1 | /* emacs edit mode for this file is -*- C++ -*- */ |
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2 | /* $Id: cf_chinese.cc,v 1.10 2005-02-08 10:28:46 Singular Exp $ */ |
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3 | |
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4 | //{{{ docu |
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5 | // |
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6 | // cf_chinese.cc - algorithms for chinese remaindering. |
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7 | // |
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8 | // Used by: cf_gcd.cc, cf_linsys.cc, sm_util.cc |
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9 | // |
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10 | // Header file: cf_algorithm.h |
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11 | // |
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12 | //}}} |
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13 | |
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14 | #include <config.h> |
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15 | |
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16 | #include "assert.h" |
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17 | #include "debug.h" |
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18 | |
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19 | #include "canonicalform.h" |
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20 | |
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21 | //{{{ void chineseRemainder ( const CanonicalForm & x1, const CanonicalForm & q1, const CanonicalForm & x2, const CanonicalForm & q2, CanonicalForm & xnew, CanonicalForm & qnew ) |
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22 | //{{{ docu |
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23 | // |
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24 | // chineseRemainder - integer chinese remaindering. |
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25 | // |
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26 | // Calculate xnew such that xnew=x1 (mod q1) and xnew=x2 (mod q2) |
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27 | // and qnew = q1*q2. q1 and q2 should be positive integers, |
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28 | // pairwise prime, x1 and x2 should be polynomials with integer |
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29 | // coefficients. If x1 and x2 are polynomials with positive |
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30 | // coefficients, the result is guaranteed to have positive |
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31 | // coefficients, too. |
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32 | // |
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33 | // Note: This algorithm is optimized for the case q1>>q2. |
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34 | // |
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35 | // This is a standard algorithm. See, for example, |
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36 | // Geddes/Czapor/Labahn - 'Algorithms for Computer Algebra', |
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37 | // par. 5.6 and 5.8, or the article of M. Lauer - 'Computing by |
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38 | // Homomorphic Images' in B. Buchberger - 'Computer Algebra - |
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39 | // Symbolic and Algebraic Computation'. |
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40 | // |
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41 | // Note: Be sure you are calculating in Z, and not in Q! |
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42 | // |
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43 | //}}} |
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44 | void |
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45 | chineseRemainder ( const CanonicalForm & x1, const CanonicalForm & q1, const CanonicalForm & x2, const CanonicalForm & q2, CanonicalForm & xnew, CanonicalForm & qnew ) |
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46 | { |
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47 | DEBINCLEVEL( cerr, "chineseRemainder" ); |
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48 | |
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49 | DEBOUTLN( cerr, "log(q1) = " << q1.ilog2() ); |
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50 | DEBOUTLN( cerr, "log(q2) = " << q2.ilog2() ); |
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51 | |
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52 | // We calculate xnew as follows: |
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53 | // xnew = v1 + v2 * q1 |
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54 | // where |
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55 | // v1 = x1 (mod q1) |
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56 | // v2 = (x2-v1)/q1 (mod q2) (*) |
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57 | // |
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58 | // We do one extra test to check whether x2-v1 vanishes (mod |
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59 | // q2) in (*) since it is not costly and may save us |
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60 | // from calculating the inverse of q1 (mod q2). |
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61 | // |
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62 | // u: v1 (mod q2) |
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63 | // d: x2-v1 (mod q2) |
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64 | // s: 1/q1 (mod q2) |
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65 | // |
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66 | CanonicalForm v2, v1; |
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67 | CanonicalForm u, d, s, dummy; |
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68 | |
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69 | v1 = mod( x1, q1 ); |
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70 | u = mod( v1, q2 ); |
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71 | d = mod( x2-u, q2 ); |
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72 | if ( d.isZero() ) { |
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73 | xnew = v1; |
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74 | qnew = q1 * q2; |
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75 | DEBDECLEVEL( cerr, "chineseRemainder" ); |
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76 | return; |
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77 | } |
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78 | (void)bextgcd( q1, q2, s, dummy ); |
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79 | v2 = mod( d*s, q2 ); |
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80 | xnew = v1 + v2*q1; |
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81 | |
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82 | // After all, calculate new modulus. It is important that |
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83 | // this is done at the very end of the algorithm, since q1 |
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84 | // and qnew may refer to the same object (same is true for x1 |
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85 | // and xnew). |
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86 | qnew = q1 * q2; |
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87 | |
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88 | DEBDECLEVEL( cerr, "chineseRemainder" ); |
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89 | } |
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90 | //}}} |
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91 | |
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92 | //{{{ void chineseRemainder ( const CFArray & x, const CFArray & q, CanonicalForm & xnew, CanonicalForm & qnew ) |
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93 | //{{{ docu |
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94 | // |
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95 | // chineseRemainder - integer chinese remaindering. |
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96 | // |
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97 | // Calculate xnew such that xnew=x[i] (mod q[i]) and qnew is the |
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98 | // product of all q[i]. q[i] should be positive integers, |
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99 | // pairwise prime. x[i] should be polynomials with integer |
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100 | // coefficients. If all coefficients of all x[i] are positive |
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101 | // integers, the result is guaranteed to have positive |
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102 | // coefficients, too. |
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103 | // |
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104 | // This is a standard algorithm, too, except for the fact that we |
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105 | // use a divide-and-conquer method instead of a linear approach |
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106 | // to calculate the remainder. |
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107 | // |
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108 | // Note: Be sure you are calculating in Z, and not in Q! |
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109 | // |
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110 | //}}} |
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111 | void |
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112 | chineseRemainder ( const CFArray & x, const CFArray & q, CanonicalForm & xnew, CanonicalForm & qnew ) |
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113 | { |
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114 | DEBINCLEVEL( cerr, "chineseRemainder( ... CFArray ... )" ); |
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115 | |
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116 | ASSERT( x.min() == q.min() && x.size() == q.size(), "incompatible arrays" ); |
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117 | CFArray X(x), Q(q); |
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118 | int i, j, n = x.size(), start = x.min(); |
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119 | |
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120 | DEBOUTLN( cerr, "array size = " << n ); |
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121 | |
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122 | while ( n != 1 ) { |
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123 | i = j = start; |
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124 | while ( i < start + n - 1 ) { |
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125 | // This is a little bit dangerous: X[i] and X[j] (and |
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126 | // Q[i] and Q[j]) may refer to the same object. But |
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127 | // xnew and qnew in the above function are modified |
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128 | // at the very end of the function, so we do not |
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129 | // modify x1 and q1, resp., by accident. |
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130 | chineseRemainder( X[i], Q[i], X[i+1], Q[i+1], X[j], Q[j] ); |
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131 | i += 2; |
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132 | j++; |
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133 | } |
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134 | |
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135 | if ( n & 1 ) { |
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136 | X[j] = X[i]; |
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137 | Q[j] = Q[i]; |
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138 | } |
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139 | // Maybe we would get some memory back at this point if |
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140 | // we would set X[j+1, ..., n] and Q[j+1, ..., n] to zero |
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141 | // at this point? |
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142 | |
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143 | n = ( n + 1) / 2; |
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144 | } |
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145 | xnew = X[start]; |
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146 | qnew = Q[q.min()]; |
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147 | |
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148 | DEBDECLEVEL( cerr, "chineseRemainder( ... CFArray ... )" ); |
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149 | } |
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150 | //}}} |
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