616 lines
18 KiB
C
616 lines
18 KiB
C
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// Copyright (c) 2008 Max-Planck-Institute Saarbruecken (Germany)
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//
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// This file is part of CGAL (www.cgal.org); you can redistribute it and/or
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// modify it under the terms of the GNU Lesser General Public License as
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// published by the Free Software Foundation; either version 3 of the License,
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// or (at your option) any later version.
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//
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// Licensees holding a valid commercial license may use this file in
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// accordance with the commercial license agreement provided with the software.
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//
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// This file is provided AS IS with NO WARRANTY OF ANY KIND, INCLUDING THE
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// WARRANTY OF DESIGN, MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE.
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//
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// $URL$
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// $Id$
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// SPDX-License-Identifier: LGPL-3.0+
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//
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//
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// Author(s) : Michael Hemmer
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//
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// ============================================================================
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// TODO: The comments are all original EXACUS comments and aren't adapted. So
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// they may be wrong now.
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#ifndef CGAL_POLYNOMIAL_BEZOUT_MATRIX_H
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#define CGAL_POLYNOMIAL_BEZOUT_MATRIX_H
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#include <algorithm>
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#include <CGAL/basic.h>
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#include <CGAL/Polynomial_traits_d.h>
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#include <CGAL/Polynomial/determinant.h>
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#include <CGAL/use.h>
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#include <vector>
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namespace CGAL {
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namespace internal {
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/*! \ingroup CGAL_resultant_matrix
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* \brief construct hybrid Bezout matrix of two polynomials
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*
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* If \c sub=0 , this function returns the hybrid Bezout matrix
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* of \c f and \c g.
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* The hybrid Bezout matrix of two polynomials \e f and \e g
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* (seen as polynomials in one variable) is a
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* square matrix of size max(deg(<I>f</I>), deg(<I>g</I>)) whose entries
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* are expressions in the polynomials' coefficients.
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* Its determinant is the resultant of \e f and \e g, maybe up to sign.
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* The function computes the same matrix as the Maple command
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* <I>BezoutMatrix</I>.
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*
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* If \c sub>0 , this function returns the matrix obtained by chopping
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* off the \c sub topmost rows and the \c sub rightmost columns.
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* Its determinant is the <I>sub</I>-th (scalar) subresultant
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* of \e f and \e g, maybe up to sign.
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*
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* If specified, \c sub must be less than the degrees of both \e f and \e g.
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*
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* See also \c CGAL::hybrid_bezout_subresultant() and \c CGAL::sylvester_matrix() .
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*
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* A formal definition of the hybrid Bezout matrix and a proof for the
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* subresultant property can be found in:
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*
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* Gema M.Diaz-Toca, Laureano Gonzalez-Vega: Various New Expressions for
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* Subresultants and Their Applications. AAECC 15, 233-266 (2004)
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*
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*/
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template <typename Polynomial_traits_d>
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typename internal::Simple_matrix< typename Polynomial_traits_d::Coefficient_type >
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hybrid_bezout_matrix(typename Polynomial_traits_d::Polynomial_d f,
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typename Polynomial_traits_d::Polynomial_d g,
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int sub = 0)
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{
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typedef typename Polynomial_traits_d::Polynomial_d Polynomial;
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typedef typename Polynomial_traits_d::Coefficient_type NT;
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typename Polynomial_traits_d::Degree degree;
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typename CGAL::Algebraic_structure_traits<Polynomial>::Is_zero is_zero;
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typename Polynomial_traits_d::Get_coefficient coeff;
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typedef typename internal::Simple_matrix<NT> Matrix;
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int n = degree(f);
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int m = degree(g);
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CGAL_precondition((n >= 0) && !is_zero(f));
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CGAL_precondition((m >= 0) && !is_zero(g));
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CGAL_precondition(n > sub || sub == 0);
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CGAL_precondition(m > sub || sub == 0);
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int i, j, k, l;
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NT s;
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if (m > n) {
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std::swap(f, g);
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std::swap(m, n);
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}
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Matrix B(n-sub);
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for (i = 1+sub; i <= m; i++) {
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for (j = 1; j <= n-sub; j++) {
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s = NT(0);
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for (k = 0; k <= i-1; k++) {
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l = n+i-j-k;
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if ((l <= n) and (l >= n-(m-i))) {
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s += coeff(f,l) * coeff(g,k);
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}
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}
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for (k = 0; k <= n-(m-i+1); k++) {
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l = n+i-j-k;
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if ((l <= m) and (l >= i)) {
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s -= coeff(f,k) * coeff(g,l);
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}
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}
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B[i-sub-1][j-1] = s;
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}
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}
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for (i = std::max(m+1, 1+sub); i <= n; i++) {
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for (j = i-m; j <= std::min(i, n-sub); j++) {
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B[i-sub-1][j-1] = coeff(g,i-j);
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}
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}
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return B; // g++ 3.1+ has NRVO, so this should not be too expensive
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}
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/*! \ingroup CGAL_resultant_matrix
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* \brief construct the symmetric Bezout matrix of two polynomials
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*
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* This function returns the (symmetric) Bezout matrix of \c f and \c g.
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* The Bezout matrix of two polynomials \e f and \e g
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* (seen as polynomials in one variable) is a
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* square matrix of size max(deg(<I>f</I>), deg(<I>g</I>)) whose entries
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* are expressions in the polynomials' coefficients.
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* Its determinant is the resultant of \e f and \e g, maybe up to sign.
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*
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*/
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template <typename Polynomial_traits_d>
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typename internal::Simple_matrix<typename Polynomial_traits_d::Coefficient_type>
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symmetric_bezout_matrix
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(typename Polynomial_traits_d::Polynomial_d f,
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typename Polynomial_traits_d::Polynomial_d g,
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int sub = 0)
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{
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// Note: The algorithm is taken from:
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// Chionh, Zhang, Goldman: Fast Computation of the Bezout and Dixon Resultant
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// Matrices. J.Symbolic Computation 33, 13-29 (2002)
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typedef typename Polynomial_traits_d::Polynomial_d Polynomial;
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typedef typename Polynomial_traits_d::Coefficient_type NT;
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typename Polynomial_traits_d::Degree degree;
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CGAL_assertion_code(typename CGAL::Algebraic_structure_traits<Polynomial>::Is_zero is_zero;)
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CGAL_USE_TYPE(Polynomial);
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typename Polynomial_traits_d::Get_coefficient coeff;
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typedef typename internal::Simple_matrix<NT> Matrix;
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int n = degree(f);
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int m = degree(g);
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CGAL_precondition((n >= 0) && !is_zero(f));
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CGAL_precondition((m >= 0) && !is_zero(g));
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int i,j,stop;
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NT sum1,sum2;
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if (m > n) {
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std::swap(f, g);
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std::swap(m, n);
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}
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CGAL_precondition((sub>=0) && sub < n);
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int d = n - sub;
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Matrix B(d);
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// 1st step: Initialisation
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for(i=0;i<d;i++) {
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for(j=i;j<d;j++) {
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sum1 = ((j+sub)+1>m) ? NT(0) : -coeff(f,i+sub)*coeff(g,(j+sub)+1);
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sum2 = ((i+sub)>m) ? NT(0) : coeff(g,i+sub)*coeff(f,(j+sub)+1);
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B[i][j]=sum1+sum2;
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}
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}
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// 2nd Step: Recursion adding
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// First, set up the first line correctly
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for(i=0;i<d-1;i++) {
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stop = (sub<d-1-i) ? sub : d-i-1;
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for(j=1;j<=stop;j++) {
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sum1 = ((i+sub+j)+1>m) ? NT(0)
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: -coeff(f,sub-j)*coeff(g,(i+sub+j)+1);
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sum2 = ((sub-j)>m) ? NT(0)
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: coeff(g,sub-j)*coeff(f,(i+sub+j)+1);
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B[0][i]+=sum1+sum2;
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}
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}
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// Now, compute the rest
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for(i=1;i<d-1;i++) {
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for(j=i;j<d-1;j++) {
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B[i][j]+=B[i-1][j+1];
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}
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}
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//3rd Step: Exploit symmetry
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for(i=1;i<d;i++) {
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for(j=0;j<i;j++) {
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B[i][j]=B[j][i];
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}
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}
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return B;
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}
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/*! \ingroup CGAL_resultant_matrix
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* \brief compute (sub)resultant as Bezout matrix determinant
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*
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* This function returns the determinant of the matrix returned
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* by <TT>hybrid_bezout_matrix(f, g, sub)</TT> which is the
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* resultant of \c f and \c g, maybe up to sign;
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* or rather the <I>sub</I>-th (scalar) subresultant, if a non-zero third
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* argument is given.
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*
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* If specified, \c sub must be less than the degrees of both \e f and \e g.
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*
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* This function might be faster than \c CGAL::Polynomial<..>::resultant() ,
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* which computes the resultant from a subresultant remainder sequence.
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* See also \c CGAL::sylvester_subresultant().
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*/
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template <class Polynomial_traits_d>
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typename Polynomial_traits_d::Coefficient_type hybrid_bezout_subresultant(
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typename Polynomial_traits_d::Polynomial_d f,
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typename Polynomial_traits_d::Polynomial_d g,
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int sub = 0
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) {
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typedef typename Polynomial_traits_d::Polynomial_d Polynomial;
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typedef typename Polynomial_traits_d::Coefficient_type NT;
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typename Polynomial_traits_d::Degree degree;
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typename CGAL::Algebraic_structure_traits<Polynomial>::Is_zero is_zero;
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typedef internal::Simple_matrix<NT> Matrix;
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CGAL_precondition((degree(f) >= 0));
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CGAL_precondition((degree(g) >= 0));
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if (is_zero(f) || is_zero(g)) return NT(0);
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Matrix S = hybrid_bezout_matrix<Polynomial_traits_d>(f, g, sub);
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CGAL_assertion(S.row_dimension() == S.column_dimension());
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if (S.row_dimension() == 0) {
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return NT(1);
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} else {
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return internal::determinant(S);
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}
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}
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// Transforms the minors of the symmetric bezout matrix into the subresultant.
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// Needs the appropriate power of the leading coedfficient of f and the
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// degrees of f and g
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template<class InputIterator,class OutputIterator,class NT>
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void symmetric_minors_to_subresultants(InputIterator in,
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OutputIterator out,
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NT divisor,
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int n,
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int m,
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bool swapped) {
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typename CGAL::Algebraic_structure_traits<NT>::Integral_division idiv;
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for(int i=0;i<m;i++) {
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bool negate = ((n-m+i+1) & 2)>>1; // (n-m+i+1)==2 or 3 mod 4
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negate=negate ^ (swapped & ((n-m+i+1)*(i+1)));
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//...XOR (swapped AND (n-m+i+1)* (i+1) is odd)
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*out = idiv(*in, negate ? -divisor : divisor);
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in++;
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out++;
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}
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}
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/*! \ingroup CGAL_resultant_matrix
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* \brief compute the principal subresultant coefficients as minors
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* of the symmetric Bezout matrix.
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*
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* Returns the sequence sres<sub>0</sub>,..,sres<sub>m</sub>, where
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* sres<sub>i</sub> denotes the ith principal subresultant coefficient
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*
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* The function uses an extension of the Berkowitz method to compute the
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* determinant
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* See also \c CGAL::minors_berkowitz
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*/
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template<class Polynomial_traits_d,class OutputIterator>
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OutputIterator symmetric_bezout_subresultants(
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typename Polynomial_traits_d::Polynomial_d f,
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typename Polynomial_traits_d::Polynomial_d g,
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OutputIterator sres)
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{
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typedef typename Polynomial_traits_d::Polynomial_d Polynomial;
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typedef typename Polynomial_traits_d::Coefficient_type NT;
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typename Polynomial_traits_d::Degree degree;
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typename CGAL::Algebraic_structure_traits<Polynomial>::Is_zero is_zero;
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typename Polynomial_traits_d::Leading_coefficient lcoeff;
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typedef typename internal::Simple_matrix<NT> Matrix;
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int n = degree(f);
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int m = degree(g);
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bool swapped=false;
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if(n < m) {
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std::swap(f,g);
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std::swap(n,m);
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swapped=true;
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}
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Matrix B = symmetric_bezout_matrix<Polynomial_traits_d>(f,g);
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// Compute a_0^{n-m}
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NT divisor=ipower(lcoeff(f),n-m);
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std::vector<NT> minors;
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minors_berkowitz(B,std::back_inserter(minors),n,m);
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CGAL::internal::symmetric_minors_to_subresultants(minors.begin(),sres,
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divisor,n,m,swapped);
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return sres;
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}
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/*
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* Return a modified version of the hybrid bezout matrix such that the minors
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* from the last k rows and columns give the subresultants
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*/
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template<class Polynomial_traits_d>
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typename internal::Simple_matrix<typename Polynomial_traits_d::Coefficient_type>
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modified_hybrid_bezout_matrix
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(typename Polynomial_traits_d::Polynomial_d f,
|
||
|
|
typename Polynomial_traits_d::Polynomial_d g) {
|
||
|
|
|
||
|
|
typedef typename Polynomial_traits_d::Coefficient_type NT;
|
||
|
|
|
||
|
|
typedef typename internal::Simple_matrix<NT> Matrix;
|
||
|
|
|
||
|
|
typename Polynomial_traits_d::Degree degree;
|
||
|
|
|
||
|
|
int n = degree(f);
|
||
|
|
int m = degree(g);
|
||
|
|
|
||
|
|
int i,j;
|
||
|
|
|
||
|
|
bool negate, swapped=false;
|
||
|
|
|
||
|
|
if(n < m) {
|
||
|
|
std::swap(f,g); //(*)
|
||
|
|
std::swap(n,m);
|
||
|
|
swapped=true;
|
||
|
|
}
|
||
|
|
|
||
|
|
Matrix B = CGAL::internal::hybrid_bezout_matrix<Polynomial_traits_d>(f,g);
|
||
|
|
|
||
|
|
|
||
|
|
// swap columns
|
||
|
|
i=0;
|
||
|
|
while(i<n-i-1) {
|
||
|
|
B.swap_columns(i,n-i-1); // (**)
|
||
|
|
i++;
|
||
|
|
}
|
||
|
|
for(i=0;i<n;i++) {
|
||
|
|
negate=(n-i-1) & 1; // Negate every second column because of (**)
|
||
|
|
negate=negate ^ (swapped & (n-m+1)); // XOR negate everything because of(*)
|
||
|
|
if(negate) {
|
||
|
|
for(j=0;j<n;j++) {
|
||
|
|
B[j][i] *= -1;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
}
|
||
|
|
return B;
|
||
|
|
}
|
||
|
|
|
||
|
|
/*! \ingroup CGAL_resultant_matrix
|
||
|
|
* \brief compute the principal subresultant coefficients as minors
|
||
|
|
* of the hybrid Bezout matrix.
|
||
|
|
*
|
||
|
|
* Returns the sequence sres<sub>0</sub>,...,sres<sub>m</sub>$, where
|
||
|
|
* sres<sub>i</sub> denotes the ith principal subresultant coefficient
|
||
|
|
*
|
||
|
|
* The function uses an extension of the Berkowitz method to compute the
|
||
|
|
* determinant
|
||
|
|
* See also \c CGAL::minors_berkowitz
|
||
|
|
*/
|
||
|
|
template<class Polynomial_traits_d,class OutputIterator>
|
||
|
|
OutputIterator hybrid_bezout_subresultants(
|
||
|
|
typename Polynomial_traits_d::Polynomial_d f,
|
||
|
|
typename Polynomial_traits_d::Polynomial_d g,
|
||
|
|
OutputIterator sres)
|
||
|
|
{
|
||
|
|
|
||
|
|
typedef typename Polynomial_traits_d::Coefficient_type NT;
|
||
|
|
typename Polynomial_traits_d::Degree degree;
|
||
|
|
|
||
|
|
typedef typename internal::Simple_matrix<NT> Matrix;
|
||
|
|
|
||
|
|
int n = degree(f);
|
||
|
|
int m = degree(g);
|
||
|
|
|
||
|
|
Matrix B = CGAL::internal::modified_hybrid_bezout_matrix<Polynomial_traits_d>
|
||
|
|
(f,g);
|
||
|
|
|
||
|
|
if(n<m) {
|
||
|
|
std::swap(n,m);
|
||
|
|
}
|
||
|
|
|
||
|
|
return minors_berkowitz(B,sres,n,m);
|
||
|
|
}
|
||
|
|
|
||
|
|
|
||
|
|
// Swap entry A_ij with A_(n-i)(n-j) for square matrix A of dimension n
|
||
|
|
template<class NT>
|
||
|
|
void swap_entries(typename internal::Simple_matrix<NT> & A) {
|
||
|
|
CGAL_precondition(A.row_dimension()==A.column_dimension());
|
||
|
|
int n = A.row_dimension();
|
||
|
|
int i=0;
|
||
|
|
while(i<n-i-1) {
|
||
|
|
A.swap_rows(i,n-i-1);
|
||
|
|
A.swap_columns(i,n-i-1);
|
||
|
|
i++;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
// Produce S-matrix with the given matrix and integers.
|
||
|
|
template<class NT,class InputIterator>
|
||
|
|
typename internal::Simple_matrix<NT> s_matrix(
|
||
|
|
const typename internal::Simple_matrix<NT>& B,
|
||
|
|
InputIterator num,int size)
|
||
|
|
{
|
||
|
|
typename internal::Simple_matrix<NT> S(size);
|
||
|
|
CGAL_precondition_code(int n = B.row_dimension();)
|
||
|
|
CGAL_precondition(n==(int)B.column_dimension());
|
||
|
|
int curr_num;
|
||
|
|
bool negate;
|
||
|
|
|
||
|
|
for(int i=0;i<size;i++) {
|
||
|
|
curr_num=(*num);
|
||
|
|
num++;
|
||
|
|
negate = curr_num<0;
|
||
|
|
if(curr_num<0) {
|
||
|
|
curr_num=-curr_num;
|
||
|
|
}
|
||
|
|
for(int j=0;j<size;j++) {
|
||
|
|
|
||
|
|
S[j][i]=negate ? -B[j][curr_num-1] : B[j][curr_num-1];
|
||
|
|
|
||
|
|
}
|
||
|
|
}
|
||
|
|
return S;
|
||
|
|
}
|
||
|
|
|
||
|
|
// Produces the integer sequence for the S-matrix, where c is the first entry
|
||
|
|
// of the sequence, s the number of desired diagonals and n the dimension
|
||
|
|
// of the base matrix
|
||
|
|
template<class OutputIterator>
|
||
|
|
OutputIterator s_matrix_integer_sequence(OutputIterator it,
|
||
|
|
int c,int s,int n) {
|
||
|
|
CGAL_precondition(0<s);
|
||
|
|
CGAL_precondition(s<=n);
|
||
|
|
// c is interpreted modulo s wrt to the representants {1,..,s}
|
||
|
|
c=c%s;
|
||
|
|
if(c==0) {
|
||
|
|
c=s;
|
||
|
|
}
|
||
|
|
|
||
|
|
int i, p=0, q=c;
|
||
|
|
while(q<=n) {
|
||
|
|
*it = q;
|
||
|
|
it++;
|
||
|
|
for(i=p+1;i<q;i++) {
|
||
|
|
*it = -i;
|
||
|
|
it++;
|
||
|
|
}
|
||
|
|
p = q;
|
||
|
|
q = q+s;
|
||
|
|
}
|
||
|
|
return it;
|
||
|
|
}
|
||
|
|
|
||
|
|
/*! \ingroup CGAL_resultant_matrix
|
||
|
|
* \brief computes the coefficients of the polynomial subresultant sequence
|
||
|
|
*
|
||
|
|
* Returns an upper triangular matrix <I>A</I> such that A<sub>i,j</sub> is
|
||
|
|
* the coefficient of <I>x<sup>j-1</sup></I> in the <I>i</I>th polynomial
|
||
|
|
* subresultant. In particular, the main diagonal contains the scalar
|
||
|
|
* subresultants.
|
||
|
|
*
|
||
|
|
* If \c d > 0 is specified, only the first \c d diagonals of <I>A</I> are
|
||
|
|
* computed. In particular, setting \c d to one yields exactly the same
|
||
|
|
* result as applying \c hybrid_subresultants or \c symmetric_subresultants
|
||
|
|
* (except the different output format).
|
||
|
|
*
|
||
|
|
* These coefficients are computed as special minors of the hybrid Bezout matrix.
|
||
|
|
* See also \c CGAL::minors_berkowitz
|
||
|
|
*/
|
||
|
|
template<typename Polynomial_traits_d>
|
||
|
|
typename internal::Simple_matrix<typename Polynomial_traits_d::Coefficient_type>
|
||
|
|
polynomial_subresultant_matrix(typename Polynomial_traits_d::Polynomial_d f,
|
||
|
|
typename Polynomial_traits_d::Polynomial_d g,
|
||
|
|
int d=0) {
|
||
|
|
|
||
|
|
typedef typename Polynomial_traits_d::Coefficient_type NT;
|
||
|
|
typename Polynomial_traits_d::Degree degree;
|
||
|
|
typename Polynomial_traits_d::Leading_coefficient lcoeff;
|
||
|
|
|
||
|
|
int n = degree(f);
|
||
|
|
int m = degree(g);
|
||
|
|
|
||
|
|
CGAL_precondition(n>=0);
|
||
|
|
CGAL_precondition(m>=0);
|
||
|
|
CGAL_precondition(d>=0);
|
||
|
|
|
||
|
|
typedef internal::Simple_matrix<NT> Matrix;
|
||
|
|
|
||
|
|
bool swapped=false;
|
||
|
|
|
||
|
|
if(n < m) {
|
||
|
|
std::swap(f,g);
|
||
|
|
std::swap(n,m);
|
||
|
|
swapped=true;
|
||
|
|
}
|
||
|
|
|
||
|
|
if(d==0) {
|
||
|
|
d=m;
|
||
|
|
};
|
||
|
|
|
||
|
|
|
||
|
|
Matrix B = CGAL::internal::symmetric_bezout_matrix<Polynomial_traits_d>(f,g);
|
||
|
|
|
||
|
|
// For easier notation, we swap all entries:
|
||
|
|
internal::swap_entries<NT>(B);
|
||
|
|
|
||
|
|
// Compute the S-matrices and collect the minors
|
||
|
|
std::vector<Matrix> s_mat(m);
|
||
|
|
std::vector<std::vector<NT> > coeffs(d);
|
||
|
|
for(int i = 1; i<=d;i++) {
|
||
|
|
std::vector<int> intseq;
|
||
|
|
internal::s_matrix_integer_sequence(std::back_inserter(intseq),i,d,n);
|
||
|
|
|
||
|
|
Matrix S = internal::s_matrix<NT>(B,intseq.begin(),(int)intseq.size());
|
||
|
|
internal::swap_entries<NT>(S);
|
||
|
|
//std::cout << S << std::endl;
|
||
|
|
int Sdim = S.row_dimension();
|
||
|
|
int number_of_minors=(Sdim < m) ? Sdim : Sdim;
|
||
|
|
|
||
|
|
internal::minors_berkowitz(S,std::back_inserter(coeffs[i-1]),
|
||
|
|
Sdim,number_of_minors);
|
||
|
|
|
||
|
|
}
|
||
|
|
|
||
|
|
// Now, rearrange the minors in the matrix
|
||
|
|
|
||
|
|
Matrix Ret(m,m,NT(0));
|
||
|
|
|
||
|
|
for(int i = 0; i < d; i++) {
|
||
|
|
for(int j = 0;j < m-i ; j++) {
|
||
|
|
int s_index=(n-m+j+i+1)%d;
|
||
|
|
if(s_index==0) {
|
||
|
|
s_index=d;
|
||
|
|
}
|
||
|
|
s_index--;
|
||
|
|
Ret[j][j+i]=coeffs[s_index][n-m+j];
|
||
|
|
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
typename CGAL::Algebraic_structure_traits<NT>::Integral_division idiv;
|
||
|
|
|
||
|
|
NT divisor = ipower(lcoeff(f),n-m);
|
||
|
|
|
||
|
|
int bit_mask = swapped ? 1 : 0;
|
||
|
|
// Divide through the divisor and set the correct sign
|
||
|
|
for(int i=0;i<m;i++) {
|
||
|
|
for(int j = i;j<m;j++) {
|
||
|
|
int negate = ((n-m+i+1) & 2)>>1; // (n-m+i+1)==2 or 3 mod 4
|
||
|
|
negate^=(bit_mask & ((n-m+i+1)*(i+1)));
|
||
|
|
//...XOR (swapped AND (n-m+i+1)* (i+1) is odd)
|
||
|
|
Ret[i][j] = idiv(Ret[i][j], negate>0 ? -divisor : divisor);
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
return Ret;
|
||
|
|
}
|
||
|
|
|
||
|
|
}
|
||
|
|
|
||
|
|
} //namespace CGAL
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
#endif // CGAL_POLYNOMIAL_BEZOUT_MATRIX_H
|
||
|
|
// EOF
|