 268 25 "F2:=evalf(F2/nor
m(F2,2)):" }}}{EXCHG {PARA 261 "> " 0 "" {MPLTEXT 1 268 11 "st:=time()
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#stG$\"'5+@!\"$" }}}{EXCHG 
{PARA 261 "" 0 "" {TEXT 272 0 "" }}}{EXCHG {PARA 261 "> " 0 "" 
{MPLTEXT 1 268 37 "fac_Ver_f2:=multifac1(F2,[x,y,z],10);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#%hnthe~gap~is~too~small,~we~think~the~polynomial
~is~irreducible.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%hnthe~gap~is~too
~small,~we~think~the~polynomial~is~irreducible.G" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%+fac_Ver_f2G7$7$,du**^$$\"5@z623:%G!>B!#@$!5KEC'fQ%G
\\ohF,\"\"\")%\"yG\"\"$F/)%\"zG\"\"#F/%\"xGF/F/**^$$\"5<C4#4Q@M2Z\"!#?
$\"5)R%ov]*p3pS\"F;F/)F1F5F/)F4F2F/F6F/F/**^$$!5(edq\"GRN%e/#F;$!5%*G1
@)z\"=<u?F;F/F0F/)F6F5F/F4F/F/**^$$\"5j[eza!eW$RcF,$\"57S%RGq\\XYh'F,F
/)F1\"\"%F/F4F/F6F/F/**^$$!5n**41x5cx8GF;$!5B+-\\&G'zGaSF,F/F?F/FFF/F1
F/F/**^$$\"5'eEkN>evN/\"F;$!5S,([36j9Ro#F,F/)F4FNF/F1F/F6F/F/**^$$\"5<
1a**)y\"Q^jEF;$!5TrJ&)y>+@JUF,F/F3F/FFF/F>F/F/**^$$\"5or@L'yN$Rk$)F,$
\"5t@b1o\"[bky$F,F/)F6F2F/F3F/F1F/F/**^$$\"5Z\\/'\\\"3')*GG\"F;$!5#R-
\"[C\"3:)eIF,F/FboF/F>F/F4F/F/**^$$!5W/06F=-#H<\"F;$!5\\?)f=quE%=xF,F/
)F6FNF/F1F/F4F/F/**^$$\"51S<uDOW(*RyF,$\"5>j)Q\"4z3G&f$F,F/FFF/F3F/F1F
/F/**^$$!53r;z9yB(Rq#F;$\"5xSVf*\\wIeG\"F;F/F?F/F1F/F6F/F/**^$$\"5`[`h
A^av5yF,$\"5aev+xf!pNN\"F;F/F>F/F3F/F6F/F/**^$$!5xwRp.E<+16F;$\"5pVi]&
R%4K(z&F,F/F0F/F6F/F4F/F/**^$$!5,\"RNy2;\"**=zF,$\"5JgEy3#yEfM%F,F/F>F
/F4F/F6F/F/**^$$!5[S)HfO8T!z:F;$\"5)z`*fcAN&[3\"F;F/FboF/F1F/F4F/F/**^
$$\"5AP/5)4v^lT#F;$!5:5mZT5%=-!fF,F/FFF/F>F/F4F/F/**^$$!5GH:(>XJ@$Q%*F
,$\"5_^Bd;(eeo(pF,F/FFF/F1F/F4F/F/**^$$!5&)Gyic0iL\"p#F;$!5YV\\(*Hggp&
p\"F;F/F3F/F1F/F6F/F/*&^$$\"5IfB&3N.8xt&F,$\"5eyrg9hM/5BF,F/FenF/F/*&^
$$\"5D*G!R-R;U*>\"!#A$!54HV'3+v%*)\\T!#BF/FMF/F/*&^$$!5#*pVN\"G*>UJ;F,
$!5M<3aJJ\"pT<\"F,F/)F6\"\"&F/F/*&^$$!5^6kM(\\VOmt%F`t$!5'pBwEM0RR0(F`
tF/)F4F[uF/F/*&^$$!5dh'3l%Q4G$f#F`t$\"5&o/8d&)\\QbH\"F,F/)F1F[uF/F/*(^
$$!5\\+4bVv!fw+(F,$\"5O;%)3jQJe<=F`tF/FenF/F>F/F/*(^$$!5U\")GIn3%yFL#F
,$\"5w;N%QL:*ouNF,F/F?F/F0F/F/*&^$$!5>M3M\"=)*\\3(QF`t$\"5F.Qt/GN5[dF`
tF/)F1\"\"'F/F/*&^$$!5%)zDD_z#>2;\"F,$\"549Z'pEQ$e*H\"F,F/F?F/F/*&^$$!
5$RxnQm@sc6\"F,$\"5oWOp7[fM3OF,F/F0F/F/*(^$$!5\"p)[s77`U2HF,$!5&H6nJjW
fy)HF,F/FbuF/F1F/F/*&^$$!5))f4QvN^-4^F`t$\"5UG(p*p3'R%)G#F,F/F_pF/F/*(
^$$\"5Pa!\\/_5F8d'F`t$\"5fc6I?14Q#f)F`tF/FbuF/F6F/F/*(^$$\"5&fJFp0N$f=
WF,$\"5?ewW^Y')RzXF,F/FFF/FMF/F/*&^$$!5wTd+GGY1;\\F`t$!5*)[<B5xt9=qF`t
F/)F6F]wF/F/*(^$$\"5pDx6]%p%\\)f&F,$!5<WVrpAH*)y\"*F,F/F>F/F?F/F/*(^$$
\"5Z\"Hm7wR@'=8F;$!5nA1wuIB([L'F,F/FFF/F?F/F/*(^$$!5-%3f&HMQ$*3GF,$\"5
:@sf$)f!Q3U%F,F/FMF/F6F/F/*(^$$\"5j3XI8kH4:xF,$\"5LGr[8kx$4H#F,F/F_pF/
F4F/F/*(^$$!5tG.edY2SySF,$\"5D[m1c5v'=X&F,F/F3F/FboF/F/*(^$$\"5oNub]Z?
cF:F,$!5A#3)HzUsE:5F;F/FenF/F1F/F/*(^$$\"5$[.JNf_**QI%F`t$\"5c?-&p;1\\
r\")*F`tF/FjtF/F4F/F/*(^$$!5b]rk76*)*3%=F`t$!5uQg!4$3uP2QF,F/F_pF/F3F/
F/*(^$$!5@ANK<WIO8lF,$\"5m]X*4%4T#Q.&F,F/FboF/F0F/F/*(^$$\"5vfH1W1]$)z
&)F,$\"5o)*G*R\"Q^Ln')F,F/F?F/FboF/F/*(^$$\"5Q8g&=_Oe;X$F,$\"5))f\"H_W
jyDl#F,F/FjtF/F1F/F/*(^$$\"5y!z>37!=:/5F,$\"51(px#H+=([C\"F,F/F_pF/F>F
/F/*(^$$\"5ZW8%3#*=/Yi#F,$\"5l0O4\">Dzdo#F,F/FMF/F4F/F/*(^$$\"5E&[)e%y
\"4n\"G#F,$!5&oFG&=-PF()eF,F/F3F/F0F/F/*(^$$!5(yX\"\\WI')QjnF,$\"5I\\F
^P'eD)e=F,F/FFF/FenF/F/*(^$$!5ZSI2#Q-(f=6F,$\"5\"R<2Zj`ux_\"F,F/FiuF/F
6F/F/*&^$$!5L')y&p\"eGUwtF`t$!5O]6$oEh6d+\"F,F/)F4F]wF/F/*&^$$\"5jNi.'
*o7#=T$F,$\"5/!zh*zx'RW4\"F,F/FboF/F/*(^$$\"5^fQcuh=E)f(F`t$!5\">dS)R5
.\"*p=F`tF/FiuF/F4F/F/*(^$$\"5U'f()==_P&H8F,$!5k1+Z&QNLfX\"F,F/F3F/FMF
/F/*(^$$!5(*ou^aRj;o@F,$!5*)*f(zV=e(*HoF,F/F?F/F6F/F/*(^$$\"5>Ni8(p*4H
>VF,$!5k%\\J5M$)4f*QF,F/F_pF/F1F/F/*(^$$\"5!f(Gl&oP)4OXF,$\"54')f)*QK>
8VBF,F/F0F/F4F/F/*(^$$\"5UKT90N+k(H\"F;$\"5&fXtj;9%RPlF,F/F3F/F>F/F/*(
^$$\"5!>uf0MVLrC\"F;$\"5%R5AOMi0eZ%F,F/F?F/F1F/F/*(^$$!5r*)yST5TzY;F,$
!5Bg/8xi>r'H&F,F/F0F/F6F/F/*(^$$\"5(*pdp1efD/cF,$!54yzB%*f%ocr%F,F/Fbo
F/F4F/F/*(^$$\"5OE/<_/6*oH\"F;$\"5d,Q!pdFf*Q$)F,F/FFF/F>F/F/*(^$$\"5Ke
7jSfpKwOF,$\"5KgNH,t7WD8F;F/FFF/F3F/F/*(^$$\"5#)**H\"HMl)yVVF,$\"5VnQC
\"*eq!Go\"F,F/FFF/F0F/F/*(^$$\"5EkbV\\2y/TQF,$!59.Pn.g=tVLF,F/F>F/FboF
/F/*&^$$!5/yfYg?du#=\"F,$\"5Kq!49D&)GZz'F`tF/F3F/F/*(^$$!57H6dkr&*43>F
,$\"5g6uZY*fC+9\"F;F/FenF/F6F/F/*&^$$\"5)*\\$f#GO#p:%=F,$!5F58I1)e4g.#
F,F/F1F/F/^$$!5_WY\\A'3;#RnF`t$\"5YY%RFOq3\\L\"F,F/*&^$$!58lj?cO6B4:F,
$\"5>j.$f'R-tu')F`tF/F6F/F/*&^$$!5A%Ry=++&y@UF,$!5'3G'G#R%Hg*y\"F,F/F>
F/F/*&^$$\"5ccjMV.@\"=l#F,$!5ot)4I)HareTF,F/FFF/F/*&^$$!5%Q\\P;z`T=L&F
`t$\"5u=\\#))GMVV'HF,F/F4F/F/*(^$$\"5hMuKvx#f\"y8F;$!5v(HkTVho9K&F,F/F
3F/F6F/F/*(^$$!5Bj^ea=e&e0)F,$\"5#*offU>p$y1%F,F/FFF/F1F/F/*(^$$!5b[6m
:TV9bCF,$\"5KT'R)R9hDw8F,F/FFF/F4F/F/*(^$$!5O%RDv+.jy,#F`t$\"5[&)y-$os
G?-#F,F/F>F/F6F/F/*(^$$\"59BfH5T!Hk_\"F,$!5\\)HBsc85'))>F`tF/F1F/F4F/F
/*(^$$\"5lwr)e(*p*oQLF,$!5-Iwb%HBGr]%F,F/F1F/F6F/F/*(^$$\"5pV()3a`!QKl
#F,$\"5OR@SuY\")zwHF`tF/F4F/F6F/F/*(^$$!5U!z!fye#>_z%F,$\"5KTYq#fmY!f5
F,F/F3F/F1F/F/*(^$$!530l`ETHiAUF,$!5XQgQn+DPrJF,F/FboF/F1F/F/*(^$$\"5g
BZBviMX9BF,$\"57k!*GB%oS]0&F,F/F>F/F4F/F/**^$$\"51>?dJ$>\\pB$F,$!5/J)
\\\"f(\\!=9$)F,F/F1F/F6F/F4F/F/F/7$,6*&$\"5&)z**************\\F;F/F3F/
F/*($\"5cAoFcr!))p'Q!#gF/F1F/F4F/F/*&$\"5l-%*************\\F;F/F>F/F/*
($\"5vHbLvGFc*)R!#RF/F4F/F6F/!\"\"*($\"5,I&*)\\())R.=6!#SF/F1F/F6F/Fe]
m*&$\"5Od/++++++]F;F/FFF/F/*&$\"5*3-R$)=#o!R(**!#hF/F4F/F/*&$\"5$RXQ^(
*e;C5%!#iF/F1F/F/*&$\"5a*)*************\\#!#TF/F6F/Fe]m$\"57g,++++++]F
;F/F2" }}}{EXCHG {PARA 261 "" 0 "" {TEXT 272 32 "Total time for the fa
ctorization" }}}{EXCHG {PARA 261 "> " 0 "" {MPLTEXT 1 268 17 "TIME2:=t
ime()-st;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&TIME2G$\"'MK=!\"$" }}}
{EXCHG {PARA 261 "" 0 "" {TEXT 272 26 "Check the backward error :" }}}
{EXCHG {PARA 261 "> " 0 "" {MPLTEXT 1 268 53 "result2:=expand(fac_Ver_
f2[1][1]*fac_Ver_f2[2][1]^3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 51 "berror_Ver_F2:=backward_error(result2,F2, [x,y,z]);" }}{PARA 6 "
" 1 "" {TEXT -1 50 "Approximate factorization is not a real polynomial
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"cG^$$\"59t54%)pG!)pB!#>$\"5\"y
]W&\\R%RIX%!#R" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%.berror_Ver_F2G$\"
5N#=(y&RcOHv$!#N" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "ceil(lo
g10(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!#:" }}}{EXCHG {PARA 261 "
> " 0 "" {MPLTEXT 1 268 75 "save berror_Ver_F2,TIME2,fac_Ver_f2, cat(p
ath,\"verschelde/svw_F2-factors\");" }}}}{SECT 0 {PARA 266 "" 0 "" 
{TEXT 275 9 "Example 3" }}{PARA 261 "" 0 "" {TEXT 272 47 "In addition \+
we set q[1] and q[2] equal to zero:" }}{EXCHG {PARA 261 "> " 0 "" 
{MPLTEXT 1 268 11 "Digits:=20;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'D
igitsG\"#?" }}}{EXCHG {PARA 261 "> " 0 "" {MPLTEXT 1 268 37 "read cat(
path,\"verschelde/svw_exf3\"):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 7 "f3:=ff:" }}}{EXCHG {PARA 261 "> " 0 "" {MPLTEXT 1 268 22 "ord:=
[op(indets(f3))];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ordG7$&%\"qG6#
\"\"!&F'6#\"\"$" }}}{EXCHG {PARA 261 "> " 0 "" {MPLTEXT 1 268 29 "F3:=
subs(\{q[0]=x,q[3]=y\},f3);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#F3G,
V*&^#$\"5(ykY(p5GCo]!#<\"\"\")%\"yG\"#7F+F+*&^#$!5E())4eVY\"Q=QF*F+)%
\"xGF.F+F+*(^#$\"59I[ao#>eA!=!#;F+)F4\"\"(F+)F-\"\"&F+F+*(^#$\"5!>6^\"
zY2iV<F9F+)F4\"\")F+)F-\"\"%F+F+*(^#$\"5m<pWQ76ZD6F9F+)F-\"#6F+F4F+F+*
(^#$\"5tahS'eur+`$F9F+)F4\"\"'F+)F-FQF+F+*(^#$!5*\\cZ(Hkj;yEF*F+)F4FKF
+F-F+F+*(^#$!54Aa$eUzv.5$F*F+)F4\"\"*F+)F-\"\"$F+F+*(^#$\"5ay\")QJ!pL)
)e%F9F+)F4F=F+)F-F;F+F+*(^#$\"5d%>$[xb![%y<F9F+)F-\"#5F+)F4\"\"#F+F+*(
^#$\"5E![\"o,nc#)pQF9F+)F-FgnF+)F4FinF+F+*(^#$\"54@KrT0=XrJF9F+)F4FEF+
)F-FCF+F+*(^#$!4BS8jz96Yo#F9F+)F-FgoF+)F4FeoF+F+*($\"5$o\"[ZX.`l;7F9F+
FfoF+FdoF+!\"\"*($\"5J[KI^<n?4@F9F+F^oF+F_oF+F]q*($\"4j')\\4V\"*yuO'F*
F+FbpF+FcpF+F+*($\"5nTBrnC8*z/\"F9F+F]pF+F\\pF+F]q*($\"5%[bH$*)f$p>!)*
F*F+FfnF+FhnF+F]q*($\"5l3+bGepN)z\"F*F+FWF+F-F+F]q*($\"5x\\%fL-Ba#Q8F9
F+FhpF+FipF+F+*($\"5\"H'z9\"RDQV-#F*F+FJF+F4F+F]q*($\"5YdOyQe/,k?F9F+F
<F+F:F+F]q*&$\"5T)Qc+\"oOg+GF*F+F3F+F]q*&$\"5%\\IHRZhv'eDF*F+F,F+F]q*(
$\"5)o:;GDFsgC&F9F+FBF+FDF+F+*($\"5G`C%3m??Al%F9F+FPF+FRF+F+" }}}
{EXCHG {PARA 261 "> " 0 "" {MPLTEXT 1 268 11 "Digits:=10;" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%'DigitsG\"#5" }}}{EXCHG {PARA 261 "> " 0 "" 
{MPLTEXT 1 268 11 "st:=time();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#s
tG$\"'vRR!\"$" }}}{EXCHG {PARA 261 "" 0 "" {TEXT 272 53 "This example \+
is simple, there are only two variables." }}}{EXCHG {PARA 261 "> " 0 "
" {MPLTEXT 1 268 33 "fac_Ver_f3:=multifac(F3,[x,y],5);" }}{PARA 6 "" 
1 "" {TEXT -1 108 "` the biggest gap, the last r-th singular values an
d the number of factors `, 60478803.59, .1323067709e-9, 8" }}{PARA 6 "
" 1 "" {TEXT -1 58 "` The time for computing the number of factors****
*`, .530" }}{PARA 6 "" 1 "" {TEXT -1 53 "`The time for the entire fact
orization******`, 46.386" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%+fac_Ver
_f3G7*7$,(*&^$$\"39>$fhv(3jE!#=$!38T3-Y<tMA!#<\"\"\"%\"yGF0F0*&^$$!3&Q
![ELio@KF,$\"3DFpc]$o&>7F/F0%\"xGF0F0^$$!+)3;J:#!#C$!+(Q3TG\"F<F0F07$,
(*&^$$!3%Qew_A(o`7F/$\"3W&f0>zZ6R)F,F0F1F0F0*&^$$!3_\\Um=P&yY&F,$\"3\\
mMHH/y0=F,F0F8F0F0^$$\"+XQ)=g#F<$!+;t\\d@!#DF0F07$,(*&^$$!3(*[aIQLN-[F
,$\"3YikFZ:?oCF/F0F1F0F0*&^$$!3\\eS\")o(H:?$F/$!3Q#\\Bo`)\\88F/F0F8F0F
0^$$!+Vk^:>F<$\"+k?YpPF<F0F07$,(*&^$$\"3u!o#o,2y^7F/$\"3Kz'RA2)y\"f#F/
F0F1F0F0*&^$$\"3ky$3!ecJ`nF,$!3KHu;RI^4CF/F0F8F0F0^$$\"+'[5wD%F<$!+rGR
ShFRF0F07$,(*&^$$!3yS\\ad@<$f%F,$!3g*[[w*f\">(GF/F0F1F0F0*&^$$!3;JzS`S
%R.\"F/$!3:U_xXH$=A&F/F0F8F0F0^$$\"+&48Io$F<$!+HzRr9!#BF0F07$,(*&^$$\"
3eD)3LAJV4$F/$\"3>f#G&)GL2.(F,F0F1F0F0*&^$$\"3E-3'fb@>m%F/$!3q-*ysLY->
$F/F0F8F0F0^$$\"+ikm?6Ffq$\"+')*f#G[F<F0F07$,(*&^$$!3l/`t4nDHPF/$\"3op
lHf*))pW\"F/F0F1F0F0*&^$$!3]c*3-'*))pW\"F/$!37!)3\"*3nDHPF/F0F8F0F0^$$
\"+I'QF4#F<$!+(**eVx$F<F0\"\"$7$,(*&^$$!3)\\Q[K5[H3#F/$\"3gYK&H%oY![#F
/F0F1F0F0*&^$$\"3iUotXoY![#F/$\"3eO-m0\"[H3#F/F0F8F0F0^$$\"+**\\#R&GF<
$\"+&4:z)RF<F0F]t" }}}{EXCHG {PARA 261 "> " 0 "" {MPLTEXT 1 268 17 "TI
ME3:=time()-st;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&TIME3G$\"&GR&!\"
$" }}}{EXCHG {PARA 261 "" 0 "" {TEXT 272 25 "Check the backward error:
" }}}{EXCHG {PARA 261 "> " 0 "" {MPLTEXT 1 268 161 "result3:=expand(fa
c_Ver_f3[1][1]*fac_Ver_f3[2][1]*fac_Ver_f3[3][1]*fac_Ver_f3[4][1]*fac_
Ver_f3[5][1]*fac_Ver_f3[6][1]*fac_Ver_f3[7][1]^(3)*fac_Ver_f3[8][1]^(3
)):" }}}{EXCHG {PARA 261 "> " 0 "" {MPLTEXT 1 268 49 "berror_Ver_F3:=b
ackward_error(result3,F3, [x,y]);" }}{PARA 6 "" 1 "" {TEXT -1 50 "Appr
oximate factorization is not a real polynomial" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/%\"cG^$$!+ACn%f)!\"($\"+cjfo7F(" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%.berror_Ver_F3G$\"+(e^wh(!#=" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 15 "ceil(log10(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#!\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "save berror_Ver_
F3,TIME3,fac_Ver_f3, cat(path,\"verschelde/svw_F3-factors\");" }}}}
{SECT 0 {PARA 266 "" 0 "" {TEXT 275 9 "Example 4" }}{PARA 261 "" 0 "" 
{TEXT 272 56 "Finally, we let the point pt have arbitrary coordinates:
" }}{EXCHG {PARA 261 "> " 0 "" {MPLTEXT 1 268 11 "Digits:=10;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'DigitsG\"#5" }}}{EXCHG {PARA 261 ">
 " 0 "" {MPLTEXT 1 268 37 "read cat(path,\"verschelde/svw_exf4\"):" }}
}{EXCHG {PARA 261 "> " 0 "" {MPLTEXT 1 268 3 "f4:" }}}{EXCHG {PARA 
261 "> " 0 "" {MPLTEXT 1 268 22 "ord:=[op(indets(f4))];" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%$ordG7'&%\"qG6#\"\"!&F'6#\"\"$&%\"pG6#\"\"\"&F
.6#\"\"#&F.F+" }}}{EXCHG {PARA 261 "" 0 "" {TEXT 272 127 "There are fi
ve variables, but two variables have very small coefficients. Cutoff t
he small terms, we have only three variables." }}}{EXCHG {PARA 261 "> \+
" 0 "" {MPLTEXT 1 268 68 "F4:=cutoff(f4,ord,15); #EK reduced cutoff or
der on stored polynomial" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#F4G,:**
^$$\"5Yeo8#4(*e'z_!#<$\"5yEjl6z<3<BF*\"\"\")&%\"qG6#\"\"!\"\"(F-)&F06#
\"\"$\"\"&F-)&%\"pGF6F7F-F-**^$$\"5leo8#4(*e'z_!#=$\"5'oKc;\"z<3<BF@F-
)F/\"#6F-F5F-F9F-F-**^$$\"5?'*G)[jH'yX?F*$\"5-^:R$yJ!f*>#F*F-)F/\"\"%F
-)F5\"\")F-F9F-F-**^$$\"5deo8#4(*e'z_F*$\"5zEjl6z<3<BF*F-)F/F8F-)F5F3F
-F9F-F-**^$$!5O'*G)[jH'yX?F*$!54^:R$yJ!f*>#F*F-)F/FNF-)F5FLF-F9F-F-**^
$$!5/<j!zq.Hmj\"F*$!5(3C8nUDs'f<F*F-)F/\"#5F-)F5\"\"#F-F9F-F-**^$$\"5K
H%oga[H)REF*$\"5Wj\"Ge&*)3ae6F*F-)F/F7F-)F5\"\"*F-F9F-F-**^$$\"5.<j!zq
.Hmj\"F*$\"5'3C8nUDs'f<F*F-)F/FboF-)F5F`oF-F9F-F-**^$$\"5?H%oga[H)REF*
$\"5Tj\"Ge&*)3ae6F*F-)F/F[pF-)F5F7F-F9F-F-**^$$\"5neo8#4(*e'z_F@$\"5\"
pKc;\"z<3<BF@F-)F5FDF-F/F-F9F-F-*(^$$\"5i#zl(p#fs:4%F@$\"5:-JymN1=*R%F
@F-)F5\"#7F-F9F-F-*(^$$!5k#zl(p#fs:4%F@$!59-JymN1=*R%F@F-)F/FjqF-F9F-F
-" }}}{EXCHG {PARA 261 "> " 0 "" {MPLTEXT 1 268 22 "ord:=[op(indets(F4
))];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ordG7%&%\"qG6#\"\"!&F'6#\"
\"$&%\"pGF+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Digits:=10;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'DigitsG\"#5" }}}{EXCHG {PARA 
261 "> " 0 "" {MPLTEXT 1 268 23 "ord:=[op(indets(F4))];\n" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%$ordG7%&%\"qG6#\"\"!&F'6#\"\"$&%\"pGF+" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "F4:=subs(\{ord[1]=x,ord[2]=y
,ord[3]=z\},F4);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#F4G,:**^$$\"5Ye
o8#4(*e'z_!#<$\"5yEjl6z<3<BF*\"\"\")%\"xG\"\"(F-)%\"yG\"\"&F-)%\"zG\"
\"$F-F-**^$$\"5leo8#4(*e'z_!#=$\"5'oKc;\"z<3<BF;F-)F/\"#6F-F2F-F4F-F-*
*^$$\"5?'*G)[jH'yX?F*$\"5-^:R$yJ!f*>#F*F-)F/\"\"%F-)F2\"\")F-F4F-F-**^
$$\"5deo8#4(*e'z_F*$\"5zEjl6z<3<BF*F-)F/F3F-)F2F0F-F4F-F-**^$$!5O'*G)[
jH'yX?F*$!54^:R$yJ!f*>#F*F-)F/FIF-)F2FGF-F4F-F-**^$$!5/<j!zq.Hmj\"F*$!
5(3C8nUDs'f<F*F-)F/\"#5F-)F2\"\"#F-F4F-F-**^$$\"5KH%oga[H)REF*$\"5Wj\"
Ge&*)3ae6F*F-)F/F6F-)F2\"\"*F-F4F-F-**^$$\"5.<j!zq.Hmj\"F*$\"5'3C8nUDs
'f<F*F-)F/F]oF-)F2F[oF-F4F-F-**^$$\"5?H%oga[H)REF*$\"5Tj\"Ge&*)3ae6F*F
-)F/FfoF-)F2F6F-F4F-F-**^$$\"5neo8#4(*e'z_F;$\"5\"pKc;\"z<3<BF;F-)F2F?
F-F/F-F4F-F-*(^$$\"5i#zl(p#fs:4%F;$\"5:-JymN1=*R%F;F-)F2\"#7F-F4F-F-*(
^$$!5k#zl(p#fs:4%F;$!59-JymN1=*R%F;F-)F/FeqF-F4F-F-" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT 281 116 " Each terms in F4 has the the factor z^3,so w
e divide this factor out.This improves our factorization significantly
." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "F41:=coeff(F4,z,3);" }
}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$F41G,:*(^$$\"+r*e'z_!\"($\"+z<3<BF
*\"\"\")%\"xG\"\"(F-)%\"yG\"\"&F-F-*(^$$F)!\")$F,F7F-)F/\"#6F-F2F-F-*(
^$$\"+'H'yX?F*$\"+=.f*>#F*F-)F/\"\"%F-)F2\"\")F-F-*(F'F-)F/F3F-)F2F0F-
F-*(^$$!+'H'yX?F*$!+=.f*>#F*F-)F/FDF-)F2FBF-F-*(^$$!+P!Hmj\"F*$!+aAnf<
F*F-)F/\"#5F-)F2\"\"#F-F-*(^$$\"+&[H)REF*$\"+!*3ae6F*F-)F/\"\"$F-)F2\"
\"*F-F-*(^$$\"+P!Hmj\"F*$\"+aAnf<F*F-)F/FYF-)F2FWF-F-*(FenF-)F/F]oF-)F
2F[oF-F-*(F5F-)F2F:F-F/F-F-*&^$$\"+$fs:4%F7$\"+O1=*R%F7F-)F2\"#7F-F-*&
^$$!+$fs:4%F7$!+O1=*R%F7F-)F/FbpF-F-" }}}{EXCHG {PARA 261 "> " 0 "" 
{MPLTEXT 1 268 11 "st:=time();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#s
tG$\"'L#[%!\"$" }}}{EXCHG {PARA 261 "> " 0 "" {MPLTEXT 1 268 32 "facto
rF4xy:=multifac(F41,[x,y]);" }}{PARA 6 "" 1 "" {TEXT -1 108 "` the big
gest gap, the last r-th singular values and the number of factors `, 9
22115223.6, .1336144434e-9, 4" }}{PARA 6 "" 1 "" {TEXT -1 60 "` The ti
me for computing the number of factors*****`, .70e-1" }}{PARA 6 "" 1 "
" {TEXT -1 52 "`The time for the entire factorization******`, 4.146" }
}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%+factorF4xyG7&7$,(*&^$$!3K!f:$RI2]_
!#=$\"37E#yr9^$[8!#<\"\"\"%\"yGF0F0*&^$$!3#3x'z&H,/O%F,$\"3['p4Vbh#*=#
F/F0%\"xGF0F0^$$!+R*Rt\\&!#D$\"+!R'))*R#F<F0F07$,(*&^$$\"3Q$GCO'ex:EF/
$\"33s>FTH*)y5F,F0F1F0F0*&^$$!35&)HzfJed;F/$!3bh;t)=Nfj$F,F0F8F0F0^$$!
+;_:=aF<$\"+)*H<\"Q&F<F0F07$,(*&^$$\"3e$Q3N\"o<%3#F/$!3)y6!)R!QX^;F/F0
F1F0F0*&^$$\"3?3qy/QX^;F/$\"3%[*4&Q\"o<%3#F/F0F8F0F0^$$\"+MAz]$*F<$!+4
l+B[F<F0\"\"&7$,(*&^$$!3*o.[f\"yCBAF/$!33'f**=w/b[%!#>F0F1F0F0*&^$$!3]
EZw9Y]&[%Fho$\"3dsy%e\"yCBAF/F0F8F0F0^$$!+&Q5,g%F<$!+OU(QV&F<F0F_o" }}
}{EXCHG {PARA 261 "> " 0 "" {MPLTEXT 1 268 163 "result4xy:=expand(fact
orF4xy[1][1]^(factorF4xy[1][2])*factorF4xy[2][1]^(factorF4xy[2][2])*fa
ctorF4xy[3][1]^(factorF4xy[3][2])*factorF4xy[4][1]^(factorF4xy[4][2]))
:" }}}{EXCHG {PARA 261 "> " 0 "" {MPLTEXT 1 268 17 "TIME4:=time()-st;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&TIME4G$\"%#[)!\"$" }}}{EXCHG 
{PARA 261 "" 0 "" {TEXT 272 25 "Check the backward error:" }}{PARA 
261 "> " 0 "" {MPLTEXT 1 268 31 "result4:=expand(z^3*result4xy):" }}}
{EXCHG {PARA 261 "> " 0 "" {MPLTEXT 1 268 51 "berror_Ver_F4:=backward_
error(result4,F4, [x,y,z]);" }}{PARA 6 "" 1 "" {TEXT -1 50 "Approximat
e factorization is not a real polynomial" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/%\"cG^$$\"+zM2`A!\"($\"+t'*eiRF(" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%.berror_Ver_F4G$\"+LC#zO(!#=" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "ceil(log10(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!
\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "save berror_Ver_F4,T
IME4,factorF4xy, cat(path,\"verschelde/svw_F4-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 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
}}{PARA 273 "" 0 "" {TEXT -1 0 "" }}{PARA 274 "" 0 "" {TEXT -1 0 "" }}
{PARA 275 "" 0 "" {TEXT -1 0 "" }}{PARA 276 "" 0 "" {TEXT -1 0 "" }}
{PARA 277 "" 0 "" {TEXT -1 0 "" }}{PARA 278 "" 0 "" {TEXT -1 0 "" }}}
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{PAGENUMBERS 0 1 2 33 1 1 }