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{SECT 0 {EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 208 66 "# Worksheet to \+
execute the examples in Kaltofen, May, Yang, & Zhi:" }{MPLTEXT 1 208 
0 "" }{MPLTEXT 1 208 1 "\n" }{MPLTEXT 1 208 1 "\n" }{MPLTEXT 1 208 3 "
# '" }{MPLTEXT 1 208 63 "Approximate Factorization of Multivariate Pol
ynomials Using SVD" }{MPLTEXT 1 208 1 "'" }{MPLTEXT 1 208 0 "" }
{MPLTEXT 1 208 1 "\n" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 208 8 "
restart;" }{MPLTEXT 1 208 0 "" }{MPLTEXT 1 208 1 "\n" }{MPLTEXT 1 208 
1 "\n" }{MPLTEXT 1 208 25 "# Factorization routines:" }{MPLTEXT 1 208 
0 "" }{MPLTEXT 1 208 1 "\n" }{MPLTEXT 1 208 32 "read(\"../Src/multifac
_1.6.mpl\"):" }{MPLTEXT 1 208 0 "" }{MPLTEXT 1 208 1 " " }{MPLTEXT 1 
208 1 "\n" }{MPLTEXT 1 208 35 "# Set path to example subdirectory:" }
{MPLTEXT 1 208 0 "" }{MPLTEXT 1 208 1 "\n" }{MPLTEXT 1 208 19 "path:=
\"examples\\/\":" }{MPLTEXT 1 208 0 "" }{MPLTEXT 1 208 1 "\n" }
{MPLTEXT 1 208 55 "# rand_path:=\"debug_exs\\/\": # debugging the spre
adsheet" }{MPLTEXT 1 208 0 "" }{MPLTEXT 1 208 1 "\n" }{MPLTEXT 1 208 
43 "rand_path:=path: # random examples in paper" }{MPLTEXT 1 208 0 "" 
}{MPLTEXT 1 208 1 "\n" }{MPLTEXT 1 208 1 "\n" }{MPLTEXT 1 208 37 "# Fa
ctorization improvement routines:" }{MPLTEXT 1 208 0 "" }{MPLTEXT 1 
208 1 "\n" }{MPLTEXT 1 208 46 "read(\"../Src/improve_factorization-3va
r.mpl\"):" }{MPLTEXT 1 208 0 "" }{MPLTEXT 1 208 1 "\n" }{MPLTEXT 1 
208 1 "\n" }{MPLTEXT 1 208 20 "infolevel[imfac]:=1;" }{MPLTEXT 1 208 
0 "" }{MPLTEXT 1 208 1 "\n" }{MPLTEXT 1 208 30 "Digits:=floor(evalhf(D
igits));" }{MPLTEXT 1 208 0 "" }{MPLTEXT 1 208 1 "\n" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>&%*infolevelG6#%&imfacG\"\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%'DigitsG\"#9" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 
1 208 44 "# A note about backward_error(F, G, [vars]):" }{MPLTEXT 1 
208 0 "" }{MPLTEXT 1 208 1 "\n" }{MPLTEXT 1 208 99 "# We compute c so \+
that ||F-c*G||_2 is as small as possible.  The output is ||F - c*G||_2
 / ||F||_2." }{MPLTEXT 1 208 0 "" }{MPLTEXT 1 208 1 "\n" }}}{EXCHG 
{PARA 202 "> " 0 "" {MPLTEXT 1 208 14 "# Example 15a:" }{MPLTEXT 1 
208 0 "" }{MPLTEXT 1 208 1 "\n" }{MPLTEXT 1 208 33 "read(cat(rand_path
,\"exF15aK2K\")):" }{MPLTEXT 1 208 0 "" }{MPLTEXT 1 208 1 "\n" }
{MPLTEXT 1 208 46 "F15a := expand(evalf(cleanF15a + noiseF15a));:" }
{MPLTEXT 1 208 0 "" }{MPLTEXT 1 208 1 "\n" }{MPLTEXT 1 208 48 "norm(ev
alf(noiseF15a),2) / norm(expand(F15a),2);" }{MPLTEXT 1 208 0 "" }
{MPLTEXT 1 208 1 "\n" }{MPLTEXT 1 208 15 "ceil(log10(%));" }{MPLTEXT 
1 208 0 "" }{MPLTEXT 1 208 1 "\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%
%F15aG,6*&$\"#\")\"\"!\"\"\")%\"xG\"\"%F*F**&$\"#;F)F*)%\"yGF-F*F**&$
\"/++++.!['!#6F*)%\"zGF-F*!\"\"*($\"#sF)F*)F,\"\"#F*)F2F>F*F**&$\"$['F
)F*F=F*F9*&$\"$)GF)F*F?F*F9$\"%'H\"F)F**($\"/++++++?!#;F*F=F*)F8F>F*F*
*($\"/++++++5FKF*F?F*FLF*F**&$\"/++++++qFKF*FLF*F9" }}{PARA 11 "" 1 ""
 {XPPMATH 20 "6#$\"/EWpi#*3\\!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!
\"&" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 208 36 "# First Round (S
VD) of factorization" }{MPLTEXT 1 208 0 "" }{MPLTEXT 1 208 1 "\n" }
{MPLTEXT 1 208 39 "#read(cat(rand_path,\"exF15a-factors\"));" }
{MPLTEXT 1 208 0 "" }{MPLTEXT 1 208 1 "\n" }{MPLTEXT 1 208 1 "\n" }
{MPLTEXT 1 208 13 "t15a:=time():" }{MPLTEXT 1 208 0 "" }{MPLTEXT 1 
208 1 "\n" }{MPLTEXT 1 208 30 "facF15a:=appfac(F15a,[x,y,z]);" }
{MPLTEXT 1 208 0 "" }{MPLTEXT 1 208 1 "\n" }{MPLTEXT 1 208 41 "print('
number_of_factors'=nops(facF15a));" }{MPLTEXT 1 208 0 "" }{MPLTEXT 1 
208 1 "\n" }{MPLTEXT 1 208 18 "t15a:=time()-t15a;" }{MPLTEXT 1 208 0 "
" }{MPLTEXT 1 208 1 "\n" }{MPLTEXT 1 208 66 "c,berror15a:=backward_err
or(F15a, convert(facF15a, `*`), [x,y,z]);" }{MPLTEXT 1 208 0 "" }
{MPLTEXT 1 208 1 "\n" }{MPLTEXT 1 208 1 "\n" }{MPLTEXT 1 208 23 "ceil(
log10(berror15a));" }{MPLTEXT 1 208 0 "" }{MPLTEXT 1 208 1 "\n" }}
{PARA 6 "" 1 "" {TEXT 209 117 "` the biggest gap, the last r-th singul
ar values and the number of factors `, 1525179675122.9, .1717394920769
4e-13, 2" }}{PARA 6 "" 1 "" {TEXT 209 58 "` The time for computing the
 number of factors*****`, .436" }}{PARA 6 "" 1 "" {TEXT 209 52 "`The t
ime for the entire factorization******`, 2.732" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%(facF15aG7$,6*&$\"3'4r4H_*z$[(!#=\"\"\")%\"zG\"\"#F+F
+*($\"37(Hz>JGgv(!#MF+%\"yGF+F-F+F+*&$\"3Kd+*\\pof<\"F*F+)F3F.F+!\"\"*
($\"33x4m>4Uw')!#LF+%\"xGF+F-F+F+*($\"3=#Hu>L/l!QF<F+F=F+F3F+F+*&$\"33
+Q4?\"Gfk#F*F+)F=F.F+F8*&$\"3p\")R]e(H.U\"!#KF+F-F+F+*&$\"33d\"3_>#)QG
%F2F+F3F+F+*&$\"3#fCEZgvqP$!#NF+F=F+F8$\"3yOSeq1Pe5!#<F+,6*&$\"3'y?4I
\\TQ[(F*F+F,F+F8*($\"37)QT45/gv(F2F+F3F+F-F+F8*&$\"3Wxt)3ujf<\"F*F+F7F
+F8*($\"3auWOVw#yP)F<F+F=F+F-F+F8*($\"3-Ps%>[Aob$F<F+F=F+F3F+F8*&$\"35
Pfig)>fk#F*F+FDF+F8*&$\"3hVb$[KD.U\"FHF+F-F+F8*&$\"3'z%3uA)oQG%F2F+F3F
+F8*&$\"3vu;vmt*4$HF2F+F=F+F8$\"3wb?q@&o$e5FRF+" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/%2number_of_factorsG\"\"#" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%t15aG$\"%ZF!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
6$%\"cG%*berror15aG6$$\"/DGE=*p:\"!#5$\"/IOS%R4)*)!#E" }}{PARA 11 "" 1
 "" {XPPMATH 20 "6#!#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "e
valf(facF15a,10);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7$,6*&$\"+B&*z$[(
!#5\"\"\")%\"zG\"\"#F)F)*($\"+7$Ggv(!#EF)%\"yGF)F+F)F)*&$\"+&pof<\"F(F
))F1F,F)!\"\"*($\"+?4Uw')!#DF)%\"xGF)F+F)F)*($\"+KV]1QF:F)F;F)F1F)F)*&
$\"+?\"Gfk#F(F))F;F,F)F6*&$\"+f(H.U\"!#CF)F+F)F)*&$\"+&>#)QG%F0F)F1F)F
)*&$\"+0c2xL!#FF)F;F)F6$\"+r1Pe5!\"*F),6*&$\"+$\\TQ[(F(F)F*F)F6*($\"+,
T+cxF0F)F1F)F+F)F6*&$\"+TP'f<\"F(F)F5F)F6*($\"+Vw#yP)F:F)F;F)F+F)F6*($
\"+#[Aob$F:F)F;F)F1F)F6*&$\"+h)>fk#F(F)FBF)F6*&$\"+D`K?9FFF)F+F)F6*&$
\"+B)oQG%F0F)F1F)F6*&$\"+nt*4$HF0F)F;F)F6$\"+A&o$e5FPF)" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 208 30 "Digits:=floor(evalhf(Digits));" }
{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'DigitsG\"#9" }}
}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 208 51 "# Improvement Round (Ga
uss-Newton) of Factorization" }{MPLTEXT 1 208 0 "" }{MPLTEXT 1 208 1 "
\n" }{MPLTEXT 1 208 1 "\n" }{MPLTEXT 1 208 14 "t15an:=time():" }
{MPLTEXT 1 208 0 "" }{MPLTEXT 1 208 1 "\n" }{MPLTEXT 1 208 1 "\n" }
{MPLTEXT 1 208 54 "facF15a:=improve_factorization(F15a,facF15a,[x,y,z]
);:" }{MPLTEXT 1 208 0 "" }{MPLTEXT 1 208 1 "\n" }{MPLTEXT 1 208 27 "t
15an:=time()-t15an + t15a;" }{MPLTEXT 1 208 0 "" }{MPLTEXT 1 208 1 "\n
" }{MPLTEXT 1 208 66 "c,berror15a:=backward_error(F15a, convert(facF15
a, `*`), [x,y,z]);" }{MPLTEXT 1 208 0 "" }{MPLTEXT 1 208 1 "\n" }
{MPLTEXT 1 208 23 "ceil(log10(berror15a));" }{MPLTEXT 1 208 0 "" }
{MPLTEXT 1 208 1 "\n" }}{PARA 6 "" 1 "" {TEXT 209 22 "improve_factoriz
ation:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*org_errorG$\"/-D4a\"3\"*)
!#E" }}{PARA 6 "" 1 "" {TEXT 209 22 "improve_factorization:" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/%&itersG\"\"#" }}{PARA 6 "" 1 "" {TEXT 209 
22 "improve_factorization:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%*new_e
rrorG$\"/R\\(oIoT)!#F/%,improvementG$\"/$R.6!pe5!#7" }}{PARA 12 "" 1 "
" {XPPMATH 20 "6#>%(facF15aG7$,6$\"/!oYl.+g$!#7\"\"\"*&$\",DvZw!G!#RF*
%\"zGF*F**&$\"-L!QqD7\"!#SF*%\"yGF*!\"\"*&$\"-/$=@Ez\"F3F*%\"xGF*F**&$
\"/U&o;$eXDF)F*)F/\"\"#F*F**($\"-Fk?0[XF3F*F4F*F/F*F5*&$\"/NfD%3++%!#8
F*)F4F>F*F5*($\",@U;*3XF.F*F9F*F/F*F**($\"-\"ocd/'>F.F*F9F*F4F*F**&$\"
/t7ZS,+!*FEF*)F9F>F*F5,6$\"/eLXj***f$F)F**&$\",*[d17:F.F*F/F*F**&$\"-W
3K_H9F3F*F4F*F**&$\".DB#*fl&H!#TF*F9F*F**&$\"/$3GW(fXDF)F*F=F*F5*($\"-
&Gjjk@$F3F*F4F*F/F*F**&$\"/+Uu:****RFEF*FFF*F5*($\",dE*RaaF.F*F9F*F/F*
F5*($\",6dqEv)F.F*F9F*F4F*F5*&$\"/>*G&f)****)FEF*FPF*F5" }}{PARA 11 ""
 1 "" {XPPMATH 20 "6#>%&t15anG$\"%rG!\"$" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>6$%\"cG%*berror15aG6$$\"/**************!#9$\"/h5Kw\"))f$!#F" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#!#8" }}}{EXCHG {PARA 202 "> " 0 "" 
{MPLTEXT 1 208 0 "" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 208 67 "s
ave facF15a, t15an, berror15a, cat(rand_path,\"exF15a-factors-im\");" 
}{MPLTEXT 1 208 0 "" }{MPLTEXT 1 208 1 "\n" }}}{EXCHG {PARA 202 "> " 0
 "" {MPLTEXT 1 208 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 206 "" 0 "" {TEXT 202 0 "" }}{PARA 206 
"" 0 "" {TEXT 210 0 "" }}}
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