489 lines
14 KiB
C
489 lines
14 KiB
C
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// Copyright (c) 2008 Max-Planck-Institute Saarbruecken (Germany).
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// All rights reserved.
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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 Kerber <mkerber@mpi-inf.mpg.de>
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//
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// ============================================================================
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#ifndef CGAL_POLYNOMIAL_STURM_HABICHT
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#define CGAL_POLYNOMIAL_STURM_HABICHT 1
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#include <vector>
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#include <algorithm>
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#include <CGAL/Polynomial/bezout_matrix.h>
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#include <CGAL/Polynomial/subresultants.h>
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namespace CGAL {
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namespace internal {
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/*!
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* \brief compute the leading coefficients of the Sturm-Habicht sequence of
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* the polynomial <I>P</I>
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*
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* The principal Sturm-Habicht sequence is obtained by computing the scalar
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* subresultant sequence of <I>P</I> and its derivative, extended
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* by <I>P</I> and <I>P'</I> and some sign changes.
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*
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* For details, see: Gonzalez-Vega,Lombardi,Recio,Roy: Determinants and Real
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* Roots of Univariate Polynomials. Texts and Monographs in Symbolic
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* Computation. Springer (1999) 300-316.
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* Only the special case Q=1 is implemented
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*/
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template<typename Polynomial_traits_d,typename OutputIterator>
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OutputIterator prs_principal_sturm_habicht_sequence
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(typename Polynomial_traits_d::Polynomial_d P,
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OutputIterator out) {
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typedef typename Polynomial_traits_d::Coefficient_type NT;
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typename Polynomial_traits_d::Get_coefficient coeff;
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typename Polynomial_traits_d::Degree degree;
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std::vector<typename Polynomial_traits_d::Polynomial_d> stha;
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CGAL::internal::sturm_habicht_sequence<Polynomial_traits_d>
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(P,std::back_inserter(stha));
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for(int i=0; i<static_cast<int>(stha.size()); i++) {
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int d = degree(stha[i]);
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CGAL_assertion(d<=i);
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if(d<i) {
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*out++ = NT(0);
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} else {
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*out++ = coeff(stha[i],i);
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}
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}
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return out;
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}
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/*!
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* \brief compute the leading coefficients of the Sturm-Habicht sequence of
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* the polynomial <I>P</I>
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*
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* The principal Sturm-Habicht sequence is obtained by computing the scalar
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* subresultant sequence of <I>P</I> and its derivative, extended
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* by <I>P</I> and <I>P'</I> and some sign changes.
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*
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* For details, see: Gonzalez-Vega,Lombardi,Recio,Roy: Determinants and Real
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* Roots of Univariate Polynomials. Texts and Monographs in Symbolic
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* Computation. Springer (1999) 300-316.
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* Only the special case Q=1 is implemented
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*/
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template<typename Polynomial_traits_d,typename OutputIterator>
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OutputIterator bezout_principal_sturm_habicht_sequence
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(typename Polynomial_traits_d::Polynomial_d P,
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OutputIterator out) {
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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::Leading_coefficient lcoeff;
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typename Polynomial_traits_d::Differentiate diff;
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Polynomial Px = diff(P);
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CGAL::internal::Simple_matrix<NT> M
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= CGAL::internal::polynomial_subresultant_matrix<Polynomial_traits_d>
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(P,Px,1);
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int n = static_cast<int>(M.row_dimension());
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for(int i=0; i<n; i++) {
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if((n-1-i)%4==0 || (n-1-i)%4==1) {
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*out++ = -M[n-1-i][n-1-i];
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} else {
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*out++ = M[n-1-i][n-1-i];
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}
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}
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*out++=lcoeff(Px);
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*out++=lcoeff(P);
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return out;
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}
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/*!
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* \brief compute the principal and coprincipal Sturm-Habicht sequence
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*/
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template<typename Polynomial_traits_d,
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typename OutputIterator1,
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typename OutputIterator2>
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void prs_first_two_sturm_habicht_coefficients
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(typename Polynomial_traits_d::Polynomial_d P,
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OutputIterator1 pstha,
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OutputIterator2 copstha) {
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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::Get_coefficient coeff;
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typename Polynomial_traits_d::Degree degree;
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std::vector<Polynomial> stha;
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int n = degree(P);
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sturm_habicht_sequence<Polynomial_traits_d>(P,std::back_inserter(stha));
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CGAL_assertion(static_cast<int>(stha.size())==n+1);
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for(int i=0;i<=n;i++) {
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int d = degree(stha[i]);
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CGAL_assertion(d<=i);
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if(d<i) {
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*pstha++ = NT(0);
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} else {
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*pstha++ = coeff(stha[i],i);
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}
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}
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for(int i=1;i<=n;i++) {
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int d = degree(stha[i]);
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CGAL_assertion(d<=i);
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if(d<i-1) {
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*copstha++ = NT(0);
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} else {
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*copstha++ = coeff(stha[i],i-1);
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}
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}
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return;
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}
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/*! \brief compute the principal and coprincipal Sturm-Habicht sequence
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*/
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template<typename Polynomial_traits_d,
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typename OutputIterator1,
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typename OutputIterator2>
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void bezout_first_two_sturm_habicht_coefficients
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(typename Polynomial_traits_d::Polynomial_d P,
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OutputIterator1 pstha,
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OutputIterator2 copstha) {
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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::Get_coefficient coeff;
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typename Polynomial_traits_d::Degree degree;
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typename Polynomial_traits_d::Leading_coefficient lcoeff;
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typename Polynomial_traits_d::Differentiate diff;
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Polynomial Px=diff(P);
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CGAL::internal::Simple_matrix<NT> M
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= CGAL::internal::polynomial_subresultant_matrix<Polynomial_traits_d>
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(P,Px,2);
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int n = static_cast<int>(M.row_dimension());
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for(int i=0; i<n; i++) {
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if((n-1-i)%4==0 || (n-1-i)%4==1) {
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*pstha++ = -M[n-1-i][n-1-i];
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} else {
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*pstha++ = M[n-1-i][n-1-i];
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}
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}
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*pstha++ = lcoeff(Px);
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*pstha++ = lcoeff(P);
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for(int i=1; i<n; i++) {
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if(n-i-1%4==0 || n-i-1%4==1) {
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*copstha++ = -M[n-i-1][n-i];
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} else {
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*copstha++ = M[n-1-i][n-i];
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}
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}
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*copstha++ = coeff(Px,degree(Px)-1);
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*copstha++ = coeff(P,degree(P)-1);
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}
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// the general function for CGAL::Integral_domain_without_division_tag
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template < typename Polynomial_traits_d,
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typename OutputIterator1,
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typename OutputIterator2 > inline
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void
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first_two_sturm_habicht_coefficients_
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(typename Polynomial_traits_d::Polynomial_d A,
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OutputIterator1 pstha,
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OutputIterator2 copstha,
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CGAL::Integral_domain_without_division_tag){
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bezout_first_two_sturm_habicht_coefficients<Polynomial_traits_d>
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(A,pstha,copstha);
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}
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// the general function for CGAL::Integral_domain_tag
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template < typename Polynomial_traits_d,
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typename OutputIterator1,
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typename OutputIterator2 > inline
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void
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first_two_sturm_habicht_coefficients_
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(typename Polynomial_traits_d::Polynomial_d A,
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OutputIterator1 pstha,
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OutputIterator2 copstha,
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CGAL::Integral_domain_tag) {
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return prs_first_two_sturm_habicht_coefficients<Polynomial_traits_d>
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(A,pstha,copstha);
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}
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template < typename Polynomial_traits_d,
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typename OutputIterator1,
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typename OutputIterator2 > inline
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void
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first_two_sturm_habicht_coefficients_
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(typename Polynomial_traits_d::Polynomial_d A,
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OutputIterator1 pstha,
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OutputIterator2 copstha) {
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typedef typename Polynomial_traits_d::Coefficient_type NT;
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typedef typename
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CGAL::Algebraic_structure_traits<NT>::Algebraic_category
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Algebraic_category;
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first_two_sturm_habicht_coefficients_<Polynomial_traits_d>
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(A,pstha,copstha,Algebraic_category());
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}
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// the general function for CGAL::Integral_domain_without_division_tag
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template <typename Polynomial_traits_d,typename OutputIterator> inline
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OutputIterator
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principal_sturm_habicht_sequence_
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(typename Polynomial_traits_d::Polynomial_d A,
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OutputIterator out,
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CGAL::Integral_domain_without_division_tag) {
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return bezout_principal_sturm_habicht_sequence<Polynomial_traits_d>
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(A,out);
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}
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// the specialization for CGAL::Integral_domain_tag
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template <typename Polynomial_traits_d,typename OutputIterator> inline
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OutputIterator
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principal_sturm_habicht_sequence_
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(typename Polynomial_traits_d::Polynomial_d A,
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OutputIterator out,
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CGAL::Integral_domain_tag) {
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return prs_principal_sturm_habicht_sequence<Polynomial_traits_d>(A,out);
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}
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template <typename Polynomial_traits_d,typename OutputIterator> inline
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OutputIterator principal_sturm_habicht_sequence_
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(typename Polynomial_traits_d::Polynomial_d A,
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OutputIterator out) {
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typedef typename Polynomial_traits_d::Coefficient_type NT;
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typedef typename
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CGAL::Algebraic_structure_traits<NT>::Algebraic_category
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Algebraic_category;
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return principal_sturm_habicht_sequence_<Polynomial_traits_d>
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(A,out,Algebraic_category());
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}
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/*!
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* \brief computes the first two coefficients of each polynomial of
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* the Sturm-Habicht sequence.
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*
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* This function is needed in Curve_analysis_2 for certain genericity checks
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*/
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template < typename Polynomial_traits_d,
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typename OutputIterator1,
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typename OutputIterator2 > inline
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void first_two_sturm_habicht_coefficients
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(typename Polynomial_traits_d::Polynomial_d A,
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OutputIterator1 pstha,
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OutputIterator2 copstha){
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return CGAL::internal::first_two_sturm_habicht_coefficients_
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<Polynomial_traits_d> (A,pstha,copstha);
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}
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/*!
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* \brief compute the sequence of
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* principal Sturm-Habicht coefficients
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*/
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template <typename Polynomial_traits_d,typename OutputIterator> inline
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OutputIterator
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principal_sturm_habicht_sequence
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(typename Polynomial_traits_d::Polynomial_d A,
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OutputIterator out){
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return CGAL::internal::principal_sturm_habicht_sequence_
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<Polynomial_traits_d>(A,out);
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}
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/*!
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* \brief compute the Sturm-Habicht sequence
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*/
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template<typename Polynomial_traits_d,typename OutputIterator> OutputIterator
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sturm_habicht_sequence(typename Polynomial_traits_d::Polynomial_d P,
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OutputIterator out) {
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typedef typename Polynomial_traits_d::Polynomial_d Polynomial;
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typename Polynomial_traits_d::Degree degree;
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typename Polynomial_traits_d::Differentiate diff;
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int p = degree(P);
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Polynomial P_x = diff(P);
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std::vector<Polynomial> stha;
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CGAL::internal::polynomial_subresultants<Polynomial_traits_d>
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(P,P_x,std::back_inserter(stha));
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stha.push_back(P);
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CGAL_assertion(static_cast<int>(stha.size())==p+1);
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for(int i=0;i<=p; i++) {
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if((p-i)%4==0 || (p-i)%4==1) {
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*out++ = stha[i];
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} else {
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*out++ = -stha[i];
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}
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}
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return out;
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|
|
}
|
||
|
|
|
||
|
|
/*!
|
||
|
|
* \brief compute the Sturm-Habicht sequence with cofactors
|
||
|
|
*/
|
||
|
|
template<typename Polynomial_traits_d,
|
||
|
|
typename OutputIterator1,
|
||
|
|
typename OutputIterator2,
|
||
|
|
typename OutputIterator3>
|
||
|
|
OutputIterator1
|
||
|
|
sturm_habicht_sequence_with_cofactors
|
||
|
|
(typename Polynomial_traits_d::Polynomial_d P,
|
||
|
|
OutputIterator1 stha_out,
|
||
|
|
OutputIterator2 cof_out,
|
||
|
|
OutputIterator3 cofx_out) {
|
||
|
|
|
||
|
|
typedef typename Polynomial_traits_d::Polynomial_d Polynomial;
|
||
|
|
typename Polynomial_traits_d::Degree degree;
|
||
|
|
typename Polynomial_traits_d::Differentiate diff;
|
||
|
|
typename Polynomial_traits_d::Construct_polynomial construct;
|
||
|
|
|
||
|
|
int p = degree(P);
|
||
|
|
|
||
|
|
Polynomial P_x = diff(P);
|
||
|
|
|
||
|
|
std::vector<Polynomial> stha,co_f,co_fx;
|
||
|
|
|
||
|
|
CGAL::internal::polynomial_subresultants_with_cofactors<Polynomial_traits_d>
|
||
|
|
(P,P_x,
|
||
|
|
std::back_inserter(stha),
|
||
|
|
std::back_inserter(co_f),
|
||
|
|
std::back_inserter(co_fx));
|
||
|
|
|
||
|
|
stha.push_back(P);
|
||
|
|
co_f.push_back(construct(1));
|
||
|
|
co_fx.push_back(construct(0));
|
||
|
|
|
||
|
|
CGAL_assertion(static_cast<int>(stha.size())==p+1);
|
||
|
|
|
||
|
|
for(int i=0;i<=p; i++) {
|
||
|
|
if((p-i)%4==0 || (p-i)%4==1) {
|
||
|
|
*stha_out++ = stha[i];
|
||
|
|
*cof_out++ = co_f[i];
|
||
|
|
*cofx_out++ = co_fx[i];
|
||
|
|
} else {
|
||
|
|
*stha_out++ = -stha[i];
|
||
|
|
*cof_out++ = -co_f[i];
|
||
|
|
*cofx_out++ = -co_fx[i];
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
return stha_out;
|
||
|
|
}
|
||
|
|
|
||
|
|
} // namespace internal
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
/*!
|
||
|
|
* \brief returns the number of roots of a polynomial with given
|
||
|
|
* principal Sturm-Habicht sequence (counted without multiplicity)
|
||
|
|
*/
|
||
|
|
template<typename InputIterator>
|
||
|
|
int number_of_real_roots(InputIterator start,InputIterator end) {
|
||
|
|
if(start==end) {
|
||
|
|
return 0;
|
||
|
|
}
|
||
|
|
int m = 0;
|
||
|
|
|
||
|
|
CGAL::Sign last_non_zero=CGAL::ZERO; //marks the starting point
|
||
|
|
CGAL::Sign curr_sign;
|
||
|
|
int k;
|
||
|
|
InputIterator el=start;
|
||
|
|
|
||
|
|
//std::cout << "Sign of." << (*el) << std::endl;
|
||
|
|
|
||
|
|
curr_sign=CGAL::sign(*el);
|
||
|
|
|
||
|
|
while(curr_sign==CGAL::ZERO && el!=end) {
|
||
|
|
el++;
|
||
|
|
curr_sign=CGAL::sign(*el);
|
||
|
|
}
|
||
|
|
if(el==end) return 0;
|
||
|
|
|
||
|
|
last_non_zero=curr_sign;
|
||
|
|
k=0;
|
||
|
|
el++;
|
||
|
|
|
||
|
|
while(el!=end) {
|
||
|
|
curr_sign=CGAL::sign(*el);
|
||
|
|
|
||
|
|
el++;
|
||
|
|
|
||
|
|
if(curr_sign==CGAL::ZERO) {
|
||
|
|
k++;
|
||
|
|
}
|
||
|
|
else {
|
||
|
|
if(k%2==0) { // k is even
|
||
|
|
k=k/2;
|
||
|
|
int pm_one = (curr_sign==last_non_zero ? 1 : -1);
|
||
|
|
pm_one = (k%2==1) ? -pm_one : pm_one;
|
||
|
|
m+=pm_one;
|
||
|
|
}
|
||
|
|
k=0;
|
||
|
|
last_non_zero=curr_sign;
|
||
|
|
}
|
||
|
|
|
||
|
|
}
|
||
|
|
return m;
|
||
|
|
}
|
||
|
|
|
||
|
|
/*!
|
||
|
|
* \brief returns the number of roots of a polynomial
|
||
|
|
*/
|
||
|
|
template<typename Polynomial_d>
|
||
|
|
int number_of_real_roots(Polynomial_d f) {
|
||
|
|
|
||
|
|
typedef CGAL::Polynomial_traits_d<Polynomial_d> Poly_traits_d;
|
||
|
|
typedef typename Poly_traits_d::Coefficient_type Coeff;
|
||
|
|
std::vector<Coeff> stha;
|
||
|
|
typename Poly_traits_d::Principal_sturm_habicht_sequence()
|
||
|
|
(f,std::back_inserter(stha));
|
||
|
|
return CGAL::number_of_real_roots(stha.begin(),stha.end());
|
||
|
|
}
|
||
|
|
|
||
|
|
} //namespace CGAL
|
||
|
|
|
||
|
|
#endif // CGAL_POLYNOMIAL_STURM_HABICHT
|