1 | /* emacs edit mode for this file is -*- C++ -*- */ |
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2 | /* $Id: cf_gcd.cc,v 1.17 1998-02-02 08:58:43 schmidt 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 "cf_map.h" |
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16 | #include "sm_sparsemod.h" |
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17 | #include "fac_util.h" |
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18 | #ifdef macintosh |
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19 | #include <:templates:ftmpl_functions.h> |
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20 | #else |
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21 | #include "templates/ftmpl_functions.h" |
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22 | #endif |
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23 | |
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24 | static CanonicalForm gcd_poly( const CanonicalForm & f, const CanonicalForm& g, bool modularflag ); |
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25 | |
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26 | bool |
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27 | gcd_test_one ( const CanonicalForm & f, const CanonicalForm & g, bool swap ) |
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28 | { |
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29 | int count = 0; |
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30 | // assume polys have same level; |
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31 | CFRandom * sample = CFRandomFactory::generate(); |
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32 | REvaluation e( 2, tmax( f.level(), g.level() ), *sample ); |
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33 | delete sample; |
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34 | CanonicalForm lcf, lcg; |
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35 | if ( swap ) { |
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36 | lcf = swapvar( LC( f ), Variable(1), f.mvar() ); |
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37 | lcg = swapvar( LC( g ), Variable(1), f.mvar() ); |
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38 | } |
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39 | else { |
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40 | lcf = LC( f, Variable(1) ); |
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41 | lcg = LC( g, Variable(1) ); |
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42 | } |
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43 | while ( ( e( lcf ).isZero() || e( lcg ).isZero() ) && count < 100 ) { |
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44 | e.nextpoint(); |
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45 | count++; |
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46 | } |
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47 | if ( count == 100 ) |
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48 | return false; |
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49 | CanonicalForm F, G; |
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50 | if ( swap ) { |
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51 | F=swapvar( f, Variable(1), f.mvar() ); |
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52 | G=swapvar( g, Variable(1), g.mvar() ); |
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53 | } |
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54 | else { |
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55 | F = f; |
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56 | G = g; |
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57 | } |
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58 | if ( e(F).taildegree() > 0 && e(G).taildegree() > 0 ) |
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59 | return false; |
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60 | return gcd( e( F ), e( G ) ).degree() < 1; |
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61 | } |
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62 | |
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63 | //{{{ static CanonicalForm maxnorm ( const CanonicalForm & f ) |
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64 | //{{{ docu |
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65 | // |
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66 | // maxnorm() - return the maximum of the absolute values of all |
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67 | // coefficients of f. |
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68 | // |
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69 | // The absolute value and the maximum are calculated with respect |
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70 | // to operator < on canonical forms which is most meaningful for |
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71 | // rational numbers and integers. |
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72 | // |
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73 | // Used by gcd_poly_univar0(). |
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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; |
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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 | |
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87 | //{{{ static CanonicalForm balance ( const CanonicalForm & f, const CanonicalForm & q ) |
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88 | //{{{ docu |
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89 | // |
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90 | // balance() - map f from positive to symmetric representation |
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91 | // mod q. |
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92 | // |
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93 | // This makes sense for univariate polynomials over Z only. |
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94 | // q should be an integer. |
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95 | // |
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96 | // Used by gcd_poly_univar0(). |
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97 | // |
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98 | //}}} |
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99 | static CanonicalForm |
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100 | balance ( const CanonicalForm & f, const CanonicalForm & q ) |
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101 | { |
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102 | Variable x = f.mvar(); |
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103 | CanonicalForm result = 0, qh = q / 2; |
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104 | CanonicalForm c; |
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105 | CFIterator i; |
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106 | for ( i = f; i.hasTerms(); i++ ) { |
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107 | c = mod( i.coeff(), q ); |
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108 | if ( c > qh ) |
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109 | result += power( x, i.exp() ) * (c - q); |
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110 | else |
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111 | result += power( x, i.exp() ) * c; |
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112 | } |
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113 | return result; |
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114 | } |
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115 | //}}} |
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116 | |
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117 | //{{{ static CanonicalForm icontent ( const CanonicalForm & f, const CanonicalForm & c ) |
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118 | //{{{ docu |
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119 | // |
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120 | // icontent() - return gcd of c and all coefficients of f which |
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121 | // are in a coefficient domain. |
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122 | // |
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123 | // Used by icontent(). |
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124 | // |
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125 | //}}} |
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126 | static CanonicalForm |
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127 | icontent ( const CanonicalForm & f, const CanonicalForm & c ) |
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128 | { |
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129 | if ( f.inCoeffDomain() ) |
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130 | return gcd( f, c ); |
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131 | else { |
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132 | CanonicalForm g = c; |
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133 | for ( CFIterator i = f; i.hasTerms() && ! g.isOne(); i++ ) |
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134 | g = icontent( i.coeff(), g ); |
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135 | return g; |
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136 | } |
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137 | } |
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138 | //}}} |
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139 | |
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140 | //{{{ CanonicalForm icontent ( const CanonicalForm & f ) |
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141 | //{{{ docu |
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142 | // |
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143 | // icontent() - return gcd over all coefficients of f which are |
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144 | // in a coefficient domain. |
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145 | // |
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146 | //}}} |
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147 | CanonicalForm |
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148 | icontent ( const CanonicalForm & f ) |
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149 | { |
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150 | return icontent( f, 0 ); |
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151 | } |
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152 | //}}} |
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153 | |
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154 | //{{{ CanonicalForm extgcd ( const CanonicalForm & f, const CanonicalForm & g, CanonicalForm & a, CanonicalForm & b ) |
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155 | //{{{ docu |
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156 | // |
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157 | // extgcd() - returns polynomial extended gcd of f and g. |
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158 | // |
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159 | // Returns gcd(f, g) and a and b sucht that f*a+g*b=gcd(f, g). |
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160 | // The gcd is calculated using an extended euclidean polynomial |
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161 | // remainder sequence, so f and g should be polynomials over an |
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162 | // euclidean domain. Normalizes result. |
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163 | // |
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164 | // Note: be sure that f and g have the same level! |
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165 | // |
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166 | //}}} |
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167 | CanonicalForm |
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168 | extgcd ( const CanonicalForm & f, const CanonicalForm & g, CanonicalForm & a, CanonicalForm & b ) |
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169 | { |
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170 | CanonicalForm contf = content( f ); |
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171 | CanonicalForm contg = content( g ); |
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172 | |
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173 | CanonicalForm p0 = f / contf, p1 = g / contg; |
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174 | CanonicalForm f0 = 1, f1 = 0, g0 = 0, g1 = 1, q, r; |
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175 | |
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176 | while ( ! p1.isZero() ) { |
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177 | divrem( p0, p1, q, r ); |
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178 | p0 = p1; p1 = r; |
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179 | r = g0 - g1 * q; |
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180 | g0 = g1; g1 = r; |
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181 | r = f0 - f1 * q; |
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182 | f0 = f1; f1 = r; |
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183 | } |
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184 | CanonicalForm contp0 = content( p0 ); |
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185 | a = f0 / ( contf * contp0 ); |
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186 | b = g0 / ( contg * contp0 ); |
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187 | p0 /= contp0; |
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188 | if ( p0.sign() < 0 ) { |
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189 | p0 = -p0; |
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190 | a = -a; |
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191 | b = -b; |
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192 | } |
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193 | return p0; |
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194 | } |
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195 | //}}} |
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196 | |
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197 | static CanonicalForm |
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198 | gcd_poly_univar0( const CanonicalForm & F, const CanonicalForm & G, bool primitive ) |
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199 | { |
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200 | CanonicalForm f, g, c, cg, cl, BB, B, M, q, Dp, newD, D, newq; |
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201 | int p, i, n; |
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202 | |
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203 | if ( primitive ) { |
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204 | f = F; |
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205 | g = G; |
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206 | c = 1; |
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207 | } |
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208 | else { |
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209 | CanonicalForm cF = content( F ), cG = content( G ); |
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210 | f = F / cF; |
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211 | g = G / cG; |
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212 | c = bgcd( cF, cG ); |
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213 | } |
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214 | cg = gcd( f.lc(), g.lc() ); |
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215 | cl = ( f.lc() / cg ) * g.lc(); |
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216 | // B = 2 * cg * tmin( |
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217 | // maxnorm(f)*power(CanonicalForm(2),f.degree())*isqrt(f.degree()+1), |
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218 | // maxnorm(g)*power(CanonicalForm(2),g.degree())*isqrt(g.degree()+1) |
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219 | // )+1; |
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220 | M = tmin( maxnorm(f), maxnorm(g) ); |
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221 | BB = power(CanonicalForm(2),tmin(f.degree(),g.degree()))*M; |
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222 | q = 0; |
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223 | i = cf_getNumSmallPrimes() - 1; |
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224 | while ( true ) { |
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225 | B = BB; |
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226 | while ( i >= 0 && q < B ) { |
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227 | p = cf_getSmallPrime( i ); |
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228 | i--; |
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229 | while ( i >= 0 && mod( cl, p ) == 0 ) { |
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230 | p = cf_getSmallPrime( i ); |
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231 | i--; |
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232 | } |
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233 | setCharacteristic( p ); |
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234 | Dp = gcd( mapinto( f ), mapinto( g ) ); |
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235 | Dp = ( Dp / Dp.lc() ) * mapinto( cg ); |
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236 | setCharacteristic( 0 ); |
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237 | if ( Dp.degree() == 0 ) return c; |
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238 | if ( q.isZero() ) { |
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239 | D = mapinto( Dp ); |
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240 | q = p; |
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241 | B = power(CanonicalForm(2),D.degree())*M+1; |
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242 | } |
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243 | else { |
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244 | if ( Dp.degree() == D.degree() ) { |
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245 | chineseRemainder( D, q, mapinto( Dp ), p, newD, newq ); |
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246 | q = newq; |
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247 | D = newD; |
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248 | } |
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249 | else if ( Dp.degree() < D.degree() ) { |
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250 | // all previous p's are bad primes |
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251 | q = p; |
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252 | D = mapinto( Dp ); |
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253 | B = power(CanonicalForm(2),D.degree())*M+1; |
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254 | } |
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255 | // else p is a bad prime |
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256 | } |
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257 | } |
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258 | if ( i >= 0 ) { |
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259 | // now balance D mod q |
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260 | D = pp( balance( D, q ) ); |
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261 | if ( divides( D, f ) && divides( D, g ) ) |
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262 | return D * c; |
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263 | else |
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264 | q = 0; |
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265 | } |
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266 | else { |
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267 | return gcd_poly( F, G, false ); |
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268 | } |
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269 | DEBOUTLN( cerr, "another try ..." ); |
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270 | } |
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271 | } |
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272 | |
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273 | static CanonicalForm |
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274 | gcd_poly1( const CanonicalForm & f, const CanonicalForm & g, bool modularflag ) |
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275 | { |
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276 | CanonicalForm C, Ci, Ci1, Hi, bi, pi, pi1, pi2; |
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277 | int delta; |
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278 | Variable v = f.mvar(); |
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279 | |
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280 | if ( f.degree( v ) >= g.degree( v ) ) { |
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281 | pi = f; pi1 = g; |
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282 | } |
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283 | else { |
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284 | pi = g; pi1 = f; |
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285 | } |
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286 | Ci = content( pi ); Ci1 = content( pi1 ); |
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287 | C = gcd( Ci, Ci1 ); |
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288 | pi1 = pi1 / Ci1; pi = pi / Ci; |
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289 | if ( pi.isUnivariate() && pi1.isUnivariate() ) { |
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290 | if ( modularflag ) |
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291 | return gcd_poly_univar0( pi, pi1, true ) * C; |
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292 | } |
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293 | else |
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294 | if ( gcd_test_one( pi1, pi, true ) ) |
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295 | return C; |
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296 | delta = degree( pi, v ) - degree( pi1, v ); |
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297 | Hi = power( LC( pi1, v ), delta ); |
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298 | if ( (delta+1) % 2 ) |
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299 | bi = 1; |
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300 | else |
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301 | bi = -1; |
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302 | while ( degree( pi1, v ) > 0 ) { |
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303 | pi2 = psr( pi, pi1, v ); |
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304 | pi2 = pi2 / bi; |
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305 | pi = pi1; pi1 = pi2; |
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306 | if ( degree( pi1, v ) > 0 ) { |
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307 | delta = degree( pi, v ) - degree( pi1, v ); |
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308 | if ( (delta+1) % 2 ) |
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309 | bi = LC( pi, v ) * power( Hi, delta ); |
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310 | else |
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311 | bi = -LC( pi, v ) * power( Hi, delta ); |
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312 | Hi = power( LC( pi1, v ), delta ) / power( Hi, delta-1 ); |
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313 | } |
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314 | } |
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315 | if ( degree( pi1, v ) == 0 ) |
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316 | return C; |
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317 | else { |
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318 | return C * pp( pi ); |
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319 | } |
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320 | } |
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321 | |
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322 | //{{{ static CanonicalForm gcd_poly ( const CanonicalForm & f, const CanonicalForm & g, bool modularflag ) |
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323 | //{{{ docu |
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324 | // |
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325 | // gcd_poly() - calculate polynomial gcd. |
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326 | // |
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327 | // This is the dispatcher for polynomial gcd calculation. We call either |
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328 | // ezgcd(), sparsemod() or gcd_poly1() in dependecy on the current |
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329 | // characteristic and settings of SW_USE_EZGCD and SW_USE_SPARSEMOD, resp. |
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330 | // |
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331 | // modularflag is reached down to gcd_poly1() without change in case of |
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332 | // zero characteristic. |
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333 | // |
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334 | // Used by gcd() and gcd_poly_univar0(). |
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335 | // |
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336 | //}}} |
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337 | static CanonicalForm |
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338 | gcd_poly ( const CanonicalForm & f, const CanonicalForm & g, bool modularflag ) |
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339 | { |
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340 | if ( getCharacteristic() != 0 ) { |
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341 | return gcd_poly1( f, g, false ); |
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342 | } |
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343 | else if ( isOn( SW_USE_EZGCD ) && ! ( f.isUnivariate() && g.isUnivariate() ) ) { |
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344 | CFMap M, N; |
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345 | compress( f, g, M, N ); |
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346 | return N( ezgcd( M(f), M(g) ) ); |
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347 | } |
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348 | else if ( isOn( SW_USE_SPARSEMOD ) && ! ( f.isUnivariate() && g.isUnivariate() ) ) { |
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349 | return sparsemod( f, g ); |
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350 | } |
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351 | else { |
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352 | return gcd_poly1( f, g, modularflag ); |
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353 | } |
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354 | } |
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355 | //}}} |
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356 | |
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357 | //{{{ static CanonicalForm cf_content ( const CanonicalForm & f, const CanonicalForm & g ) |
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358 | //{{{ docu |
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359 | // |
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360 | // cf_content() - return gcd(g, content(f)). |
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361 | // |
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362 | // content(f) is calculated with respect to f's main variable. |
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363 | // |
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364 | // Used by gcd(), content(), content( CF, Variable ). |
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365 | // |
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366 | //}}} |
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367 | static CanonicalForm |
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368 | cf_content ( const CanonicalForm & f, const CanonicalForm & g ) |
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369 | { |
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370 | if ( f.inPolyDomain() || ( f.inExtension() && ! getReduce( f.mvar() ) ) ) { |
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371 | CFIterator i = f; |
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372 | CanonicalForm result = g; |
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373 | while ( i.hasTerms() && ! result.isOne() ) { |
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374 | result = gcd( result, i.coeff() ); |
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375 | i++; |
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376 | } |
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377 | return result; |
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378 | } |
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379 | else |
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380 | if ( f.sign() < 0 ) |
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381 | return -f; |
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382 | else |
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383 | return f; |
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384 | } |
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385 | //}}} |
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386 | |
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387 | //{{{ CanonicalForm content ( const CanonicalForm & f ) |
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388 | //{{{ docu |
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389 | // |
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390 | // content() - return content(f) with respect to main variable. |
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391 | // |
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392 | // Normalizes result. |
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393 | // |
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394 | //}}} |
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395 | CanonicalForm |
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396 | content ( const CanonicalForm & f ) |
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397 | { |
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398 | return cf_content( f, 0 ); |
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399 | } |
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400 | //}}} |
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401 | |
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402 | //{{{ CanonicalForm content ( const CanonicalForm & f, const Variable & x ) |
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403 | //{{{ docu |
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404 | // |
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405 | // content() - return content(f) with respect to x. |
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406 | // |
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407 | // x should be a polynomial variable. |
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408 | // |
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409 | //}}} |
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410 | CanonicalForm |
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411 | content ( const CanonicalForm & f, const Variable & x ) |
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412 | { |
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413 | ASSERT( x.level() > 0, "cannot calculate content with respect to algebraic variable" ); |
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414 | Variable y = f.mvar(); |
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415 | |
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416 | if ( y == x ) |
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417 | return cf_content( f, 0 ); |
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418 | else if ( y < x ) |
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419 | return f; |
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420 | else |
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421 | return swapvar( content( swapvar( f, y, x ), y ), y, x ); |
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422 | } |
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423 | //}}} |
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424 | |
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425 | //{{{ CanonicalForm vcontent ( const CanonicalForm & f, const Variable & x ) |
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426 | //{{{ docu |
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427 | // |
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428 | // vcontent() - return content of f with repect to variables >= x. |
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429 | // |
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430 | // The content is recursively calculated over all coefficients in |
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431 | // f having level less than x. x should be a polynomial |
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432 | // variable. |
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433 | // |
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434 | //}}} |
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435 | CanonicalForm |
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436 | vcontent ( const CanonicalForm & f, const Variable & x ) |
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437 | { |
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438 | ASSERT( x.level() > 0, "cannot calculate vcontent with respect to algebraic variable" ); |
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439 | |
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440 | if ( f.mvar() <= x ) |
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441 | return content( f, x ); |
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442 | else { |
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443 | CFIterator i; |
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444 | CanonicalForm d = 0; |
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445 | for ( i = f; i.hasTerms() && ! d.isOne(); i++ ) |
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446 | d = gcd( d, vcontent( i.coeff(), x ) ); |
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447 | return d; |
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448 | } |
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449 | } |
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450 | //}}} |
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451 | |
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452 | //{{{ CanonicalForm pp ( const CanonicalForm & f ) |
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453 | //{{{ docu |
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454 | // |
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455 | // pp() - return primitive part of f. |
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456 | // |
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457 | // Returns zero if f equals zero, otherwise f / content(f). |
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458 | // |
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459 | //}}} |
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460 | CanonicalForm |
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461 | pp ( const CanonicalForm & f ) |
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462 | { |
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463 | if ( f.isZero() ) |
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464 | return f; |
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465 | else |
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466 | return f / content( f ); |
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467 | } |
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468 | //}}} |
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469 | |
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470 | CanonicalForm |
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471 | gcd ( const CanonicalForm & f, const CanonicalForm & g ) |
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472 | { |
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473 | if ( f.isZero() ) |
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474 | if ( g.lc().sign() < 0 ) |
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475 | return -g; |
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476 | else |
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477 | return g; |
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478 | else if ( g.isZero() ) |
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479 | if ( f.lc().sign() < 0 ) |
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480 | return -f; |
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481 | else |
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482 | return f; |
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483 | else if ( f.inBaseDomain() ) |
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484 | if ( g.inBaseDomain() ) |
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485 | return bgcd( f, g ); |
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486 | else |
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487 | return cf_content( g, f ); |
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488 | else if ( g.inBaseDomain() ) |
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489 | return cf_content( f, g ); |
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490 | else if ( f.mvar() == g.mvar() ) |
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491 | if ( f.inExtension() && getReduce( f.mvar() ) ) |
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492 | return 1; |
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493 | else { |
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494 | if ( divides( f, g ) ) |
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495 | if ( f.lc().sign() < 0 ) |
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496 | return -f; |
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497 | else |
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498 | return f; |
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499 | else if ( divides( g, f ) ) |
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500 | if ( g.lc().sign() < 0 ) |
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501 | return -g; |
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502 | else |
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503 | return g; |
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504 | if ( getCharacteristic() == 0 && isOn( SW_RATIONAL ) ) { |
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505 | CanonicalForm cdF = common_den( f ); |
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506 | CanonicalForm cdG = common_den( g ); |
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507 | Off( SW_RATIONAL ); |
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508 | CanonicalForm l = lcm( cdF, cdG ); |
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509 | On( SW_RATIONAL ); |
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510 | CanonicalForm F = f * l, G = g * l; |
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511 | Off( SW_RATIONAL ); |
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512 | l = gcd_poly( F, G, true ); |
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513 | On( SW_RATIONAL ); |
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514 | if ( l.lc().sign() < 0 ) |
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515 | return -l; |
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516 | else |
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517 | return l; |
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518 | } |
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519 | else { |
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520 | CanonicalForm d = gcd_poly( f, g, getCharacteristic()==0 ); |
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521 | if ( d.lc().sign() < 0 ) |
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522 | return -d; |
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523 | else |
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524 | return d; |
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525 | } |
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526 | } |
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527 | else if ( f.mvar() > g.mvar() ) |
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528 | return cf_content( f, g ); |
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529 | else |
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530 | return cf_content( g, f ); |
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531 | } |
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532 | |
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533 | //{{{ CanonicalForm lcm ( const CanonicalForm & f, const CanonicalForm & g ) |
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534 | //{{{ docu |
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535 | // |
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536 | // lcm() - return least common multiple of f and g. |
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537 | // |
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538 | // The lcm is calculated using the formula lcm(f, g) = f * g / gcd(f, g). |
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539 | // |
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540 | // Returns zero if one of f or g equals zero. |
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541 | // |
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542 | //}}} |
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543 | CanonicalForm |
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544 | lcm ( const CanonicalForm & f, const CanonicalForm & g ) |
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545 | { |
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546 | if ( f.isZero() || g.isZero() ) |
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547 | return f; |
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548 | else |
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549 | return ( f / gcd( f, g ) ) * g; |
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550 | } |
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551 | //}}} |
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