# source:git/factory/cf_chinese.cc@08a6ebb

spielwiese
Last change on this file since 08a6ebb was 08a6ebb, checked in by Hans Schönemann <hannes@…>, 16 years ago
*hannes: farey fixed git-svn-id: file:///usr/local/Singular/svn/trunk@10312 2c84dea3-7e68-4137-9b89-c4e89433aadc
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1/* emacs edit mode for this file is -*- C++ -*- */
2/* \$Id: cf_chinese.cc,v 1.12 2007-09-26 14:31:12 Singular Exp \$ */
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//
11//
12//}}}
13
14#include <config.h>
15
16#include "assert.h"
17#include "debug.h"
18
19#include "canonicalform.h"
20#include "cf_iter.h"
21
22
23//{{{ void chineseRemainder ( const CanonicalForm & x1, const CanonicalForm & q1, const CanonicalForm & x2, const CanonicalForm & q2, CanonicalForm & xnew, CanonicalForm & qnew )
24//{{{ docu
25//
26// chineseRemainder - integer chinese remaindering.
27//
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.
34//
35// Note: This algorithm is optimized for the case q1>>q2.
36//
37// This is a standard algorithm.  See, for example,
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'.
42//
43// Note: Be sure you are calculating in Z, and not in Q!
44//
45//}}}
46void
47chineseRemainder ( const CanonicalForm & x1, const CanonicalForm & q1, const CanonicalForm & x2, const CanonicalForm & q2, CanonicalForm & xnew, CanonicalForm & qnew )
48{
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 );
74    if ( d.isZero() )
75    {
76        xnew = v1;
77        qnew = q1 * q2;
78        DEBDECLEVEL( cerr, "chineseRemainder" );
79        return;
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).
89    qnew = q1 * q2;
90
91    DEBDECLEVEL( cerr, "chineseRemainder" );
92}
93//}}}
94
95//{{{ void chineseRemainder ( const CFArray & x, const CFArray & q, CanonicalForm & xnew, CanonicalForm & qnew )
96//{{{ docu
97//
98// chineseRemainder - integer chinese remaindering.
99//
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.
106//
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//
111// Note: Be sure you are calculating in Z, and not in Q!
112//
113//}}}
114void
115chineseRemainder ( const CFArray & x, const CFArray & q, CanonicalForm & xnew, CanonicalForm & qnew )
116{
117    DEBINCLEVEL( cerr, "chineseRemainder( ... CFArray ... )" );
118
119    ASSERT( x.min() == q.min() && x.size() == q.size(), "incompatible arrays" );
120    CFArray X(x), Q(q);
121    int i, j, n = x.size(), start = x.min();
122
123    DEBOUTLN( cerr, "array size = " << n );
124
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;
150    }
151    xnew = X[start];
152    qnew = Q[q.min()];
153
154    DEBDECLEVEL( cerr, "chineseRemainder( ... CFArray ... )" );
155}
156//}}}
157
158CanonicalForm 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);
164   // assume !isOn(SW_RATIONAL): mod is a no-op otherwise
165   if (N<0) N +=P;
166   CanonicalForm A,B,C,D,E;
167   E=P;
168   B=1;
169   while (!N.isZero())
170   {
171        if (2*N*N<P)
172        {
173           On(SW_RATIONAL);
174           N /=B;
175           Off(SW_RATIONAL);
176           return(N);
177        }
178        D=mod(E , N);
179        C=A-(E-mod(E , N))/N*B;
180        E=N;
181        N=D;
182        A=B;
183        B=C;
184   }
185   return(0);
186}
187
188CanonicalForm Farey ( const CanonicalForm & f, const CanonicalForm & q )
189{
190    int is_rat=isOn(SW_RATIONAL);
191    Off(SW_RATIONAL);
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    }
206    if (is_rat) On(SW_RATIONAL);
207    return result;
208}
209
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