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rand_path:=\" debug_exs\\/\": # debugging the spreadsheet\n" }{MPLTEXT 1 202 43 "ran d_path:=path: # random examples in paper" }}}{EXCHG {PARA 208 "" 0 "" {TEXT 214 43 "A note about backward_error(F, G, [vars]):\n" }{TEXT 214 97 "We compute 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 212 58 "Examples from previous papers on approximate factorizati on" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "read(cat(path,\"exLit \")):" }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 30 "Digits:=floor( evalhf(Digits));" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"#9" }}}{SECT 0 {PARA 202 "" 0 "" {TEXT 204 32 "Example 1: Nagasaka in ISSAC'02 " }} {EXCHG {PARA 208 "" 0 "" {TEXT 206 20 "Example taken from:\n" }{TEXT 206 118 "Nagasaka, Kosaku. Towards certified irreducibility testing of bivariate approximate polynomials. In Mora, T., editor " }{TEXT 213 89 "ISSAC 2002 Procedings of 2002 International Symposium on Symbolic \+ and Algebraic Computing" }{TEXT 206 69 ", pages 192-199, New York, N.Y ., 2002. ACM Press. ISBN 1-58113-484-3." }}{PARA 208 "" 0 "" {TEXT 215 0 "" }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 5 "Na02;" }} {PARA 11 "" 1 "" {XPPMATH 20 ",@*$)I\"xG6\"\"\"&\"\"\"F(*&)F%\"\"$F()I \"uGF&\"\"#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 208 "> " 0 "" {MPLTEXT 1 202 12 "t1:=time():\n" }{MPLTEXT 1 202 29 "facNa02:=appfac(Na02,[x,u]);\n" }{MPLTEXT 1 202 14 "t1:=time()-t1; " }}{PARA 6 "" 1 "" {TEXT 216 117 "` the biggest gap, the last r-th si ngular values and the number of factors `, 10.666976822227, 0.43625314 862261e-3, 2" }}{PARA 6 "" 1 "" {TEXT 216 59 "` The time for computing the number of factors*****`, 1.813" }}{PARA 6 "" 1 "" {TEXT 216 52 "` The time for the entire factorization******`, 4.106" }}{PARA 11 "" 1 " " {XPPMATH 20 "7$,6*&,&$\"3F&Gm>FA5$R!#@\"\"\"*&$\"\"!F,F)^#F)F)F)F))I \"uG6\"\"\"$F)F)*(,&$\"3nsX,HvC6F!#=!\"\"F*F7F))F/\"\"#F)I\"xGF0F)F)*( ,&$\"32*[#4bA/W=F(F)F*F)F))F:F9F)F/F)F)*&,&$\"3:\"*4ATI-AFF6F7F*F7F))F :F1F)F)*&,&$\"3e.d7$[g`p)!#?F)F*F)F)F8F)F)*(,&$\"3Eu2JC(y(=7!#>F7F*F)F )F/F)F:F)F)*&,&$\"3!on#ygZ_#>\"FNF)F*F)F)F?F)F)*&,&$\"3c4,+Rmu%o#F6F)F *F)F)F/F)F)*&,&$\"3S$4#e34(*z&*FIF)F*F)F)F:F)F),&$\"/u@GB^x=!#8F7F*F)F ),.*&,&$\"3*R[0I@7$=qF(F7F*F)F)F8F)F)*(,&$\"3r8==0%GV$\\F6F7F*F)F)F/F) F:F)F)*&,&$\"3[])HG3iZ)[F6F7F*F)F)F?F)F)*&,&$\"3u:^$f,(***y*F6F7F*F)F) F/F)F)*&,&$\"35Sh\\9nT^8FNF7F*F)F)F:F)F),&$\"/]iCHv'*[!#9F)F*F)F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "$\"%OT!\"$" }}}{EXCHG {PARA 208 "> " 0 " " {MPLTEXT 1 202 64 "berror1:=backward_error(Na02, convert(facNa02, `* `), [x,u])[2];\n" }{MPLTEXT 1 202 17 "ceil(log[10](%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"/kmvYNE5!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "!\" \"" }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 53 "save facNa02, t1, berror1, cat(path,\"exF01-factors\");" }}}}{SECT 0 {PARA 202 "" 0 "" {TEXT 204 46 "Example 2: Sasaki's second example in ISSAC'01" }} {EXCHG {PARA 208 "" 0 "" {TEXT 206 20 "Example taken from:\n" }{TEXT 206 280 "Sasaki, Tateaki. Approximate multivariate polynomial factoriz ation based on zero-sum relations. In Mourrain, B., editor ISSAC 2001 Procedings of 2001 International Symposium on Symbolic and Algebraic \+ Computing, pages 284-291, New York, N.Y., 2001. ACM Press. ISBN 1-5811 3-417-7.\n" }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 7 "Sa01_2;" } }{PARA 11 "" 1 "" {XPPMATH 20 ",jr*&$\"\"*\"\"!\"\"\"I\"xG6\"F'F'*($\" #=F&F'I\"uGF)F'F(F'!\"\"*&$\"\"'F&F')F(\"\"#F'F'*&F0F')F(\"\"$F'F'*($ \"+++++O!\")F'F5F')F-F3F'F'*(F$F'F2F'F-F'F'*($\"#:F&F'F-F')F(\"\"%F'F' *($\"#FF&F')F-F6F'F2F'F.*($F6F&F'F-F'F5F'F.*($\"+++++7F:F'FEF'F(F'F'*( $\"+++++?!\"*F'F-F')F(F%F'F'*($\"#?F&F'F2F'F;F'F'*($\"#AF&F'F(F')F-FAF 'F'*&$FAF&F')F(\"\"&F'F.*&$\"#7F&F'F@F'F'*&F$F'FEF'F'*&FTF'FVF'F'*&F$F ')F-FZF'F'*&FXF')F(F1F'F.*&F+F')F-\"\"(F'F.*$)F(\"#5F'F'*$FOF'F'*&F0F' )F(\"\")F'F.*&$F`oF&F')F(F`oF'F.*&$F3F&F')F-FcoF'F'*&FXF')F-F%F'F'*&FG F')F-FgoF'F.*&$FgoF&F')F-F1F'F.*&FfoF'F;F'F'*(F\\pF'FfoF'F-F'F.*&F]oF' FVF'F'*($\"+++++!*FNF'F]oF'FEF'F'*($\"#9F&F'F]oF'F;F'F.*($\"+++++qFNF' F]oF'F-F'F'*($FcoF&F'FYF'F[oF'F'*(FcpF'FYF'FVF'F.*($FZF&F'FYF'F;F'F.*( F\\pF'FEF'FjoF'F'*(FfnF'F;F'FjoF'F.*($F'F&F'F-F'FjoF'F.*(FeqF'FEF'FYF' F.*&FdpF'F@F'F'*(F0F'F[oF'F@F'F.*(FbqF'FEF'F@F'F.*(FfnF'FYF'F-F'F'*(Fi qF'F@F'FVF'F.*(FioF'F@F'F;F'F.*(F0F'F5F'F_oF'F'*(FbqF'F5F'FdpF'F.*(Feq F'F5F'FVF'F'*($\"+++++GF:F'F5F'F[oF'F.*(FXF'FapF'F2F'F'*(F\\pF'F_oF'F2 F'F'*(FGF'F[oF'F2F'F.*(F+F'F5F'FEF'F'*($\"#8F&F'F2F'FdpF'F.*(F\\qF'F2F 'FVF'F'*(FeqF'F(F'F_pF'F'*(FioF'F(F'FapF'F'*($\"#CF&F'F(F'FdpF'F.*($\" +++++;F:F'F_oF'F(F'F.*(FioF'F[oF'F(F'F'F$F." }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 12 "t2:=time():\n" }{MPLTEXT 1 202 33 "facSa01_2:= appfac(Sa01_2,[x,u]);\n" }{MPLTEXT 1 202 14 "t2:=time()-t2;" }}{PARA 6 "" 1 "" {TEXT 216 110 "` the biggest gap, the last r-th singular valu es and the number of factors `, 516493663.5, 0.1781973899e-10, 2" } {TEXT 216 1 "\n" }{TEXT 216 59 "` The time for computing the number of factors*****`, 1.172" }}{PARA 6 "" 1 "" {TEXT 216 52 "`The time for t he entire factorization******`, 3.555" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$,L*&,&$\"3)*H8-vp'o6r#GTJF7F- F*F-F-F8F-)F0\"\"#F-F-*&,&$\"3_')zCc8kq:F7F-F*F-F-)F0\"\"&F-F-*&,&$\"3 hO*f'4FGTJF7F-F*F-F-)F9\"\"%F-F-*&,&$\"3#*)yONUlDG'F7F)F*F-F-F>F-F-*(, &$\"3Knv3Dac#G'F7F)F*F-F-F>F-F9F-F-*(,&$\"3;'yB1r#GTJF7F-F*F-F-F9F-)F0 FKF-F-*(,&$\"3UsIjlS#>r%F7F)F*F-F-F0F-F8F-F-*(,&$\"3c,aE#psO=\"F(F)F*F -F-F/F-)F9F?F-F-*&,&$\"3](30s#\\K(y\"F(F-F*F-F-F0F-F-*(,&$\"3u(HbWNT1d \"F7F-F*F-F-F0F-FJF-F-*&,&$\"3%Qk;dt5i&RF(F-F*F-F-F9F-F-*(,&$\"3k)fyAc y@m&!#GF)F*F-F-F[oF-F0F-F-*&,&$\"34(pAI!4&>/#F(F)F*F-F-F[oF-F-*(,&$\"3 95M&o1C>r%F7F-F*F-F-F/F-F9F-F-*(,&$\"3r:eokS#>r%F7F)F*F-F-F0F-F9F-F-*& ,&$\"3e,U1c8kq:F7F-F*F-F-)F9FEF-F-*(,&$\"3h))=)ROZ]!HF\\pF)F*F-F-F[oF- F>F-F-*&,&$\"3B*yc4r#GTJF7F-F*F-F-FXF-F-,&$\"+nS#>r%!#5F)F*F)F-,L*&,&$ \"3)4*[35ie;ZF7F-F*F-F-F/F-F-*&,&$\"3OF^$)3ie;ZF7F-F*F-F-F8F-F-*(,&$\" 3kO)Ga!3RWJF7F-F*F-F-F8F-F>F-F-*&,&$\"36\"G;_!3RWJF7F)F*F-F-FDF-F-*&,& $\"3En<..a>s:F7F-F*F-F-FJF-F-*&,&$\"3'fQ?\\g\"y)G'F7F-F*F-F-F>F-F-*(,& $\"3g\">U_B-;T\"F(F)F*F)F-F>F-F9F-F-*(,&$\"3-j5^.a>s:F7F)F*F-F-F9F-FXF -F-*(,&$\"33'HQsgO*))>F\\pF-F*F-F-F0F-F8F-F-*(,&$\"3P3:=03RWJF7F)F*F-F -F/F-F[oF-F-*&,&$\"3nVA88M0NZF(F)F*F-F-F0F-F-*(,&$\"3'32@/S&>s:F7F)F*F -F-F0F-FJF-F-*&,&$\"3'Hw=B@'e;ZF7F-F*F-F-F9F-F-*(,&$\"3E==%y$GO`GF(F-F *F-F-F[oF-F0F-F-*&,&$\"3eTj\"[!3RWJF7F-F*F-F-F[oF-F-*(,&$\"3;L$o$)R&>s :F7F-F*F-F-F/F-F9F-F-*(,&$\"3-;c4+ie;ZF7F)F*F-F-F0F-F9F-F-*&,&$\"31S]_ 0a>s:F7F)F*F-F-F]qF-F-*(,&$\"3xPl/5ie;ZF7F-F*F-F-F[oF-F>F-F-*&,&$\"3A+ ??!4y\"ykF(F-F*F-F-FXF-F-,&$\"+-ie;ZFiqF)F*F)F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"%vN!\"$" }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 68 "berror2:=backward_error(Sa01_2, convert(facSa01_2, `*`), [x,u])[2] ;\n" }{MPLTEXT 1 202 17 "ceil(log[10](%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "/I\"cG6\"$!/Qm*HS'\\S!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"/IwSP?b6!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "!\")" }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 55 "save facSa01_2, t2, berror2, ca t(path,\"exF02-factors\");" }}}}{SECT 0 {PARA 202 "" 0 "" {TEXT 204 45 "Example 3: Sasaki's first example in ISSAC'01" }}{EXCHG {PARA 208 "" 0 "" {TEXT 206 20 "Example taken from:\n" }{TEXT 206 280 "Sasaki, T ateaki. Approximate multivariate polynomial factorization based on zer o-sum relations. In Mourrain, B., editor ISSAC 2001 Procedings of 200 1 International Symposium on Symbolic and Algebraic Computing, pages 2 84-291, New York, N.Y., 2001. ACM Press. ISBN 1-58113-417-7.\n" }}} {EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 7 "Sa01_1;" }}{PARA 11 "" 1 "" {XPPMATH 20 ",`el*&$\"\"%\"\"!\"\"\"I\"xG6\"F'F'*(F$F')I\"uGF)\"\"# F'F(F'F'*&$\"#=F&F')F(F-F'!\"\"*&$\"#>F&F')F(\"\"$F'F'*($\"++I++7!\")F 'F6F'F+F'F2*($\"\")F&F'F,F')F(F%F'F'*(F4F')F,F7F'F1F'F'*($\"\"'F&F'F,F 'F6F'F2*($\"++S++:F;F'FAF'F(F'F2*($\"+++!)****!#5F'F,F')F(\"\"*F'F2*(F $F'F1F'F+F'F'*(F=F'F(F')F,F%F'F2*&$\"#5F&F')F(\"\"&F'F2*&$\"#;F&F'F?F' F2*&$FMF&F'FAF'F'*&F$F'FPF'F2*&FWF')F(FDF'F'*&$\"#7F&F')F,\"\"(F'F2*&$ \"#9F&F')F(FSF'F2*&$\"#F'F2*&F=F')F(F\\oF'F 2*&$F-F&F')F,FSF'F2*&FCF')F,FDF'F'*(F/F'FgoF'F+F'F'*(F=F'FgoF'F,F'F'*( F[pF'FgnF'FPF'F'*($\"++I++5F;F'FgnF'FAF'F2*(FCF'FgnF'F+F'F'*($\"+++++5 !#9F'FgnF'F,F'F2*(FWF'FTF')F,FUF'F2*(F$F'FTF'FPF'F2*(FWF'FTF'F+F'F'*($ FUF&F'FAF'FioF'F2*(F$F'F+F'FioF'F2*(F^oF'F,F'FioF'F'*(FWF'FAF'FTF'F'*( FinF'F^pF'F?F'F2*($F\\oF&F'F[qF'F?F'F'*(FRF'FAF'F?F'F2*(F$F'FTF'F,F'F2 *(F[pF'F?F'FPF'F2*(FinF'F?F'F+F'F'*(F$F'F6F'F[oF'F2*(FWF'F6F'F^pF'F'*( F[pF'F6F'FPF'F2*($\"++5++>F;F'F6F'F[qF'F'*(F$F')F,F>F'F1F'F'*($F7F&F'F [oF'F1F'F'*($\"#GF&F'F[qF'F1F'F2*(F^oF'F6F'FAF'F2*(FinF'F1F'F^pF'F'*(F $F'F1F'FPF'F'*($F'F&F'F(F')F,FMF'F2*(F$F'F(F'FarF'F'*(F=F'F(F'F^pF'F2* ($\"++q***>\"F;F'F[oF'F(F'F2*($\"#@F&F'F[qF'F(F'F'FCF'*&F[pF')F(\"#6F' F2*&FRF')F(FjnF'F'*&F=F')F(\"#:F'F'*&F=F')F(F_oF'F2*$)F(\"#?F'F'*&F[sF ')F(F5F'F2*&FcrF')F(FcoF'F'*&FeqF')F(\"#8F'F2*&F$F')F(F0F'F2*&FeqF')F( FXF'F2*(FRF'FfsF'F+F'F'*(F[pF'FctF'F,F'F'*(FRF'FisF'F+F'F2*(F_qF'F`oF' FPF'F2*(F=F'F`oF'F[oF'F2*(FZF'F`oF'F\\pF'F'*(F[pF'F[tF'F+F'F2*(F[pF'Fe tF'F+F'F2*&FetF'F,F'F'*(F$F'F[tF'FAF'F'*(F[pF'F^tF'F^pF'F'*(F_qF'F^tF' FPF'F'*($FgsF&F'F^tF'F+F'F'*(FCF'FisF'FarF'F'*(FjuF'FfsF'F\\sF'F'*(Fcr F'FfsF'F[oF'F'*(FZF'F,F'F`oF'F'*(F_qF'FAF'F`oF'F'*(F[pF'F+F'FjtF'F'*(F boF'FAF'FfsF'F2*(F=F'F,F'F^tF'F'*(FCF'F,F'FgtF'F2*(FcrF'F[qF'FLF'F2*(F ^oF'FarF'FLF'F2*(F[sF'F,F'FjtF'F2*(F[pF'F,F'F\\uF'F'*(FZF')F,FgsF'FLF' F'*(F_qF'FAF'F^tF'F2*(F$F'FAF'F\\uF'F2*(FCF'F+F'F\\uF'F'*(FcrF'F[oF'Fg tF'F'*(F[sF'F[qF'FgtF'F2*&FAF'FgtF'F'*(F$F'FarF'F`oF'F'*(FcrF'F`oF'F+F 'F'*(F^oF'FLF'FPF'F'*(F$F'FLF'F[oF'F'*(F$F'FLF'F\\pF'F2*(F[pF'FgtF'F^p F'F'*(F[pF'FgtF'FPF'F'*(F[sF'FgtF'F+F'F2*(FCF'FfsF'FarF'F2*(FZF'FfsF'F ^pF'F'*(F[pF'F`oF'F\\sF'F'*(FWF'FLF'F+F'F2*(FcrF'FetF'FAF'F'*($FhtF&F' FgoF'F^pF'F'*(F$F'FgoF'F\\sF'F'*(FeqF'FgoF')F,FjnF'F'*&F[tF'FPF'F'*(F[ pF'FisF'FPF'F'*(FCF'FLF'F\\sF'F'*(FcrF'FgoF'FPF'F2*(F=F'FgoF'F[oF'F'*( FCF'F[tF'F,F'F2*(F$F'F`oF'F^pF'F2*(FRF'FAF'FgoF'F2*(F^oF'F[qF'FgoF'F2* (FjuF'FAF'FLF'F2*(F[pF'FPF'F\\uF'F'*(FZF'FAF'FisF'F'*(FboF'F[oF'FioF'F 2*(F=F'F\\pF'FioF'F2*(F$F')F,FhtF'FioF'F'*(FeqF'F[qF'FisF'F'*(F[pF'F[q F'F^tF'F'*(FjuF'F[qF'FfsF'F'*&F^pF'FioF'F'*(F$F'F\\sF'FioF'F'*(F[pF'Fa xF'FioF'F'*(F[pF'FPF'FfsF'F2*(F[pF'FivF'FgoF'F2*(F$F'FioF'FPF'F'*(F$F' FLF'F^pF'F'*(FinF'F?F'F[oF'F2*($\"#DF&F'F?F'F\\sF'F'*(FinF'FTF'F[oF'F' *(F^xF'FTF'F\\sF'F2*(F^xF'F6F'FivF'F'*(F[pF'F6F')F,F_oF'F2*(FcrF'FisF' F[oF'F2*(F=F'F6F')F,FcoF'F'*(F=F'FioF'FivF'F'*(FcrF'FTF')F,F\\tF'F'*(F _qF'FTF'FayF'F'*(FjuF'F?F')F,FXF'F'*&F?F'FdzF'F'*(F/F'FioF'F[qF'F'*(F= F'F6F'F\\sF'F2*(FinF'F6F'FaxF'F'*(FeqF'F`oF'F[qF'F2*&F6F'FjzF'F'*(F[sF 'FgoF'FarF'F2*(FZF'FTF'FivF'F'*(F^oF'F?F'FaxF'F'*(F[pF'FarF'F6F'F'*(F_ qF'F,F'FisF'F'*(FeoF'F\\pF'F?F'F2*(FRF'FarF'F?F'F'*(F[pF'F6F'F\\pF'F'* (FeqF'FTF'FarF'F'*(F4F'FTF'F^pF'F'*(FCF'FarF'FioF'F'*(FcrF'FarF'FgnF'F 2*(FRF'FaxF'F1F'F2*(FCF'FjzF'F1F'F2*(F=F')F,F0F'F1F'F'*(F$F'FdzF'FgnF' F'*(F[sF'FaxF'FgnF'F2*(FCF'F\\pF'FgnF'F'*(FWF'F[oF'FgnF'F'*(FeqF'FivF' F1F'F'*($FatF&F'FdzF'F1F'F'*&FCF'FdzF'F2*&FZF'FgzF'F2*&FcrF')F,FatF'F' *&FcrF'FayF'F2*&F[pF'F][lF'F'*(F[sF'FgzF'F1F'F2*(F[pF'FayF'FgnF'F2*(F$ F'FivF'FgnF'F'*(F_qF'F\\sF'FgnF'F'*(F[sF'FjzF'F?F'F2*(F$F'FayF'F?F'F2* (FcrF'F][lF'F6F'F2*(F$F'F\\pF'F1F'F2*(FcrF'FayF'F1F'F'*(FWF'F[qF'FgnF' F'*(FcrF'FivF'F?F'F'*(F_qF'FgnF'F^pF'F2*(Fj\\lF'F1F'F\\sF'F'*(F_qF'F(F 'FayF'F2*(FcrF'F(F'F][lF'F2*(FRF'F(F')F,F5F'F'*(FjuF'F1F'F][lF'F'*(Fju F'FjzF'F(F'F'*(F_qF'FayF'F6F'F'*(F=F'F\\pF'F(F'F2*(FCF'F\\pF'FTF'F'*(F ZF'FaxF'FTF'F'*(FcrF'FgzF'F(F'F2" }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 14 "t3 := time():\n" }{MPLTEXT 1 202 33 "facSa01_1:=app fac(Sa01_1,[x,u]):\n" }{MPLTEXT 1 202 16 "t3 := t3-time();" }}{PARA 6 "" 1 "" {TEXT 216 109 "` the biggest gap, the last r-th singular value s and the number of factors `, 149706.7995, 0.3264602952e-7, 2" }{TEXT 216 1 "\n" }{TEXT 216 60 "` The time for computing the number of fact ors*****`, 38.205" }}{PARA 6 "" 1 "" {TEXT 216 54 "`The time for the e ntire factorization******`, 100.014" }}{PARA 11 "" 1 "" {XPPMATH 20 "$ !'C+5!\"$" }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 68 "berror3:=b ackward_error(Sa01_1, convert(facSa01_1, `*`), [x,u])[2];\n" }{MPLTEXT 1 202 17 "ceil(log[10](%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"cG $!/Uap/7ia!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(berror3G$\"/\"zPC( RX(*!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"'" }}}{EXCHG {PARA 208 " > " 0 "" {MPLTEXT 1 202 55 "save facSa01_1, t3, berror3, cat(path,\"ex F03-factors\");" }}}}{SECT 0 {PARA 202 "" 0 "" {TEXT 204 70 "Example 4 : Corless, Giesbrech, van Hoeij, Kotsireas & Watt in ISSAC'01" }} {EXCHG {PARA 208 "" 0 "" {TEXT 206 20 "Example taken from:\n" }{TEXT 206 333 "Corless, Robert M., Geisbrecht, Mark W., van Hoeij, Mark, Kot sireas, Ilias S, and Watt, Stephen M. Towards factoring bivariate appr oximate polynomials. In Mourrain, B., editor ISSAC 2001 Procedings of 2001 International Symposium on Symbolic and Algebraic Computing, pag es 85-92, New York, N.Y., 2001. ACM Press. ISBN 1-58113-417-7." }}} {EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 6 "Cor01;" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#,\\\\l*&$\"%S\"*\"\"!\"\"\"%\"yGF(!\"\"*&$\"%O8F'F(% \"xGF(F(*&$\"$\"eF'F()F.\"\"#F(F**&$\"&AP\"F'F()F.\"\"$F(F**&$\"%S$)F' F()F.\"\"&F(F**&$\"&**F'F(FbpF(F(*&$\"%IEF'F(F]qF(F**&$\"&GX\"F'F(FhqF(F(*&$\"%qg F'F()F)FVF(F**&$\"%!y&F'F(FaqF(F(*&$\"%_()F'F(F\\rF(F**&$\"%waF'F(FipF (F**($\"%5uF'F(F2F(F\\rF(F(*($\"&EW#F'F(FAF(F)F(F**($\"%[7F'F(F)F(FKF( F**($\"&G/\"F'F(F)F(FPF(F(*($\"%z7F'F(F)F(FUF(F**($\"%'*=F'F(F)F(FinF( F(*($\"%$\\#F'F(F)F(F^oF(F**($\"%0KF'F(F)F(FhoF(F(*($\"%s$)F'F(F)F(F]p F(F**($\"&P%=F'F(FaqF(FKF(F(*($\"&GS\"F'F(FaqF(FPF(F(*($\"&*o9F'F(FaqF (FUF(F(*($\"&U6#F'F(FaqF(FZF(F**($\"%CFF'F(FaqF(FinF(F(*($\"&D5\"F'F(F aqF(F^oF(F**($\"%!f'F'F(FaqF(F]pF(F(*($\"%:vF'F(F]qF(FFF(F(*($\"%5eF'F (F]qF(FKF(F(*($\"%(o(F'F(F]qF(FPF(F**($\"%ZVF'F(F]qF(FUF(F**($\"&)>9F' F(F]qF(FZF(F(*($\"%'f\"F'F(F]qF(FinF(F**($\"$k%F'F(F]qF(F^oF(F**($\"%c SF'F(FipF(F$F'F(FhqF(F2F(F**($\"&W.#F'F(FhqF(F7F(F**($\" %#o&F'F(FhqF(F8F'F(FhqF(FZF(F(*($\"%)f(F'F(F_vF(F2F(F(* ($\"%N$)F'F(F_vF(F7F(F**($\"&Nt$F'F(F_vF(FF'F(F`clF(F.F(F**($\"& b/\"F'F(F`clF(F2F(F(*($\"%saF'F(F`clF(F7F(F**($\"%cgF'F(F`clF(FF'F(FdclF(F2F(F(*($\"$B'F'F(FdclF(FAF(F(*($\"%QFF'F(FhclF( F7F(F**($\"&`s\"F'F(FhclF(F.F(F(*($\"&ED\"F'F(FhclF(F2F(F**($\"%OCF'F( F\\dlF(F.F(F(*($\"%\"H&F'F(F\\dlF(F2F(F**($\"%eBF'F(F`dlF(F.F(F*" }}} {EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 12 "t4:=time():\n" }{MPLTEXT 1 202 31 "facCor01:=appfac(Cor01,[x,y]):\n" }{MPLTEXT 1 202 14 "t4:=t ime()-t4;" }}{PARA 6 "" 1 "" {TEXT 216 116 "` the biggest gap, the las t r-th singular values and the number of factors `, 11326122.692735, . 59336683859219e-9, 2" }}{PARA 6 "" 1 "" {TEXT 216 59 "` The time for c omputing the number of factors*****`, 9.531" }}{PARA 6 "" 1 "" {TEXT 216 53 "`The time for the entire factorization******`, 32.859" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#t4G$\"&\"*G$!\"$" }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 66 "berror4:=backward_error(Cor01, conver t(facCor01, `*`), [x,y])[2];\n" }{MPLTEXT 1 202 17 "ceil(log[10](%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"cG$!/Q&=Y09o$!\"*" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%(berror4G$\"/6'*fywP>!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"(" }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 54 " save facCor01, t4, berror4, cat(path,\"exF04-factors\");" }}}}{SECT 0 {PARA 202 "" 0 "" {TEXT 204 58 "Example 5: Corless, Galligo, Kotsireas & Watt in ISSAC'02:" }}{EXCHG {PARA 208 "" 0 "" {TEXT 206 20 "Example taken from:\n" }{TEXT 206 344 "Corless, Robert M., Galligo, Andre, Ko tsireas, Ilias S., and Watt, Stephen M. A geometric-numeric algorithm \+ for absolute factorization of multivariate polynomials. In Mora, T., e ditor ISSAC 2002 Procedings of 2002 International Symposium on Symbol ic and Algebraic Computing, pages 192-199, New York, N.Y., 2002. ACM P ress. ISBN 1-58113-484-3.\n" }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 6 "Cor02;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,ho\"\"$!\"\"*&\"\"( \"\"\"%\"yGF(F(*&\"#;F(%\"xGF(F%*&\"#?F()F,\"\"#F(F%*&\"#6F()F,F$F(F%* &\"\"%F()F,\"\"&F(F%*&F0F()F,F5F(F%*&F$F()F,\"\"*F(F%*&\"\")F()F,F'F(F %*&F/F()F)F7F(F(*(F5F(F/F()F)F5F(F(*(F7F(F/F()F)F$F(F(*&F/F()F)F0F(F%* (F+F(F/F(F)F(F%*(F5F(F,F(FCF(F(*(F>F(F,F(FEF(F(*(F5F(F,F(FGF(F(*(F+F(F 3F(F)F(F%*(F5F(F?F(F)F(F%*(F>F()F,\"\"'F(F)F(F(*&F6F(FGF(F%*(F5F(F6F(F )F(F%*(F$F(F9F(FCF(F(*(FPF(F9F(FEF(F(*(F$F(F9F(FGF(F(*(F2F(F3F(FGF(F%* (\"#5F(F3F(FCF(F%*(F7F(F3F(FEF(F%*&F$F(FAF(F(*&F>F(FEF(F(*&F$F()F)F'F( F(*&F$F(FGF(F(*&F$F()F)FPF(F(*&FPF(FCF(F(*(F7F(F9F(F)F(F(*(F>F(FEF(FOF (F(*(F7F(FjnF(F3F(F%*$)F)F " 0 "" {MPLTEXT 1 202 12 "t5:=time():\n" }{MPLTEXT 1 202 31 "facCor02:=appfac (Cor02,[x,y]):\n" }{MPLTEXT 1 202 14 "t5:=time()-t5;" }}{PARA 6 "" 1 " " {TEXT 216 117 "` the biggest gap, the last r-th singular values and \+ the number of factors `, 1979360653719.1, .24942569555666e-14, 3" }} {PARA 6 "" 1 "" {TEXT 216 58 "` The time for computing the number of f actors*****`, .578" }}{PARA 6 "" 1 "" {TEXT 216 52 "`The time for the \+ entire factorization******`, 8.531" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#t5G$\"%j&)!\"$" }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 65 "be rror5:=backward_error(Cor02, convert(facCor02, `*`), [x,y])[2];" }} {PARA 6 "" 1 "" {TEXT 216 50 "Approximate factorization is not a real \+ polynomial" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"cG,&$\"/N#p%(berror5G$\"/C@S*>DC\"!#E" }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 54 "save facCor02, t5, berror5, cat(path,\"exF05-factors\");" }} }}}{SECT 0 {PARA 203 "" 0 "" {TEXT 212 31 "New Randomly Generated Exam ples" }}{SECT 0 {PARA 202 "" 0 "" {TEXT 204 54 "Example 6: Three Facto rs (6, 6, 10), Moderate Noise -5" }}{EXCHG {PARA 208 "" 0 "" {TEXT 206 177 " This example is a polynomial with three factors with coeffic ients in [-5, 5], two factors with total degree 6 and one with total d egree 10; the added noise is of order 10^(-5)." }}{PARA 208 "" 0 "" {TEXT 215 0 "" }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 29 "read(c at(rand_path,\"exF06\")):" }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 39 "F6 := expand(evalf(cleanF6 + noiseF6)):" }}}{EXCHG {PARA 208 " > " 0 "" {MPLTEXT 1 202 37 "norm(evalf(noiseF6),2) / norm(F6,2);\n" } {MPLTEXT 1 202 15 "ceil(log10(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# $\"/A7Me******!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"&" }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 12 "t6:=time():\n" }{MPLTEXT 1 202 25 "facF6:=appfac(F6,[x,y]):\n" }{MPLTEXT 1 202 14 "t6:=time()-t6;" }} {PARA 6 "" 1 "" {TEXT 216 116 "` the biggest gap, the last r-th singul ar values and the number of factors `, 1619.0264964476, .6622225844211 5e-6, 3" }}{PARA 6 "" 1 "" {TEXT 216 60 "` The time for computing the \+ number of factors*****`, 84.234" }}{PARA 6 "" 1 "" {TEXT 216 55 "`The \+ time for the entire factorization******`, 1304.267" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#t6G$\"(XeI\"!\"$" }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 60 "berror6:=backward_error(F6, convert(facF6, `*`), [x ,y])[2];\n" }{MPLTEXT 1 202 15 "ceil(log10(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"cG$\"/Gi`4wu6!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%(berror6G$\"/-J!))HpY\"!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"$" }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 56 "save facF6, t6, berr or6, cat(rand_path,\"exF06-factors\");" }}}}{SECT 0 {PARA 202 "" 0 "" {TEXT 204 49 "Example 7: Two Factors (9, 7), Moderate Noise -4" }} {EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 29 "read(cat(rand_path,\"exF 07\")):" }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 39 "F7 := expand (evalf(cleanF7 + noiseF7)):" }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 37 "norm(evalf(noiseF7),2) / norm(F7,2);\n" }{MPLTEXT 1 202 15 "ce il(log10(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"/!omt+++\"!#<" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#!\"$" }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 12 "t7:=time():\n" }{MPLTEXT 1 202 25 "facF7:=appfac(F7 ,[x,y]):\n" }{MPLTEXT 1 202 14 "t7:=time()-t7;" }}{PARA 6 "" 1 "" {TEXT 216 116 "` the biggest gap, the last r-th singular values and th e number of factors `, 485.93235239264, .15289929645422e-4, 2" }} {PARA 6 "" 1 "" {TEXT 216 60 "` The time for computing the number of f actors*****`, 13.062" }}{PARA 6 "" 1 "" {TEXT 216 53 "`The time for th e entire factorization******`, 65.186" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#t7G$\"&-_'!\"$" }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 60 "berror7:=backward_error(F7, convert(facF7, `*`), [x,y])[2];\n" } {MPLTEXT 1 202 15 "ceil(log10(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /%\"cG$\"/'yO$4178!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(berror7G$ \"/(Rza>-=#!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"$" }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 56 "save facF7, t7, berror7, cat(ra nd_path,\"exF07-factors\");" }}}}{SECT 0 {PARA 202 "" 0 "" {TEXT 204 54 "Example 8: Five Factors (4,4,4,4,4), Moderate Noise -5" }}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 29 "read(cat(rand_path,\"exF08\")): " }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 39 "F8 := expand(evalf( cleanF8 + noiseF8)):" }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 37 "norm(evalf(noiseF8),2) / norm(F8,2);\n" }{MPLTEXT 1 202 15 "ceil(log1 0(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"/rlq&)******!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"&" }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 12 "t8:=time():\n" }{MPLTEXT 1 202 25 "facF8:=appfac(F8,[x,y]): \n" }{MPLTEXT 1 202 14 "t8:=time()-t8;" }}{PARA 6 "" 1 "" {TEXT 216 116 "` the biggest gap, the last r-th singular values and the number o f factors `, 472.86103612118, .30931375992914e-6, 5" }}{PARA 6 "" 1 "" {TEXT 216 60 "` The time for computing the number of factors*****`, 4 8.657" }}{PARA 6 "" 1 "" {TEXT 216 55 "`The time for the entire factor ization******`, 2420.355" }}{PARA 6 "" 1 "" {TEXT 216 116 "` the bigge st gap, the last r-th singular values and the number of factors `, 472 .86103572027, .30931376019090e-6, 5" }}{PARA 6 "" 1 "" {TEXT 216 60 "` The time for computing the number of factors*****`, 53.250" }}{PARA 6 "" 1 "" {TEXT 216 55 "`The time for the entire factorization******`, \+ 2816.606" }}{PARA 6 "" 1 "" {TEXT 216 116 "` the biggest gap, the last r-th singular values and the number of factors `, 105.06854657482, .1 4442485197142e-4, 3" }}{PARA 6 "" 1 "" {TEXT 216 59 "` The time for co mputing the number of factors*****`, 4.688" }}{PARA 6 "" 1 "" {TEXT 216 53 "`The time for the entire factorization******`, 93.501" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#t8G$\"(r?L&!\"$" }}}{EXCHG {PARA 208 "" 0 "" {TEXT 215 0 "" }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 60 "berror8:=backward_error(F8, convert(facF8, `*`), [x,y])[2];\n" }{MPLTEXT 1 202 15 "ceil(log10(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/%\"cG$\"/yyCVMD9!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(berror8G$ \"/O\">#zJWL!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"#" }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 56 "save facF8, t8, berror8, cat(ra nd_path,\"exF08-factors\");" }}}}{SECT 0 {PARA 202 "" 0 "" {TEXT 204 48 "Example 9: Three Factors (3,3,3), Large Noise -1" }}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 29 "read(cat(rand_path,\"exF09\")):" }}} {EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 39 "F9 := expand(evalf(clean F9 + noiseF9)):" }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 45 "norm (evalf(noiseF9),2) / norm(expand(F9),2);\n" }{MPLTEXT 1 202 15 "ceil(l og10(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"/K<,`GD)*!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"\"" }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 12 "t9:=time():\n" }{MPLTEXT 1 202 25 "facF9:=appfac(F9 ,[x,y]):\n" }{MPLTEXT 1 202 14 "t9:=time()-t9;" }}{PARA 6 "" 1 "" {TEXT 216 116 "` the biggest gap, the last r-th singular values and th e number of factors `, 1.7044363233015, .44900371187706e-2, 2" }} {PARA 6 "" 1 "" {TEXT 216 58 "` The time for computing the number of f actors*****`, .750" }}{PARA 6 "" 1 "" {TEXT 216 53 "`The time for the \+ entire factorization******`, 25.438" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%#t9G$\"&Qa#!\"$" }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 59 "b error9:=backward_error(F9, convert(facF9, `*`), [x,y])[2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"cG$\"/&o&GsI))H!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(berror9G$\"/xmu+IL!)!#9" }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 56 "save facF9, t9, berror9, cat(rand_path,\"exF09-f actors\");" }}}}{PARA 202 "" 0 "" {TEXT 217 0 "" }}{SECT 0 {PARA 202 " " 0 "" {TEXT 204 51 "Example 13: Two Factors (5, 5^2), Moderate Noise \+ -5" }}{EXCHG {PARA 208 "" 0 "" {TEXT 206 42 "This example should have \+ a very small gap." }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 29 "re ad(cat(rand_path,\"exF13\")):" }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 42 "F13 := expand(evalf(cleanF13 + noiseF13)):" }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 39 "norm(evalf(noiseF13),2) / norm(F13,2) ;\n" }{MPLTEXT 1 202 15 "ceil(log10(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"/!=s`ynC)!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"&" }}} {EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 13 "t13:=time():\n" } {MPLTEXT 1 202 31 "facF13:=multifac(F13,[x,y],1):\n" }{MPLTEXT 1 202 16 "t13:=time()-t13;" }}{PARA 6 "" 1 "" {TEXT 216 116 "` the biggest g ap, the last r-th singular values and the number of factors `, 8979.13 99593767, .11846143594456e-5, 2" }}{PARA 6 "" 1 "" {TEXT 216 59 "` The time for computing the number of factors*****`, 1.343" }}{PARA 6 "" 1 "" {TEXT 216 52 "`The time for the entire factorization******`, 7.063 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$t13G$\"&/A%!\"$" }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 92 "berror13:=backward_error(F13, m ul(facF13[i][1]^facF13[i][2], i=1..nops(facF13)), [x,y])[2];\n" } {MPLTEXT 1 202 15 "ceil(log10(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /%\"cG$!/'*)y%G*4H$!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)berror13G $\"/AN[#*\\No!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"%" }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 59 "save facF13, t13, berror13, cat (rand_path,\"exF13-factors\");" }}}}{SECT 0 {PARA 202 "" 0 "" {TEXT 204 62 "Example 15: Two Factors (5, 5), Moderate Noise -5, three vars " }}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 29 "read(cat(rand_path, \"exF15\")):" }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 42 "F15 := \+ expand(evalf(cleanF15 + noiseF15)):" }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 39 "norm(evalf(noiseF15),2) / norm(F15,2);\n" }{MPLTEXT 1 202 15 "ceil(log10(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"/MZ*> +++\"!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"%" }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 13 "t15:=time():\n" }{MPLTEXT 1 202 29 "facF1 5:=appfac(F15,[x,y,z]):\n" }{MPLTEXT 1 202 16 "t15:=time()-t15;" }} {PARA 6 "" 1 "" {TEXT 216 116 "` the biggest gap, the last r-th singul ar values and the number of factors `, 11933.547270212, .2013889738077 1e-5, 2" }}{PARA 6 "" 1 "" {TEXT 216 61 "` The time for computing the \+ number of factors*****`, 240.563" }}{PARA 6 "" 1 "" {TEXT 216 54 "`The time for the entire factorization******`, 559.264" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$t15G$\"'&Hf&!\"$" }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 65 "berror15:=backward_error(F15, convert(facF15, `*`), [x,y,z])[2];\n" }{MPLTEXT 1 202 15 "ceil(log10(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"cG$\"/elPxzdC!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)berror15G$\"/o#*Q-B^:!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\" %" }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 59 "save facF15, t15, \+ berror15, cat(rand_path,\"exF15-factors\");" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 208 "" 0 "" {TEXT 215 0 "" }}}} {SECT 0 {PARA 202 "" 0 "" {TEXT 204 91 "Example 15a: Two Factors (2,2) , three vars, Kaltofen challenge JSC 29/6, pp. 891-919 (2000)" }} {EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 30 "read(cat(rand_path,\"exF 15a\")):" }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 45 "F15a := exp and(evalf(cleanF15a + noiseF15a)):" }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 41 "norm(evalf(noiseF15a),2) / norm(F15a,2);\n" } {MPLTEXT 1 202 15 "ceil(log10(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# $\"/_xlco'e%!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"%" }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 14 "t15a:=time():\n" }{MPLTEXT 1 202 31 "facF15a:=appfac(F15a,[x,y,z]):\n" }{MPLTEXT 1 202 18 "t15a:=ti me()-t15a;" }}{PARA 6 "" 1 "" {TEXT 216 117 "` the biggest gap, the la st r-th singular values and the number of factors `, 16725770752.442, \+ .15660565689187e-11, 2" }}{PARA 6 "" 1 "" {TEXT 216 58 "` The time for computing the number of factors*****`, .282" }}{PARA 6 "" 1 "" {TEXT 216 52 "`The time for the entire factorization******`, 1.610" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%t15aG$\"%5;!\"$" }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 68 "berror15a:=backward_error(F15a, convert(facF1 5a, `*`), [x,y,z])[2];\n" }{MPLTEXT 1 202 15 "ceil(log10(%));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"cG$!/E:a2jc6!#5" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%*berror15aG$\"/\"e_([nM')!#C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!#5" }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 63 "s ave facF15a, t15a, berror15a, cat(rand_path,\"exF15a-factors\");" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT 218 0 "" }}{SECT 0 {PARA 202 "" 0 "" {TEXT 204 111 "Example 15b: Two F actors (2,2), three vars, Pseudofactors of Multivariate Polynomials I SSAC2000, pp. 161-168 ." }}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 30 "read(cat(rand_path,\"exF15b\")):" }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 6 "F15b :" }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 0 "" }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 14 "t15b:=time() :\n" }{MPLTEXT 1 202 31 "facF15b:=appfac(F15b,[x,y,z]):\n" }{MPLTEXT 1 202 18 "t15b:=time()-t15b;" }}{PARA 6 "" 1 "" {TEXT 216 116 "` the b iggest gap, the last r-th singular values and the number of factors `, 258.21220386592, .15607617615691e-3, 2" }}{PARA 6 "" 1 "" {TEXT 216 59 "` The time for computing the number of factors*****`, 1.000" }} {PARA 6 "" 1 "" {TEXT 216 52 "`The time for the entire factorization** ****`, 7.406" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%t15bG$\"%@u!\"$" }} }{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 68 "berror15b:=backward_err or(F15b, convert(facF15b, `*`), [x,y,z])[2];\n" }{MPLTEXT 1 202 15 "ce il(log10(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"cG$!/*eXN]Jg%!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*berror15bG$\"/iB/id$*o!#<" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#!\"$" }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 202 63 "save facF15b, t15b, berror15b, cat(rand_path,\"exF1 5b-factors\");" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 12 "" 1 "" {TEXT 219 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT 218 0 "" }}{PARA 205 "" 0 "" {TEXT 220 0 "" }}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }