[493c477] | 1 | /* emacs edit mode for this file is -*- C++ -*- */ |
---|
[341696] | 2 | /* $Id$ */ |
---|
[66e0d2] | 3 | |
---|
| 4 | //{{{ docu |
---|
| 5 | // |
---|
| 6 | // cf_chinese.cc - algorithms for chinese remaindering. |
---|
| 7 | // |
---|
| 8 | // Used by: cf_gcd.cc, cf_linsys.cc, sm_util.cc |
---|
| 9 | // |
---|
[030c681] | 10 | // Header file: cf_algorithm.h |
---|
| 11 | // |
---|
[66e0d2] | 12 | //}}} |
---|
[2dd068] | 13 | |
---|
[cc1367] | 14 | #include <config.h> |
---|
| 15 | |
---|
[2dd068] | 16 | #include "assert.h" |
---|
[d57f19] | 17 | #include "debug.h" |
---|
[cc1367] | 18 | |
---|
[2dd068] | 19 | #include "canonicalform.h" |
---|
[6f62c3] | 20 | #include "cf_iter.h" |
---|
| 21 | |
---|
[2dd068] | 22 | |
---|
[c2b784] | 23 | //{{{ void chineseRemainder ( const CanonicalForm & x1, const CanonicalForm & q1, const CanonicalForm & x2, const CanonicalForm & q2, CanonicalForm & xnew, CanonicalForm & qnew ) |
---|
[66e0d2] | 24 | //{{{ docu |
---|
| 25 | // |
---|
[c12372] | 26 | // chineseRemainder - integer chinese remaindering. |
---|
[66e0d2] | 27 | // |
---|
[c12372] | 28 | // Calculate xnew such that xnew=x1 (mod q1) and xnew=x2 (mod q2) |
---|
| 29 | // and qnew = q1*q2. q1 and q2 should be positive integers, |
---|
| 30 | // pairwise prime, x1 and x2 should be polynomials with integer |
---|
| 31 | // coefficients. If x1 and x2 are polynomials with positive |
---|
| 32 | // coefficients, the result is guaranteed to have positive |
---|
| 33 | // coefficients, too. |
---|
[66e0d2] | 34 | // |
---|
[d57f19] | 35 | // Note: This algorithm is optimized for the case q1>>q2. |
---|
| 36 | // |
---|
[030c681] | 37 | // This is a standard algorithm. See, for example, |
---|
[d57f19] | 38 | // Geddes/Czapor/Labahn - 'Algorithms for Computer Algebra', |
---|
| 39 | // par. 5.6 and 5.8, or the article of M. Lauer - 'Computing by |
---|
| 40 | // Homomorphic Images' in B. Buchberger - 'Computer Algebra - |
---|
| 41 | // Symbolic and Algebraic Computation'. |
---|
[030c681] | 42 | // |
---|
[d57f19] | 43 | // Note: Be sure you are calculating in Z, and not in Q! |
---|
[66e0d2] | 44 | // |
---|
| 45 | //}}} |
---|
[c12372] | 46 | void |
---|
[c2b784] | 47 | chineseRemainder ( const CanonicalForm & x1, const CanonicalForm & q1, const CanonicalForm & x2, const CanonicalForm & q2, CanonicalForm & xnew, CanonicalForm & qnew ) |
---|
[2dd068] | 48 | { |
---|
[d57f19] | 49 | DEBINCLEVEL( cerr, "chineseRemainder" ); |
---|
| 50 | |
---|
| 51 | DEBOUTLN( cerr, "log(q1) = " << q1.ilog2() ); |
---|
| 52 | DEBOUTLN( cerr, "log(q2) = " << q2.ilog2() ); |
---|
| 53 | |
---|
| 54 | // We calculate xnew as follows: |
---|
| 55 | // xnew = v1 + v2 * q1 |
---|
| 56 | // where |
---|
| 57 | // v1 = x1 (mod q1) |
---|
| 58 | // v2 = (x2-v1)/q1 (mod q2) (*) |
---|
| 59 | // |
---|
| 60 | // We do one extra test to check whether x2-v1 vanishes (mod |
---|
| 61 | // q2) in (*) since it is not costly and may save us |
---|
| 62 | // from calculating the inverse of q1 (mod q2). |
---|
| 63 | // |
---|
| 64 | // u: v1 (mod q2) |
---|
| 65 | // d: x2-v1 (mod q2) |
---|
| 66 | // s: 1/q1 (mod q2) |
---|
| 67 | // |
---|
| 68 | CanonicalForm v2, v1; |
---|
| 69 | CanonicalForm u, d, s, dummy; |
---|
| 70 | |
---|
| 71 | v1 = mod( x1, q1 ); |
---|
| 72 | u = mod( v1, q2 ); |
---|
| 73 | d = mod( x2-u, q2 ); |
---|
[6f62c3] | 74 | if ( d.isZero() ) |
---|
| 75 | { |
---|
| 76 | xnew = v1; |
---|
| 77 | qnew = q1 * q2; |
---|
[d2cdd65] | 78 | DEBDECLEVEL( cerr, "chineseRemainder" ); |
---|
[6f62c3] | 79 | return; |
---|
[d57f19] | 80 | } |
---|
| 81 | (void)bextgcd( q1, q2, s, dummy ); |
---|
| 82 | v2 = mod( d*s, q2 ); |
---|
| 83 | xnew = v1 + v2*q1; |
---|
| 84 | |
---|
| 85 | // After all, calculate new modulus. It is important that |
---|
| 86 | // this is done at the very end of the algorithm, since q1 |
---|
| 87 | // and qnew may refer to the same object (same is true for x1 |
---|
| 88 | // and xnew). |
---|
[2dd068] | 89 | qnew = q1 * q2; |
---|
[d57f19] | 90 | |
---|
| 91 | DEBDECLEVEL( cerr, "chineseRemainder" ); |
---|
[2dd068] | 92 | } |
---|
[66e0d2] | 93 | //}}} |
---|
[2dd068] | 94 | |
---|
[030c681] | 95 | //{{{ void chineseRemainder ( const CFArray & x, const CFArray & q, CanonicalForm & xnew, CanonicalForm & qnew ) |
---|
[66e0d2] | 96 | //{{{ docu |
---|
| 97 | // |
---|
| 98 | // chineseRemainder - integer chinese remaindering. |
---|
| 99 | // |
---|
[c12372] | 100 | // Calculate xnew such that xnew=x[i] (mod q[i]) and qnew is the |
---|
| 101 | // product of all q[i]. q[i] should be positive integers, |
---|
| 102 | // pairwise prime. x[i] should be polynomials with integer |
---|
| 103 | // coefficients. If all coefficients of all x[i] are positive |
---|
| 104 | // integers, the result is guaranteed to have positive |
---|
| 105 | // coefficients, too. |
---|
[66e0d2] | 106 | // |
---|
[ad2a147] | 107 | // This is a standard algorithm, too, except for the fact that we |
---|
| 108 | // use a divide-and-conquer method instead of a linear approach |
---|
| 109 | // to calculate the remainder. |
---|
| 110 | // |
---|
[d57f19] | 111 | // Note: Be sure you are calculating in Z, and not in Q! |
---|
[66e0d2] | 112 | // |
---|
| 113 | //}}} |
---|
[c12372] | 114 | void |
---|
[030c681] | 115 | chineseRemainder ( const CFArray & x, const CFArray & q, CanonicalForm & xnew, CanonicalForm & qnew ) |
---|
[2dd068] | 116 | { |
---|
[d57f19] | 117 | DEBINCLEVEL( cerr, "chineseRemainder( ... CFArray ... )" ); |
---|
| 118 | |
---|
[e074407] | 119 | ASSERT( x.min() == q.min() && x.size() == q.size(), "incompatible arrays" ); |
---|
| 120 | CFArray X(x), Q(q); |
---|
[d57f19] | 121 | int i, j, n = x.size(), start = x.min(); |
---|
| 122 | |
---|
| 123 | DEBOUTLN( cerr, "array size = " << n ); |
---|
[c12372] | 124 | |
---|
[6f62c3] | 125 | while ( n != 1 ) |
---|
| 126 | { |
---|
| 127 | i = j = start; |
---|
| 128 | while ( i < start + n - 1 ) |
---|
| 129 | { |
---|
| 130 | // This is a little bit dangerous: X[i] and X[j] (and |
---|
| 131 | // Q[i] and Q[j]) may refer to the same object. But |
---|
| 132 | // xnew and qnew in the above function are modified |
---|
| 133 | // at the very end of the function, so we do not |
---|
| 134 | // modify x1 and q1, resp., by accident. |
---|
| 135 | chineseRemainder( X[i], Q[i], X[i+1], Q[i+1], X[j], Q[j] ); |
---|
| 136 | i += 2; |
---|
| 137 | j++; |
---|
| 138 | } |
---|
| 139 | |
---|
| 140 | if ( n & 1 ) |
---|
| 141 | { |
---|
| 142 | X[j] = X[i]; |
---|
| 143 | Q[j] = Q[i]; |
---|
| 144 | } |
---|
| 145 | // Maybe we would get some memory back at this point if |
---|
| 146 | // we would set X[j+1, ..., n] and Q[j+1, ..., n] to zero |
---|
| 147 | // at this point? |
---|
| 148 | |
---|
| 149 | n = ( n + 1) / 2; |
---|
[2dd068] | 150 | } |
---|
[d57f19] | 151 | xnew = X[start]; |
---|
[e074407] | 152 | qnew = Q[q.min()]; |
---|
[d57f19] | 153 | |
---|
| 154 | DEBDECLEVEL( cerr, "chineseRemainder( ... CFArray ... )" ); |
---|
[2dd068] | 155 | } |
---|
[66e0d2] | 156 | //}}} |
---|
[6f62c3] | 157 | |
---|
| 158 | CanonicalForm Farey_n (CanonicalForm N, const CanonicalForm P) |
---|
| 159 | //"USAGE: Farey_n (N,P); P, N number; |
---|
| 160 | //RETURN: a rational number a/b such that a/b=N mod P |
---|
| 161 | // and |a|,|b|<(P/2)^{1/2} |
---|
| 162 | { |
---|
| 163 | //assume(P>0); |
---|
[806c18] | 164 | // assume !isOn(SW_RATIONAL): mod is a no-op otherwise |
---|
[08a6ebb] | 165 | if (N<0) N +=P; |
---|
[6f62c3] | 166 | CanonicalForm A,B,C,D,E; |
---|
| 167 | E=P; |
---|
| 168 | B=1; |
---|
[08a6ebb] | 169 | while (!N.isZero()) |
---|
[6f62c3] | 170 | { |
---|
| 171 | if (2*N*N<P) |
---|
| 172 | { |
---|
[806c18] | 173 | On(SW_RATIONAL); |
---|
| 174 | N /=B; |
---|
| 175 | Off(SW_RATIONAL); |
---|
[08a6ebb] | 176 | return(N); |
---|
[6f62c3] | 177 | } |
---|
[08a6ebb] | 178 | D=mod(E , N); |
---|
| 179 | C=A-(E-mod(E , N))/N*B; |
---|
[6f62c3] | 180 | E=N; |
---|
| 181 | N=D; |
---|
| 182 | A=B; |
---|
| 183 | B=C; |
---|
| 184 | } |
---|
| 185 | return(0); |
---|
| 186 | } |
---|
| 187 | |
---|
| 188 | CanonicalForm Farey ( const CanonicalForm & f, const CanonicalForm & q ) |
---|
| 189 | { |
---|
[08a6ebb] | 190 | int is_rat=isOn(SW_RATIONAL); |
---|
| 191 | Off(SW_RATIONAL); |
---|
[6f62c3] | 192 | Variable x = f.mvar(); |
---|
| 193 | CanonicalForm result = 0; |
---|
| 194 | CanonicalForm c; |
---|
| 195 | CFIterator i; |
---|
| 196 | for ( i = f; i.hasTerms(); i++ ) |
---|
| 197 | { |
---|
| 198 | c = i.coeff(); |
---|
| 199 | if ( c.inCoeffDomain()) |
---|
| 200 | { |
---|
| 201 | result += power( x, i.exp() ) * Farey_n(c,q); |
---|
| 202 | } |
---|
| 203 | else |
---|
| 204 | result += power( x, i.exp() ) * Farey(c,q); |
---|
| 205 | } |
---|
[08a6ebb] | 206 | if (is_rat) On(SW_RATIONAL); |
---|
[6f62c3] | 207 | return result; |
---|
| 208 | } |
---|
| 209 | |
---|