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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 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "read \"../Src/multifac_1.6.mpl\":\n " }{MPLTEXT 1 0 19 "path:=\"examples\\/\":" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 30 "Digits:=floor(evalhf(Digits));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'DigitsG\"#9" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 206 20 "Zeng's sixth example" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "p1 :=x^2+x*y-y^2+1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p1G,**$)%\"xG\" \"#\"\"\"F**&F(F*%\"yGF*F**$)F,F)F*!\"\"F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "p2:=expand((x^3+1)*(y^2-2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p2G,**&)%\"xG\"\"$\"\"\")%\"yG\"\"#F*F**&F-F*F'F*!\" \"*$F+F*F*F-F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "p3:=x^3-y ^3-3*x*y^2+2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p3G,**$)%\"xG\"\"$ \"\"\"F**$)%\"yGF)F*!\"\"*(F)F*F(F*)F-\"\"#F*F.F1F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "pp:=randpoly([x,y],degree=4,coeffs=rand(-5 ..5),terms=10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ppG,4\"\"%!\"\"* &\"\"&\"\"\"%\"xGF*F**&\"\"#F*%\"yGF*F'*&F+F*F.F*F'*(F)F*)F+F-F*F.F*F* *(F)F*F+F*)F.F-F*F'*&F-F*)F+F&F*F**&)F+\"\"$F*F.F*F'*(F&F*F+F*)F.F8F*F *" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "cleanF14:=p1^4*p2^3*p3 :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "noiseF14:='`*`'(10^(-8 ),pp):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "save cleanF14, no iseF14, cat(path,\"exF14\");" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "zengex6:=evalf(expand(cleanF14+noiseF14)):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 206 68 "The example has univariate multiple factors (x^3 +1)^3, (y^2-2)^3. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "t14:= time():\n" }{MPLTEXT 1 0 37 "res:=multifac_cont(zengex6,[x,y],8);\n" } {MPLTEXT 1 0 16 "t14:=time()-t14;" }}{PARA 6 "" 1 "" {TEXT 207 117 "` \+ the biggest gap, the last r-th singular values and the number of facto rs `, 417417718.55256, .29977124647286e-10, 5" }}{PARA 6 "" 1 "" {TEXT 207 58 "` The time for computing the number of factors*****`, .483" } }{PARA 6 "" 1 "" {TEXT 207 53 "`The time for the entire factorization* *****`, 16.546" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$resG7(7$,0$\"/&o! HlP([&!#8\"\"\"*&$\"3mZ?Oh1Afo!#=F+)%\"yG\"\"'F+!\"\"*&$\"3Ft\"oK1#*3^ %!#IF+)F1\"\"&F+F3*&$\"39v()z'RKb6%!#e$HwYpE)F7F+F1F +F3F+7$,6*&,&$\"3:\"y&fc&4C_%F/F3*&$\"\"!FVF+^#F+F+F+F+FDF+F+*(,&$\"3# =B:s'Gsc8F=F3FTF3F+%\"xGF+FIF+F+*(,&$\"3mrlIF?l!p\"!#FF3FTF+F+)FfnFJF+ F1F+F+*&,&$\"3!RxW+c4C_%F/F+FTF+F+)FfnFEF+F+*&,&$\"3%4>3oe5&zw!#GF3FTF 3F+FIF+F+*(,&$\"3#*)3E\">.9!y#F[oF+FTF+F+FfnF+F1F+F+*&,&$\"3Lq_5N-9?gF foF+FTF+F+F\\oF+F+*&,&$\"3A\"faYfc61#F[oF+FTF+F+F1F+F+*&,&$\"3C;1t[/*!#9F+FTF+F+F+7$,(*&,&$\"3IH2qaN!)4KF [oF3FTF+F+F1F+F+*&,&$\"3#y\"4'=zG3!QF=F3FTF3F+FfnF+F+,&$\"/l$QzG3!QF*F 3FTF+F+FE7$,(*&,&$\"3qIL\\(R[$R\\F[oF+*&$\"3BkLESKQN8F[oF+FWF+F+F+F1F+ F+*&,&$\"3c)))p\"o5?\"o'F/F3*&$\"3?Eq5#yy!=>F=F+FWF+F3F+FfnF+F+,&$\"/s A)*[/F8F*F3*&$\"/yvv$[w`\"F*F+FWF+F+F+FE7$,(*&,&$\"3%)p;\"Gz')))4$F[oF 3*&$\"3!R#39IZdrSF[oF+FWF+F+F+F1F+F+*&,&$\"3s\"*o#y7$)zS'F/F3*&$\"3hkr ,'eyt#>F=F+FWF+F3F+FfnF+F+,&$\"/t8uzb*)>F*F+*&$\"/(o0v;u3%FjpF+FWF+F+F +FE7$,.*&,&$\"3'*=\\lI;[a:F=F+FTF+F+FIF+F+*(,&$\"3yX]1J;[a:F=F3FTF+F+F fnF+F1F+F+*&,&$\"3+z@()G;[a:F=F3FTF+F+F\\oF+F+*&,&$\"3M]L#H1@(fE!#EF3F TF+F+F1F+F+*&,&$\"3w91CkdvbLFfuF+FTF+F+FfnF+F+,&$\"/p>H;[a:F*F3FTF3F+F ?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$t14G$\"&ks'!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "result:=expand(res[1][1]*res[2][1]* res[3][1]^3*res[4][1]^3*res[5][1]^3*res[6][1]^4):" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 40 "backward_error(result,zengex6,[x,y])[2];" }} {PARA 6 "" 1 "" {TEXT 207 50 "Approximate factorization is not a real \+ polynomial" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"cG,&$\"/7bDtE/F!#5\" \"\"*&$\"/*[hh%yPkF(F)^#F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"/ h5!=]_B$!#A" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "h1:=res[6][1 ];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#h1G,.*&,&$\"3'*=\\lI;[a:!#<\" \"\"*&$\"\"!F.F+^#F+F+F+F+)%\"yG\"\"#F+F+*(,&$\"3yX]1J;[a:F*!\"\"F,F+F +%\"xGF+F1F+F+*&,&$\"3+z@()G;[a:F*F7F,F+F+)F8F2F+F+*&,&$\"3M]L#H1@(fE! #EF7F,F+F+F1F+F+*&,&$\"3w91CkdvbLFBF+F,F+F+F8F+F+,&$\"/p>H;[a:!#8F7F,F 7F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "backward_error(h1,p1 ,[x,y])[2];" }}{PARA 6 "" 1 "" {TEXT 207 50 "Approximate factorization is not a real polynomial" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"cG$!/ s%*H;[a:!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"/ZD&GV>]\"!#A" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "h2:=expand(res[1][1]*res[3][ 1]^3*res[4][1]^3*res[5][1]^3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "backward_error(h2,expand(p2^3),[x,y])[2];" }}{PARA 6 "" 1 "" {TEXT 207 50 "Approximate factorization is not a real polynomial" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"cG,&$\"/YM#=)3C5!#5\"\"\"*&$\"/FCJ l%zV#F(F)^#F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"/ " 0 "" {MPLTEXT 1 0 14 "h3:=res[2][1];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#h3G,6*&,&$\"3:\"y&fc&4C_%!#=!\"\"*&$\"\"!F. \"\"\"^#F/F/F/F/)%\"yG\"\"$F/F/*(,&$\"3#=B:s'Gsc8!#3oe5&zw!#GF+F,F+F/F:F/F/*(,&$\"3#*)3E\"> .9!y#F@F/F,F/F/F9F/F2F/F/*&,&$\"3Lq_5N-9?gFKF/F,F/F/FAF/F/*&,&$\"3A\"f aYfc61#F@F/F,F/F/F2F/F/*&,&$\"3C;1t[/ *!#9F/F,F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "backward_er ror(h3,p3,[x,y])[2];" }}{PARA 6 "" 1 "" {TEXT 207 50 "Approximate fact orization is not a real polynomial" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /%\"cG$\"/`hd&4C_%!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"/\"pz!H;qF !#B" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "facF14:=res:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "save facF14, t14, cat(path, \"exF14-factors\");" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 15 10 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }