1 | /* emacs edit mode for this file is -*- C++ -*- */ |
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2 | /* $Id: cf_gcd.cc,v 1.61 2008-05-16 13:11:31 Singular Exp $ */ |
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3 | |
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4 | #include <config.h> |
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5 | |
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6 | #include "assert.h" |
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7 | #include "debug.h" |
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8 | |
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9 | #include "cf_defs.h" |
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10 | #include "canonicalform.h" |
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11 | #include "cf_iter.h" |
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12 | #include "cf_reval.h" |
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13 | #include "cf_primes.h" |
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14 | #include "cf_algorithm.h" |
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15 | #include "fac_util.h" |
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16 | #include "ftmpl_functions.h" |
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17 | #include "ffreval.h" |
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18 | |
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19 | #ifdef HAVE_NTL |
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20 | #include <NTL/ZZX.h> |
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21 | #include "NTLconvert.h" |
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22 | bool isPurePoly(const CanonicalForm & ); |
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23 | static CanonicalForm gcd_univar_ntl0( const CanonicalForm &, const CanonicalForm & ); |
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24 | static CanonicalForm gcd_univar_ntlp( const CanonicalForm &, const CanonicalForm & ); |
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25 | #endif |
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26 | |
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27 | static CanonicalForm cf_content ( const CanonicalForm &, const CanonicalForm & ); |
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28 | static bool gcd_avoid_mtaildegree ( CanonicalForm &, CanonicalForm &, CanonicalForm & ); |
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29 | static void cf_prepgcd( const CanonicalForm &, const CanonicalForm &, int &, int &, int & ); |
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30 | |
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31 | void out_cf(char *s1,const CanonicalForm &f,char *s2); |
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32 | |
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33 | CanonicalForm chinrem_gcd(const CanonicalForm & FF,const CanonicalForm & GG); |
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34 | CanonicalForm newGCD(CanonicalForm A, CanonicalForm B); |
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35 | |
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36 | bool |
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37 | gcd_test_one ( const CanonicalForm & f, const CanonicalForm & g, bool swap ) |
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38 | { |
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39 | int count = 0; |
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40 | // assume polys have same level; |
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41 | CFRandom * sample = CFRandomFactory::generate(); |
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42 | REvaluation e( 2, tmax( f.level(), g.level() ), *sample ); |
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43 | delete sample; |
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44 | CanonicalForm lcf, lcg; |
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45 | if ( swap ) |
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46 | { |
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47 | lcf = swapvar( LC( f ), Variable(1), f.mvar() ); |
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48 | lcg = swapvar( LC( g ), Variable(1), f.mvar() ); |
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49 | } |
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50 | else |
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51 | { |
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52 | lcf = LC( f, Variable(1) ); |
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53 | lcg = LC( g, Variable(1) ); |
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54 | } |
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55 | #define TEST_ONE_MAX 50 |
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56 | while ( ( e( lcf ).isZero() || e( lcg ).isZero() ) && count < TEST_ONE_MAX ) |
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57 | { |
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58 | e.nextpoint(); |
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59 | count++; |
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60 | } |
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61 | if ( count == TEST_ONE_MAX ) |
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62 | return false; |
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63 | CanonicalForm F, G; |
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64 | if ( swap ) |
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65 | { |
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66 | F=swapvar( f, Variable(1), f.mvar() ); |
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67 | G=swapvar( g, Variable(1), g.mvar() ); |
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68 | } |
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69 | else |
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70 | { |
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71 | F = f; |
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72 | G = g; |
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73 | } |
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74 | if ( e(F).taildegree() > 0 && e(G).taildegree() > 0 ) |
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75 | return false; |
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76 | return gcd( e( F ), e( G ) ).degree() < 1; |
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77 | } |
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78 | |
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79 | //{{{ static CanonicalForm icontent ( const CanonicalForm & f, const CanonicalForm & c ) |
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80 | //{{{ docu |
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81 | // |
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82 | // icontent() - return gcd of c and all coefficients of f which |
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83 | // are in a coefficient domain. |
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84 | // |
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85 | // Used by icontent(). |
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86 | // |
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87 | //}}} |
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88 | static CanonicalForm |
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89 | icontent ( const CanonicalForm & f, const CanonicalForm & c ) |
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90 | { |
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91 | if ( f.inBaseDomain() ) |
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92 | { |
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93 | if (c.isZero()) return abs(f); |
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94 | return bgcd( f, c ); |
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95 | } |
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96 | //else if ( f.inCoeffDomain() ) |
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97 | // return gcd(f,c); |
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98 | else |
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99 | { |
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100 | CanonicalForm g = c; |
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101 | for ( CFIterator i = f; i.hasTerms() && ! g.isOne(); i++ ) |
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102 | g = icontent( i.coeff(), g ); |
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103 | return g; |
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104 | } |
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105 | } |
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106 | //}}} |
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107 | |
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108 | //{{{ CanonicalForm icontent ( const CanonicalForm & f ) |
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109 | //{{{ docu |
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110 | // |
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111 | // icontent() - return gcd over all coefficients of f which are |
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112 | // in a coefficient domain. |
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113 | // |
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114 | //}}} |
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115 | CanonicalForm |
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116 | icontent ( const CanonicalForm & f ) |
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117 | { |
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118 | return icontent( f, 0 ); |
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119 | } |
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120 | //}}} |
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121 | |
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122 | //{{{ CanonicalForm extgcd ( const CanonicalForm & f, const CanonicalForm & g, CanonicalForm & a, CanonicalForm & b ) |
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123 | //{{{ docu |
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124 | // |
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125 | // extgcd() - returns polynomial extended gcd of f and g. |
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126 | // |
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127 | // Returns gcd(f, g) and a and b sucht that f*a+g*b=gcd(f, g). |
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128 | // The gcd is calculated using an extended euclidean polynomial |
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129 | // remainder sequence, so f and g should be polynomials over an |
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130 | // euclidean domain. Normalizes result. |
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131 | // |
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132 | // Note: be sure that f and g have the same level! |
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133 | // |
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134 | //}}} |
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135 | CanonicalForm |
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136 | extgcd ( const CanonicalForm & f, const CanonicalForm & g, CanonicalForm & a, CanonicalForm & b ) |
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137 | { |
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138 | #ifdef HAVE_NTL |
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139 | if (isOn(SW_USE_NTL_GCD_P) && ( getCharacteristic() > 0 ) |
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140 | && isPurePoly(f) && isPurePoly(g)) |
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141 | { |
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142 | if (fac_NTL_char!=getCharacteristic()) |
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143 | { |
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144 | fac_NTL_char=getCharacteristic(); |
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145 | #ifdef NTL_ZZ |
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146 | ZZ r; |
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147 | r=getCharacteristic(); |
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148 | ZZ_pContext ccc(r); |
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149 | #else |
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150 | zz_pContext ccc(getCharacteristic()); |
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151 | #endif |
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152 | ccc.restore(); |
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153 | #ifdef NTL_ZZ |
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154 | ZZ_p::init(r); |
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155 | #else |
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156 | zz_p::init(getCharacteristic()); |
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157 | #endif |
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158 | } |
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159 | #ifdef NTL_ZZ |
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160 | ZZ_pX F1=convertFacCF2NTLZZpX(f); |
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161 | ZZ_pX G1=convertFacCF2NTLZZpX(g); |
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162 | ZZ_pX R; |
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163 | ZZ_pX A,B; |
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164 | XGCD(R,A,B,F1,G1); |
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165 | a=convertNTLZZpX2CF(A,f.mvar()); |
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166 | b=convertNTLZZpX2CF(B,f.mvar()); |
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167 | return convertNTLZZpX2CF(R,f.mvar()); |
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168 | #else |
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169 | zz_pX F1=convertFacCF2NTLzzpX(f); |
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170 | zz_pX G1=convertFacCF2NTLzzpX(g); |
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171 | zz_pX R; |
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172 | zz_pX A,B; |
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173 | XGCD(R,A,B,F1,G1); |
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174 | a=convertNTLzzpX2CF(A,f.mvar()); |
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175 | b=convertNTLzzpX2CF(B,f.mvar()); |
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176 | return convertNTLzzpX2CF(R,f.mvar()); |
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177 | #endif |
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178 | } |
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179 | #endif |
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180 | CanonicalForm contf = content( f ); |
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181 | CanonicalForm contg = content( g ); |
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182 | |
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183 | CanonicalForm p0 = f / contf, p1 = g / contg; |
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184 | CanonicalForm f0 = 1, f1 = 0, g0 = 0, g1 = 1, q, r; |
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185 | |
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186 | while ( ! p1.isZero() ) |
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187 | { |
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188 | divrem( p0, p1, q, r ); |
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189 | p0 = p1; p1 = r; |
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190 | r = g0 - g1 * q; |
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191 | g0 = g1; g1 = r; |
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192 | r = f0 - f1 * q; |
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193 | f0 = f1; f1 = r; |
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194 | } |
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195 | CanonicalForm contp0 = content( p0 ); |
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196 | a = f0 / ( contf * contp0 ); |
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197 | b = g0 / ( contg * contp0 ); |
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198 | p0 /= contp0; |
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199 | if ( p0.sign() < 0 ) |
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200 | { |
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201 | p0 = -p0; |
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202 | a = -a; |
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203 | b = -b; |
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204 | } |
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205 | return p0; |
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206 | } |
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207 | //}}} |
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208 | |
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209 | //{{{ static CanonicalForm balance ( const CanonicalForm & f, const CanonicalForm & q ) |
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210 | //{{{ docu |
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211 | // |
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212 | // balance() - map f from positive to symmetric representation |
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213 | // mod q. |
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214 | // |
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215 | // This makes sense for univariate polynomials over Z only. |
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216 | // q should be an integer. |
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217 | // |
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218 | // Used by gcd_poly_univar0(). |
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219 | // |
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220 | //}}} |
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221 | static CanonicalForm |
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222 | balance ( const CanonicalForm & f, const CanonicalForm & q ) |
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223 | { |
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224 | Variable x = f.mvar(); |
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225 | CanonicalForm result = 0, qh = q / 2; |
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226 | CanonicalForm c; |
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227 | CFIterator i; |
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228 | for ( i = f; i.hasTerms(); i++ ) { |
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229 | c = mod( i.coeff(), q ); |
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230 | if ( c > qh ) |
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231 | result += power( x, i.exp() ) * (c - q); |
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232 | else |
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233 | result += power( x, i.exp() ) * c; |
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234 | } |
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235 | return result; |
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236 | } |
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237 | //}}} |
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238 | |
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239 | static CanonicalForm |
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240 | gcd_poly_univar0( const CanonicalForm & F, const CanonicalForm & G, bool primitive ) |
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241 | { |
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242 | CanonicalForm f, g, c, cg, cl, BB, B, M, q, Dp, newD, D, newq; |
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243 | int p, i, n; |
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244 | |
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245 | if ( primitive ) |
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246 | { |
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247 | f = F; |
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248 | g = G; |
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249 | c = 1; |
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250 | } |
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251 | else |
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252 | { |
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253 | CanonicalForm cF = content( F ), cG = content( G ); |
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254 | f = F / cF; |
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255 | g = G / cG; |
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256 | c = bgcd( cF, cG ); |
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257 | } |
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258 | cg = gcd( f.lc(), g.lc() ); |
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259 | cl = ( f.lc() / cg ) * g.lc(); |
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260 | // B = 2 * cg * tmin( |
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261 | // maxnorm(f)*power(CanonicalForm(2),f.degree())*isqrt(f.degree()+1), |
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262 | // maxnorm(g)*power(CanonicalForm(2),g.degree())*isqrt(g.degree()+1) |
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263 | // )+1; |
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264 | M = tmin( maxNorm(f), maxNorm(g) ); |
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265 | BB = power(CanonicalForm(2),tmin(f.degree(),g.degree()))*M; |
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266 | q = 0; |
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267 | i = cf_getNumSmallPrimes() - 1; |
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268 | while ( true ) |
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269 | { |
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270 | B = BB; |
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271 | while ( i >= 0 && q < B ) |
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272 | { |
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273 | p = cf_getSmallPrime( i ); |
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274 | i--; |
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275 | while ( i >= 0 && mod( cl, p ) == 0 ) |
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276 | { |
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277 | p = cf_getSmallPrime( i ); |
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278 | i--; |
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279 | } |
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280 | setCharacteristic( p ); |
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281 | Dp = gcd( mapinto( f ), mapinto( g ) ); |
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282 | Dp = ( Dp / Dp.lc() ) * mapinto( cg ); |
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283 | setCharacteristic( 0 ); |
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284 | if ( Dp.degree() == 0 ) |
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285 | return c; |
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286 | if ( q.isZero() ) |
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287 | { |
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288 | D = mapinto( Dp ); |
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289 | q = p; |
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290 | B = power(CanonicalForm(2),D.degree())*M+1; |
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291 | } |
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292 | else |
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293 | { |
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294 | if ( Dp.degree() == D.degree() ) |
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295 | { |
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296 | chineseRemainder( D, q, mapinto( Dp ), p, newD, newq ); |
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297 | q = newq; |
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298 | D = newD; |
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299 | } |
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300 | else if ( Dp.degree() < D.degree() ) |
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301 | { |
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302 | // all previous p's are bad primes |
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303 | q = p; |
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304 | D = mapinto( Dp ); |
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305 | B = power(CanonicalForm(2),D.degree())*M+1; |
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306 | } |
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307 | // else p is a bad prime |
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308 | } |
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309 | } |
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310 | if ( i >= 0 ) |
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311 | { |
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312 | // now balance D mod q |
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313 | D = pp( balance( D, q ) ); |
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314 | if ( fdivides( D, f ) && fdivides( D, g ) ) |
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315 | return D * c; |
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316 | else |
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317 | q = 0; |
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318 | } |
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319 | else |
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320 | return gcd_poly( F, G ); |
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321 | DEBOUTLN( cerr, "another try ..." ); |
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322 | } |
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323 | } |
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324 | |
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325 | static CanonicalForm |
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326 | gcd_poly_p( const CanonicalForm & f, const CanonicalForm & g ) |
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327 | { |
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328 | CanonicalForm pi, pi1; |
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329 | CanonicalForm C, Ci, Ci1, Hi, bi, pi2; |
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330 | bool bpure; |
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331 | int delta = degree( f ) - degree( g ); |
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332 | |
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333 | if ( delta >= 0 ) |
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334 | { |
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335 | pi = f; pi1 = g; |
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336 | } |
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337 | else |
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338 | { |
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339 | pi = g; pi1 = f; delta = -delta; |
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340 | } |
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341 | Ci = content( pi ); Ci1 = content( pi1 ); |
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342 | pi1 = pi1 / Ci1; pi = pi / Ci; |
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343 | C = gcd( Ci, Ci1 ); |
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344 | if ( !( pi.isUnivariate() && pi1.isUnivariate() ) ) |
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345 | { |
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346 | //out_cf("F:",f,"\n"); |
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347 | //out_cf("G:",g,"\n"); |
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348 | //out_cf("newGCD:",newGCD(f,g),"\n"); |
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349 | if (isOn(SW_USE_GCD_P) && (getCharacteristic()>0)) |
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350 | { |
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351 | return newGCD(f,g); |
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352 | } |
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353 | if ( gcd_test_one( pi1, pi, true ) ) |
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354 | { |
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355 | C=abs(C); |
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356 | //out_cf("GCD:",C,"\n"); |
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357 | return C; |
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358 | } |
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359 | bpure = false; |
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360 | } |
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361 | else |
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362 | { |
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363 | bpure = isPurePoly(pi) && isPurePoly(pi1); |
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364 | #ifdef HAVE_NTL |
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365 | if ( isOn(SW_USE_NTL_GCD_P) && bpure ) |
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366 | return gcd_univar_ntlp(pi, pi1 ) * C; |
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367 | #endif |
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368 | } |
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369 | Variable v = f.mvar(); |
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370 | Hi = power( LC( pi1, v ), delta ); |
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371 | if ( (delta+1) % 2 ) |
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372 | bi = 1; |
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373 | else |
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374 | bi = -1; |
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375 | while ( degree( pi1, v ) > 0 ) { |
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376 | pi2 = psr( pi, pi1, v ); |
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377 | pi2 = pi2 / bi; |
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378 | pi = pi1; pi1 = pi2; |
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379 | if ( degree( pi1, v ) > 0 ) { |
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380 | delta = degree( pi, v ) - degree( pi1, v ); |
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381 | if ( (delta+1) % 2 ) |
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382 | bi = LC( pi, v ) * power( Hi, delta ); |
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383 | else |
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384 | bi = -LC( pi, v ) * power( Hi, delta ); |
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385 | Hi = power( LC( pi1, v ), delta ) / power( Hi, delta-1 ); |
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386 | } |
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387 | } |
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388 | if ( degree( pi1, v ) == 0 ) |
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389 | { |
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390 | C=abs(C); |
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391 | //out_cf("GCD:",C,"\n"); |
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392 | return C; |
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393 | } |
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394 | pi /= content( pi ); |
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395 | if ( bpure ) |
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396 | pi /= pi.lc(); |
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397 | C=abs(C*pi); |
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398 | //out_cf("GCD:",C,"\n"); |
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399 | return C; |
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400 | } |
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401 | |
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402 | static CanonicalForm |
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403 | gcd_poly_0( const CanonicalForm & f, const CanonicalForm & g ) |
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404 | { |
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405 | CanonicalForm pi, pi1; |
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406 | CanonicalForm C, Ci, Ci1, Hi, bi, pi2; |
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407 | int delta = degree( f ) - degree( g ); |
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408 | |
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409 | if ( delta >= 0 ) |
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410 | { |
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411 | pi = f; pi1 = g; |
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412 | } |
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413 | else |
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414 | { |
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415 | pi = g; pi1 = f; delta = -delta; |
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416 | } |
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417 | Ci = content( pi ); Ci1 = content( pi1 ); |
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418 | pi1 = pi1 / Ci1; pi = pi / Ci; |
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419 | C = gcd( Ci, Ci1 ); |
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420 | if ( pi.isUnivariate() && pi1.isUnivariate() ) |
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421 | { |
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422 | #ifdef HAVE_NTL |
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423 | if ( isOn(SW_USE_NTL_GCD_0) && isPurePoly(pi) && isPurePoly(pi1) ) |
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424 | return gcd_univar_ntl0(pi, pi1 ) * C; |
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425 | #endif |
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426 | return gcd_poly_univar0( pi, pi1, true ) * C; |
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427 | } |
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428 | else if ( gcd_test_one( pi1, pi, true ) ) |
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429 | return C; |
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430 | Variable v = f.mvar(); |
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431 | Hi = power( LC( pi1, v ), delta ); |
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432 | if ( (delta+1) % 2 ) |
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433 | bi = 1; |
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434 | else |
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435 | bi = -1; |
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436 | while ( degree( pi1, v ) > 0 ) |
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437 | { |
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438 | pi2 = psr( pi, pi1, v ); |
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439 | pi2 = pi2 / bi; |
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440 | pi = pi1; pi1 = pi2; |
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441 | if ( degree( pi1, v ) > 0 ) |
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442 | { |
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443 | delta = degree( pi, v ) - degree( pi1, v ); |
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444 | if ( (delta+1) % 2 ) |
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445 | bi = LC( pi, v ) * power( Hi, delta ); |
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446 | else |
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447 | bi = -LC( pi, v ) * power( Hi, delta ); |
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448 | Hi = power( LC( pi1, v ), delta ) / power( Hi, delta-1 ); |
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449 | } |
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450 | } |
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451 | if ( degree( pi1, v ) == 0 ) |
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452 | return C; |
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453 | else |
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454 | return C * pp( pi ); |
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455 | } |
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456 | |
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457 | //{{{ CanonicalForm gcd_poly ( const CanonicalForm & f, const CanonicalForm & g ) |
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458 | //{{{ docu |
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459 | // |
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460 | // gcd_poly() - calculate polynomial gcd. |
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461 | // |
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462 | // This is the dispatcher for polynomial gcd calculation. We call either |
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463 | // ezgcd(), sparsemod() or gcd_poly1() in dependecy on the current |
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464 | // characteristic and settings of SW_USE_EZGCD and SW_USE_SPARSEMOD, resp. |
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465 | // |
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466 | // Used by gcd() and gcd_poly_univar0(). |
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467 | // |
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468 | //}}} |
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469 | #if 0 |
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470 | int si_factor_reminder=1; |
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471 | #endif |
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472 | CanonicalForm gcd_poly ( const CanonicalForm & f, const CanonicalForm & g ) |
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473 | { |
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474 | CanonicalForm fc, gc, d1; |
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475 | int mp, cc, p1, pe; |
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476 | mp = f.level()+1; |
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477 | bool fc_isUnivariate=f.isUnivariate(); |
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478 | bool gc_isUnivariate=g.isUnivariate(); |
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479 | bool fc_and_gc_Univariate=fc_isUnivariate && gc_isUnivariate; |
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480 | #if 1 |
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481 | if (( getCharacteristic() == 0 ) |
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482 | && (f.level() >4) |
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483 | && (g.level() >4) |
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484 | && isOn( SW_USE_CHINREM_GCD) |
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485 | && (!fc_and_gc_Univariate) |
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486 | && (isPurePoly_m(f)) |
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487 | && (isPurePoly_m(g)) |
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488 | ) |
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489 | { |
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490 | return chinrem_gcd( f, g ); |
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491 | } |
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492 | #endif |
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493 | cf_prepgcd( f, g, cc, p1, pe); |
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494 | if ( cc != 0 ) |
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495 | { |
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496 | if ( cc > 0 ) |
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497 | { |
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498 | fc = replacevar( f, Variable(cc), Variable(mp) ); |
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499 | gc = g; |
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500 | } |
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501 | else |
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502 | { |
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503 | fc = replacevar( g, Variable(-cc), Variable(mp) ); |
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504 | gc = f; |
---|
505 | } |
---|
506 | return cf_content( fc, gc ); |
---|
507 | } |
---|
508 | // now each appearing variable is in f and g |
---|
509 | fc = f; |
---|
510 | gc = g; |
---|
511 | if( gcd_avoid_mtaildegree ( fc, gc, d1 ) ) |
---|
512 | return d1; |
---|
513 | if ( getCharacteristic() != 0 ) |
---|
514 | { |
---|
515 | if (isOn( SW_USE_EZGCD_P ) && (!fc_and_gc_Univariate)) |
---|
516 | { |
---|
517 | if ( pe == 1 ) |
---|
518 | fc = fin_ezgcd( fc, gc ); |
---|
519 | else if ( pe > 0 )// no variable at position 1 |
---|
520 | { |
---|
521 | fc = replacevar( fc, Variable(pe), Variable(1) ); |
---|
522 | gc = replacevar( gc, Variable(pe), Variable(1) ); |
---|
523 | fc = replacevar( fin_ezgcd( fc, gc ), Variable(1), Variable(pe) ); |
---|
524 | } |
---|
525 | else |
---|
526 | { |
---|
527 | pe = -pe; |
---|
528 | fc = swapvar( fc, Variable(pe), Variable(1) ); |
---|
529 | gc = swapvar( gc, Variable(pe), Variable(1) ); |
---|
530 | fc = swapvar( fin_ezgcd( fc, gc ), Variable(1), Variable(pe) ); |
---|
531 | } |
---|
532 | } |
---|
533 | else if (isOn(SW_USE_GCD_P)) |
---|
534 | { |
---|
535 | fc=newGCD(fc,gc); |
---|
536 | } |
---|
537 | else if ( p1 == fc.level() ) |
---|
538 | fc = gcd_poly_p( fc, gc ); |
---|
539 | else |
---|
540 | { |
---|
541 | fc = replacevar( fc, Variable(p1), Variable(mp) ); |
---|
542 | gc = replacevar( gc, Variable(p1), Variable(mp) ); |
---|
543 | fc = replacevar( gcd_poly_p( fc, gc ), Variable(mp), Variable(p1) ); |
---|
544 | } |
---|
545 | } |
---|
546 | else if (!fc_and_gc_Univariate) |
---|
547 | { |
---|
548 | if ( |
---|
549 | isOn(SW_USE_CHINREM_GCD) |
---|
550 | && (gc.level() >5) |
---|
551 | && (fc.level() >5) |
---|
552 | && (isPurePoly_m(fc)) && (isPurePoly_m(gc)) |
---|
553 | ) |
---|
554 | { |
---|
555 | #if 0 |
---|
556 | if ( p1 == fc.level() ) |
---|
557 | fc = chinrem_gcd( fc, gc ); |
---|
558 | else |
---|
559 | { |
---|
560 | fc = replacevar( fc, Variable(p1), Variable(mp) ); |
---|
561 | gc = replacevar( gc, Variable(p1), Variable(mp) ); |
---|
562 | fc = replacevar( chinrem_gcd( fc, gc ), Variable(mp), Variable(p1) ); |
---|
563 | } |
---|
564 | #else |
---|
565 | fc = chinrem_gcd( fc, gc); |
---|
566 | #endif |
---|
567 | } |
---|
568 | if ( isOn( SW_USE_EZGCD ) ) |
---|
569 | { |
---|
570 | if ( pe == 1 ) |
---|
571 | fc = ezgcd( fc, gc ); |
---|
572 | else if ( pe > 0 )// no variable at position 1 |
---|
573 | { |
---|
574 | fc = replacevar( fc, Variable(pe), Variable(1) ); |
---|
575 | gc = replacevar( gc, Variable(pe), Variable(1) ); |
---|
576 | fc = replacevar( ezgcd( fc, gc ), Variable(1), Variable(pe) ); |
---|
577 | } |
---|
578 | else |
---|
579 | { |
---|
580 | pe = -pe; |
---|
581 | fc = swapvar( fc, Variable(pe), Variable(1) ); |
---|
582 | gc = swapvar( gc, Variable(pe), Variable(1) ); |
---|
583 | fc = swapvar( ezgcd( fc, gc ), Variable(1), Variable(pe) ); |
---|
584 | } |
---|
585 | } |
---|
586 | else if ( |
---|
587 | isOn(SW_USE_CHINREM_GCD) |
---|
588 | && (isPurePoly_m(fc)) && (isPurePoly_m(gc)) |
---|
589 | ) |
---|
590 | { |
---|
591 | #if 0 |
---|
592 | if ( p1 == fc.level() ) |
---|
593 | fc = chinrem_gcd( fc, gc ); |
---|
594 | else |
---|
595 | { |
---|
596 | fc = replacevar( fc, Variable(p1), Variable(mp) ); |
---|
597 | gc = replacevar( gc, Variable(p1), Variable(mp) ); |
---|
598 | fc = replacevar( chinrem_gcd( fc, gc ), Variable(mp), Variable(p1) ); |
---|
599 | } |
---|
600 | #else |
---|
601 | fc = chinrem_gcd( fc, gc); |
---|
602 | #endif |
---|
603 | } |
---|
604 | else |
---|
605 | { |
---|
606 | fc = gcd_poly_0( fc, gc ); |
---|
607 | } |
---|
608 | } |
---|
609 | else |
---|
610 | { |
---|
611 | fc = gcd_poly_0( fc, gc ); |
---|
612 | } |
---|
613 | if ( d1.degree() > 0 ) |
---|
614 | fc *= d1; |
---|
615 | return fc; |
---|
616 | } |
---|
617 | //}}} |
---|
618 | |
---|
619 | //{{{ static CanonicalForm cf_content ( const CanonicalForm & f, const CanonicalForm & g ) |
---|
620 | //{{{ docu |
---|
621 | // |
---|
622 | // cf_content() - return gcd(g, content(f)). |
---|
623 | // |
---|
624 | // content(f) is calculated with respect to f's main variable. |
---|
625 | // |
---|
626 | // Used by gcd(), content(), content( CF, Variable ). |
---|
627 | // |
---|
628 | //}}} |
---|
629 | static CanonicalForm |
---|
630 | cf_content ( const CanonicalForm & f, const CanonicalForm & g ) |
---|
631 | { |
---|
632 | if ( f.inPolyDomain() || ( f.inExtension() && ! getReduce( f.mvar() ) ) ) |
---|
633 | { |
---|
634 | CFIterator i = f; |
---|
635 | CanonicalForm result = g; |
---|
636 | while ( i.hasTerms() && ! result.isOne() ) |
---|
637 | { |
---|
638 | result = gcd( i.coeff(), result ); |
---|
639 | i++; |
---|
640 | } |
---|
641 | return result; |
---|
642 | } |
---|
643 | else |
---|
644 | return abs( f ); |
---|
645 | } |
---|
646 | //}}} |
---|
647 | |
---|
648 | //{{{ CanonicalForm content ( const CanonicalForm & f ) |
---|
649 | //{{{ docu |
---|
650 | // |
---|
651 | // content() - return content(f) with respect to main variable. |
---|
652 | // |
---|
653 | // Normalizes result. |
---|
654 | // |
---|
655 | //}}} |
---|
656 | CanonicalForm |
---|
657 | content ( const CanonicalForm & f ) |
---|
658 | { |
---|
659 | if ( f.inPolyDomain() || ( f.inExtension() && ! getReduce( f.mvar() ) ) ) |
---|
660 | { |
---|
661 | CFIterator i = f; |
---|
662 | CanonicalForm result = abs( i.coeff() ); |
---|
663 | i++; |
---|
664 | while ( i.hasTerms() && ! result.isOne() ) |
---|
665 | { |
---|
666 | result = gcd( i.coeff(), result ); |
---|
667 | i++; |
---|
668 | } |
---|
669 | return result; |
---|
670 | } |
---|
671 | else |
---|
672 | return abs( f ); |
---|
673 | } |
---|
674 | //}}} |
---|
675 | |
---|
676 | //{{{ CanonicalForm content ( const CanonicalForm & f, const Variable & x ) |
---|
677 | //{{{ docu |
---|
678 | // |
---|
679 | // content() - return content(f) with respect to x. |
---|
680 | // |
---|
681 | // x should be a polynomial variable. |
---|
682 | // |
---|
683 | //}}} |
---|
684 | CanonicalForm |
---|
685 | content ( const CanonicalForm & f, const Variable & x ) |
---|
686 | { |
---|
687 | ASSERT( x.level() > 0, "cannot calculate content with respect to algebraic variable" ); |
---|
688 | Variable y = f.mvar(); |
---|
689 | |
---|
690 | if ( y == x ) |
---|
691 | return cf_content( f, 0 ); |
---|
692 | else if ( y < x ) |
---|
693 | return f; |
---|
694 | else |
---|
695 | return swapvar( content( swapvar( f, y, x ), y ), y, x ); |
---|
696 | } |
---|
697 | //}}} |
---|
698 | |
---|
699 | //{{{ CanonicalForm vcontent ( const CanonicalForm & f, const Variable & x ) |
---|
700 | //{{{ docu |
---|
701 | // |
---|
702 | // vcontent() - return content of f with repect to variables >= x. |
---|
703 | // |
---|
704 | // The content is recursively calculated over all coefficients in |
---|
705 | // f having level less than x. x should be a polynomial |
---|
706 | // variable. |
---|
707 | // |
---|
708 | //}}} |
---|
709 | CanonicalForm |
---|
710 | vcontent ( const CanonicalForm & f, const Variable & x ) |
---|
711 | { |
---|
712 | ASSERT( x.level() > 0, "cannot calculate vcontent with respect to algebraic variable" ); |
---|
713 | |
---|
714 | if ( f.mvar() <= x ) |
---|
715 | return content( f, x ); |
---|
716 | else { |
---|
717 | CFIterator i; |
---|
718 | CanonicalForm d = 0; |
---|
719 | for ( i = f; i.hasTerms() && ! d.isOne(); i++ ) |
---|
720 | d = gcd( d, vcontent( i.coeff(), x ) ); |
---|
721 | return d; |
---|
722 | } |
---|
723 | } |
---|
724 | //}}} |
---|
725 | |
---|
726 | //{{{ CanonicalForm pp ( const CanonicalForm & f ) |
---|
727 | //{{{ docu |
---|
728 | // |
---|
729 | // pp() - return primitive part of f. |
---|
730 | // |
---|
731 | // Returns zero if f equals zero, otherwise f / content(f). |
---|
732 | // |
---|
733 | //}}} |
---|
734 | CanonicalForm |
---|
735 | pp ( const CanonicalForm & f ) |
---|
736 | { |
---|
737 | if ( f.isZero() ) |
---|
738 | return f; |
---|
739 | else |
---|
740 | return f / content( f ); |
---|
741 | } |
---|
742 | //}}} |
---|
743 | |
---|
744 | CanonicalForm |
---|
745 | gcd ( const CanonicalForm & f, const CanonicalForm & g ) |
---|
746 | { |
---|
747 | bool b = f.isZero(); |
---|
748 | if ( b || g.isZero() ) |
---|
749 | { |
---|
750 | if ( b ) |
---|
751 | return abs( g ); |
---|
752 | else |
---|
753 | return abs( f ); |
---|
754 | } |
---|
755 | if ( f.inPolyDomain() || g.inPolyDomain() ) |
---|
756 | { |
---|
757 | if ( f.mvar() != g.mvar() ) |
---|
758 | { |
---|
759 | if ( f.mvar() > g.mvar() ) |
---|
760 | return cf_content( f, g ); |
---|
761 | else |
---|
762 | return cf_content( g, f ); |
---|
763 | } |
---|
764 | if ( f.inExtension() && getReduce( f.mvar() ) ) |
---|
765 | return 1; |
---|
766 | else |
---|
767 | { |
---|
768 | if ( fdivides( f, g ) ) |
---|
769 | return abs( f ); |
---|
770 | else if ( fdivides( g, f ) ) |
---|
771 | return abs( g ); |
---|
772 | if ( !( getCharacteristic() == 0 && isOn( SW_RATIONAL ) ) ) |
---|
773 | { |
---|
774 | CanonicalForm d; |
---|
775 | #if 1 |
---|
776 | do{ d = gcd_poly( f, g ); } |
---|
777 | while ((!fdivides(d,f)) || (!fdivides(d,g))); |
---|
778 | #else |
---|
779 | while(1) |
---|
780 | { |
---|
781 | d = gcd_poly( f, g ); |
---|
782 | if ((fdivides(d,f)) && (fdivides(d,g))) break; |
---|
783 | printf("g"); fflush(stdout); |
---|
784 | } |
---|
785 | #endif |
---|
786 | return abs( d ); |
---|
787 | } |
---|
788 | else |
---|
789 | { |
---|
790 | CanonicalForm cdF = bCommonDen( f ); |
---|
791 | CanonicalForm cdG = bCommonDen( g ); |
---|
792 | Off( SW_RATIONAL ); |
---|
793 | CanonicalForm l = lcm( cdF, cdG ); |
---|
794 | On( SW_RATIONAL ); |
---|
795 | CanonicalForm F = f * l, G = g * l; |
---|
796 | Off( SW_RATIONAL ); |
---|
797 | do { l = gcd_poly( F, G ); } |
---|
798 | while ((!fdivides(l,F)) || (!fdivides(l,G))); |
---|
799 | On( SW_RATIONAL ); |
---|
800 | return abs( l ); |
---|
801 | } |
---|
802 | } |
---|
803 | } |
---|
804 | if ( f.inBaseDomain() && g.inBaseDomain() ) |
---|
805 | return bgcd( f, g ); |
---|
806 | else |
---|
807 | return 1; |
---|
808 | } |
---|
809 | |
---|
810 | //{{{ CanonicalForm lcm ( const CanonicalForm & f, const CanonicalForm & g ) |
---|
811 | //{{{ docu |
---|
812 | // |
---|
813 | // lcm() - return least common multiple of f and g. |
---|
814 | // |
---|
815 | // The lcm is calculated using the formula lcm(f, g) = f * g / gcd(f, g). |
---|
816 | // |
---|
817 | // Returns zero if one of f or g equals zero. |
---|
818 | // |
---|
819 | //}}} |
---|
820 | CanonicalForm |
---|
821 | lcm ( const CanonicalForm & f, const CanonicalForm & g ) |
---|
822 | { |
---|
823 | if ( f.isZero() || g.isZero() ) |
---|
824 | return 0; |
---|
825 | else |
---|
826 | return ( f / gcd( f, g ) ) * g; |
---|
827 | } |
---|
828 | //}}} |
---|
829 | |
---|
830 | #ifdef HAVE_NTL |
---|
831 | |
---|
832 | static CanonicalForm |
---|
833 | gcd_univar_ntl0( const CanonicalForm & F, const CanonicalForm & G ) |
---|
834 | { |
---|
835 | ZZX F1=convertFacCF2NTLZZX(F); |
---|
836 | ZZX G1=convertFacCF2NTLZZX(G); |
---|
837 | ZZX R=GCD(F1,G1); |
---|
838 | return convertNTLZZX2CF(R,F.mvar()); |
---|
839 | } |
---|
840 | |
---|
841 | static CanonicalForm |
---|
842 | gcd_univar_ntlp( const CanonicalForm & F, const CanonicalForm & G ) |
---|
843 | { |
---|
844 | if (fac_NTL_char!=getCharacteristic()) |
---|
845 | { |
---|
846 | fac_NTL_char=getCharacteristic(); |
---|
847 | #ifdef NTL_ZZ |
---|
848 | ZZ r; |
---|
849 | r=getCharacteristic(); |
---|
850 | ZZ_pContext ccc(r); |
---|
851 | #else |
---|
852 | zz_pContext ccc(getCharacteristic()); |
---|
853 | #endif |
---|
854 | ccc.restore(); |
---|
855 | #ifdef NTL_ZZ |
---|
856 | ZZ_p::init(r); |
---|
857 | #else |
---|
858 | zz_p::init(getCharacteristic()); |
---|
859 | #endif |
---|
860 | } |
---|
861 | #ifdef NTL_ZZ |
---|
862 | ZZ_pX F1=convertFacCF2NTLZZpX(F); |
---|
863 | ZZ_pX G1=convertFacCF2NTLZZpX(G); |
---|
864 | ZZ_pX R=GCD(F1,G1); |
---|
865 | return convertNTLZZpX2CF(R,F.mvar()); |
---|
866 | #else |
---|
867 | zz_pX F1=convertFacCF2NTLzzpX(F); |
---|
868 | zz_pX G1=convertFacCF2NTLzzpX(G); |
---|
869 | zz_pX R=GCD(F1,G1); |
---|
870 | return convertNTLzzpX2CF(R,F.mvar()); |
---|
871 | #endif |
---|
872 | } |
---|
873 | |
---|
874 | #endif |
---|
875 | |
---|
876 | static bool |
---|
877 | gcd_avoid_mtaildegree ( CanonicalForm & f1, CanonicalForm & g1, CanonicalForm & d1 ) |
---|
878 | { |
---|
879 | bool rdy = true; |
---|
880 | int df = f1.taildegree(); |
---|
881 | int dg = g1.taildegree(); |
---|
882 | |
---|
883 | d1 = d1.genOne(); |
---|
884 | if ( dg == 0 ) |
---|
885 | { |
---|
886 | if ( df == 0 ) |
---|
887 | return false; |
---|
888 | else |
---|
889 | { |
---|
890 | if ( f1.degree() == df ) |
---|
891 | d1 = cf_content( g1, LC( f1 ) ); |
---|
892 | else |
---|
893 | { |
---|
894 | f1 /= power( f1.mvar(), df ); |
---|
895 | rdy = false; |
---|
896 | } |
---|
897 | } |
---|
898 | } |
---|
899 | else |
---|
900 | { |
---|
901 | if ( df == 0) |
---|
902 | { |
---|
903 | if ( g1.degree() == dg ) |
---|
904 | d1 = cf_content( f1, LC( g1 ) ); |
---|
905 | else |
---|
906 | { |
---|
907 | g1 /= power( g1.mvar(), dg ); |
---|
908 | rdy = false; |
---|
909 | } |
---|
910 | } |
---|
911 | else |
---|
912 | { |
---|
913 | if ( df > dg ) |
---|
914 | d1 = power( f1.mvar(), dg ); |
---|
915 | else |
---|
916 | d1 = power( f1.mvar(), df ); |
---|
917 | if ( f1.degree() == df ) |
---|
918 | { |
---|
919 | if (g1.degree() == dg) |
---|
920 | d1 *= gcd( LC( f1 ), LC( g1 ) ); |
---|
921 | else |
---|
922 | { |
---|
923 | g1 /= power( g1.mvar(), dg); |
---|
924 | d1 *= cf_content( g1, LC( f1 ) ); |
---|
925 | } |
---|
926 | } |
---|
927 | else |
---|
928 | { |
---|
929 | f1 /= power( f1.mvar(), df ); |
---|
930 | if ( g1.degree() == dg ) |
---|
931 | d1 *= cf_content( f1, LC( g1 ) ); |
---|
932 | else |
---|
933 | { |
---|
934 | g1 /= power( g1.mvar(), dg ); |
---|
935 | rdy = false; |
---|
936 | } |
---|
937 | } |
---|
938 | } |
---|
939 | } |
---|
940 | return rdy; |
---|
941 | } |
---|
942 | |
---|
943 | /* |
---|
944 | * compute positions p1 and pe of optimal variables: |
---|
945 | * pe is used in "ezgcd" and |
---|
946 | * p1 in "gcd_poly1" |
---|
947 | */ |
---|
948 | static |
---|
949 | void optvalues ( const int * df, const int * dg, const int n, int & p1, int &pe ) |
---|
950 | { |
---|
951 | int i, o1, oe; |
---|
952 | if ( df[n] > dg[n] ) |
---|
953 | { |
---|
954 | o1 = df[n]; oe = dg[n]; |
---|
955 | } |
---|
956 | else |
---|
957 | { |
---|
958 | o1 = dg[n]; oe = df[n]; |
---|
959 | } |
---|
960 | i = n-1; |
---|
961 | while ( i > 0 ) |
---|
962 | { |
---|
963 | if ( df[i] != 0 ) |
---|
964 | { |
---|
965 | if ( df[i] > dg[i] ) |
---|
966 | { |
---|
967 | if ( o1 > df[i]) { o1 = df[i]; p1 = i; } |
---|
968 | if ( oe <= dg[i]) { oe = dg[i]; pe = i; } |
---|
969 | } |
---|
970 | else |
---|
971 | { |
---|
972 | if ( o1 > dg[i]) { o1 = dg[i]; p1 = i; } |
---|
973 | if ( oe <= df[i]) { oe = df[i]; pe = i; } |
---|
974 | } |
---|
975 | } |
---|
976 | i--; |
---|
977 | } |
---|
978 | } |
---|
979 | |
---|
980 | /* |
---|
981 | * make some changes of variables, see optvalues |
---|
982 | */ |
---|
983 | static void |
---|
984 | cf_prepgcd( const CanonicalForm & f, const CanonicalForm & g, int & cc, int & p1, int &pe ) |
---|
985 | { |
---|
986 | int i, k, n; |
---|
987 | n = f.level(); |
---|
988 | cc = 0; |
---|
989 | p1 = pe = n; |
---|
990 | if ( n == 1 ) |
---|
991 | return; |
---|
992 | int * degsf = new int[n+1]; |
---|
993 | int * degsg = new int[n+1]; |
---|
994 | for ( i = n; i > 0; i-- ) |
---|
995 | { |
---|
996 | degsf[i] = degsg[i] = 0; |
---|
997 | } |
---|
998 | degsf = degrees( f, degsf ); |
---|
999 | degsg = degrees( g, degsg ); |
---|
1000 | |
---|
1001 | k = 0; |
---|
1002 | for ( i = n-1; i > 0; i-- ) |
---|
1003 | { |
---|
1004 | if ( degsf[i] == 0 ) |
---|
1005 | { |
---|
1006 | if ( degsg[i] != 0 ) |
---|
1007 | { |
---|
1008 | cc = -i; |
---|
1009 | break; |
---|
1010 | } |
---|
1011 | } |
---|
1012 | else |
---|
1013 | { |
---|
1014 | if ( degsg[i] == 0 ) |
---|
1015 | { |
---|
1016 | cc = i; |
---|
1017 | break; |
---|
1018 | } |
---|
1019 | else k++; |
---|
1020 | } |
---|
1021 | } |
---|
1022 | |
---|
1023 | if ( ( cc == 0 ) && ( k != 0 ) ) |
---|
1024 | optvalues( degsf, degsg, n, p1, pe ); |
---|
1025 | if ( ( pe != 1 ) && ( degsf[1] != 0 ) ) |
---|
1026 | pe = -pe; |
---|
1027 | |
---|
1028 | delete [] degsf; |
---|
1029 | delete [] degsg; |
---|
1030 | } |
---|
1031 | |
---|
1032 | |
---|
1033 | static CanonicalForm |
---|
1034 | balance_p ( const CanonicalForm & f, const CanonicalForm & q ) |
---|
1035 | { |
---|
1036 | Variable x = f.mvar(); |
---|
1037 | CanonicalForm result = 0, qh = q / 2; |
---|
1038 | CanonicalForm c; |
---|
1039 | CFIterator i; |
---|
1040 | for ( i = f; i.hasTerms(); i++ ) |
---|
1041 | { |
---|
1042 | c = i.coeff(); |
---|
1043 | if ( c.inCoeffDomain()) |
---|
1044 | { |
---|
1045 | if ( c > qh ) |
---|
1046 | result += power( x, i.exp() ) * (c - q); |
---|
1047 | else |
---|
1048 | result += power( x, i.exp() ) * c; |
---|
1049 | } |
---|
1050 | else |
---|
1051 | result += power( x, i.exp() ) * balance_p(c,q); |
---|
1052 | } |
---|
1053 | return result; |
---|
1054 | } |
---|
1055 | |
---|
1056 | CanonicalForm chinrem_gcd ( const CanonicalForm & FF, const CanonicalForm & GG ) |
---|
1057 | { |
---|
1058 | CanonicalForm f, g, cg, cl, q, Dp, newD, D, newq; |
---|
1059 | int p, i, dp_deg, d_deg;; |
---|
1060 | |
---|
1061 | CanonicalForm cd = bCommonDen( FF ); |
---|
1062 | f=cd*FF; |
---|
1063 | f /=vcontent(f,Variable(1)); |
---|
1064 | //cd = bCommonDen( f ); f *=cd; |
---|
1065 | //f /=vcontent(f,Variable(1)); |
---|
1066 | |
---|
1067 | cd = bCommonDen( GG ); |
---|
1068 | g=cd*GG; |
---|
1069 | g /=vcontent(g,Variable(1)); |
---|
1070 | //cd = bCommonDen( g ); g *=cd; |
---|
1071 | //g /=vcontent(g,Variable(1)); |
---|
1072 | |
---|
1073 | q = 0; |
---|
1074 | i = cf_getNumBigPrimes() - 1; |
---|
1075 | cl = f.lc()* g.lc(); |
---|
1076 | |
---|
1077 | while ( true ) |
---|
1078 | { |
---|
1079 | p = cf_getBigPrime( i ); |
---|
1080 | i--; |
---|
1081 | while ( i >= 0 && mod( cl, p ) == 0 ) |
---|
1082 | { |
---|
1083 | p = cf_getBigPrime( i ); |
---|
1084 | i--; |
---|
1085 | } |
---|
1086 | //printf("try p=%d\n",p); |
---|
1087 | setCharacteristic( p ); |
---|
1088 | Dp = gcd_poly( mapinto( f ), mapinto( g ) ); |
---|
1089 | Dp /=Dp.lc(); |
---|
1090 | setCharacteristic( 0 ); |
---|
1091 | dp_deg=totaldegree(Dp); |
---|
1092 | if ( dp_deg == 0 ) |
---|
1093 | { |
---|
1094 | //printf(" -> 1\n"); |
---|
1095 | return CanonicalForm(1); |
---|
1096 | } |
---|
1097 | if ( q.isZero() ) |
---|
1098 | { |
---|
1099 | D = mapinto( Dp ); |
---|
1100 | d_deg=dp_deg; |
---|
1101 | q = p; |
---|
1102 | } |
---|
1103 | else |
---|
1104 | { |
---|
1105 | if ( dp_deg == d_deg ) |
---|
1106 | { |
---|
1107 | chineseRemainder( D, q, mapinto( Dp ), p, newD, newq ); |
---|
1108 | q = newq; |
---|
1109 | D = newD; |
---|
1110 | } |
---|
1111 | else if ( dp_deg < d_deg ) |
---|
1112 | { |
---|
1113 | //printf(" were all bad, try more\n"); |
---|
1114 | // all previous p's are bad primes |
---|
1115 | q = p; |
---|
1116 | D = mapinto( Dp ); |
---|
1117 | d_deg=dp_deg; |
---|
1118 | } |
---|
1119 | else |
---|
1120 | { |
---|
1121 | //printf(" was bad, try more\n"); |
---|
1122 | } |
---|
1123 | //else dp_deg > d_deg: bad prime |
---|
1124 | } |
---|
1125 | if ( i >= 0 ) |
---|
1126 | { |
---|
1127 | CanonicalForm Dn= Farey(D,q); |
---|
1128 | int is_rat=isOn(SW_RATIONAL); |
---|
1129 | On(SW_RATIONAL); |
---|
1130 | CanonicalForm cd = bCommonDen( Dn ); // we need On(SW_RATIONAL) |
---|
1131 | if (!is_rat) Off(SW_RATIONAL); |
---|
1132 | Dn *=cd; |
---|
1133 | //Dn /=vcontent(Dn,Variable(1)); |
---|
1134 | if ( fdivides( Dn, f ) && fdivides( Dn, g ) ) |
---|
1135 | { |
---|
1136 | //printf(" -> success\n"); |
---|
1137 | return Dn; |
---|
1138 | } |
---|
1139 | //else: try more primes |
---|
1140 | } |
---|
1141 | else |
---|
1142 | { // try other method |
---|
1143 | //printf("try other gcd\n"); |
---|
1144 | Off(SW_USE_CHINREM_GCD); |
---|
1145 | D=gcd_poly( f, g ); |
---|
1146 | On(SW_USE_CHINREM_GCD); |
---|
1147 | return D; |
---|
1148 | } |
---|
1149 | } |
---|
1150 | } |
---|