{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 2 1 2 0 0 0 1 }{CSTYLE "2D Output" -1 20 "Times" 0 1 0 0 255 1 0 0 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle1" -1 206 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle2" -1 207 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 0 0 0 1 }{CSTYLE "_cstyle3" -1 208 "Times" 1 14 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle4" -1 209 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle5" -1 210 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle6" -1 211 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "Line Printed \+ Output" -1 6 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "Left Justified Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "_pstyle1" -1 203 1 {CSTYLE "" -1 -1 "Times " 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 2 0 2 0 2 2 0 1 } {PSTYLE "_pstyle2" -1 204 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "_pstyle3" -1 205 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 2 0 2 0 2 2 0 1 }{PSTYLE "_pstyle7" -1 208 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 203 "" 0 "" {TEXT 206 153 "Worksheet to execute t he examples in Gao, Kaltofen, May, Yang, & Zhi:\n\n'Approximate Factor ization of Multivariate polynomials via differential Equations'" }}} {EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 207 51 "# Factorization routines :\nread(\"multifac_1.3.mpl\"):" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 207 156 "# Set path to example subdirectory:\npath:=\"examp les\\/\": \n# rand_path:=\"debug_exs\\/\": # debugging the spreadsheet \nrand_path:=path: # random examples in paper" }}}{EXCHG {PARA 204 "" 0 "" {TEXT 208 140 "A note about backward_error(F, G, [vars]):\nWe com pute c so that ||F-c*G||_2 is as small as possible. The output is ||F - c*G||_2 / ||F||_2." }}}{SECT 0 {PARA 203 "" 0 "" {TEXT 206 58 "Exam ples from previous papers on approximate factorization" }}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 207 24 "read(cat(path,\"exLit\")):" }}} {SECT 0 {PARA 205 "" 0 "" {TEXT 209 32 "Example 1: Nagasaka in ISSAC'0 2 " }}{EXCHG {PARA 204 "" 0 "" {TEXT 210 138 "Example taken from:\nNag asaka, Kosaku. Towards certified irreducibility testing of bivariate a pproximate polynomials. In Mora, T., editor " }{TEXT 211 89 "ISSAC 20 02 Procedings of 2002 International Symposium on Symbolic and Algebrai c Computing" }{TEXT 210 69 ", pages 192-199, New York, N.Y., 2002. ACM Press. ISBN 1-58113-484-3." }}{PARA 204 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 207 5 "Na02;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,@*$)%\"xG\"\"&\"\"\"F(*&)F&\"\"$F()%\"uG\"\"#F(F(*&) F&F.F(F-F(!\"\"*&\"\"(F(F0F(F(*&F-F()F&\"\"%F(F(*&)F-F+F(F0F(F(*(F.F(F ,F(F&F(F1*(F3F(F-F(F&F(F(*(F.F(F-F(F*F(F(*(F.F(F8F(F&F(F(*&F.F(F,F(F1* &\"#:F(F-F(F(*$F*F(F1F3F1*&$F.F1F(F&F(F(" }}}{EXCHG {PARA 204 "> " 0 " " {MPLTEXT 1 207 55 "t1:=time():\nfacNa02:=appfac(Na02,[x,u]);\nt1:=ti me()-t1;" }}{PARA 6 "" 1 "" {TEXT -1 108 "` the biggest gap, the last \+ r-th singular values and the number of factors `, 10.66697691, .436253 1450e-3, 2" }}{PARA 6 "" 1 "" {TEXT -1 59 "` The time for computing th e number of factors*****`, 5.728" }}{PARA 6 "" 1 "" {TEXT -1 53 "`The \+ time for the entire factorization******`, 14.561" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(facNa02G7$,6*&^$$\"3b&*)=pMX7T$!#@$\"\"!F-\"\"\")%\" uG\"\"$F.F.*(^$$!3Sh>!QL%4XH!#=F,F.)F0\"\"#F.%\"xGF.F.*(^$$!3VMIZCs*y8 %!#AF,F.)F9F8F.F0F.F.*&^$$!3[\"*4o/B(>'HF6F,F.)F9F1F.F.*&^$$\"3--a!QN' *Gr*!#?F,F.F7F.F.*(^$$!3.h!36)=DF9!#>F,F.F0F.F9F.F.*&^$$\"3KL63[t.!H\" FNF,F.F?F.F.*&^$$\"3G<1w+()49HF6F,F.F0F.F.*&^$$\"333gz-mT$4\"FNF,F.F9F .F.^$$!+\\TvQ?!\"*F,F.,.*&^$$\"3!>q4'zv=ksF+F,F.F7F.F.*(^$$\"3+(*[mz'3 y=&F6F,F.F0F.F9F.F.*&^$$\"33C&oaG+d8&F6F,F.F?F.F.*&^$$\"3B:A5$>*GH5!#< F,F.F0F.F.*&^$$\"35(>McaP$=9FNF,F.F9F.F.^$$!+R9D[^!#5F,F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#t1G$\"&JY\"!\"$" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 207 78 "berror1:=backward_error(Na02, convert(facNa02, ` *`), [x,u]);\nceil(log[10](%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/% \"cG$!+*)>&pn'!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(berror1G$\"+T vA%3\"!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"\"" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 207 53 "save facNa02, t1, berror1, cat(path, \"exF01-factors\");" }}}}{SECT 0 {PARA 205 "" 0 "" {TEXT 209 46 "Examp le 2: Sasaki's second example in ISSAC'01" }}{EXCHG {PARA 204 "" 0 "" {TEXT 210 300 "Example taken from:\nSasaki, Tateaki. Approximate multi variate polynomial factorization based on zero-sum relations. In Mourr ain, B., editor ISSAC 2001 Procedings of 2001 International Symposium on Symbolic and Algebraic Computing, pages 284-291, New York, N.Y., 2 001. ACM Press. ISBN 1-58113-417-7.\n" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 207 7 "Sa01_2;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,jr*&$\" \"*\"\"!\"\"\"%\"xGF(F(*&$\"\"%F'F()F)\"\"'F(!\"\"*&$\"#=F'F()%\"uG\" \"(F(F/*&F%F()F4\"\"&F(F(*($\"+++++7!\")F()F4\"\"$F(F)F(F(*($F>F'F(F4F ()F)F>F(F/*($\"#AF'F(F)F()F4F,F(F(*($F2F'F(F4F(F)F(F/*($\"+++++OFF'F(F7F(FNF(F/*($F2F'F(FAF(F=F(F(*$)F)FdpF(F(*$F_ oF(F(*&$F.F'F(FdoF(F/*&$F5F'F(F[qF(F/*&FjpF()F4FdpF(F(*&FfrF()F4F&F(F( *&$F>F'F(FgrF(F/*&$FeoF'F(FcqF(F/$F&F'F/*($\"#8F'F(FNF(FcqF(F/*($F^pF' F(FNF(FEF(F(*(FarF(F)F(FgsF(F(*($F5F'F(F)F(FgrF(F(*($\"#CF'F(F)F(FcqF( F/*($\"+++++;F \+ " 0 "" {MPLTEXT 1 207 59 "t2:=time():\nfacSa01_2:=appfac(Sa01_2,[x,u]) ;\nt2:=time()-t2;" }}{PARA 6 "" 1 "" {TEXT -1 109 "` the biggest gap, \+ the last r-th singular values and the number of factors `, 516494277.3 , .1781971781e-10, 2" }}{PARA 6 "" 1 "" {TEXT -1 58 "` The time for co mputing the number of factors*****`, .471" }}{PARA 6 "" 1 "" {TEXT -1 52 "`The time for the entire factorization******`, 5.198" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#>%*facSa01_2G7$,L*&^$$!354ou9@w;=!#=$\"\"!F-\"\" \")%\"xG\"\"&F.F.*(^$$\"3qFg\\cn\"4R\"!#FF,F.)F0\"\"#F.%\"uGF.F.*(^$$! 3gs)))GM'G]aF+F,F.)F9\"\"$F.F0F.F.*&^$$\"3;`4$4yP.q)!#GF,F.F7F.F.*&^$$ !366/o%>R3,#F6F,F.F9F.F.*(^$$!3+u(fvACNj$F+F,F.)F0F?F.)F9F8F.F.*&^$$\" 39#*>7bM\\c5F6F,F.F>F.F.*(^$$!3J9)>xACNj$F+F,F.F0F.)F9\"\"%F.F.*(^$$\" 3GY/Zc%[qE(F+F,F.FNF.F0F.F.*(^$$!3Yq&*=![Y4V#F6F,F.F7F.FNF.F.*&^$$!3)) >v$\\6in\"=F+F,F.)F9F1F.F.*(^$$\"3CJ()e>RpX:F6F,F.F>F.F7F.F.*&^$$\"3Q+ =o7@w;=F+F,F.FMF.F.*(^$$\"3-A3RTjG]aF+F,F.F9F.F0F.F.*&^$$!3iB:TEU_LOF+ F,F.)F0FXF.F.*&^$$\"3O,2)fX[qE(F+F,F.FNF.F.*(^$$!3'[vGR6in\"=F+F,F.F9F .F`pF.F.*&^$$!3Z]%)[wP9N>F6F,F.F0F.F.*&^$$!3oE()*pACNj$F+F,F.FWF.F.*(^ $$\"3AuF7TjG]aF+F,F.F9F.FMF.F.^$$\"+TjG]a!#5F,F.,L*&^$$!3.aGe38AC=F+F, F.F/F.F.*(^$$!3())fOm!H6SKF6F,F.F7F.F9F.F.*(^$$\"31#)Q:58AC=F+F,F.F>F. F0F.F.*&^$$\"35-D!)=EW[OF+F,F.F7F.F.*&^$$\"3%y.%*Gk-jQ&FDF,F.F9F.F.*(^ $$\"3cNpC?EW[OF+F,F.FMF.FNF.F.*&^$$\"3]UNwFRmsaF+F,F.F>F.F.*(^$$!3aleb 28AC=F+F,F.F0F.FWF.F.*(^$$\"3un@I[T)QD\"F6F,F.FNF.F0F.F.*(^$$\"3IEv+HR msaF+F,F.F7F.FNF.F.*&^$$!3/Wo\"*>EW[OF+F,F.F_oF.F.*(^$$!3Q\"y6%>EW[OF+ F,F.F>F.F7F.F.*&^$$\"3!RQ5w#RmsaF+F,F.FMF.F.*(^$$!3cNR-JRmsaF+F,F.F9F. F0F.F.*&^$$\"34::S58AC=F+F,F.F`pF.F.*&^$$\"3Aq3QQ_)oH(F+F,F.FNF.F.*(^$ $!3yg9`48AC=F+F,F.F9F.F`pF.F.*&^$$\"3Yq/EFRmsaF+F,F.F0F.F.*&^$$!3o>k,k p2'p#F6F,F.FWF.F.*(^$$!3c\\cXEDxcCF6F,F.F9F.FMF.F.^$$!+JRmsaFhqF,F." } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#t2G$\"%e_!\"$" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 207 82 "berror2:=backward_error(Sa01_2, conve rt(facSa01_2, `*`), [x,u]);\nceil(log[10](%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"cG$\"+=6M%(berror2G$\"+$)f'HI)!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"*" }}} {EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 207 55 "save facSa01_2, t2, berr or2, cat(path,\"exF02-factors\");" }}}}{SECT 0 {PARA 205 "" 0 "" {TEXT 209 45 "Example 3: Sasaki's first example in ISSAC'01" }}{EXCHG {PARA 204 "" 0 "" {TEXT 210 300 "Example taken from:\nSasaki, Tateaki. Approximate multivariate polynomial factorization based on zero-sum r elations. In Mourrain, B., editor ISSAC 2001 Procedings of 2001 Inter national Symposium on Symbolic and Algebraic Computing, pages 284-291, New York, N.Y., 2001. ACM Press. ISBN 1-58113-417-7.\n" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 207 7 "Sa01_1;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,`el*&$\"\"%\"\"!\"\"\"%\"xGF(F(*&$\"#;F'F()F)\"\"'F(F( *&$\"#7F'F()%\"uG\"\"(F(!\"\"*($\"++S++:!\")F()F3\"\"$F(F)F(F5*($F.F'F (F3F()F)F;F(F5*($\"\")F'F(F)F()F3F&F(F5*(F%F()F3\"\"#F(F)F(F(*($\"++I+ +7F9F(F>F(FDF(F5*($FAF'F(F3F()F)F&F(F(*($\"#>F'F(F:F()F)FEF(F(*&$\"#=F 'F(FOF(F5*&FMF(F>F(F(*&$\"#5F'F()F)\"\"&F(F5*&$F,F'F(FKF(F5*&$\"\"*F'F (F:F(F(*&$F&F'F(FBF(F5*($\"+++!)****!#5F(F3F()F)FgnF(F5*(F%F(FOF(FDF(F (*($FRF'F()F)FAF(FDF(F(*(FJF(FboF(F3F(F(*($FEF'F(F-F(FBF(F(*($\"++I++5 F9F(F-F(F:F(F5*($F.F'F(F-F(FDF(F(*($\"+++++5!#9F(F-F(F3F(F5*($F,F'F(FW F()F3FXF(F5*($F&F'F(FWF(FBF(F5*(F+F(FWF(FDF(F(*($FXF'F(F:F()F)F4F(F5*( $F&F'F(FDF(FgpF(F5*($\"#9F'F(F3F(FgpF(F(*(F+F(F:F(FWF(F(*($F1F'F()F3F. F(FKF(F5*($F4F'F(FapF(FKF(F(*($FVF'F(F:F(FKF(F5*($F&F'F(FWF(F3F(F5*($F EF'F(FKF(FBF(F5*($F1F'F(FKF(FDF(F(*($F&F'F(F>F(F2F(F5*(F+F(F>F(F`qF(F( *($FEF'F(F>F(FBF(F5*($\"++5++>F9F(F>F(FapF(F(*(F%F()F3FAF(FOF(F(*($F;F 'F(F2F(FOF(F(*($\"#GF'F(FapF(FOF(F5*($F\\qF'F(F>F(F:F(F5*&$F\\qF'F()F) FVF(F5*&$\"#\"F9F(F2F (F)F(F5*($\"#@F'F(FapF(F)F(F(*&$FEF'F()F)\"#6F(F5*&$FVF'F()F)F1F(F(*&F JF()F)\"#:F(F(*&$FAF'F()F)F\\qF(F5*$)F)\"#?F(F(*&$F(F'F()F)FNF(F5*&Ffr F()F)FasF(F(*&$F4F'F()F)\"#8F(F5*&$F&F'F()F)FRF(F5*&$F4F'F()F)F,F(F5*( $FAF'F(F^sF(F2F(F5*(F^uF(F[uF(FDF(F(*($FXF'F(F^sF(FBF(F5*(FeoF(F[vF(F3 F(F(*($FVF'F(F_uF(FDF(F5*($FEF'F(FauF(FDF(F5*($FEF'F(F]vF(FDF(F5*&F]vF (F3F(F(*(F%F(FauF(F:F(F(*(FeoF(FeuF(F`qF(F(*($FXF'F(FeuF(FBF(F(*($F\\u F'F(FeuF(FDF(F(*(FjoF(F_uF(FdrF(F(*(FfnF(F^sF(FisF(F(*(FjwF(F[uF(F_tF( F(*(FfrF(F[uF(F2F(F(*(FfnF(F3F(F^sF(F(*($F\\qF'F(FdrF(F^oF(F5*(FhwF(F: F(F^sF(F(*(FJF(F3F(FeuF(F(*($F.F'F(F3F(F`vF(F5*($F;F'F(FapF(F^oF(F5*(F eoF(FDF(FdvF(F(*($FasF'F(F:F(F[uF(F5*($FXF'F(F:F(FeuF(F5*($F&F'F(F:F(F gvF(F5*(FjoF(FDF(FgvF(F(*(FfrF(F2F(F`vF(F(*($F(F'F(FapF(F`vF(F5*&F:F(F `vF(F(*($F(F'F(F3F(FdvF(F5*(FeoF(F3F(FgvF(F(*(FfnF()F3F\\uF(F^oF(F(*(F %F(FdrF(F^sF(F(*(F%F(F^oF(F2F(F(*(FfrF(F^sF(FDF(F(*(F[qF(F^oF(FBF(F(*( FeoF(F`vF(F`qF(F(*(FeoF(F`vF(FBF(F(*($F(F'F(F`vF(FDF(F5*($F.F'F(F[uF(F drF(F5*($F&F'F(F^oF(FisF(F5*(FfnF(F[uF(F`qF(F(*(FeoF(F^sF(F_tF(F(*($F, F'F(F^oF(FDF(F5*(F%F(FboF(F_tF(F(*($FavF'F(FboF(F`qF(F(*(FfrF(F]vF(F:F (F(*&FauF(FBF(F(*(FeoF(F_uF(FBF(F(*(FbqF(FboF()F3F1F(F(*(FjoF(F^oF(F_t F(F(*(FJF(FboF(F2F(F(*($F;F'F(FboF(FBF(F5*($F.F'F(FauF(F3F(F5*($F&F'F( F^sF(F`qF(F5*($FVF'F(F:F(FboF(F5*($FAF'F(FisF(FgpF(F5*($F\\qF'F(FapF(F boF(F5*($F\\uF'F(F:F(F^oF(F5*(FfnF(F:F(F_uF(F(*($FasF'F(F2F(FgpF(F5*(F eoF(FBF(FgvF(F(*(FbqF(FapF(F_uF(F(*(FeoF(FapF(FeuF(F(*(FjwF(FapF(F[uF( F(*(F%F()F3FavF(FgpF(F(*(F%F(F_tF(FgpF(F(*&F`qF(FgpF(F(*($FEF'F(FBF(F[ uF(F5*(FeoF(F`[lF(FgpF(F(*($FEF'F(FhyF(FboF(F5*(F%F(FgpF(FBF(F(*(F%F(F ^oF(F`qF(F(*($F1F'F(FKF(F2F(F5*($FEF'F(F>F()F3F\\qF(F5*($\"#DF'F(FKF(F _tF(F(*(FjqF(FWF(F2F(F(*(F[[lF(F>F(FhyF(F(*($FavF'F(FWF(F_tF(F5*(FJF(F gpF(FhyF(F(*(FfrF(FWF()F3FbuF(F(*($F;F'F(F_uF(F2F(F5*(FJF(F>F()F3FasF( F(*(FhwF(FWF(Fi\\lF(F(*(FjwF(FKF()F3F,F(F(*&FKF(Fg]lF(F(*(FjqF(F>F(F`[ lF(F(*(FaoF(FgpF(FapF(F(*($FAF'F(F>F(F_tF(F5*($F(F'F(FboF(FdrF(F5*($F4 F'F(F^sF(FapF(F5*&F>F(Fa^lF(F(*(FfnF(FWF(FhyF(F(*(F[qF(FKF(F`[lF(F(*(F eoF(FdrF(F>F(F(*(FhwF(F3F(F_uF(F(*($FdsF'F(FisF(FKF(F5*(F^uF(FdrF(FKF( F(*(FbqF(FWF(FdrF(F(*(FeoF(F>F(FisF(F(*(FMF(FWF(F`qF(F(*($F.F'F(Fa^lF( FOF(F5*(FjoF(FdrF(FgpF(F(*($F;F'F(FdrF(F-F(F5*($FVF'F(F`[lF(FOF(F5*(F% F(Fg]lF(F-F(F(*($F(F'F(F`[lF(F-F(F5*(FjoF(FisF(F-F(F(*(FJF()F3FRF(FOF( F(*($FhuF'F(Fg]lF(FOF(F(*(F+F(F2F(F-F(F(*(FbqF(FhyF(FOF(F(*($FEF'F(Fi \\lF(F-F(F5*(F%F(FhyF(F-F(F(*(FhwF(F_tF(F-F(F(*($F(F'F(Fa^lF(FKF(F5*($ F(F'F(Fe^lF(FOF(F5*($F&F'F(Fi\\lF(FKF(F5*($F;F'F(Fh^lF(F>F(F5*($F&F'F( FisF(FOF(F5*(F+F(FapF(F-F(F(*(FfrF(FhyF(FKF(F(*(FfrF(Fi\\lF(FOF(F(*($F XF'F(F-F(F`qF(F5*(F[alF(FOF(F_tF(F(*($F;F'F(F)F(Fh^lF(F5*($FXF'F(F)F(F i\\lF(F5*(F^uF(F)F()F3FNF(F(*&$FgnF'F(Fe^lF(F5*&$F.F'F(Fg]lF(F5*&FfrF( )F3FhuF(F(*&$F;F'F(Fi\\lF(F5*&FeoF(Fh^lF(F(*(FjwF(FOF(Fh^lF(F(*(FhwF(F i\\lF(F>F(F(*(FjwF(Fa^lF(F)F(F(*($FAF'F(FisF(F)F(F5*(FjoF(FisF(FWF(F(* (FfnF(F`[lF(FWF(F(*($F;F'F(Fe^lF(F)F(F5" }}}{EXCHG {PARA 204 "> " 0 " " {MPLTEXT 1 207 63 "t3 := time():\nfacSa01_1:=appfac(Sa01_1,[x,u]):\n t3 := t3-time();" }}{PARA 6 "" 1 "" {TEXT -1 108 "` the biggest gap, t he last r-th singular values and the number of factors `, 149706.7995, .3264602951e-7, 2" }}{PARA 6 "" 1 "" {TEXT -1 60 "` The time for comp uting the number of factors*****`, 16.454" }}{PARA 6 "" 1 "" {TEXT -1 53 "`The time for the entire factorization******`, 85.904" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#t3G$!&kf)!\"$" }}}{EXCHG {PARA 204 "> " 0 " " {MPLTEXT 1 207 82 "berror3:=backward_error(Sa01_1, convert(facSa01_1 , `*`), [x,u]);\nceil(log[10](%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /%\"cG$!+F&3q-*!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(berror3G$\"+ znN[5!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"&" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 207 55 "save facSa01_1, t3, berror3, cat(path,\"e xF03-factors\");" }}}}{SECT 0 {PARA 205 "" 0 "" {TEXT 209 70 "Example \+ 4: Corless, Giesbrech, van Hoeij, Kotsireas & Watt in ISSAC'01" }} {EXCHG {PARA 204 "" 0 "" {TEXT 210 353 "Example taken from:\nCorless, \+ Robert M., Geisbrecht, Mark W., van Hoeij, Mark, Kotsireas, Ilias S, a nd Watt, Stephen M. Towards factoring bivariate approximate polynomial s. In Mourrain, B., editor ISSAC 2001 Procedings of 2001 Internationa l Symposium on Symbolic and Algebraic Computing, pages 85-92, New York , N.Y., 2001. ACM Press. ISBN 1-58113-417-7." }}}{EXCHG {PARA 204 "> \+ " 0 "" {MPLTEXT 1 207 6 "Cor01;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,\\ \\l*&$\"%O8\"\"!\"\"\"%\"xGF(F(*&$\"%S\"*F'F(%\"yGF(!\"\"*($\"&,n\"F'F ()F)\"\"%F()F-F3F(F.*($\"&vG\"F'F(F2F()F-\"\"#F(F.*($\"%k**F'F(F2F()F- \"\"$F(F.*($\"%:vF'F()F)\"\"'F(F=F(F(*($\"&ld\"F'F(FBF()F-\"\")F(F.*($ \"&2W#F'F(FBF()F-\"\"(F(F.*($\"&ED\"F'F()F)F9F()F-\"#7F(F.*($\"%d;F'F( F)F(FLF(F(*($\"%,7F'F(F)F(F4F(F(*($\"%+&)F'F(F)F()F-FCF(F(*($\"&v$GF'F ()F)F>F(F=F(F(*($\"&y\"=F'F(FBF(F4F(F.*($\"&HI\"F'F(FBF(FgnF(F(*($\"%L @F'F(F)F()F-\"\"&F(F.*($\"&pD\"F'F(F)F(F-F(F(*($\"&0B\"F'F(FBF(F8F(F(* ($\"&l3\"F'F()F)FfoF(F8F(F.*($\"&P\"=F'F(F`pF(F=F(F.*($\"&\"[;F'F(F[oF (F4F(F(*($\"%KVF'F(F[oF(F-F(F.*($\"&`s\"F'F(F)F(FRF(F(*&$\"%KNF'F(FBF( F.*($\"%pWF'F(F)F(F=F(F.*($\"&JI#F'F()F)FMF(F-F(F.*($\"%*[)F'F(F)F(F8F (F.*($\"%AcF'F(F2F(FGF(F(*($\"%]6F'F(F2F(FgnF(F.*($\"%YuF'F(FBF(F-F(F( *($\"&!\\6F'F(F[oF(FgnF(F(*($\"&#p**F'F(FeoF(F(*&$\"%IEF'F(F=F(F. *&$\"&GX\"F'F(FLF(F(*&$\"%qgF'F(FGF(F.*&$\"%!y&F'F(F8F(F(*&$\"%_()F'F( FgnF(F.*&$\"%waF'F(F4F(F.*($\"&c\\#F'F(FQF(FeoF(F.*($\"%xXF'F(FQF(F4F( F.*($\"&i/\"F'F(FQF(F8F(F(*($\"%Q^F'F(FQF(F-F(F(*($\"%#*zF'F(F`pF(F-F( F(*($\"&P[\"F'F(FQF(F=F(F(*($\"&EW#F'F(F2F(F-F(F.*($\"%lcF'F(F)F(FGF(F (*($\"%OCF'F(F)F()F-F]wF(F(*($\"&l4\"F'F(F)F(FirF(F(*($\"%L>F'F(F)F()F -F[uF(F.*($\"%eBF'F(F)F()F-FhvF(F.*($\"%7RF'F(F)F()F-FjuF(F(*($\"%(>$F 'F(FQF(FLF(F.*($\"%)f(F'F(FQF(FGF(F(*($\"%\"H&F'F(FQF(F^zF(F.*($\"%EgF 'F(FQF(FirF(F(*($\"&b/\"F'F(FQF(FezF(F(*($\"&6'>F'F(FQF(F][lF(F(*($\"% QFF'F(F[oF(FRF(F.*($\"&W.#F'F(F[oF(FLF(F.*($\"%N$)F'F(F[oF(FGF(F.*&$\" %QNF'F()F-FcvF(F(*&$\"%m6F'F(F^zF(F(*&$\"%QwF'F(FirF(F.*&$\"%0:F'F(Fez F(F(*&$\"%CoF'F(FizF(F(*&$\"%z$)F'F(F][lF(F.*($\"%saF'F(F[oF(FezF(F.*( $\"%%)zF'F(F2F(FeoF(F(*($\"&h!QF'F(F2F(FLF(F(*($\"&`;\"F'F(F[oF(F][lF( F(*($\"&)HAF'F(F2F(FirF(F.*($\"%^\\F'F(F2F(FezF(F(*($\"$B'F'F(F2F(F][l F(F(*($\"%z\"*F'F(F`pF(FeoF(F.*($\"&Nt$F'F(F`pF(FGF(F.*($\"%#o&F'F(F`p F(FLF(F(*($\"%@lF'F(F`pF(FgnF(F.*($\"%cSF'F(F`pF(F4F(F.*($\"&Eg#F'F(F` pF(FirF(F.*($\"%cgF'F(F`pF(FezF(F(*($\"%4oF'F(FBF(FirF(F(*($\"&O(GF'F( FBF(FeoF(F.*($\"&Jh\"F'F(FfqF(FeoF(F.*($\"&)>9F'F(FfqF(F=F(F(*($\"&#>8 F'F(FfqF(FLF(F(*($\"%j'*F'F(FfqF(FGF(F(*($\"&U6#F'F(FfqF(F8F(F.*($\"%` aF'F(FfqF(FgnF(F(*($\"%yCF'F(FfqF(F4F(F.*($\"&?O$F'F(F^sF(FeoF(F(*($\" %ZVF'F(F^sF(F=F(F.*($\"%y))F'F(F^sF(FLF(F.*($\"&*o9F'F(F^sF(F8F(F(*($ \"%MbF'F(F^sF(FgnF(F.*($\"&T;$F'F(F^sF(F4F(F.*($\"&G/\"F'F(F_uF(F-F(F( *($\"&o3#F'F(F_uF(FeoF(F(*($\"%(o(F'F(F_uF(F=F(F.*($\"&GS\"F'F(F_uF(F8 F(F(*($\"%lvF'F(F_uF(FgnF(F.*($\"&p_\"F'F(F_uF(F4F(F(*($\"%[7F'F(FjtF( F-F(F.*($\"%a&)F'F(FjtF(FeoF(F.*($\"%5eF'F(FjtF(F=F(F(*($\"&P%=F'F(Fjt F(F8F(F(*($\"&P\"HF'F(FjtF(F4F(F(*($\"%CFF'F(FiuF(F8F(F(*($\"%r " 0 "" {MPLTEXT 1 207 57 "t4:=time( ):\nfacCor01:=appfac(Cor01,[x,y]):\nt4:=time()-t4;" }}{PARA 6 "" 1 "" {TEXT -1 108 "` the biggest gap, the last r-th singular values and the number of factors `, 11274956.99, .5960595343e-9, 2" }}{PARA 6 "" 1 " " {TEXT -1 59 "` The time for computing the number of factors*****`, 3 .615" }}{PARA 6 "" 1 "" {TEXT -1 53 "`The time for the entire factoriz ation******`, 19.568" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#t4G$\"&G'>! \"$" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 207 80 "berror4:=backwar d_error(Cor01, convert(facCor01, `*`), [x,y]);\nceil(log[10](%));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"cG$!+6B&e+'!\"&" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%(berror4G$\"+ZT*)49!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"(" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 207 54 " save facCor01, t4, berror4, cat(path,\"exF04-factors\");" }}}}{SECT 0 {PARA 205 "" 0 "" {TEXT 209 58 "Example 5: Corless, Galligo, Kotsireas & Watt in ISSAC'02:" }}{EXCHG {PARA 204 "" 0 "" {TEXT 210 364 "Exampl e taken from:\nCorless, Robert M., Galligo, Andre, Kotsireas, Ilias S. , and Watt, Stephen M. A geometric-numeric algorithm for absolute fact orization of multivariate polynomials. In Mora, T., editor ISSAC 2002 Procedings of 2002 International Symposium on Symbolic and Algebraic \+ Computing, pages 192-199, New York, N.Y., 2002. ACM Press. ISBN 1-5811 3-484-3.\n" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 207 6 "Cor02;" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#,ho\"\"$!\"\"*&\"#;\"\"\"%\"xGF(F%*&\" \"(F(%\"yGF(F(*(F$F()F)\"\"%F()F,F/F(F(*(F$F(F.F()F,\"\"#F(F(*(\"\"'F( F.F()F,F$F(F(*(\"\")F()F)F5F(F6F(F(*(F/F(F)F(F0F(F(*(\"\"&F()F)F$F(F6F (F%*&)F)F " 0 "" {MPLTEXT 1 207 57 "t5:=tim e():\nfacCor02:=appfac(Cor02,[x,y]):\nt5:=time()-t5;" }}{PARA 6 "" 1 " " {TEXT -1 109 "` the biggest gap, the last r-th singular values and t he number of factors `, 262965857.3, .1877443001e-10, 3" }}{PARA 6 "" 1 "" {TEXT -1 58 "` The time for computing the number of factors*****` , .231" }}{PARA 6 "" 1 "" {TEXT -1 52 "`The time for the entire factor ization******`, 9.184" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#t5G$\"%M#* !\"$" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 207 62 "berror5:=backwa rd_error(Cor02, convert(facCor02, `*`), [x,y]);" }}{PARA 6 "" 1 "" {TEXT -1 50 "Approximate factorization is not a real polynomial" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"cG^$$\"+&Ryc.*!#5$!+mn'HX'!\"*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%(berror5G$\"+ " 0 "" {MPLTEXT 1 207 54 "save facCor02, t5, berror5, cat (path,\"exF05-factors\");" }}}}}{SECT 0 {PARA 203 "" 0 "" {TEXT 206 31 "New Randomly Generated Examples" }}{SECT 0 {PARA 205 "" 0 "" {TEXT 209 54 "Example 6: Three Factors (6, 6, 10), Moderate Noise -5" }}{EXCHG {PARA 204 "" 0 "" {TEXT 210 177 " This example is a polynomia l with three factors with coefficients in [-5, 5], two factors with to tal degree 6 and one with total degree 10; the added noise is of order 10^(-5)." }}{PARA 204 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 207 29 "read(cat(rand_path,\"exF06\")):" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 207 39 "F6 := expand(evalf(cleanF6 + no iseF6)):" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 207 52 "norm(evalf( noiseF6),2) / norm(F6,2);\nceil(log10(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Ne******!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"& " }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 207 51 "t6:=time():\nfacF6: =appfac(F6,[x,y]):\nt6:=time()-t6;" }}{PARA 6 "" 1 "" {TEXT -1 108 "` \+ the biggest gap, the last r-th singular values and the number of facto rs `, 1619.016618, .6622266238e-6, 3" }}{PARA 6 "" 1 "" {TEXT -1 60 "` The time for computing the number of factors*****`, 30.363" }}{PARA 6 "" 1 "" {TEXT -1 54 "`The time for the entire factorization******`, \+ 539.616" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#t6G$\"'w'R&!\"$" }}} {EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 207 72 "berror6:=backward_error( F6, convert(facF6, `*`), [x,y]);\nceil(log10(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"cG$!+v`V %(berror6G$\"+hOjtC!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"$" }}} {EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 207 56 "save facF6, t6, berror6, cat(rand_path,\"exF06-factors\");" }}}}{SECT 0 {PARA 205 "" 0 "" {TEXT 209 49 "Example 7: Two Factors (9, 7), Moderate Noise -4" }} {EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 207 29 "read(cat(rand_path,\"exF 07\")):" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 207 39 "F7 := expand (evalf(cleanF7 + noiseF7)):" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 207 52 "norm(evalf(noiseF7),2) / norm(F7,2);\nceil(log10(%));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+P2++5!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"$" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 207 51 " t7:=time():\nfacF7:=appfac(F7,[x,y]):\nt7:=time()-t7;" }}{PARA 6 "" 1 "" {TEXT -1 108 "` the biggest gap, the last r-th singular values and \+ the number of factors `, 485.9325000, .1528992500e-4, 2" }}{PARA 6 "" 1 "" {TEXT -1 59 "` The time for computing the number of factors*****` , 5.127" }}{PARA 6 "" 1 "" {TEXT -1 53 "`The time for the entire facto rization******`, 43.773" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#t7G$\"&B Q%!\"$" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 207 72 "berror7:=back ward_error(F7, convert(facF7, `*`), [x,y]);\nceil(log10(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"cG$!+(ph`]\"!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(berror7G$\"+[u9O@!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"$" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 207 56 "save facF 7, t7, berror7, cat(rand_path,\"exF07-factors\");" }}}}{SECT 0 {PARA 205 "" 0 "" {TEXT 209 54 "Example 8: Five Factors (4,4,4,4,4), Moderat e Noise -5" }}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 207 29 "read(cat(r and_path,\"exF08\")):" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 207 39 "F8 := expand(evalf(cleanF8 + noiseF8)):" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 207 52 "norm(evalf(noiseF8),2) / norm(F8,2);\nceil(log 10(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+u&)******!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"&" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 207 51 "t8:=time():\nfacF8:=appfac(F8,[x,y]):\nt8:=time()-t 8;" }}{PARA 6 "" 1 "" {TEXT -1 108 "` the biggest gap, the last r-th s ingular values and the number of factors `, 472.8585859, .3093153669e- 6, 5" }}{PARA 6 "" 1 "" {TEXT -1 60 "` The time for computing the numb er of factors*****`, 17.795" }}{PARA 6 "" 1 "" {TEXT -1 55 "`The time \+ for the entire factorization******`, 1348.065" }}{PARA 6 "" 1 "" {TEXT -1 108 "` the biggest gap, the last r-th singular values and the number of factors `, 472.8585859, .3093153669e-6, 5" }}{PARA 6 "" 1 " " {TEXT -1 60 "` The time for computing the number of factors*****`, 1 8.376" }}{PARA 6 "" 1 "" {TEXT -1 55 "`The time for the entire factori zation******`, 1690.841" }}{PARA 6 "" 1 "" {TEXT -1 108 "` the biggest gap, the last r-th singular values and the number of factors `, 109.0 779874, .1410754845e-4, 3" }}{PARA 6 "" 1 "" {TEXT -1 59 "` The time f or computing the number of factors*****`, 1.482" }}{PARA 6 "" 1 "" {TEXT -1 53 "`The time for the entire factorization******`, 59.525" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#t8G$\"(s))4$!\"$" }}}{EXCHG {PARA 204 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 207 72 "berror8:=backward_error(F8, convert(facF8, `*`), [x,y]);\nceil (log10(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"cG$\"+A9J$p%!\"'" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(berror8G$\"+l\\/98!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"#" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 207 56 "save facF8, t8, berror8, cat(rand_path,\"exF08-factors\");" }} }}{SECT 0 {PARA 205 "" 0 "" {TEXT 209 48 "Example 9: Three Factors (3, 3,3), Large Noise -1" }}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 207 29 " read(cat(rand_path,\"exF09\")):" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 207 39 "F9 := expand(evalf(cleanF9 + noiseF9)):" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 207 60 "norm(evalf(noiseF9),2) / norm(e xpand(F9),2);\nceil(log10(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+ 3`GD)*!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"\"" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 207 51 "t9:=time():\nfacF9:=appfac(F9,[x,y]): \nt9:=time()-t9;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%hnthe~gap~is~too~ small,~we~think~the~polynomial~is~irreducible.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#t9G$\"$T%!\"$" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 207 72 "berror9:=backward_error(F9, convert(facF9, `*`), [x ,y]);\nceil(log10(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"cG$\"+%* ********!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(berror9G$\"++1oDh!#> " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"*" }}}{EXCHG {PARA 204 "> " 0 " " {MPLTEXT 1 207 56 "save facF9, t9, berror9, cat(rand_path,\"exF09-fa ctors\");" }}}}{PARA 205 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 205 "" 0 "" {TEXT 209 51 "Example 13: Two Factors (5, 5^2), Moderate Noise -5 " }}{EXCHG {PARA 204 "" 0 "" {TEXT 210 42 "This example should have a \+ very small gap." }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 207 29 "read (cat(rand_path,\"exF13\")):" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 207 42 "F13 := expand(evalf(cleanF13 + noiseF13)):" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 207 54 "norm(evalf(noiseF13),2) / norm(F13,2) ;\nceil(log10(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+N&ynC)!#:" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"&" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 207 60 "t13:=time():\nfacF13:=multifac(F13,[x,y],1):\nt13:= time()-t13;" }}{PARA 6 "" 1 "" {TEXT -1 108 "` the biggest gap, the la st r-th singular values and the number of factors `, 8979.180166, .118 4609054e-5, 2" }}{PARA 6 "" 1 "" {TEXT -1 58 "` The time for computing the number of factors*****`, .561" }}{PARA 6 "" 1 "" {TEXT -1 52 "`Th e time for the entire factorization******`, 6.009" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$t13G$\"&!GM!\"$" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 207 104 "berror13:=backward_error(F13, mul(facF13[i][1]^fac F13[i][2], i=1..nops(facF13)), [x,y]);\nceil(log10(%));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/%\"cG$!+FBKS^!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)berror13G$\"+D0Jvp!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"% " }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 207 59 "save facF13, t13, b error13, cat(rand_path,\"exF13-factors\");" }}}}{SECT 0 {PARA 205 "" 0 "" {TEXT 209 62 "Example 15: Two Factors (5, 5), Moderate Noise -5, three vars" }}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 207 29 "read(cat( rand_path,\"exF15\")):" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 207 42 "F15 := expand(evalf(cleanF15 + noiseF15)):" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 207 54 "norm(evalf(noiseF15),2) / norm(F15,2);\ncei l(log10(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"++-++5!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"%" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 207 58 "t15:=time():\nfacF15:=appfac(F15,[x,y,z]):\nt15:=ti me()-t15;" }}{PARA 6 "" 1 "" {TEXT -1 108 "` the biggest gap, the last r-th singular values and the number of factors `, 11933.53114, .20138 92459e-5, 2" }}{PARA 6 "" 1 "" {TEXT -1 61 "` The time for computing t he number of factors*****`, 117.178" }}{PARA 6 "" 1 "" {TEXT -1 54 "`T he time for the entire factorization******`, 332.948" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$t15G$\"'))HL!\"$" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 207 77 "berror15:=backward_error(F15, convert(facF15, `*`), [x,y,z]);\nceil(log10(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"cG$ !+E'>bF%!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)berror15G$\"+41:> " 0 "" {MPLTEXT 1 207 59 "save facF15, t15, berror15, cat(rand_path,\"ex F15-factors\");" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 204 "" 0 " " {TEXT -1 0 "" }}}}{SECT 0 {PARA 205 "" 0 "" {TEXT 209 91 "Example 15 a: Two Factors (2,2), three vars, Kaltofen challenge JSC 29/6, pp. 891 -919 (2000)" }}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 207 29 "read(cat( rand_path,\"exF19\")):" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 207 42 "F19 := expand(evalf(cleanF19 + noiseF19)):" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 207 54 "norm(evalf(noiseF19),2) / norm(F19,2);\ncei l(log10(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+nco'e%!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#!\"%" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 207 58 "t19:=time():\nfacF19:=appfac(F19,[x,y,z]):\nt19:=ti me()-t19;" }}{PARA 6 "" 1 "" {TEXT -1 109 "` the biggest gap, the last r-th singular values and the number of factors `, 1563747223., .16750 47140e-10, 2" }}{PARA 6 "" 1 "" {TEXT -1 59 "` The time for computing \+ the number of factors*****`, 5.809" }}{PARA 6 "" 1 "" {TEXT -1 53 "`Th e time for the entire factorization******`, 12.969" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%$t19G$\"&4I\"!\"$" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 207 77 "berror19:=backward_error(F19, convert(facF19, `*`), [x,y,z]);\nceil(log10(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"cG$ !+A:jc6!\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)berror19G$\"+>VU<5!# =" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\")" }}}{EXCHG {PARA 204 "> " 0 "" {MPLTEXT 1 207 59 "save facF19, t19, berror19, cat(rand_path,\"exF1 9-factors\");" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 12 "" 1 "" {TEXT -1 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 208 "" 0 "" {TEXT -1 0 "" }}}{MARK "5 8 \+ 0 0" 11 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }