1 | // emacs edit mode for this file is -*- C++ -*- |
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2 | // $Id: cf_gcd.cc,v 1.3 1996-06-18 12:22:54 stobbe Exp $ |
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3 | |
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4 | /* |
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5 | $Log: not supported by cvs2svn $ |
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6 | Revision 1.2 1996/06/13 08:18:34 stobbe |
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7 | "balance: Now balaces polynomials even if the coefficient sizes are higher |
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8 | than the bound. |
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9 | gcd: Now returns the results with positive leading coefficient. |
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10 | The isOne test is now performed if pi or pi1 is multivariate. |
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11 | " |
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12 | |
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13 | Revision 1.1 1996/06/03 08:32:56 stobbe |
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14 | "gcd_poly: now uses new function gcd_poly_univar0 to compute univariate |
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15 | polynomial gcd's over Z. |
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16 | gcd_poly_univar0: computes univariate polynomial gcd's in characteristic 0 |
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17 | via chinese remaindering. |
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18 | maxnorm: computes the maximum norm of all coefficients of a polynomial. |
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19 | " |
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20 | |
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21 | Revision 1.0 1996/05/17 11:56:37 stobbe |
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22 | Initial revision |
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23 | |
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24 | */ |
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25 | |
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26 | #include "assert.h" |
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27 | #include "cf_defs.h" |
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28 | #include "canonicalform.h" |
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29 | #include "cf_iter.h" |
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30 | #include "cf_reval.h" |
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31 | #include "cf_primes.h" |
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32 | #include "cf_chinese.h" |
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33 | #include "templates/functions.h" |
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34 | |
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35 | static CanonicalForm gcd_poly( const CanonicalForm & f, const CanonicalForm& g, bool modularflag ); |
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36 | |
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37 | |
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38 | static int |
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39 | isqrt ( int a ) |
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40 | { |
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41 | int h, x0, x1 = a; |
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42 | do { |
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43 | x0 = x1; |
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44 | h = x0 * x0 + a - 1; |
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45 | if ( h % (2 * x0) == 0 ) |
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46 | x1 = h / (2 * x0); |
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47 | else |
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48 | x1 = (h - 1) / (2 * x0); |
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49 | } while ( x1 < x0 ); |
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50 | return x1; |
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51 | } |
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52 | |
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53 | static bool |
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54 | gcd_test_one ( const CanonicalForm & f, const CanonicalForm & g ) |
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55 | { |
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56 | int count = 0; |
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57 | // assume polys have same level; |
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58 | CFRandom * sample = CFRandomFactory::generate(); |
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59 | REvaluation e( 2, f.level(), *sample ); |
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60 | delete sample; |
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61 | CanonicalForm lcf = swapvar( LC( f ), Variable(1), f.mvar() ); |
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62 | CanonicalForm lcg = swapvar( LC( g ), Variable(1), f.mvar() ); |
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63 | while ( ( e( lcf ).isZero() || e( lcg ).isZero() ) && count < 100 ) { |
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64 | e.nextpoint(); |
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65 | count++; |
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66 | } |
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67 | if ( count == 100 ) |
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68 | return false; |
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69 | CanonicalForm F=swapvar( f, Variable(1), f.mvar() ); |
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70 | CanonicalForm G=swapvar( g, Variable(1), g.mvar() ); |
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71 | if ( e(F).taildegree() > 0 && e(G).taildegree() > 0 ) |
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72 | return false; |
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73 | return gcd( e( F ), e( G ) ).degree() < 1; |
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74 | } |
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75 | |
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76 | static CanonicalForm |
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77 | maxnorm ( const CanonicalForm & f ) |
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78 | { |
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79 | CanonicalForm m = 0, h; |
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80 | CFIterator i; |
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81 | for ( i = f; i.hasTerms(); i++ ) |
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82 | m = tmax( m, abs( i.coeff() ) ); |
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83 | return m; |
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84 | } |
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85 | |
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86 | static void |
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87 | chinesePoly ( const CanonicalForm & f1, const CanonicalForm & q1, const CanonicalForm & f2, const CanonicalForm & q2, CanonicalForm & f, CanonicalForm & q ) |
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88 | { |
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89 | CFIterator i1 = f1, i2 = f2; |
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90 | CanonicalForm c; |
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91 | Variable x = f1.mvar(); |
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92 | f = 0; |
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93 | while ( i1.hasTerms() && i2.hasTerms() ) { |
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94 | if ( i1.exp() == i2.exp() ) { |
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95 | chineseRemainder( i1.coeff(), q1, i2.coeff(), q2, c, q ); |
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96 | f += power( x, i1.exp() ) * c; |
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97 | i1++; i2++; |
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98 | } |
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99 | else if ( i1.exp() > i2.exp() ) { |
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100 | chineseRemainder( 0, q1, i2.coeff(), q2, c, q ); |
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101 | f += power( x, i2.exp() ) * c; |
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102 | i2++; |
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103 | } |
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104 | else { |
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105 | chineseRemainder( i1.coeff(), q1, 0, q2, c, q ); |
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106 | f += power( x, i1.exp() ) * c; |
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107 | i1++; |
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108 | } |
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109 | } |
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110 | while ( i1.hasTerms() ) { |
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111 | chineseRemainder( i1.coeff(), q1, 0, q2, c, q ); |
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112 | f += power( x, i1.exp() ) * c; |
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113 | i1++; |
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114 | } |
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115 | while ( i2.hasTerms() ) { |
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116 | chineseRemainder( 0, q1, i2.coeff(), q2, c, q ); |
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117 | f += power( x, i2.exp() ) * c; |
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118 | i2++; |
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119 | } |
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120 | } |
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121 | |
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122 | static CanonicalForm |
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123 | balance ( const CanonicalForm & f, const CanonicalForm & q ) |
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124 | { |
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125 | CFIterator i; |
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126 | CanonicalForm result = 0, qh = q / 2; |
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127 | CanonicalForm c; |
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128 | Variable x = f.mvar(); |
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129 | for ( i = f; i.hasTerms(); i++ ) { |
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130 | c = i.coeff() % q; |
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131 | if ( c > qh ) |
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132 | result += power( x, i.exp() ) * (c - q); |
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133 | else |
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134 | result += power( x, i.exp() ) * c; |
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135 | } |
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136 | return result; |
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137 | } |
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138 | |
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139 | CanonicalForm |
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140 | igcd ( const CanonicalForm & f, const CanonicalForm & g ) |
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141 | { |
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142 | CanonicalForm a, b, c, dummy; |
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143 | |
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144 | if ( f.inZ() && g.inZ() && ! isOn( SW_RATIONAL ) ) { |
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145 | if ( f.sign() < 0 ) a = -f; else a = f; |
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146 | if ( g.sign() < 0 ) b = -g; else b = g; |
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147 | while ( ! b.isZero() ) { |
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148 | divrem( a, b, dummy, c ); |
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149 | a = b; |
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150 | b = c; |
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151 | } |
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152 | return a; |
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153 | } |
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154 | else |
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155 | return 1; |
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156 | } |
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157 | |
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158 | static CanonicalForm |
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159 | icontent ( const CanonicalForm & f, const CanonicalForm & c ) |
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160 | { |
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161 | if ( f.inCoeffDomain() ) |
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162 | return gcd( f, c ); |
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163 | else { |
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164 | CanonicalForm g = c; |
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165 | for ( CFIterator i = f; i.hasTerms() && ! g.isOne(); i++ ) |
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166 | g = icontent( i.coeff(), g ); |
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167 | return g; |
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168 | } |
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169 | } |
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170 | |
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171 | CanonicalForm |
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172 | icontent ( const CanonicalForm & f ) |
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173 | { |
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174 | return icontent( f, 0 ); |
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175 | } |
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176 | |
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177 | CanonicalForm |
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178 | iextgcd ( const CanonicalForm & f, const CanonicalForm & g, CanonicalForm & a, CanonicalForm & b ) |
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179 | { |
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180 | CanonicalForm p0 = f, p1 = g; |
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181 | CanonicalForm f0 = 1, f1 = 0, g0 = 0, g1 = 1, q, r; |
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182 | |
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183 | while ( ! p1.isZero() ) { |
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184 | divrem( p0, p1, q, r ); |
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185 | p0 = p1; p1 = r; |
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186 | r = g0 - g1 * q; |
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187 | g0 = g1; g1 = r; |
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188 | r = f0 - f1 * q; |
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189 | f0 = f1; f1 = r; |
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190 | } |
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191 | a = f0; |
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192 | b = g0; |
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193 | if ( p0.sign() < 0 ) { |
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194 | p0 = -p0; |
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195 | a = -a; |
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196 | b = -b; |
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197 | } |
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198 | return p0; |
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199 | } |
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200 | |
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201 | CanonicalForm |
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202 | extgcd ( const CanonicalForm & f, const CanonicalForm & g, CanonicalForm & a, CanonicalForm & b ) |
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203 | { |
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204 | CanonicalForm contf = content( f ); |
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205 | CanonicalForm contg = content( g ); |
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206 | |
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207 | CanonicalForm p0 = f / contf, p1 = g / contg; |
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208 | CanonicalForm f0 = 1, f1 = 0, g0 = 0, g1 = 1, q, r; |
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209 | |
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210 | while ( ! p1.isZero() ) { |
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211 | divrem( p0, p1, q, r ); |
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212 | p0 = p1; p1 = r; |
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213 | r = g0 - g1 * q; |
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214 | g0 = g1; g1 = r; |
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215 | r = f0 - f1 * q; |
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216 | f0 = f1; f1 = r; |
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217 | } |
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218 | CanonicalForm contp0 = content( p0 ); |
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219 | a = f0 / ( contf * contp0 ); |
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220 | b = g0 / ( contg * contp0 ); |
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221 | p0 /= contp0; |
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222 | if ( p0.sign() < 0 ) { |
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223 | p0 = -p0; |
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224 | a = -a; |
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225 | b = -b; |
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226 | } |
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227 | return p0; |
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228 | } |
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229 | |
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230 | static CanonicalForm |
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231 | gcd_poly_univar0( const CanonicalForm & F, const CanonicalForm & G, bool primitive ) |
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232 | { |
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233 | CanonicalForm f, g, c, cg, cl, BB, B, M, q, Dp, newD, D, newq; |
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234 | int p, i, n; |
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235 | |
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236 | if ( primitive ) { |
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237 | f = F; |
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238 | g = G; |
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239 | c = 1; |
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240 | } |
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241 | else { |
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242 | CanonicalForm cF = content( F ), cG = content( G ); |
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243 | f = F / cF; |
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244 | g = G / cG; |
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245 | c = igcd( cF, cG ); |
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246 | } |
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247 | cg = gcd( f.lc(), g.lc() ); |
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248 | cl = ( f.lc() / cg ) * g.lc(); |
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249 | // B = 2 * cg * tmin( |
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250 | // maxnorm(f)*power(CanonicalForm(2),f.degree())*isqrt(f.degree()+1), |
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251 | // maxnorm(g)*power(CanonicalForm(2),g.degree())*isqrt(g.degree()+1) |
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252 | // )+1; |
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253 | M = tmin( maxnorm(f), maxnorm(g) ); |
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254 | BB = power(CanonicalForm(2),tmin(f.degree(),g.degree()))*M; |
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255 | q = 0; |
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256 | i = cf_getNumSmallPrimes() - 1; |
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257 | while ( true ) { |
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258 | B = BB; |
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259 | while ( i >= 0 && q < B ) { |
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260 | p = cf_getSmallPrime( i ); |
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261 | i--; |
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262 | while ( i >= 0 && mod( cl, p ) == 0 ) { |
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263 | p = cf_getSmallPrime( i ); |
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264 | i--; |
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265 | } |
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266 | setCharacteristic( p ); |
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267 | Dp = gcd( mapinto( f ), mapinto( g ) ); |
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268 | Dp = ( Dp / Dp.lc() ) * mapinto( cg ); |
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269 | setCharacteristic( 0 ); |
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270 | if ( Dp.degree() == 0 ) return c; |
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271 | if ( q.isZero() ) { |
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272 | D = mapinto( Dp ); |
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273 | q = p; |
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274 | B = power(CanonicalForm(2),D.degree())*M+1; |
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275 | } |
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276 | else { |
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277 | if ( Dp.degree() == D.degree() ) { |
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278 | chinesePoly( D, q, mapinto( Dp ), p, newD, newq ); |
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279 | q = newq; |
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280 | D = newD; |
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281 | } |
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282 | else if ( Dp.degree() < D.degree() ) { |
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283 | // all previous p's are bad primes |
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284 | q = p; |
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285 | D = mapinto( Dp ); |
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286 | B = power(CanonicalForm(2),D.degree())*M+1; |
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287 | } |
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288 | // else p is a bad prime |
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289 | } |
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290 | } |
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291 | if ( i >= 0 ) { |
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292 | // now balance D mod q |
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293 | D = pp( balance( cg * D, q ) ); |
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294 | if ( divides( D, f ) && divides( D, g ) ) |
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295 | return D * c; |
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296 | else |
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297 | q = 0; |
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298 | } |
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299 | else { |
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300 | return gcd_poly( F, G, false ); |
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301 | } |
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302 | } |
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303 | } |
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304 | |
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305 | |
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306 | static CanonicalForm |
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307 | gcd_poly( const CanonicalForm & f, const CanonicalForm& g, bool modularflag ) |
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308 | { |
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309 | CanonicalForm C, Ci, Ci1, Hi, bi, pi, pi1, pi2; |
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310 | int delta; |
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311 | Variable v = f.mvar(); |
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312 | |
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313 | if ( f.degree( v ) >= g.degree( v ) ) { |
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314 | pi = f; pi1 = g; |
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315 | } |
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316 | else { |
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317 | pi = g; pi1 = f; |
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318 | } |
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319 | Ci = content( pi ); Ci1 = content( pi1 ); |
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320 | C = gcd( Ci, Ci1 ); |
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321 | pi1 = pi1 / Ci1; pi = pi / Ci; |
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322 | if ( pi.isUnivariate() && pi1.isUnivariate() ) { |
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323 | if ( modularflag ) |
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324 | return gcd_poly_univar0( pi, pi1, true ) * C; |
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325 | } |
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326 | else |
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327 | if ( gcd_test_one( pi1, pi ) ) |
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328 | return C; |
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329 | delta = degree( pi, v ) - degree( pi1, v ); |
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330 | Hi = power( LC( pi1, v ), delta ); |
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331 | if ( (delta+1) % 2 ) |
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332 | bi = 1; |
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333 | else |
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334 | bi = -1; |
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335 | while ( degree( pi1, v ) > 0 ) { |
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336 | pi2 = psr( pi, pi1, v ); |
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337 | pi2 = pi2 / bi; |
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338 | pi = pi1; pi1 = pi2; |
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339 | if ( degree( pi1, v ) > 0 ) { |
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340 | delta = degree( pi, v ) - degree( pi1, v ); |
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341 | if ( (delta+1) % 2 ) |
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342 | bi = LC( pi, v ) * power( Hi, delta ); |
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343 | else |
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344 | bi = -LC( pi, v ) * power( Hi, delta ); |
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345 | Hi = power( LC( pi1, v ), delta ) / power( Hi, delta-1 ); |
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346 | } |
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347 | } |
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348 | if ( degree( pi1, v ) == 0 ) |
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349 | return C; |
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350 | else { |
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351 | return C * pp( pi ); |
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352 | } |
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353 | } |
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354 | |
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355 | |
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356 | static CanonicalForm |
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357 | cf_content ( const CanonicalForm & f, const CanonicalForm & g ) |
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358 | { |
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359 | if ( f.inPolyDomain() || ( f.inExtension() && ! getReduce( f.mvar() ) ) ) { |
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360 | CFIterator i = f; |
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361 | CanonicalForm result = g; |
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362 | while ( i.hasTerms() && ! result.isOne() ) { |
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363 | result = gcd( result, i.coeff() ); |
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364 | i++; |
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365 | } |
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366 | return result; |
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367 | } |
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368 | else |
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369 | if ( f.sign() < 0 ) |
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370 | return -f; |
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371 | else |
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372 | return f; |
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373 | } |
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374 | |
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375 | CanonicalForm |
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376 | content ( const CanonicalForm & f ) |
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377 | { |
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378 | return cf_content( f, 0 ); |
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379 | } |
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380 | |
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381 | CanonicalForm |
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382 | content ( const CanonicalForm & f, const Variable & x ) |
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383 | { |
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384 | if ( f.mvar() == x ) |
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385 | return cf_content( f, 0 ); |
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386 | else if ( f.mvar() < x ) |
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387 | return f; |
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388 | else |
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389 | return swapvar( content( swapvar( f, f.mvar(), x ), f.mvar() ), f.mvar(), x ); |
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390 | } |
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391 | |
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392 | CanonicalForm |
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393 | vcontent ( const CanonicalForm & f, const Variable & x ) |
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394 | { |
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395 | if ( f.mvar() <= x ) |
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396 | return content( f, x ); |
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397 | else { |
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398 | CFIterator i; |
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399 | CanonicalForm d = 0; |
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400 | for ( i = f; i.hasTerms() && ! d.isOne(); i++ ) |
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401 | d = gcd( d, vcontent( i.coeff(), x ) ); |
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402 | return d; |
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403 | } |
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404 | } |
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405 | |
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406 | CanonicalForm |
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407 | pp ( const CanonicalForm & f ) |
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408 | { |
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409 | if ( f.isZero() ) |
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410 | return f; |
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411 | else |
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412 | return f / content( f ); |
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413 | } |
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414 | |
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415 | CanonicalForm |
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416 | gcd ( const CanonicalForm & f, const CanonicalForm & g ) |
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417 | { |
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418 | if ( f.isZero() ) |
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419 | if ( g.lc().sign() < 0 ) |
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420 | return -g; |
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421 | else |
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422 | return g; |
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423 | else if ( g.isZero() ) |
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424 | if ( f.lc().sign() < 0 ) |
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425 | return -f; |
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426 | else |
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427 | return f; |
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428 | else if ( f.inBaseDomain() ) |
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429 | if ( g.inBaseDomain() ) |
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430 | return igcd( f, g ); |
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431 | else |
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432 | return cf_content( g, f ); |
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433 | else if ( g.inBaseDomain() ) |
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434 | return cf_content( f, g ); |
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435 | else if ( f.mvar() == g.mvar() ) |
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436 | if ( f.inExtension() && getReduce( f.mvar() ) ) |
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437 | return 1; |
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438 | else { |
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439 | if ( divides( f, g ) ) |
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440 | if ( f.lc().sign() < 0 ) |
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441 | return -f; |
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442 | else |
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443 | return f; |
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444 | else if ( divides( g, f ) ) |
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445 | if ( g.lc().sign() < 0 ) |
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446 | return -g; |
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447 | else |
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448 | return g; |
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449 | if ( getCharacteristic() == 0 && isOn( SW_RATIONAL ) ) { |
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450 | Off( SW_RATIONAL ); |
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451 | CanonicalForm l = lcm( common_den( f ), common_den( g ) ); |
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452 | On( SW_RATIONAL ); |
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453 | CanonicalForm F = f * l, G = g * l; |
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454 | Off( SW_RATIONAL ); |
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455 | l = gcd_poly( F, G, true ); |
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456 | On( SW_RATIONAL ); |
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457 | if ( l.lc().sign() < 0 ) |
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458 | return -l; |
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459 | else |
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460 | return l; |
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461 | } |
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462 | else { |
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463 | CanonicalForm d = gcd_poly( f, g, getCharacteristic()==0 ); |
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464 | if ( d.lc().sign() < 0 ) |
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465 | return -d; |
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466 | else |
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467 | return d; |
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468 | } |
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469 | } |
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470 | else if ( f.mvar() > g.mvar() ) |
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471 | return cf_content( f, g ); |
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472 | else |
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473 | return cf_content( g, f ); |
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474 | } |
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475 | |
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476 | CanonicalForm |
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477 | lcm ( const CanonicalForm & f, const CanonicalForm & g ) |
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478 | { |
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479 | return ( f / gcd( f, g ) ) * g; |
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480 | } |
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