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2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 207 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 208 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{PSTYLE "" -1 211 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 8 2 2 0 2 0 2 2 -1 1 }} {SECT 0 {EXCHG {PARA 3 "" 0 "" {TEXT 209 73 "Worksheet to generate th e random examples in Kaltofen, May, Yang, & Zhi:" }}{PARA 3 "" 0 "" {TEXT 209 1 "'" }{TEXT 209 63 "Approximate Factorization of Multivaria te Polynomials Using SVD" }{TEXT 209 1 "'" }{MPLTEXT 1 0 0 "" }}} {SECT 0 {PARA 3 "" 0 "" {TEXT 209 48 "The routine used to generate ran dom polynomials " }}{EXCHG {PARA 0 "" 0 "" {TEXT 210 276 "This routine generates a polynomial with random dense factors of the given degrees with integer coefficents smaller than N. Another random dense polyno mial (noise) with the same degree as the product and coeffients smalle r than 10^(-e) is generated then added to the product." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "randp:=proc(vars::list \+ # the variables \n" }{MPLTEXT 1 0 80 " ,list_deg_factor::l ist # the list of the degree of the factors \n" }{MPLTEXT 1 0 87 " ,N::integer # absolute maximum of coefficien ts of the factors\n" }{MPLTEXT 1 0 70 " ,e::integer \+ # the perturbation size: 10^(-e)\n" }{MPLTEXT 1 0 153 " \+ ,dn) # Optional: number between 0 and 1 giving \+ the how dense the perturbation should be compared to product default = 1.\n" }{MPLTEXT 1 0 2 " \n" }{MPLTEXT 1 0 50 " local i,f0,fac,p ert_pol,res, deg_pert, dd;\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 51 "i f nops([args]) = 4 then dd:=1 else dd:=dn end if;\n" }{MPLTEXT 1 0 1 " \n" }{MPLTEXT 1 0 50 " deg_pert := convert(list_deg_factor, `+`) ;\n" }{MPLTEXT 1 0 8 " \n" }{MPLTEXT 1 0 14 " f0:=1;\n" } {MPLTEXT 1 0 192 " for i from 1 to nops(list_deg_factor) do \+ fa c:=randpoly(vars,degree=list_deg_factor[i],coeffs=rand(-N..N),dense); \n" }{MPLTEXT 1 0 22 " f0:=f0*fac;\n" }{MPLTEXT 1 0 15 " \+ end do;\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 46 "dd:=round(nops(con vert(expand(f0),list))*dd);\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 27 " print('noise_terms' = dd);\n" }{MPLTEXT 1 0 89 " pert_pol:=randp oly(vars,degree=deg_pert,coeffs=rand(-10^(e)..10^(e)), terms=dd); \n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 80 " return(f0,'`*`'(10^(-e)* norm(expand(f0),2)/norm(pert_pol, 2),pert_pol));\n" }{MPLTEXT 1 0 21 " #return(res);\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 9 "end proc :" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 207 10 "The Random" }{TEXT 211 1 " " }{TEXT 208 8 "Examples" }}{SECT 0 {PARA 4 "" 0 "" {TEXT 211 71 "Se t the path to the subdirectory where the example files will be stored" }}{EXCHG {PARA 209 "> " 0 "" {MPLTEXT 1 0 19 "path:=\"examples\\/\";" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%pathGQ*examples/6\"" }}}}{SECT 0 {PARA 211 "" 0 "" {TEXT 200 54 "Example 6: Three Factors (6, 6, 10), \+ Moderate Noise -5" }}{EXCHG {PARA 0 "" 0 "" {TEXT 210 177 "This exampl e is a polynomial with three factors with coefficients in [-5, 5], two factors with total degree 6 and one with total degree 10; the added n oise is of order 10^(-5).\n" }{TEXT 210 1 " " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "cleanF6, noiseF6:=randp([x,y],[6,6,10],5,5,.25):\n" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "save cleanF6, noiseF6, cat(path,\" exF06\");" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%,noise_termsG\"#p" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 200 9 "Example 7" }{TEXT 211 40 ": Two Factors (9, 7), Moderate Noise -4" }}{EXCHG {PARA 0 "" 0 "" {TEXT 210 172 "This example is a polynom ial with two factors with coefficients in [-5, 5], one factor with tot al degree 9and one with total degree 7; the added noise is of order 10 ^(-4).\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "cleanF7, noiseF 7:=randp([x,y],[9,7],5,4,.25):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%,n oise_termsG\"#Q" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "save cle anF7, noiseF7, cat(path,\"exF07\");" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 211 1 " " }{TEXT 200 54 "Example 8: Five Factors (4,4,4,4,4), Moderate Noise -5" }} {EXCHG {PARA 0 "" 0 "" {TEXT 210 138 "This example is a polynomial wit h five factors with coefficients in [-5, 5] all of total degree 4; t he added noise is of order 10^(-5).\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "cleanF8, noiseF8:=randp([x,y],[4,4,4,4,4],5,5,.25):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%,noise_termsG\"#d" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "save cleanF8, noiseF8, cat(path,\"e xF08\");" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 200 9 "Example 9" }{TEXT 211 39 ": Three Factors (3,3,3), Large Noise -1" }}{EXCHG {PARA 0 "" 0 "" {TEXT 210 139 "This example is a polynomial with three factors with coefficients in [-5, 5] all of total degree 3; the added noise is of order 10^(-1).\n" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "cleanF9, noiseF9:=randp([x, y],[3,3,3],5,1,.25):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%,noise_terms G\"#9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "save cleanF9, nois eF9, cat(path,\"exF09\");" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 211 74 " Examples 10, 11 & 12: On e polynomial (12, 7, 5) with perturbations added:" }}{EXCHG {PARA 0 "" 0 "" {TEXT 210 166 "This example we generate a polynomial with three \+ factors, then add the same noise first with order -2 then with order - 1 to generate examples 11 and 12 respectively.\n" }}}{SECT 0 {PARA 5 " " 0 "" {TEXT 212 52 "Example 10: Three Factors (12, 7, 5), Large Noise -5" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "cleanF10, noiseF10 := randp([x,y], [12,7,5], 5, 5, .05):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/%,noise_termsG\"#;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "sav e cleanF10, noiseF10, cat(path,\"exF10\");" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 0 "" 0 "" {TEXT 200 52 "Example \+ 11: Three Factors (12, 7, 5), Large Noise -5" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "cleanF11, noiseF11 := randp([x,y], [12,7,5], 5, 5, .50):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%,noise_termsG\"$h\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "cleanF11 := cleanF10:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "save cleanF11, noiseF11, cat (path,\"exF11\");" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 0 {PARA 0 "" 0 "" {TEXT 200 53 "Example 12: Three Factors (12, 7 , 5), Larger Noise -3" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "cle anF12 := cleanF11:\n" }{MPLTEXT 1 0 34 "noiseF12 := '`*`'(10^2, noiseF 11):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 43 "save cleanF12, noiseF12, cat(path,\"exF12\");" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 0 {PARA 4 "" 0 " " {TEXT 200 10 "Example 13" }{TEXT 211 41 ": Two Factors (5, 5^2), Mod erate Noise -5" }}{EXCHG {PARA 0 "" 0 "" {TEXT 210 138 "This example i s a polynomial with two factors with coefficients in [-5, 5] both of \+ total degree 8; the added noise is of order 10^(-1).\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "cleanF13, noiseF13:=randp([x,y],[5, 5,5],5,5,.25):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%,noise_termsG\"#M" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "cleanF13:=op(1,cleanF13) *op(2,cleanF13)^2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "save \+ cleanF13, noiseF13, cat(path,\"exF13\");" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 211 62 "Example 15 : Two Factors (5, 5), Moderate Noise -5, three vars" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "cleanF15, noiseF15 := randp([x,y,z], [5,5] , 5, 5, .25):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%,noise_termsG\"#q" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "save cleanF15, noiseF15, cat(path,\"exF15\");" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 " " {TEXT 200 10 "Example 16" }{TEXT 211 42 ": Two Factors (18, 18), Mo derate Noise -6" }}{EXCHG {PARA 0 "" 0 "" {TEXT 210 139 "This example \+ is a polynomial with two factors with coefficients in [-5, 5] both of total degree 18; the added noise is of order 10^(-6).\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "cleanF16, noiseF16:=randp([x,y],[18 ,18],5,6,.25):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%,noise_termsG\"$v \"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "save cleanF16, noiseF 16, cat(path,\"exF16\");" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 200 10 "Ex ample 17" }{TEXT 211 62 ": Two Factors (18, 18), Moderate Noise -6 in 99% of the terms" }}{EXCHG {PARA 0 "" 0 "" {TEXT 210 139 "This exampl e is a polynomial with two factors with coefficients in [-5, 5] both \+ of total degree 18; the added noise is of order 10^(-6).\n" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "cleanF17, noiseF17:=randp([x ,y],[18,18],5,6,.99):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%,noise_term sG\"$\"p" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "save cleanF17, \+ noiseF17, cat(path,\"exF17\");" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 211 65 "Example 18: Complex example : Two Factors(6,6), Moderate Noise -6" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "fa18:=randpoly([x,y],degree=6,coeffs=rand(-5..5) ,dense)\n" }{MPLTEXT 1 0 59 " +I*randpoly([x,y],degree=6,coeffs=r and(-5..5),dense):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "ga18: =randpoly([x,y],degree=6,coeffs=rand(-5..5),dense)\n" }{MPLTEXT 1 0 59 " +I*randpoly([x,y],degree=6,coeffs=rand(-5..5),dense):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "cleanF18:=fa18*ga18:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "noiseF18:='`*`'(10^(-6),rand poly([x,y],degree=12,coeffs=rand(-5..5),dense)\n" }{MPLTEXT 1 0 66 " \+ +I*randpoly([x,y],degree=12,coeffs=rand(-5..5),dense)):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "save cleanF18, noiseF18, cat (path,\"exF18\");" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT 210 0 "" }}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }