songsenand
/
cgal
mirror of https://github.com/CGAL/cgal
1
0
Fork 0
Code Issues Packages Projects Releases Wiki Activity
cgal/Polynomial/include/CGAL/prs_resultant.h

194 lines
6.4 KiB
C++
Raw Blame History

// TODO: Add licence
//
// This file is provided AS IS with NO WARRANTY OF ANY KIND, INCLUDING THE
// WARRANTY OF DESIGN, MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE.
//
// $URL$
// $Id$
//
//
// Author(s) : Arno Eigenwillig <arno@mpi-inf.mpg.de>
//
// ============================================================================
// TODO: The comments are all original EXACUS comments and aren't adapted. So
// they may be wrong now.
/*! \file NiX/prs_resultant.h
* \brief Resultant computation via polynomial remainder sequences (PRS)
*
*/
#include <CGAL/basic.h>
#include <CGAL/Polynomial.h>
#ifndef CGAL_PRS_RESULTANT_H
#define CGAL_PRS_RESULTANT_H
CGAL_BEGIN_NAMESPACE
template <class NT> inline
NT prs_resultant_ufd(Polynomial<NT> A, Polynomial<NT> B) {
// implemented using the subresultant algorithm for resultant computation
// see [Cohen, 1993], algorithm 3.3.7
if (A.is_zero() || B.is_zero()) return NT(0);
int signflip;
if (A.degree() < B.degree()) {
Polynomial<NT> T = A; A = B; B = T;
signflip = (A.degree() & B.degree() & 1);
} else {
signflip = 0;
}
NT a = A.content(), b = B.content();
NT g(1), h(1), t = ipower(a, B.degree()) * ipower(b, A.degree());
Polynomial<NT> Q, R; NT d;
int delta;
A /= a; B /= b;
do {
signflip ^= (A.degree() & B.degree() & 1);
Polynomial<NT>::pseudo_division(A, B, Q, R, d);
delta = A.degree() - B.degree();
CGAL_expensive_assertion(
typename CGAL::Algebraic_structure_traits<NT>::Is_exact == CGAL_Tag_false
|| d == ipower(B.lcoeff(), delta + 1) );
A = B;
B = R / (g * ipower(h, delta));
g = A.lcoeff();
// h = h^(1-delta) * g^delta
INTERN_POLYNOMIAL::hgdelta_update(h, g, delta);
} while (B.degree() > 0);
// h = h^(1-deg(A)) * lcoeff(B)^deg(A)
delta = A.degree();
g = B.lcoeff();
INTERN_POLYNOMIAL::hgdelta_update(h, g, delta);
h = signflip ? -(t*h) : t*h;
typename Algebraic_structure_traits<NT>::Simplify simplify;
simplify(h);
return h;
}
template <class NT> inline
NT prs_resultant_field(Polynomial<NT> A, Polynomial<NT> B) {
// implemented using the Euclidean algorithm for resultant computation
// compare [Cox et al, 1997], p.157
if (A.is_zero() || B.is_zero()) return NT(0);
int signflip;
if (A.degree() < B.degree()) {
Polynomial<NT> T = A; A = B; B = T;
signflip = (A.degree() & B.degree() & 1);
} else {
signflip = 0;
}
NT res(1);
Polynomial<NT> Q, R;
while (B.degree() > 0) {
signflip ^= (A.degree() & B.degree() & 1);
Polynomial<NT>::euclidean_division(A, B, Q, R);
res *= ipower(B.lcoeff(), A.degree() - R.degree());
A = B;
B = R;
}
res = ipower(B.lcoeff(), A.degree()) * (signflip ? -res : res);
typename Algebraic_structure_traits<NT>::Simplify simplify;
simplify(res);
return res;
}
// definition follows below
template <class NT> inline
NT prs_resultant_decompose(Polynomial<NT> A, Polynomial<NT> B);
namespace INTERN_PRS_RESULTANT {
template <class NT> inline
NT prs_resultant_(Polynomial<NT> A, Polynomial<NT> B, ::CGAL::Tag_false) {
return prs_resultant_field(A, B);
}
template <class NT> inline
NT prs_resultant_(Polynomial<NT> A, Polynomial<NT> B, ::CGAL::Tag_true) {
return prs_resultant_decompose(A, B);
}
template <class NT> inline
NT prs_resultant_(Polynomial<NT> A, Polynomial<NT> B, Field_tag) {
typedef typename Fraction_traits<NT>::Is_fraction Is_decomposable;
return prs_resultant_(A, B, Is_decomposable());
}
template <class NT> inline
NT prs_resultant_(Polynomial<NT> A, Polynomial<NT> B, Unique_factorization_domain_tag) {
return prs_resultant_ufd(A, B);
}
} // namespace Intern
template <class NT> inline
NT prs_resultant_decompose(Polynomial<NT> A, Polynomial<NT> B){
typedef Polynomial<NT> POLY;
typedef typename Fraction_traits<POLY>::Numerator_type INTPOLY;
typedef typename Fraction_traits<POLY>::Denominator_type DENOM;
typedef typename INTPOLY::NT RES;
DENOM a, b;
A.simplify_coefficients();
B.simplify_coefficients();
INTPOLY A0 = integralize_polynomial(A, a);
INTPOLY B0 = integralize_polynomial(B, b);
DENOM c = ipower(a, B.degree()) * ipower(b, A.degree());
typedef typename Algebraic_structure_traits<RES>::Algebraic_category Algebraic_category;
RES res0 = INTERN_PRS_RESULTANT::prs_resultant_(A0, B0, Algebraic_category());
typename Fraction_traits<NT>::Compose comp_frac;
NT res = comp_frac(res0, c);
typename Algebraic_structure_traits<NT>::Simplify simplify;
simplify(res);
return res;
}
/*! \ingroup NiX_Polynomial
* \relates NiX::Polynomial
* \brief compute the resultant of polynomials \c A and \c B
*
* The resultant of two polynomials is computed from their
* polynomial remainder sequence (PRS), in the Euclidean or
* subresultant version. This depends on the coefficient type:
* If \c NT is a \c UFDomain , the subresultant PRS is formed.
* If \c NT is a \c Field that is not decomposable (see
* \c NiX::Fraction_traits ), then a Euclidean PRS is formed.
* If \c NT is a \c Field that is decomposable, then the
* \c Numerator must be a \c UFDomain, and the subresultant
* PRS is formed for the decomposed polynomials.
*
* Using \c NiX::hybrid_bezout_subresultant() may be faster in some cases
* and works for non-UFDomains, too.
* Using \c NiX::resultant() from \c NiX/resultant.h
* chooses automatically among these alternative methods of resultant
* computation for you.
*
* For the benefit of those who want to do their own template
* metaprogramming to choose the method of resultant computation,
* the three variants of resultant computation from a PRS
* can be called directly as \c prs_resultant_field() ,
* \c prs_resultant_ufd() and \c prs_resultant_decompose() .
* <b>Do not use them directly unless you know what you are doing!</b>
*
*/
template <class NT> inline
NT prs_resultant(Polynomial<NT> A, Polynomial<NT> B) {
typedef typename Algebraic_structure_traits<NT>::Algebraic_category
Algebraic_category;
return INTERN_PRS_RESULTANT::prs_resultant_(A, B, Algebraic_category());
}
CGAL_END_NAMESPACE
#endif // CGAL_PRS_RESULTANT_H
// EOF
Reference in New Issue View Git Blame Copy Permalink