cf_chinese.cc File Reference

algorithms for chinese remaindering and rational reconstruction More...

#include "config.h"
#include "NTLconvert.h"
#include "FLINTconvert.h"
#include "cf_assert.h"
#include "debug.h"
#include "canonicalform.h"
#include "cf_iter.h"
#include "cf_util.h"

Go to the source code of this file.

Functions
void	chineseRemainder (const CanonicalForm &x1, const CanonicalForm &q1, const CanonicalForm &x2, const CanonicalForm &q2, CanonicalForm &xnew, CanonicalForm &qnew)
	void chineseRemainder ( const CanonicalForm & x1, const CanonicalForm & q1, const CanonicalForm & x2, const CanonicalForm & q2, CanonicalForm & xnew, CanonicalForm & qnew ) More...
void	chineseRemainder (const CFArray &x, const CFArray &q, CanonicalForm &xnew, CanonicalForm &qnew)
	void chineseRemainder ( const CFArray & x, const CFArray & q, CanonicalForm & xnew, CanonicalForm & qnew ) More...
CanonicalForm	Farey (const CanonicalForm &f, const CanonicalForm &q)
	Farey rational reconstruction. More...
static CanonicalForm	chin_mul_inv (const CanonicalForm a, const CanonicalForm b, int ind, CFArray &inv)
void	out_cf (const char *s1, const CanonicalForm &f, const char *s2)
	cf_algorithm.cc - simple mathematical algorithms. More...
void	chineseRemainderCached (const CFArray &a, const CFArray &n, CanonicalForm &xnew, CanonicalForm &prod, CFArray &inv)
void	chineseRemainderCached (const CanonicalForm &a, const CanonicalForm &q1, const CanonicalForm &b, const CanonicalForm &q2, CanonicalForm &xnew, CanonicalForm &qnew, CFArray &inv)

Detailed Description

algorithms for chinese remaindering and rational reconstruction

Used by: cf_gcd.cc, cf_linsys.cc

Header file: cf_algorithm.h

Definition in file cf_chinese.cc.

Function Documentation

chin_mul_inv()

static CanonicalForm chin_mul_inv ( const CanonicalForm  a, const CanonicalForm  b, int  ind, CFArray &  inv  )

inlinestatic

Definition at line 276 of file cf_chinese.cc.

{
  if (inv[ind].isZero())
  {
    CanonicalForm s,dummy;
    (void)bextgcd( a, b, s, dummy );
    inv[ind]=s;
    return s;
  }
  else
    return inv[ind];
}

chineseRemainder() [1/2]

void chineseRemainder	(	const CanonicalForm &	x1,
		const CanonicalForm &	q1,
		const CanonicalForm &	x2,
		const CanonicalForm &	q2,
		CanonicalForm &	xnew,
		CanonicalForm &	qnew
	)

void chineseRemainder ( const CanonicalForm & x1, const CanonicalForm & q1, const CanonicalForm & x2, const CanonicalForm & q2, CanonicalForm & xnew, CanonicalForm & qnew )

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!

Definition at line 57 of file cf_chinese.cc.

{
    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" );
}

chineseRemainder() [2/2]

void chineseRemainder	(	const CFArray &	x,
		const CFArray &	q,
		CanonicalForm &	xnew,
		CanonicalForm &	qnew
	)

void chineseRemainder ( const CFArray & x, const CFArray & q, CanonicalForm & xnew, CanonicalForm & qnew )

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!

Definition at line 124 of file cf_chinese.cc.

{
    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 ... )" );
}

chineseRemainderCached() [1/2]

void chineseRemainderCached	(	const CanonicalForm &	a,
		const CanonicalForm &	q1,
		const CanonicalForm &	b,
		const CanonicalForm &	q2,
		CanonicalForm &	xnew,
		CanonicalForm &	qnew,
		CFArray &	inv
	)

Definition at line 308 of file cf_chinese.cc.

{
  CFArray A(2); A[0]=a; A[1]=b;
  CFArray Q(2); Q[0]=q1; Q[1]=q2;
  chineseRemainderCached(A,Q,xnew,qnew,inv);
}

chineseRemainderCached() [2/2]

void chineseRemainderCached	(	const CFArray &	a,
		const CFArray &	n,
		CanonicalForm &	xnew,
		CanonicalForm &	prod,
		CFArray &	inv
	)

Definition at line 290 of file cf_chinese.cc.

{
  CanonicalForm p, sum=0L; prod=1L;
  int i;
  int len=n.size();
 
  for (i = 0; i < len; i++) prod *= n[i];
 
  for (i = 0; i < len; i++)
  {
    p = prod / n[i];
    sum += a[i] * chin_mul_inv(p, n[i], i, inv) * p;
  }
 
  xnew = mod(sum , prod);
}

Farey()

CanonicalForm Farey	(	const CanonicalForm &	f,
		const CanonicalForm &	q
	)

Farey rational reconstruction.

If NTL is available it uses the fast algorithm from NTL, i.e. Encarnacion, Collins.

Definition at line 202 of file cf_chinese.cc.

{
  int is_rat=isOn(SW_RATIONAL);
  Off(SW_RATIONAL);
  Variable x = f.mvar();
  CanonicalForm result = 0;
  CanonicalForm c;
  CFIterator i;
#ifdef HAVE_FLINT
  fmpz_t FLINTq;
  fmpz_init(FLINTq);
  convertCF2initFmpz(FLINTq,q);
  fmpz_t FLINTc;
  fmpz_init(FLINTc);
  fmpq_t FLINTres;
  fmpq_init(FLINTres);
#elif defined(HAVE_NTL)
  ZZ NTLq= convertFacCF2NTLZZ (q);
  ZZ bound;
  SqrRoot (bound, NTLq/2);
#else
  factoryError("NTL/FLINT missing:Farey");
#endif
  for ( i = f; i.hasTerms(); i++ )
  {
    c = i.coeff();
    if ( c.inCoeffDomain())
    {
#ifdef HAVE_FLINT
      if (c.inZ())
      {
        convertCF2initFmpz(FLINTc,c);
        fmpq_reconstruct_fmpz(FLINTres,FLINTc,FLINTq);
        result += power (x, i.exp())*(convertFmpq2CF(FLINTres));
      }
#elif defined(HAVE_NTL)
      if (c.inZ())
      {
        ZZ NTLc= convertFacCF2NTLZZ (c);
        bool lessZero= (sign (NTLc) == -1);
        if (lessZero)
          NTL::negate (NTLc, NTLc);
        ZZ NTLnum, NTLden;
        if (ReconstructRational (NTLnum, NTLden, NTLc, NTLq, bound, bound))
        {
          if (lessZero)
            NTL::negate (NTLnum, NTLnum);
          CanonicalForm num= convertNTLZZX2CF (to_ZZX (NTLnum), Variable (1));
          CanonicalForm den= convertNTLZZX2CF (to_ZZX (NTLden), Variable (1));
          On (SW_RATIONAL);
          result += power (x, i.exp())*(num/den);
          Off (SW_RATIONAL);
        }
      }
#else
      if (c.inZ())
        result += power (x, i.exp()) * Farey_n(c,q);
#endif
      else
        result += power( x, i.exp() ) * Farey(c,q);
    }
    else
      result += power( x, i.exp() ) * Farey(c,q);
  }
  if (is_rat) On(SW_RATIONAL);
#ifdef HAVE_FLINT
  fmpq_clear(FLINTres);
  fmpz_clear(FLINTc);
  fmpz_clear(FLINTq);
#endif
  return result;
}

out_cf()

void out_cf	(	const char *	s1,
		const CanonicalForm &	f,
		const char *	s2
	)

cf_algorithm.cc - simple mathematical algorithms.

Hierarchy: mathematical algorithms on canonical forms

Developers note:

A "mathematical" algorithm is an algorithm which calculates some mathematical function in contrast to a "structural" algorithm which gives structural information on polynomials.

Compare these functions to the functions in ‘cf_ops.cc’, which are structural algorithms.

Definition at line 99 of file cf_factor.cc.

{
  printf("%s",s1);
  if (f.isZero()) printf("+0");
  //else if (! f.inCoeffDomain() )
  else if (! f.inBaseDomain() )
  {
    int l = f.level();
    for ( CFIterator i = f; i.hasTerms(); i++ )
    {
      int e=i.exp();
      if (i.coeff().isOne())
      {
        printf("+");
        if (e==0) printf("1");
        else
        {
          printf("%c",'a'+l-1);
          if (e!=1) printf("^%d",e);
        }
      }
      else
      {
        out_cf("+(",i.coeff(),")");
        if (e!=0)
        {
          printf("*%c",'a'+l-1);
          if (e!=1) printf("^%d",e);
        }
      }
    }
  }
  else
  {
    if ( f.isImm() )
    {
      if (CFFactory::gettype()==GaloisFieldDomain)
      {
         long a= imm2int (f.getval());
         if ( a == gf_q )
           printf ("+%ld", a);
         else  if ( a == 0L )
           printf ("+1");
         else  if ( a == 1L )
           printf ("+%c",gf_name);
         else
         {
           printf ("+%c",gf_name);
           printf ("^%ld",a);
         }
      }
      else
      {
        long l=f.intval();
        if (l<0) printf("%ld",l);
        else     printf("+%ld",l);
      }
    }
    else
    {
    #ifdef NOSTREAMIO
      if (f.inZ())
      {
        mpz_t m;
        gmp_numerator(f,m);
        char * str = new char[mpz_sizeinbase( m, 10 ) + 2];
        str = mpz_get_str( str, 10, m );
        puts(str);
        delete[] str;
        mpz_clear(m);
      }
      else if (f.inQ())
      {
        mpz_t m;
        gmp_numerator(f,m);
        char * str = new char[mpz_sizeinbase( m, 10 ) + 2];
        str = mpz_get_str( str, 10, m );
        while(str[strlen(str)]<' ') { str[strlen(str)]='\0'; }
        puts(str);putchar('/');
        delete[] str;
        mpz_clear(m);
        gmp_denominator(f,m);
        str = new char[mpz_sizeinbase( m, 10 ) + 2];
        str = mpz_get_str( str, 10, m );
        while(str[strlen(str)]<' ') { str[strlen(str)]='\0'; }
        puts(str);
        delete[] str;
        mpz_clear(m);
      }
    #else
       std::cout << f;
    #endif
    }
    //if (f.inZ()) printf("(Z)");
    //else if (f.inQ()) printf("(Q)");
    //else if (f.inFF()) printf("(FF)");
    //else if (f.inPP()) printf("(PP)");
    //else if (f.inGF()) printf("(PP)");
    //else
    if (f.inExtension()) printf("E(%d)",f.level());
  }
  printf("%s",s2);
}