{VERSION 6 0 "Windows XP" "6.0" }
{USTYLETAB {PSTYLE "Heading 4" -1 20 1 {CSTYLE "" -1 -1 "Times" 1 10 
0 0 0 1 1 1 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Or
dered List 5" -1 200 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2
 2 1 0 0 1 }1 1 0 0 3 3 2 144 2 0 2 2 -1 1 }{PSTYLE "Ordered List 1" 
-1 201 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 
1 0 0 3 3 2 0 2 0 2 2 -1 1 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1
 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 
2 2 -1 1 }{PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 
0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 1 }{PSTYLE "Le
ft Justified Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0
 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Help" 
-1 10 1 {CSTYLE "" -1 -1 "Courier" 1 9 0 0 255 1 2 2 2 2 2 1 1 0 0 1 }
1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Author" -1 19 1 {CSTYLE "" -1 
-1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 8 8 2 0 2 0 2 2 
-1 1 }{PSTYLE "Diagnostic" -1 9 1 {CSTYLE "" -1 -1 "Courier" 1 10 64 
128 64 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "H
eading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 0
 0 1 }1 1 0 0 8 2 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle1" -1 202 1 
{CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 8 4
 2 0 2 0 2 2 -1 1 }{PSTYLE "Ordered List 3" -1 203 1 {CSTYLE "" -1 -1 
"Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 72 2 0 2 2 -1 
1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 
2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle8" 
-1 204 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 
1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "
Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 
}{PSTYLE "Annotation Title" -1 205 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0
 0 1 2 1 2 2 2 2 1 0 0 1 }3 1 0 0 12 12 2 0 2 0 2 2 -1 1 }{PSTYLE "Das
h Item" -1 16 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 
0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 1 }{PSTYLE "Heading 3" -1 5 1 {CSTYLE 
"" -1 -1 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0
 2 2 -1 1 }{PSTYLE "Ordered List 4" -1 206 1 {CSTYLE "" -1 -1 "Times" 
1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 108 2 0 2 2 -1 1 }
{PSTYLE "_pstyle2" -1 207 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 
2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Maple Output" 
-1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1
 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle5" -1 208 1 {CSTYLE "" -1 
-1 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 
-1 1 }{PSTYLE "List Item" -1 14 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 
1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 1 }{PSTYLE "Line Pri
nted Output" -1 6 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 
2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Fixed Width" -1 17
 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 
0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Cou
rier" 1 10 0 0 255 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1
 }{PSTYLE "Error" -1 8 1 {CSTYLE "" -1 -1 "Courier" 1 10 255 0 255 1 2
 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Heading 1" 
-1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 
0 0 8 4 2 0 2 0 2 2 -1 1 }{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Ti
mes" 1 18 0 0 0 1 2 1 1 2 2 2 1 0 0 1 }3 1 0 0 12 12 2 0 2 0 2 2 -1 1 
}{PSTYLE "_pstyle4" -1 209 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1
 2 2 2 2 1 0 0 1 }1 1 0 0 8 2 2 0 2 0 2 2 -1 1 }{PSTYLE "Ordered List \+
2" -1 210 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 
}1 1 0 0 3 3 2 36 2 0 2 2 -1 1 }{CSTYLE "Help Variable" -1 25 "Courier
" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Text" -1 200 "Times" 1 12
 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle2" -1 201 "Courier" 1 12
 255 0 0 1 2 1 2 2 1 2 0 0 0 1 }{CSTYLE "Help Bold" -1 39 "Times" 1 12
 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Page Number" -1 33 "Times" 1 12
 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small" -1 202 "T
imes" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Nonterminal" -1 
24 "Courier" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Default" -1 38
 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle4" -1 203 "
Times" 1 14 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Maple Comment" -1 21
 "Courier" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle5" -1 204
 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Maple Input" -1 0 
"Courier" 1 12 255 0 0 1 2 1 2 2 1 2 0 0 0 1 }{CSTYLE "2D Math Small" 
-1 7 "Times" 1 1 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Inert Output
" -1 205 "Times" 1 12 144 144 144 1 2 2 2 2 1 2 0 0 0 1 }{CSTYLE "Help
 Fixed" -1 23 "Courier" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Pop
up" -1 31 "Times" 1 12 0 128 128 1 1 2 1 2 2 2 0 0 0 1 }{CSTYLE "Plot \+
Title" -1 27 "Times" 1 10 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "2D Inp
ut" -1 19 "Times" 1 12 255 0 0 1 2 2 2 2 1 2 0 0 0 1 }{CSTYLE "Copyrig
ht" -1 34 "Times" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Maple Inp
ut Placeholder" -1 206 "Courier" 1 12 200 0 200 1 2 1 2 2 1 2 0 0 0 1 
}{CSTYLE "2D Math Bold Small" -1 10 "Times" 1 1 0 0 0 1 2 1 2 2 2 2 0 
0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1
 }{CSTYLE "2D Math Italic Small202" -1 207 "Times" 1 1 0 0 0 1 1 2 2 2
 2 2 0 0 0 1 }{CSTYLE "Annotation Text" -1 208 "Times" 1 12 0 0 0 1 2 
2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Notes" -1 37 "Times" 1 12 0 0 0 1 2 1
 2 2 2 2 0 0 0 1 }{CSTYLE "Help Underlined Bold" -1 41 "Times" 1 12 0 
0 0 1 1 1 2 2 2 2 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "Times" 1 12 0 
128 128 1 2 2 1 2 2 2 0 0 0 1 }{CSTYLE "2D Math Symbol 2" -1 16 "Times
" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Equation Label" -1 209 "C
ourier" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Plot Text" -1 28 "T
imes" 1 8 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle1" -1 210 "Time
s" 1 18 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Help Italic" -1 42 "Time
s" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle6" -1 211 "Times"
 1 12 0 0 0 1 1 1 2 2 2 2 0 0 0 1 }{CSTYLE "Help Heading" -1 26 "Times
" 1 14 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Output Labels" -1 29 "Tim
es" 1 8 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Normal" -1 30 "Time
s" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Comment" -1 18 "Times
" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Maple Name" -1 35 "T
imes" 1 12 104 64 92 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "2D Output" -1 20 
"Times" 1 12 0 0 255 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Dictionary Hyperl
ink" -1 45 "Times" 1 12 147 0 15 1 2 2 1 2 2 2 0 0 0 1 }{CSTYLE "Help \+
Emphasized" -1 22 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "H
elp Italic Bold" -1 40 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }
{CSTYLE "LaTeX" -1 32 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }
{CSTYLE "Help Menus" -1 36 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }
{CSTYLE "Prompt" -1 1 "Courier" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }
{CSTYLE "Help Underlined" -1 44 "Times" 1 12 0 0 0 1 2 2 1 2 2 2 0 0 0
 1 }{CSTYLE "Help Underlined Italic" -1 43 "Times" 1 12 0 0 0 1 1 2 1 
2 2 2 0 0 0 1 }{CSTYLE "2D Math Bold" -1 5 "Times" 1 12 0 0 0 1 2 1 2 
2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic" -1 3 "Times" 1 12 0 0 0 1 1 2 
2 2 2 2 0 0 0 1 }}
{SECT 0 {EXCHG {PARA 202 "" 0 "" {TEXT 210 64 "Worksheet to execute t
he examples in Kaltofen, May, Yang, & Zhi:" }{TEXT 210 1 "\n" }{TEXT 
210 1 "\n" }{TEXT 210 1 "'" }{TEXT 210 63 "Approximate Factorization o
f Multivariate Polynomials Using SVD" }{TEXT 210 1 "'" }}}{EXCHG 
{PARA 207 "> " 0 "" {MPLTEXT 1 201 25 "# Factorization routines:" }
{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 32 "read(\"../Src/multifac_1.6.m
pl\"):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Digits:=floor(eva
lhf(Digits));" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"#9" }}}{EXCHG {PARA 
207 "> " 0 "" {MPLTEXT 1 201 57 "# rand_path:=\"debug_exs\\/\": # for \+
debugging the worksheet" }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 48 "ra
nd_path:=\"examples\\/\": # examples in the paper" }}}{EXCHG {PARA 207
 "> " 0 "" {MPLTEXT 1 201 23 "infolevel[multifac]:=2;" }}{PARA 11 "" 1
 "" {XPPMATH 20 "\"\"#" }}}{EXCHG {PARA 207 "" 0 "" {TEXT 203 42 "A no
te about backward_error(F, G, [vars]):" }{TEXT 203 1 "\n" }{TEXT 203 
97 "We compute c so that ||F-c*G||_2 is as small as possible.  The out
put is ||F - c*G||_2 / ||F||_2." }{TEXT 203 1 "\n" }{TEXT 203 1 "\n" }
{TEXT 203 48 "Example 16 has been moved to it's own worksheet." }}}
{SECT 0 {PARA 209 "" 0 "" {TEXT 204 105 "Examples 10, 11 & 12: One pol
ynomial (12, 7, 5) with perturbations of different orders (-5 and -1) \+
added:" }}{SECT 0 {PARA 208 "" 0 "" {TEXT 211 75 "Example 10: Three Fa
ctors (12, 7, 5), Large Noise -5, very sparse noise: 5%" }}{EXCHG 
{PARA 207 "> " 0 "" {MPLTEXT 1 201 29 "read(cat(rand_path,\"exF10\")):
" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 201 42 "F10 := expand(evalf
(cleanF10 + noiseF10)):" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 201 
38 "norm(evalf(noiseF10),2) / norm(F10,2);" }{MPLTEXT 1 201 1 "\n" }
{MPLTEXT 1 201 15 "ceil(log10(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "$
\"/x1b_******!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "!\"&" }}}{EXCHG 
{PARA 207 "> " 0 "" {MPLTEXT 1 201 12 "t10:=time():" }{MPLTEXT 1 201 
1 "\n" }{MPLTEXT 1 201 26 "facF10:=appfac(F10,[x,y]):" }{MPLTEXT 1 
201 1 "\n" }{MPLTEXT 1 201 16 "t10:=time()-t10;" }}{PARA 6 "" 1 "" 
{TEXT 212 116 "` the biggest gap, the last r-th singular values and th
e number of factors `, 658.37353903434, .85582090298382e-6, 3" }}
{PARA 6 "" 1 "" {TEXT 212 61 "` The time for computing the number of f
actors*****`, 130.406" }}{PARA 6 "" 1 "" {TEXT 212 55 "`The time for t
he entire factorization******`, 1728.374" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%$t10G$\"(@%G<!\"$" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 
201 13 "nops(facF10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}}
{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 201 62 "berror10:=backward_error
(F10, convert(facF10, `*`), [x,y])[2];" }{MPLTEXT 1 201 1 "\n" }
{MPLTEXT 1 201 15 "ceil(log10(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
/%\"cG$!/_AX?A*e%!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)berror10G$
\"/1m;s&=;$!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"$" }}}{EXCHG 
{PARA 207 "> " 0 "" {MPLTEXT 1 201 59 "save facF10, t10, berror10, cat
(rand_path,\"exF10-factors\");" }}}}{SECT 0 {PARA 208 "" 0 "" {TEXT 
211 52 "Example 11: Three Factors (12, 7, 5), Large Noise -5" }}
{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 201 29 "read(cat(rand_path,\"exF
11\")):" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 201 42 "F11 := expan
d(evalf(cleanF11 + noiseF11)):" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT
 1 201 38 "norm(evalf(noiseF11),2) / norm(F11,2);" }{MPLTEXT 1 201 1 "
\n" }{MPLTEXT 1 201 15 "ceil(log10(%));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#$\"/#=qkf=-\"!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"%" }}}
{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 201 12 "t11:=time():" }{MPLTEXT 
1 201 1 "\n" }{MPLTEXT 1 201 26 "facF11:=appfac(F11,[x,y]):" }{MPLTEXT
 1 201 1 "\n" }{MPLTEXT 1 201 16 "t11:=time()-t11;" }}{PARA 6 "" 1 "" 
{TEXT 212 116 "` the biggest gap, the last r-th singular values and th
e number of factors `, 833.53638482292, .67589817442882e-6, 3" }}
{PARA 6 "" 1 "" {TEXT 212 61 "` The time for computing the number of f
actors*****`, 137.405" }}{PARA 6 "" 1 "" {TEXT 212 55 "`The time for t
he entire factorization******`, 2642.453" }}{PARA 6 "" 1 "" {TEXT 212 
116 "` the biggest gap, the last r-th singular values and the number o
f factors `, 1635.1608734688, .40453322177311e-5, 2" }}{PARA 6 "" 1 ""
 {TEXT 212 59 "` The time for computing the number of factors*****`, 4
.843" }}{PARA 6 "" 1 "" {TEXT 212 53 "`The time for the entire factori
zation******`, 36.984" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$t11G$\"(iN
o#!\"$" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 201 13 "nops(facF11);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}}{EXCHG {PARA 207 "> " 0 
"" {MPLTEXT 1 201 62 "berror11:=backward_error(F11, convert(facF11, `*
`), [x,y])[2];" }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 15 "ceil(log10(
%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"cG$!/(RD9(oF5!#5" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%)berror11G$\"/-i%[Hxx(!#=" }}{PARA 11 "" 1
 "" {XPPMATH 20 "6#!\"%" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 201 
59 "save facF11, t11, berror11, cat(rand_path,\"exF11-factors\");" }}}
}{SECT 0 {PARA 208 "" 0 "" {TEXT 211 53 "Example 12: Three Factors (12
, 7, 5), Larger Noise -3" }}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 201 
15 "#read(\"exF12\"):" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 201 
48 "F12 := expand(evalf(cleanF11 + 10.^2*noiseF11)):" }}}{EXCHG {PARA 
207 "> " 0 "" {MPLTEXT 1 201 45 "norm(evalf(10.^2.*noiseF11),2) / norm
(F12,2);" }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 15 "ceil(log10(%));" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"/o;3_#=-\"!#;" }}{PARA 11 "" 1 ""
 {XPPMATH 20 "6#!\"#" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 201 12 
"t12:=time():" }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 26 "facF12:=appf
ac(F12,[x,y]):" }{MPLTEXT 1 201 1 "\n" }{MPLTEXT 1 201 16 "t12:=time()
-t12;" }}{PARA 6 "" 1 "" {TEXT 212 116 "` the biggest gap, the last r-
th singular values and the number of factors `, 8.3401444211470, .6756
1577360017e-4, 3" }}{PARA 6 "" 1 "" {TEXT 212 61 "` The time for compu
ting the number of factors*****`, 139.062" }}{PARA 6 "" 1 "" {TEXT 
212 55 "`The time for the entire factorization******`, 4159.290" }}
{PARA 6 "" 1 "" {TEXT 212 116 "` the biggest gap, the last r-th singul
ar values and the number of factors `, 20.456213644050, .3245889213548
3e-3, 2" }}{PARA 6 "" 1 "" {TEXT 212 59 "` The time for computing the \+
number of factors*****`, 4.984" }}{PARA 6 "" 1 "" {TEXT 212 53 "`The t
ime for the entire factorization******`, 41.594" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$t12G$\"(4]?%!\"$" }}}{EXCHG {PARA 207 "> " 0 "" 
{MPLTEXT 1 201 13 "nops(facF12);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"
\"$" }}}{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 201 62 "berror12:=backwa
rd_error(F12, convert(facF12, `*`), [x,y])[2];" }{MPLTEXT 1 201 1 "\n"
 }{MPLTEXT 1 201 15 "ceil(log10(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#/%\"cG$!/eE0b\"o&=!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)berror12
G$\"/Q+zD.Be!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"#" }}}{EXCHG 
{PARA 207 "> " 0 "" {MPLTEXT 1 201 59 "save facF12, t12, berror12, cat
(rand_path,\"exF12-factors\");" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 207 "" 0 "" {TEXT 213 0 "" }}}}}{PARA 204 "" 0 "" {TEXT 214 0 ""
 }}}
{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2
 33 1 1 }