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{SECT 0 {EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 208 67 "# 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 15b:" }{MPLTEXT 1 
208 0 "" }{MPLTEXT 1 208 1 "\n" }{MPLTEXT 1 208 30 "read(cat(rand_path
,\"exF15b\")):" }{MPLTEXT 1 208 0 "" }{MPLTEXT 1 208 1 "\n" }}}{EXCHG 
{PARA 202 "> " 0 "" {MPLTEXT 1 208 36 "# First Round (SVD) of factoriz
ation" }{MPLTEXT 1 208 0 "" }{MPLTEXT 1 208 1 "\n" }{MPLTEXT 1 208 39 
"#read(cat(rand_path,\"exF15b-factors\"));" }{MPLTEXT 1 208 0 "" }
{MPLTEXT 1 208 1 "\n" }{MPLTEXT 1 208 1 "\n" }{MPLTEXT 1 208 13 "t15b:
=time():" }{MPLTEXT 1 208 0 "" }{MPLTEXT 1 208 1 "\n" }{MPLTEXT 1 208 
30 "facF15b:=appfac(F15b,[x,y,z]):" }{MPLTEXT 1 208 0 "" }{MPLTEXT 1 
208 1 "\n" }{MPLTEXT 1 208 41 "print('number_of_factors'=nops(facF15b)
);" }{MPLTEXT 1 208 0 "" }{MPLTEXT 1 208 1 "\n" }{MPLTEXT 1 208 18 "t1
5b:=time()-t15b;" }{MPLTEXT 1 208 0 "" }{MPLTEXT 1 208 1 "\n" }
{MPLTEXT 1 208 66 "c,berror15b:=backward_error(F15b, convert(facF15b, \+
`*`), [x,y,z]);" }{MPLTEXT 1 208 0 "" }{MPLTEXT 1 208 1 "\n" }{MPLTEXT
 1 208 1 "\n" }{MPLTEXT 1 208 23 "ceil(log10(berror15b));" }{MPLTEXT 
1 208 0 "" }{MPLTEXT 1 208 1 "\n" }}{PARA 6 "" 1 "" {TEXT 209 116 "` t
he biggest gap, the last r-th singular values and the number of factor
s `, 258.21220386595, .15607617615690e-3, 2" }}{PARA 6 "" 1 "" {TEXT 
209 58 "` The time for computing the number of factors*****`, .859" }}
{PARA 6 "" 1 "" {TEXT 209 52 "`The time for the entire factorization**
****`, 6.297" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2number_of_factorsG
\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%t15bG$\"%7j!\"$" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>6$%\"cG%*berror15bG6$$\"/,7Q2t.Y!#8$\"/!HQ
\"yCNu!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"$" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 18 "evalf(facF15b,10);" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#7$,J*&,&$\"+QQYmy!#5\"\"\"*&$\"\"!F-F*^#F*F*F*F*%\"zGF*
F**(,&$\"+MGQr:!\"*F*F+F*F*%\"xGF*F/F*F**&,&$\"+,24\\lF)F*F+F*F*F5F*F*
*(,&$\"+rn\\\"f\"!#8F*F+F*F*)F5\"\"#F*%\"yGF*F**(,&$\"+4$=wB&F)!\"\"F+
F*F*F5F*FAF*F**&,&$\"+%))fKN\"F>FFF+F*F*)F/\"\"$F*F**(,&$\"+z.?RBF>FFF
+FFF*F?F*F/F*F**(,&$\"+a$z*>DF>FFF+F*F*)FAF@F*F/F*F**&,&$\"+uB\"y$QF>F
FF+FFF*F?F*F**(,&$\"+)zuV!=F>FFF+FFF*F5F*)F/F@F*F**(,&$\"+\\:7@>!#9F*F
+F*F*FAF*FhnF*F**&,&$\"+'\\YkB&F)FFF+FFF*FhnF*F**(,&$\"+\\w&*R_F)FFF+F
FF*FAF*F/F*F**&,&$\"+N3]1uF]oF*F+F*F*)FAFLF*F**&,&$\"+c$*oR_F)FFF+FFF*
)F5FLF*F**&,&$\"+!\\;60$F)F*F+F*F*FAF*F***,&$\"+!=\")G!f!#:F*F+F*F*F5F
*FAF*F/F*F**(,&$\"+8ur@JF>FFF+F*F*F5F*FUF*F**&,&$\"+QS>vxF]oFFF+F*F*FU
F*F*,&$\"+EzCL_F)FFF+FFF*,6*&,&$\"+oKBYTF)F*F+F*F*FhnF*F**(,&$\"+z(4r(
oF>FFF+FFF*FAF*F/F*F**&,&$\"+i`GUTF)F*F+F*F*FUF*F**(,&$\"+dS7,#)F]oFFF
+FFF*F5F*F/F*F**(,&$\"+\"*)*G#z\"F>F*F+F*F*F5F*FAF*F**&,&$\"+oirV7F4F*
F+F*F*F?F*F**&,&$\"+7dMo?F)FFF+FFF*F/F*F**&,&$\"+Wm<\\TF)FFF+FFF*FAF*F
**&,&$\"+[g50$)F)FFF+F*F*F5F*F*,&$\"+o<rZTF)FFF+F*F*" }}}{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 "t15bn:=time():" }
{MPLTEXT 1 208 0 "" }{MPLTEXT 1 208 1 "\n" }{MPLTEXT 1 208 1 "\n" }
{MPLTEXT 1 208 54 "facF15b:=improve_factorization(F15b,facF15b,[x,y,z]
);:" }{MPLTEXT 1 208 0 "" }{MPLTEXT 1 208 1 "\n" }{MPLTEXT 1 208 27 "t
15bn:=time()-t15bn + t15b;" }{MPLTEXT 1 208 0 "" }{MPLTEXT 1 208 1 "\n
" }{MPLTEXT 1 208 66 "c,berror15b:=backward_error(F15b, convert(facF15
b, `*`), [x,y,z]);" }{MPLTEXT 1 208 0 "" }{MPLTEXT 1 208 1 "\n" }
{MPLTEXT 1 208 23 "ceil(log10(berror15b));" }{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$\"/e!Q\"yCNu!#
<" }}{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_erro
rG$\"/([%>%\\cx$!#</%,improvementG$\"/UG%Hj#p>!#8" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%(facF15bG7$,J*&$\"/D;C1$fa'!#9\"\"\"%\"yGF+F+*&$\"/0c
()R609!#8F+%\"xGF+F+$\"/X%)p!>L7\"F0!\"\"*&$\"/b=4FN(o\"F0F+%\"zGF+F+*
($\"/#\\h\"4lC6F0F+F,F+F8F+F4*($\"/#*RQ9USv!#=F+)F1\"\"#F+F,F+F+*($\"/
XfKEJjA!#<F+F1F+)F,FAF+F4*($\"/.t_160**F?F+F@F+F8F+F4*($\"/wn-!GDW$F?F
+FFF+F8F+F+*($\"/pJO\\#RN$FEF+F1F+)F8FAF+F4*($\"/4\\>E1#*GF?F+F,F+FPF+
F+*&$\"/Do,#)oGLFEF+F@F+F4*&$\"/A,W4jkiF?F+FFF+F+*&$\"/``[LbB6F0F+FPF+
F4*&$\"/YNkXkB6F0F+)F1\"\"$F+F4*&$\"/?*47ZPU#FEF+)F,F[oF+F+*&$\"/S!e%3
oj>F?F+)F8F[oF+F+*($\"/M+&*poB6F0F+F1F+F,F+F4*($\"/dZ*\\&*=P$F0F+F1F+F
8F+F+**$\"/q$f1*[YIFEF+F1F+F,F+F8F+F4,6$\"/L()*4mb*))F*F4*&$\"/(\\WVpw
W%F*F+F8F+F4*&$\"/`(=XQ>!*)F*F+F,F+F4*&$\"/3,u?x!y\"F0F+F1F+F4*&$\"/hb
0X[$*))F*F+FPF+F+*($\"/[BA<cHqFEF+F,F+F8F+F4*&$\"/%oVTeD*))F*F+FFF+F+*
($\"/#e*oB/yQFEF+F1F+F8F+F4*($\"/\\KeQq9NFEF+F1F+F,F+F+*&$\"/'o'>GJpEF
0F+F@F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&t15bnG$\"%(o'!\"$" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>6$%\"cG%*berror15bG6$$\"/(***********
**!#9$\"/([%>%\\cx$!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"$" }}}
{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 208 0 "" }}}{EXCHG {PARA 202 "> 
" 0 "" {MPLTEXT 1 208 67 "save facF15b, t15bn, berror15b, cat(rand_pat
h,\"exF15b-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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