source: git/factory/cf_chinese.cc @ 6f62c3

/* emacs edit mode for this file is -*- C++ -*- */
/* $Id: cf_chinese.cc,v 1.11 2007-09-26 09:17:39 Singular Exp $ */

//{{{ docu
//
// cf_chinese.cc - algorithms for chinese remaindering.
//
// Used by: cf_gcd.cc, cf_linsys.cc, sm_util.cc
//
// Header file: cf_algorithm.h
//
//}}}

#include <config.h>

#include "assert.h"
#include "debug.h"

#include "canonicalform.h"
#include "cf_iter.h"


//{{{ void chineseRemainder ( const CanonicalForm & x1, const CanonicalForm & q1, const CanonicalForm & x2, const CanonicalForm & q2, CanonicalForm & xnew, CanonicalForm & qnew )
//{{{ docu
//
// chineseRemainder - integer chinese remaindering.
//
// Calculate xnew such that xnew=x1 (mod q1) and xnew=x2 (mod q2)
// and qnew = q1*q2.  q1 and q2 should be positive integers,
// pairwise prime, x1 and x2 should be polynomials with integer
// coefficients.  If x1 and x2 are polynomials with positive
// coefficients, the result is guaranteed to have positive
// coefficients, too.
//
// Note: This algorithm is optimized for the case q1>>q2.
//
// This is a standard algorithm.  See, for example,
// Geddes/Czapor/Labahn - 'Algorithms for Computer Algebra',
// par. 5.6 and 5.8, or the article of M. Lauer - 'Computing by
// Homomorphic Images' in B. Buchberger - 'Computer Algebra -
// Symbolic and Algebraic Computation'.
//
// Note: Be sure you are calculating in Z, and not in Q!
//
//}}}
void
chineseRemainder ( const CanonicalForm & x1, const CanonicalForm & q1, const CanonicalForm & x2, const CanonicalForm & q2, CanonicalForm & xnew, CanonicalForm & qnew )
{
    DEBINCLEVEL( cerr, "chineseRemainder" );

    DEBOUTLN( cerr, "log(q1) = " << q1.ilog2() );
    DEBOUTLN( cerr, "log(q2) = " << q2.ilog2() );

    // We calculate xnew as follows:
    //     xnew = v1 + v2 * q1
    // where
    //     v1 = x1 (mod q1)
    //     v2 = (x2-v1)/q1 (mod q2)  (*)
    //
    // We do one extra test to check whether x2-v1 vanishes (mod
    // q2) in (*) since it is not costly and may save us
    // from calculating the inverse of q1 (mod q2).
    //
    // u: v1 (mod q2)
    // d: x2-v1 (mod q2)
    // s: 1/q1 (mod q2)
    //
    CanonicalForm v2, v1;
    CanonicalForm u, d, s, dummy;

    v1 = mod( x1, q1 );
    u = mod( v1, q2 );
    d = mod( x2-u, q2 );
    if ( d.isZero() )
    {
        xnew = v1;
        qnew = q1 * q2;
        DEBDECLEVEL( cerr, "chineseRemainder" );
        return;
    }
    (void)bextgcd( q1, q2, s, dummy );
    v2 = mod( d*s, q2 );
    xnew = v1 + v2*q1;

    // After all, calculate new modulus.  It is important that
    // this is done at the very end of the algorithm, since q1
    // and qnew may refer to the same object (same is true for x1
    // and xnew).
    qnew = q1 * q2;

    DEBDECLEVEL( cerr, "chineseRemainder" );
}
//}}}

//{{{ void chineseRemainder ( const CFArray & x, const CFArray & q, CanonicalForm & xnew, CanonicalForm & qnew )
//{{{ docu
//
// chineseRemainder - integer chinese remaindering.
//
// Calculate xnew such that xnew=x[i] (mod q[i]) and qnew is the
// product of all q[i].  q[i] should be positive integers,
// pairwise prime.  x[i] should be polynomials with integer
// coefficients.  If all coefficients of all x[i] are positive
// integers, the result is guaranteed to have positive
// coefficients, too.
//
// This is a standard algorithm, too, except for the fact that we
// use a divide-and-conquer method instead of a linear approach
// to calculate the remainder.
//
// Note: Be sure you are calculating in Z, and not in Q!
//
//}}}
void
chineseRemainder ( const CFArray & x, const CFArray & q, CanonicalForm & xnew, CanonicalForm & qnew )
{
    DEBINCLEVEL( cerr, "chineseRemainder( ... CFArray ... )" );

    ASSERT( x.min() == q.min() && x.size() == q.size(), "incompatible arrays" );
    CFArray X(x), Q(q);
    int i, j, n = x.size(), start = x.min();

    DEBOUTLN( cerr, "array size = " << n );

    while ( n != 1 )
    {
        i = j = start;
        while ( i < start + n - 1 )
        {
            // This is a little bit dangerous: X[i] and X[j] (and
            // Q[i] and Q[j]) may refer to the same object.  But
            // xnew and qnew in the above function are modified
            // at the very end of the function, so we do not
            // modify x1 and q1, resp., by accident.
            chineseRemainder( X[i], Q[i], X[i+1], Q[i+1], X[j], Q[j] );
            i += 2;
            j++;
        }

        if ( n & 1 )
        {
            X[j] = X[i];
            Q[j] = Q[i];
        }
        // Maybe we would get some memory back at this point if
        // we would set X[j+1, ..., n] and Q[j+1, ..., n] to zero
        // at this point?

        n = ( n + 1) / 2;
    }
    xnew = X[start];
    qnew = Q[q.min()];

    DEBDECLEVEL( cerr, "chineseRemainder( ... CFArray ... )" );
}
//}}}

CanonicalForm Farey_n (CanonicalForm N, const CanonicalForm P)
//"USAGE:  Farey_n (N,P); P, N number;
//RETURN:  a rational number a/b such that a/b=N mod P
//         and |a|,|b|<(P/2)^{1/2}
{
   //assume(P>0);
   if (N<0){N=N+P;}
   CanonicalForm A,B,C,D,E;
   E=P;
   B=1;
   while (N!=0)
   {
        if (2*N*N<P)
        {
           return(N/B);
        }
        D=E % N;
        C=A-(E-E % N)/N*B;
        E=N;
        N=D;
        A=B;
        B=C;
   }
   return(0);
}

CanonicalForm Farey ( const CanonicalForm & f, const CanonicalForm & q )
{
    Variable x = f.mvar();
    CanonicalForm result = 0;
    CanonicalForm c;
    CFIterator i;
    for ( i = f; i.hasTerms(); i++ )
    {
        c = i.coeff();
        if ( c.inCoeffDomain())
        {
          result += power( x, i.exp() ) * Farey_n(c,q);
        }
        else
          result += power( x, i.exp() ) * Farey(c,q);
    }
    return result;
}