1 | /* emacs edit mode for this file is -*- C++ -*- */ |
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2 | |
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3 | /** |
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4 | * @file cf_gcd.cc |
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5 | * |
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6 | * gcd/content/lcm of polynomials |
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7 | * |
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8 | * To compute the GCD different variants are chosen automatically |
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9 | **/ |
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10 | |
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11 | #include "config.h" |
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12 | |
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13 | |
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14 | #include "timing.h" |
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15 | #include "cf_assert.h" |
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16 | #include "debug.h" |
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17 | |
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18 | #include "cf_defs.h" |
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19 | #include "canonicalform.h" |
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20 | #include "cf_iter.h" |
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21 | #include "cf_reval.h" |
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22 | #include "cf_primes.h" |
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23 | #include "cf_algorithm.h" |
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24 | #include "cfEzgcd.h" |
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25 | #include "cfGcdAlgExt.h" |
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26 | #include "cfSubResGcd.h" |
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27 | #include "cfModGcd.h" |
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28 | #include "FLINTconvert.h" |
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29 | #include "facAlgFuncUtil.h" |
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30 | #include "templates/ftmpl_functions.h" |
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31 | |
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32 | #ifdef HAVE_NTL |
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33 | #include <NTL/ZZX.h> |
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34 | #include "NTLconvert.h" |
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35 | bool isPurePoly(const CanonicalForm & ); |
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36 | #endif |
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37 | |
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38 | void out_cf(const char *s1,const CanonicalForm &f,const char *s2); |
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39 | |
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40 | /** static CanonicalForm icontent ( const CanonicalForm & f, const CanonicalForm & c ) |
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41 | * |
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42 | * icontent() - return gcd of c and all coefficients of f which |
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43 | * are in a coefficient domain. |
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44 | * |
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45 | * @sa icontent(). |
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46 | * |
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47 | **/ |
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48 | static CanonicalForm |
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49 | icontent ( const CanonicalForm & f, const CanonicalForm & c ) |
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50 | { |
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51 | if ( f.inBaseDomain() ) |
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52 | { |
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53 | if (c.isZero()) return abs(f); |
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54 | return bgcd( f, c ); |
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55 | } |
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56 | //else if ( f.inCoeffDomain() ) |
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57 | // return gcd(f,c); |
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58 | else |
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59 | { |
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60 | CanonicalForm g = c; |
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61 | for ( CFIterator i = f; i.hasTerms() && ! g.isOne(); i++ ) |
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62 | g = icontent( i.coeff(), g ); |
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63 | return g; |
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64 | } |
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65 | } |
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66 | |
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67 | /** CanonicalForm icontent ( const CanonicalForm & f ) |
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68 | * |
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69 | * icontent() - return gcd over all coefficients of f which are |
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70 | * in a coefficient domain. |
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71 | * |
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72 | **/ |
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73 | CanonicalForm |
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74 | icontent ( const CanonicalForm & f ) |
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75 | { |
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76 | return icontent( f, 0 ); |
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77 | } |
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78 | |
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79 | #ifdef HAVE_FLINT |
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80 | static CanonicalForm |
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81 | gcd_univar_flintp (const CanonicalForm& F, const CanonicalForm& G) |
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82 | { |
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83 | nmod_poly_t F1, G1; |
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84 | convertFacCF2nmod_poly_t (F1, F); |
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85 | convertFacCF2nmod_poly_t (G1, G); |
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86 | nmod_poly_gcd (F1, F1, G1); |
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87 | CanonicalForm result= convertnmod_poly_t2FacCF (F1, F.mvar()); |
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88 | nmod_poly_clear (F1); |
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89 | nmod_poly_clear (G1); |
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90 | return result; |
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91 | } |
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92 | |
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93 | static CanonicalForm |
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94 | gcd_univar_flint0( const CanonicalForm & F, const CanonicalForm & G ) |
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95 | { |
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96 | fmpz_poly_t F1, G1; |
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97 | convertFacCF2Fmpz_poly_t(F1, F); |
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98 | convertFacCF2Fmpz_poly_t(G1, G); |
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99 | fmpz_poly_gcd (F1, F1, G1); |
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100 | CanonicalForm result= convertFmpz_poly_t2FacCF (F1, F.mvar()); |
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101 | fmpz_poly_clear (F1); |
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102 | fmpz_poly_clear (G1); |
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103 | return result; |
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104 | } |
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105 | #endif |
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106 | |
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107 | #ifdef HAVE_NTL |
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108 | #ifndef HAVE_FLINT |
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109 | static CanonicalForm |
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110 | gcd_univar_ntl0( const CanonicalForm & F, const CanonicalForm & G ) |
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111 | { |
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112 | ZZX F1=convertFacCF2NTLZZX(F); |
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113 | ZZX G1=convertFacCF2NTLZZX(G); |
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114 | ZZX R=GCD(F1,G1); |
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115 | return convertNTLZZX2CF(R,F.mvar()); |
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116 | } |
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117 | |
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118 | static CanonicalForm |
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119 | gcd_univar_ntlp( const CanonicalForm & F, const CanonicalForm & G ) |
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120 | { |
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121 | int ch=getCharacteristic(); |
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122 | if (fac_NTL_char!=ch) |
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123 | { |
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124 | fac_NTL_char=ch; |
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125 | zz_p::init(ch); |
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126 | } |
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127 | zz_pX F1=convertFacCF2NTLzzpX(F); |
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128 | zz_pX G1=convertFacCF2NTLzzpX(G); |
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129 | zz_pX R=GCD(F1,G1); |
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130 | return convertNTLzzpX2CF(R,F.mvar()); |
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131 | } |
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132 | #endif |
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133 | #endif |
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134 | |
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135 | //{{{ static CanonicalForm balance_p ( const CanonicalForm & f, const CanonicalForm & q ) |
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136 | //{{{ docu |
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137 | // |
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138 | // balance_p() - map f from positive to symmetric representation |
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139 | // mod q. |
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140 | // |
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141 | // This makes sense for univariate polynomials over Z only. |
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142 | // q should be an integer. |
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143 | // |
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144 | // Used by gcd_poly_univar0(). |
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145 | // |
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146 | //}}} |
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147 | |
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148 | static CanonicalForm |
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149 | balance_p ( const CanonicalForm & f, const CanonicalForm & q, const CanonicalForm & qh ) |
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150 | { |
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151 | Variable x = f.mvar(); |
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152 | CanonicalForm result = 0; |
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153 | CanonicalForm c; |
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154 | CFIterator i; |
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155 | for ( i = f; i.hasTerms(); i++ ) |
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156 | { |
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157 | c = i.coeff(); |
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158 | if ( c.inCoeffDomain()) |
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159 | { |
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160 | if ( c > qh ) |
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161 | result += power( x, i.exp() ) * (c - q); |
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162 | else |
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163 | result += power( x, i.exp() ) * c; |
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164 | } |
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165 | else |
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166 | result += power( x, i.exp() ) * balance_p(c,q,qh); |
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167 | } |
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168 | return result; |
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169 | } |
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170 | |
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171 | static CanonicalForm |
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172 | balance_p ( const CanonicalForm & f, const CanonicalForm & q ) |
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173 | { |
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174 | CanonicalForm qh = q / 2; |
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175 | return balance_p (f, q, qh); |
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176 | } |
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177 | |
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178 | static CanonicalForm gcd_poly_univar0( const CanonicalForm & F, const CanonicalForm & G, bool primitive ) |
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179 | { |
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180 | CanonicalForm f, g, c, cg, cl, BB, B, M, q, Dp, newD, D, newq; |
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181 | int p, i; |
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182 | |
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183 | if ( primitive ) |
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184 | { |
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185 | f = F; |
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186 | g = G; |
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187 | c = 1; |
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188 | } |
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189 | else |
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190 | { |
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191 | CanonicalForm cF = content( F ), cG = content( G ); |
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192 | f = F / cF; |
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193 | g = G / cG; |
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194 | c = bgcd( cF, cG ); |
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195 | } |
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196 | cg = gcd( f.lc(), g.lc() ); |
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197 | cl = ( f.lc() / cg ) * g.lc(); |
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198 | // B = 2 * cg * tmin( |
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199 | // maxnorm(f)*power(CanonicalForm(2),f.degree())*isqrt(f.degree()+1), |
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200 | // maxnorm(g)*power(CanonicalForm(2),g.degree())*isqrt(g.degree()+1) |
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201 | // )+1; |
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202 | M = tmin( maxNorm(f), maxNorm(g) ); |
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203 | BB = power(CanonicalForm(2),tmin(f.degree(),g.degree()))*M; |
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204 | q = 0; |
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205 | i = cf_getNumSmallPrimes() - 1; |
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206 | while ( true ) |
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207 | { |
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208 | B = BB; |
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209 | while ( i >= 0 && q < B ) |
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210 | { |
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211 | p = cf_getSmallPrime( i ); |
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212 | i--; |
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213 | while ( i >= 0 && mod( cl, p ) == 0 ) |
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214 | { |
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215 | p = cf_getSmallPrime( i ); |
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216 | i--; |
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217 | } |
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218 | setCharacteristic( p ); |
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219 | Dp = gcd( mapinto( f ), mapinto( g ) ); |
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220 | Dp = ( Dp / Dp.lc() ) * mapinto( cg ); |
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221 | setCharacteristic( 0 ); |
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222 | if ( Dp.degree() == 0 ) |
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223 | return c; |
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224 | if ( q.isZero() ) |
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225 | { |
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226 | D = mapinto( Dp ); |
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227 | q = p; |
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228 | B = power(CanonicalForm(2),D.degree())*M+1; |
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229 | } |
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230 | else |
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231 | { |
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232 | if ( Dp.degree() == D.degree() ) |
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233 | { |
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234 | chineseRemainder( D, q, mapinto( Dp ), p, newD, newq ); |
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235 | q = newq; |
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236 | D = newD; |
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237 | } |
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238 | else if ( Dp.degree() < D.degree() ) |
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239 | { |
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240 | // all previous p's are bad primes |
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241 | q = p; |
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242 | D = mapinto( Dp ); |
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243 | B = power(CanonicalForm(2),D.degree())*M+1; |
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244 | } |
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245 | // else p is a bad prime |
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246 | } |
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247 | } |
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248 | if ( i >= 0 ) |
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249 | { |
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250 | // now balance D mod q |
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251 | D = pp( balance_p( D, q ) ); |
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252 | if ( fdivides( D, f ) && fdivides( D, g ) ) |
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253 | return D * c; |
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254 | else |
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255 | q = 0; |
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256 | } |
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257 | else |
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258 | return gcd_poly( F, G ); |
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259 | DEBOUTLN( cerr, "another try ..." ); |
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260 | } |
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261 | } |
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262 | |
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263 | static CanonicalForm |
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264 | gcd_poly_p( const CanonicalForm & f, const CanonicalForm & g ) |
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265 | { |
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266 | if (f.inCoeffDomain() || g.inCoeffDomain()) //zero case should be caught by gcd |
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267 | return 1; |
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268 | CanonicalForm pi, pi1; |
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269 | CanonicalForm C, Ci, Ci1, Hi, bi, pi2; |
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270 | bool bpure, ezgcdon= isOn (SW_USE_EZGCD_P); |
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271 | int delta = degree( f ) - degree( g ); |
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272 | |
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273 | if ( delta >= 0 ) |
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274 | { |
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275 | pi = f; pi1 = g; |
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276 | } |
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277 | else |
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278 | { |
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279 | pi = g; pi1 = f; delta = -delta; |
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280 | } |
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281 | if (pi.isUnivariate()) |
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282 | Ci= 1; |
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283 | else |
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284 | { |
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285 | if (!ezgcdon) |
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286 | On (SW_USE_EZGCD_P); |
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287 | Ci = content( pi ); |
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288 | if (!ezgcdon) |
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289 | Off (SW_USE_EZGCD_P); |
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290 | pi = pi / Ci; |
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291 | } |
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292 | if (pi1.isUnivariate()) |
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293 | Ci1= 1; |
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294 | else |
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295 | { |
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296 | if (!ezgcdon) |
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297 | On (SW_USE_EZGCD_P); |
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298 | Ci1 = content( pi1 ); |
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299 | if (!ezgcdon) |
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300 | Off (SW_USE_EZGCD_P); |
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301 | pi1 = pi1 / Ci1; |
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302 | } |
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303 | C = gcd( Ci, Ci1 ); |
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304 | int d= 0; |
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305 | if ( !( pi.isUnivariate() && pi1.isUnivariate() ) ) |
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306 | { |
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307 | if ( gcd_test_one( pi1, pi, true, d ) ) |
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308 | { |
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309 | C=abs(C); |
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310 | //out_cf("GCD:",C,"\n"); |
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311 | return C; |
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312 | } |
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313 | bpure = false; |
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314 | } |
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315 | else |
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316 | { |
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317 | bpure = isPurePoly(pi) && isPurePoly(pi1); |
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318 | #ifdef HAVE_FLINT |
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319 | if (bpure && (CFFactory::gettype() != GaloisFieldDomain)) |
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320 | return gcd_univar_flintp(pi,pi1)*C; |
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321 | #else |
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322 | #ifdef HAVE_NTL |
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323 | if ( bpure && (CFFactory::gettype() != GaloisFieldDomain)) |
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324 | return gcd_univar_ntlp(pi, pi1 ) * C; |
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325 | #endif |
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326 | #endif |
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327 | } |
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328 | Variable v = f.mvar(); |
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329 | Hi = power( LC( pi1, v ), delta ); |
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330 | int maxNumVars= tmax (getNumVars (pi), getNumVars (pi1)); |
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331 | |
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332 | if (!(pi.isUnivariate() && pi1.isUnivariate())) |
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333 | { |
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334 | if (size (Hi)*size (pi)/(maxNumVars*3) > 500) //maybe this needs more tuning |
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335 | { |
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336 | On (SW_USE_FF_MOD_GCD); |
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337 | C *= gcd (pi, pi1); |
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338 | Off (SW_USE_FF_MOD_GCD); |
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339 | return C; |
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340 | } |
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341 | } |
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342 | |
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343 | if ( (delta+1) % 2 ) |
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344 | bi = 1; |
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345 | else |
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346 | bi = -1; |
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347 | CanonicalForm oldPi= pi, oldPi1= pi1, powHi; |
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348 | while ( degree( pi1, v ) > 0 ) |
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349 | { |
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350 | if (!(pi.isUnivariate() && pi1.isUnivariate())) |
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351 | { |
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352 | if (size (pi)/maxNumVars > 500 || size (pi1)/maxNumVars > 500) |
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353 | { |
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354 | On (SW_USE_FF_MOD_GCD); |
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355 | C *= gcd (oldPi, oldPi1); |
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356 | Off (SW_USE_FF_MOD_GCD); |
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357 | return C; |
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358 | } |
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359 | } |
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360 | pi2 = psr( pi, pi1, v ); |
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361 | pi2 = pi2 / bi; |
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362 | pi = pi1; pi1 = pi2; |
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363 | maxNumVars= tmax (getNumVars (pi), getNumVars (pi1)); |
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364 | if (!pi1.isUnivariate() && (size (pi1)/maxNumVars > 500)) |
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365 | { |
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366 | On (SW_USE_FF_MOD_GCD); |
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367 | C *= gcd (oldPi, oldPi1); |
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368 | Off (SW_USE_FF_MOD_GCD); |
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369 | return C; |
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370 | } |
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371 | if ( degree( pi1, v ) > 0 ) |
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372 | { |
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373 | delta = degree( pi, v ) - degree( pi1, v ); |
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374 | powHi= power (Hi, delta-1); |
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375 | if ( (delta+1) % 2 ) |
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376 | bi = LC( pi, v ) * powHi*Hi; |
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377 | else |
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378 | bi = -LC( pi, v ) * powHi*Hi; |
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379 | Hi = power( LC( pi1, v ), delta ) / powHi; |
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380 | if (!(pi.isUnivariate() && pi1.isUnivariate())) |
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381 | { |
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382 | if (size (Hi)*size (pi)/(maxNumVars*3) > 1500) //maybe this needs more tuning |
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383 | { |
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384 | On (SW_USE_FF_MOD_GCD); |
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385 | C *= gcd (oldPi, oldPi1); |
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386 | Off (SW_USE_FF_MOD_GCD); |
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387 | return C; |
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388 | } |
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389 | } |
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390 | } |
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391 | } |
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392 | if ( degree( pi1, v ) == 0 ) |
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393 | { |
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394 | C=abs(C); |
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395 | //out_cf("GCD:",C,"\n"); |
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396 | return C; |
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397 | } |
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398 | if (!pi.isUnivariate()) |
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399 | { |
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400 | if (!ezgcdon) |
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401 | On (SW_USE_EZGCD_P); |
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402 | Ci= gcd (LC (oldPi,v), LC (oldPi1,v)); |
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403 | pi /= LC (pi,v)/Ci; |
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404 | Ci= content (pi); |
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405 | pi /= Ci; |
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406 | if (!ezgcdon) |
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407 | Off (SW_USE_EZGCD_P); |
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408 | } |
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409 | if ( bpure ) |
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410 | pi /= pi.lc(); |
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411 | C=abs(C*pi); |
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412 | //out_cf("GCD:",C,"\n"); |
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413 | return C; |
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414 | } |
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415 | |
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416 | static CanonicalForm |
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417 | gcd_poly_0( const CanonicalForm & f, const CanonicalForm & g ) |
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418 | { |
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419 | CanonicalForm pi, pi1; |
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420 | CanonicalForm C, Ci, Ci1, Hi, bi, pi2; |
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421 | int delta = degree( f ) - degree( g ); |
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422 | |
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423 | if ( delta >= 0 ) |
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424 | { |
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425 | pi = f; pi1 = g; |
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426 | } |
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427 | else |
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428 | { |
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429 | pi = g; pi1 = f; delta = -delta; |
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430 | } |
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431 | Ci = content( pi ); Ci1 = content( pi1 ); |
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432 | pi1 = pi1 / Ci1; pi = pi / Ci; |
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433 | C = gcd( Ci, Ci1 ); |
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434 | int d= 0; |
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435 | if ( pi.isUnivariate() && pi1.isUnivariate() ) |
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436 | { |
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437 | #ifdef HAVE_FLINT |
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438 | if (isPurePoly(pi) && isPurePoly(pi1) ) |
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439 | return gcd_univar_flint0(pi, pi1 ) * C; |
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440 | #else |
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441 | #ifdef HAVE_NTL |
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442 | if ( isPurePoly(pi) && isPurePoly(pi1) ) |
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443 | return gcd_univar_ntl0(pi, pi1 ) * C; |
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444 | #endif |
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445 | #endif |
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446 | return gcd_poly_univar0( pi, pi1, true ) * C; |
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447 | } |
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448 | else if ( gcd_test_one( pi1, pi, true, d ) ) |
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449 | return C; |
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450 | Variable v = f.mvar(); |
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451 | Hi = power( LC( pi1, v ), delta ); |
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452 | if ( (delta+1) % 2 ) |
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453 | bi = 1; |
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454 | else |
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455 | bi = -1; |
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456 | while ( degree( pi1, v ) > 0 ) |
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457 | { |
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458 | pi2 = psr( pi, pi1, v ); |
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459 | pi2 = pi2 / bi; |
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460 | pi = pi1; pi1 = pi2; |
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461 | if ( degree( pi1, v ) > 0 ) |
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462 | { |
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463 | delta = degree( pi, v ) - degree( pi1, v ); |
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464 | if ( (delta+1) % 2 ) |
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465 | bi = LC( pi, v ) * power( Hi, delta ); |
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466 | else |
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467 | bi = -LC( pi, v ) * power( Hi, delta ); |
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468 | Hi = power( LC( pi1, v ), delta ) / power( Hi, delta-1 ); |
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469 | } |
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470 | } |
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471 | if ( degree( pi1, v ) == 0 ) |
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472 | return C; |
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473 | else |
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474 | return C * pp( pi ); |
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475 | } |
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476 | |
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477 | /** CanonicalForm gcd_poly ( const CanonicalForm & f, const CanonicalForm & g ) |
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478 | * |
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479 | * gcd_poly() - calculate polynomial gcd. |
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480 | * |
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481 | * This is the dispatcher for polynomial gcd calculation. |
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482 | * Different gcd variants get called depending the input, characteristic, and |
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483 | * on switches (cf_defs.h) |
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484 | * |
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485 | * With the current settings from Singular (i.e. SW_USE_EZGCD= on, |
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486 | * SW_USE_EZGCD_P= on, SW_USE_CHINREM_GCD= on, the EZ GCD variants are the |
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487 | * default algorithms for multivariate polynomial GCD computations) |
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488 | * |
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489 | * @sa gcd(), cf_defs.h |
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490 | * |
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491 | **/ |
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492 | CanonicalForm gcd_poly ( const CanonicalForm & f, const CanonicalForm & g ) |
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493 | { |
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494 | CanonicalForm fc, gc; |
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495 | bool fc_isUnivariate=f.isUnivariate(); |
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496 | bool gc_isUnivariate=g.isUnivariate(); |
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497 | bool fc_and_gc_Univariate=fc_isUnivariate && gc_isUnivariate; |
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498 | fc = f; |
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499 | gc = g; |
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500 | int ch=getCharacteristic(); |
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501 | if ( ch != 0 ) |
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502 | { |
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503 | if (0) {} // dummy, to be able to build without NTL and FLINT |
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504 | #if defined(HAVE_FLINT) && ( __FLINT_RELEASE >= 20503) |
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505 | if ( isOn( SW_USE_FL_GCD_P) |
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506 | && (CFFactory::gettype() != GaloisFieldDomain) |
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507 | #ifdef HAVE_NTL |
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508 | && (ch>10) // if we have NTL: it is better for char <11 |
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509 | #endif |
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510 | &&(!hasAlgVar(fc)) && (!hasAlgVar(gc))) |
---|
511 | { |
---|
512 | return gcdFlintMP_Zp(fc,gc); |
---|
513 | } |
---|
514 | #endif |
---|
515 | #ifdef HAVE_NTL |
---|
516 | if ((!fc_and_gc_Univariate) && (isOn( SW_USE_EZGCD_P ))) |
---|
517 | { |
---|
518 | fc= EZGCD_P (fc, gc); |
---|
519 | } |
---|
520 | #endif |
---|
521 | #if defined(HAVE_NTL) || defined(HAVE_FLINT) |
---|
522 | else if (isOn(SW_USE_FF_MOD_GCD) && !fc_and_gc_Univariate) |
---|
523 | { |
---|
524 | Variable a; |
---|
525 | if (hasFirstAlgVar (fc, a) || hasFirstAlgVar (gc, a)) |
---|
526 | fc=modGCDFq (fc, gc, a); |
---|
527 | else if (CFFactory::gettype() == GaloisFieldDomain) |
---|
528 | fc=modGCDGF (fc, gc); |
---|
529 | else |
---|
530 | fc=modGCDFp (fc, gc); |
---|
531 | } |
---|
532 | #endif |
---|
533 | else |
---|
534 | fc = gcd_poly_p( fc, gc ); |
---|
535 | } |
---|
536 | else if (!fc_and_gc_Univariate) /* && char==0*/ |
---|
537 | { |
---|
538 | #if defined(HAVE_FLINT) && ( __FLINT_RELEASE >= 20503) |
---|
539 | if (( isOn( SW_USE_FL_GCD_0) ) |
---|
540 | &&(!hasAlgVar(fc)) && (!hasAlgVar(gc))) |
---|
541 | { |
---|
542 | return gcdFlintMP_QQ(fc,gc); |
---|
543 | } |
---|
544 | else |
---|
545 | #endif |
---|
546 | #ifdef HAVE_NTL |
---|
547 | if ( isOn( SW_USE_EZGCD ) ) |
---|
548 | fc= ezgcd (fc, gc); |
---|
549 | else |
---|
550 | #endif |
---|
551 | #if defined(HAVE_NTL) || defined(HAVE_FLINT) |
---|
552 | if (isOn(SW_USE_CHINREM_GCD)) |
---|
553 | fc = modGCDZ( fc, gc); |
---|
554 | else |
---|
555 | #endif |
---|
556 | { |
---|
557 | fc = gcd_poly_0( fc, gc ); |
---|
558 | } |
---|
559 | } |
---|
560 | else |
---|
561 | { |
---|
562 | fc = gcd_poly_0( fc, gc ); |
---|
563 | } |
---|
564 | if ((ch>0)&&(!hasAlgVar(fc))) fc/=fc.lc(); |
---|
565 | return fc; |
---|
566 | } |
---|
567 | |
---|
568 | /** static CanonicalForm cf_content ( const CanonicalForm & f, const CanonicalForm & g ) |
---|
569 | * |
---|
570 | * cf_content() - return gcd(g, content(f)). |
---|
571 | * |
---|
572 | * content(f) is calculated with respect to f's main variable. |
---|
573 | * |
---|
574 | * @sa gcd(), content(), content( CF, Variable ). |
---|
575 | * |
---|
576 | **/ |
---|
577 | static CanonicalForm |
---|
578 | cf_content ( const CanonicalForm & f, const CanonicalForm & g ) |
---|
579 | { |
---|
580 | if ( f.inPolyDomain() || ( f.inExtension() && ! getReduce( f.mvar() ) ) ) |
---|
581 | { |
---|
582 | CFIterator i = f; |
---|
583 | CanonicalForm result = g; |
---|
584 | while ( i.hasTerms() && ! result.isOne() ) |
---|
585 | { |
---|
586 | result = gcd( i.coeff(), result ); |
---|
587 | i++; |
---|
588 | } |
---|
589 | return result; |
---|
590 | } |
---|
591 | else |
---|
592 | return abs( f ); |
---|
593 | } |
---|
594 | |
---|
595 | /** CanonicalForm content ( const CanonicalForm & f ) |
---|
596 | * |
---|
597 | * content() - return content(f) with respect to main variable. |
---|
598 | * |
---|
599 | * Normalizes result. |
---|
600 | * |
---|
601 | **/ |
---|
602 | CanonicalForm |
---|
603 | content ( const CanonicalForm & f ) |
---|
604 | { |
---|
605 | if ( f.inPolyDomain() || ( f.inExtension() && ! getReduce( f.mvar() ) ) ) |
---|
606 | { |
---|
607 | CFIterator i = f; |
---|
608 | CanonicalForm result = abs( i.coeff() ); |
---|
609 | i++; |
---|
610 | while ( i.hasTerms() && ! result.isOne() ) |
---|
611 | { |
---|
612 | result = gcd( i.coeff(), result ); |
---|
613 | i++; |
---|
614 | } |
---|
615 | return result; |
---|
616 | } |
---|
617 | else |
---|
618 | return abs( f ); |
---|
619 | } |
---|
620 | |
---|
621 | /** CanonicalForm content ( const CanonicalForm & f, const Variable & x ) |
---|
622 | * |
---|
623 | * content() - return content(f) with respect to x. |
---|
624 | * |
---|
625 | * x should be a polynomial variable. |
---|
626 | * |
---|
627 | **/ |
---|
628 | CanonicalForm |
---|
629 | content ( const CanonicalForm & f, const Variable & x ) |
---|
630 | { |
---|
631 | if (f.inBaseDomain()) return f; |
---|
632 | ASSERT( x.level() > 0, "cannot calculate content with respect to algebraic variable" ); |
---|
633 | Variable y = f.mvar(); |
---|
634 | |
---|
635 | if ( y == x ) |
---|
636 | return cf_content( f, 0 ); |
---|
637 | else if ( y < x ) |
---|
638 | return f; |
---|
639 | else |
---|
640 | return swapvar( content( swapvar( f, y, x ), y ), y, x ); |
---|
641 | } |
---|
642 | |
---|
643 | /** CanonicalForm vcontent ( const CanonicalForm & f, const Variable & x ) |
---|
644 | * |
---|
645 | * vcontent() - return content of f with repect to variables >= x. |
---|
646 | * |
---|
647 | * The content is recursively calculated over all coefficients in |
---|
648 | * f having level less than x. x should be a polynomial |
---|
649 | * variable. |
---|
650 | * |
---|
651 | **/ |
---|
652 | CanonicalForm |
---|
653 | vcontent ( const CanonicalForm & f, const Variable & x ) |
---|
654 | { |
---|
655 | ASSERT( x.level() > 0, "cannot calculate vcontent with respect to algebraic variable" ); |
---|
656 | |
---|
657 | if ( f.mvar() <= x ) |
---|
658 | return content( f, x ); |
---|
659 | else { |
---|
660 | CFIterator i; |
---|
661 | CanonicalForm d = 0; |
---|
662 | for ( i = f; i.hasTerms() && ! d.isOne(); i++ ) |
---|
663 | d = gcd( d, vcontent( i.coeff(), x ) ); |
---|
664 | return d; |
---|
665 | } |
---|
666 | } |
---|
667 | |
---|
668 | /** CanonicalForm pp ( const CanonicalForm & f ) |
---|
669 | * |
---|
670 | * pp() - return primitive part of f. |
---|
671 | * |
---|
672 | * Returns zero if f equals zero, otherwise f / content(f). |
---|
673 | * |
---|
674 | **/ |
---|
675 | CanonicalForm |
---|
676 | pp ( const CanonicalForm & f ) |
---|
677 | { |
---|
678 | if ( f.isZero() ) |
---|
679 | return f; |
---|
680 | else |
---|
681 | return f / content( f ); |
---|
682 | } |
---|
683 | |
---|
684 | CanonicalForm |
---|
685 | gcd ( const CanonicalForm & f, const CanonicalForm & g ) |
---|
686 | { |
---|
687 | bool b = f.isZero(); |
---|
688 | if ( b || g.isZero() ) |
---|
689 | { |
---|
690 | if ( b ) |
---|
691 | return abs( g ); |
---|
692 | else |
---|
693 | return abs( f ); |
---|
694 | } |
---|
695 | if ( f.inPolyDomain() || g.inPolyDomain() ) |
---|
696 | { |
---|
697 | if ( f.mvar() != g.mvar() ) |
---|
698 | { |
---|
699 | if ( f.mvar() > g.mvar() ) |
---|
700 | return cf_content( f, g ); |
---|
701 | else |
---|
702 | return cf_content( g, f ); |
---|
703 | } |
---|
704 | if (isOn(SW_USE_QGCD)) |
---|
705 | { |
---|
706 | Variable m; |
---|
707 | if ( |
---|
708 | (getCharacteristic() == 0) && |
---|
709 | (hasFirstAlgVar(f,m) || hasFirstAlgVar(g,m)) |
---|
710 | ) |
---|
711 | { |
---|
712 | bool on_rational = isOn(SW_RATIONAL); |
---|
713 | CanonicalForm r=QGCD(f,g); |
---|
714 | On(SW_RATIONAL); |
---|
715 | CanonicalForm cdF = bCommonDen( r ); |
---|
716 | if (!on_rational) Off(SW_RATIONAL); |
---|
717 | return cdF*r; |
---|
718 | } |
---|
719 | } |
---|
720 | |
---|
721 | if ( f.inExtension() && getReduce( f.mvar() ) ) |
---|
722 | return CanonicalForm(1); |
---|
723 | else |
---|
724 | { |
---|
725 | if ( fdivides( f, g ) ) |
---|
726 | return abs( f ); |
---|
727 | else if ( fdivides( g, f ) ) |
---|
728 | return abs( g ); |
---|
729 | if ( !( getCharacteristic() == 0 && isOn( SW_RATIONAL ) ) ) |
---|
730 | { |
---|
731 | CanonicalForm d; |
---|
732 | d = gcd_poly( f, g ); |
---|
733 | return abs( d ); |
---|
734 | } |
---|
735 | else |
---|
736 | { |
---|
737 | CanonicalForm cdF = bCommonDen( f ); |
---|
738 | CanonicalForm cdG = bCommonDen( g ); |
---|
739 | CanonicalForm F = f * cdF, G = g * cdG; |
---|
740 | Off( SW_RATIONAL ); |
---|
741 | CanonicalForm l = gcd_poly( F, G ); |
---|
742 | On( SW_RATIONAL ); |
---|
743 | return abs( l ); |
---|
744 | } |
---|
745 | } |
---|
746 | } |
---|
747 | if ( f.inBaseDomain() && g.inBaseDomain() ) |
---|
748 | return bgcd( f, g ); |
---|
749 | else |
---|
750 | return 1; |
---|
751 | } |
---|
752 | |
---|
753 | /** CanonicalForm lcm ( const CanonicalForm & f, const CanonicalForm & g ) |
---|
754 | * |
---|
755 | * lcm() - return least common multiple of f and g. |
---|
756 | * |
---|
757 | * The lcm is calculated using the formula lcm(f, g) = f * g / gcd(f, g). |
---|
758 | * |
---|
759 | * Returns zero if one of f or g equals zero. |
---|
760 | * |
---|
761 | **/ |
---|
762 | CanonicalForm |
---|
763 | lcm ( const CanonicalForm & f, const CanonicalForm & g ) |
---|
764 | { |
---|
765 | if ( f.isZero() || g.isZero() ) |
---|
766 | return 0; |
---|
767 | else |
---|
768 | return ( f / gcd( f, g ) ) * g; |
---|
769 | } |
---|
770 | |
---|