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{SECT 0 {EXCHG {PARA 202 "" 0 "" {TEXT 205 69 "Worksheet to execute th
e examples in Gao, Kaltofen, May, Yang, & Zhi:" }{TEXT 205 1 "\n" }
{TEXT 205 83 "\n'Approximate Factorization of Multivariate polynomials
 via differential Equations'" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 
1 206 25 "# Factorization routines:" }{MPLTEXT 1 206 26 "\nread(\"mult
ifac_1.3.mpl\"):" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 206 57 "# r
and_path:=\"debug_exs\\/\": # for debugging the worksheet" }{MPLTEXT 
1 206 49 "\nrand_path:=\"examples\\/\": # examples in the paper" }}}
{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 206 23 "infolevel[multifac]:=2;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%*infolevelG6#%)multifacG\"\"#" 
}}}{EXCHG {PARA 203 "" 0 "" {TEXT 208 42 "A note about backward_error(
F, G, [vars]):" }{TEXT 208 98 "\nWe compute c so that ||F-c*G||_2 is a
s small as possible.  The output is ||F - c*G||_2 / ||F||_2." }{TEXT 
208 1 "\n" }{TEXT 208 49 "\nExample 16 has been moved to it's own work
sheet." }}}{SECT 0 {PARA 205 "" 0 "" {TEXT 209 105 "Examples 10, 11 & \+
12: One polynomial (12, 7, 5) with perturbations of different orders (
-5 and -1) added:" }}{SECT 0 {PARA 206 "" 0 "" {TEXT 210 75 "Example 1
0: Three Factors (12, 7, 5), Large Noise -5, very sparse noise: 5%" }}
{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 206 29 "read(cat(rand_path,\"exF
10\")):" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 206 42 "F10 := expan
d(evalf(cleanF10 + noiseF10)):" }}}{EXCHG {PARA 203 "> " 0 "" 
{MPLTEXT 1 206 38 "norm(evalf(noiseF10),2) / norm(F10,2);" }{MPLTEXT 
1 206 16 "\nceil(log10(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+a_*
*****!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"&" }}}{EXCHG {PARA 203 
"> " 0 "" {MPLTEXT 1 206 12 "t10:=time():" }{MPLTEXT 1 206 27 "\nfacF1
0:=appfac(F10,[x,y]):" }{MPLTEXT 1 206 17 "\nt10:=time()-t10;" }}
{PARA 6 "" 1 "" {TEXT -1 108 "` the biggest gap, the last r-th singula
r values and the number of factors `, 658.3729016, .8558217297e-6, 3" 
}}{PARA 6 "" 1 "" {TEXT -1 60 "` The time for computing the number of \+
factors*****`, 59.605" }}{PARA 6 "" 1 "" {TEXT -1 54 "`The time for th
e entire factorization******`, 968.022" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%$t10G$\"'#4o*!\"$" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 
206 13 "nops(facF10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}}
{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 206 59 "berror10:=backward_error
(F10, convert(facF10, `*`), [x,y]);" }{MPLTEXT 1 206 16 "\nceil(log10(
%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"cG$!+P%p[1\"!\"'" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%)berror10G$\"+@<Fk:!#8" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#!\"$" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 206 
59 "save facF10, t10, berror10, cat(rand_path,\"exF10-factors\");" }}}
}{SECT 0 {PARA 206 "" 0 "" {TEXT 210 52 "Example 11: Three Factors (12
, 7, 5), Large Noise -5" }}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 206 
29 "read(cat(rand_path,\"exF11\")):" }}}{EXCHG {PARA 203 "> " 0 "" 
{MPLTEXT 1 206 42 "F11 := expand(evalf(cleanF11 + noiseF11)):" }}}
{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 206 38 "norm(evalf(noiseF11),2) \+
/ norm(F11,2);" }{MPLTEXT 1 206 16 "\nceil(log10(%));" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#$\"+Z'f=-\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
!\"%" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 206 12 "t11:=time():" }
{MPLTEXT 1 206 27 "\nfacF11:=appfac(F11,[x,y]):" }{MPLTEXT 1 206 17 "
\nt11:=time()-t11;" }}{PARA 6 "" 1 "" {TEXT -1 108 "` the biggest gap,
 the last r-th singular values and the number of factors `, 833.536703
1, .6758979173e-6, 3" }}{PARA 6 "" 1 "" {TEXT -1 60 "` The time for co
mputing the number of factors*****`, 49.741" }}{PARA 6 "" 1 "" {TEXT 
-1 55 "`The time for the entire factorization******`, 1559.803" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$t11G$\"(t)f:!\"$" }}}{EXCHG {PARA 
203 "> " 0 "" {MPLTEXT 1 206 13 "nops(facF11);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"$" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 206 59 
"berror11:=backward_error(F11, convert(facF11, `*`), [x,y]);" }
{MPLTEXT 1 206 16 "\nceil(log10(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#/%\"cG$\"+>qou[!\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)berror11G$
\"+pK3$>$!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"$" }}}{EXCHG {PARA 
203 "> " 0 "" {MPLTEXT 1 206 59 "save facF11, t11, berror11, cat(rand_
path,\"exF11-factors\");" }}}}{SECT 0 {PARA 206 "" 0 "" {TEXT 210 53 "
Example 12: Three Factors (12, 7, 5), Larger Noise -3" }}{EXCHG {PARA 
203 "> " 0 "" {MPLTEXT 1 206 15 "#read(\"exF12\"):" }}}{EXCHG {PARA 
203 "> " 0 "" {MPLTEXT 1 206 48 "F12 := expand(evalf(cleanF11 + 10.^2*
noiseF11)):" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 206 45 "norm(eva
lf(10.^2.*noiseF11),2) / norm(F12,2);" }{MPLTEXT 1 206 16 "\nceil(log1
0(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+3_#=-\"!#7" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#!\"#" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 
206 12 "t12:=time():" }{MPLTEXT 1 206 27 "\nfacF12:=appfac(F12,[x,y]):
" }{MPLTEXT 1 206 17 "\nt12:=time()-t12;" }}{PARA 6 "" 1 "" {TEXT -1 
108 "` the biggest gap, the last r-th singular values and the number o
f factors `, 8.340144257, .6756157841e-4, 3" }}{PARA 6 "" 1 "" {TEXT 
-1 60 "` The time for computing the number of factors*****`, 49.951" }
}{PARA 6 "" 1 "" {TEXT -1 55 "`The time for the entire factorization**
****`, 4337.458" }}{PARA 6 "" 1 "" {TEXT -1 108 "` the biggest gap, th
e last r-th singular values and the number of factors `, 16.13526528, \+
.4085967167e-3, 2" }}{PARA 6 "" 1 "" {TEXT -1 59 "` The time for compu
ting the number of factors*****`, 1.583" }}{PARA 6 "" 1 "" {TEXT -1 
53 "`The time for the entire factorization******`, 31.315" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%$t12G$\"(:-P%!\"$" }}}{EXCHG {PARA 203 "> " 
0 "" {MPLTEXT 1 206 13 "nops(facF12);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#\"\"$" }}}{EXCHG {PARA 203 "> " 0 "" {MPLTEXT 1 206 59 "berror12:=b
ackward_error(F12, convert(facF12, `*`), [x,y]);" }{MPLTEXT 1 206 16 "
\nceil(log10(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"cG$!+oMrAL!\"
'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)berror12G$\"+E#z.U)!#7" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"#" }}}{EXCHG {PARA 203 "> " 0 "" 
{MPLTEXT 1 206 59 "save facF12, t12, berror12, cat(rand_path,\"exF12-f
actors\");" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 203 "" 0 "" {TEXT 
-1 0 "" }}}}}{PARA 208 "" 0 "" {TEXT -1 0 "" }}}{MARK "5 3 8 0 0" 0 }
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