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// Copyright (c) 2006-2009 Max-Planck-Institute Saarbruecken (Germany). // All rights reserved. // // This file is part of CGAL (www.cgal.org) // // $URL: https://github.com/CGAL/cgal/blob/v6.1/Algebraic_kernel_d/include/CGAL/Algebraic_kernel_d/refine_zero_against.h $ // $Id: include/CGAL/Algebraic_kernel_d/refine_zero_against.h b26b07a1242 $ // SPDX-License-Identifier: LGPL-3.0-or-later OR LicenseRef-Commercial // // // Author(s) : Michael Hemmer // // ============================================================================ // TODO: The comments are all original EXACUS comments and aren't adapted. So // they may be wrong now. #ifndef CGAL_REFINE_ZERO_AGAINST_H #define CGAL_REFINE_ZERO_AGAINST_H #include #include namespace CGAL { namespace internal { /* computes an upper bound on the number of roots ]low,high[ using Descartes' * Sign Rule */ template int descartes(Polynomial& p, const Field& low,const Field& high){ // decompose interval length and upper bound CGAL_precondition(low::Numerator_type Numerator; typedef typename Fraction_traits::Denominator_type Denominator; typename Fraction_traits::Decompose decomp; //typename Algebraic_structure_traits::Simplify simplify; //simplify(low); //simplify(high); Numerator num_high, num_low_sub_high; Denominator den_high, den_low_sub_high; decomp(high, num_high, den_high); decomp(low - high, num_low_sub_high, den_low_sub_high); Coefficient tmp(num_high); // apply Descartes' rule to count zeros of q in ]low,high[ Polynomial transf = // q(high + (low-high)/(1+x)) CGAL::translate_by_one( CGAL::reversal( CGAL::scale_homogeneous( CGAL::translate_homogeneous(p ,Coefficient(num_high) ,Coefficient(den_high)) ,Coefficient(num_low_sub_high) ,Coefficient(den_low_sub_high) ) ) ); return sign_variations(transf); } /*! \ingroup \NiX_univariate_polynomial_utils * \brief refine isolating interval for \c p w.r.t. \c q * * This function refines the interval ]low, high[ * such that it does not contain any zero of \c q different from the * unique zero of \c p in ]low, high[. It is returned * whether \c q has a zero in ]low, high[ equal to * that of \c p . Note that zeroes of \c q at the boundaries are * ignored. * * This function is implemented using bisection and Descartes' Rule. * If the interval boundaries have denominators 2k, then * this property will still hold after refinement. Although this * function works similar to \c NiX::Algebraic_real<>.compare() , * it is different insofar that it always maintains an open interval * and never simplifies. * * \pre Both polynomials must be square-free. \c p must not vanish at the * interval boundaries \c low and \c high . * * \todo Provide a means to let an \c NiX::Algebraic_real benefit * from the interval refinement if it is the origin of the respective * input data. */ template bool refine_zero_against(Field& low, Field& high, Polynomial p, Polynomial q) { typename Algebraic_structure_traits::Simplify simplify; CGAL_precondition(low < high); CGAL_precondition(CGAL::degree(p) > 0); CGAL_precondition((CGAL::degree(q) >= 0) && !q.is_zero()); if (CGAL::degree(q) == 0) return false; CGAL::Sign sign_p_low = p.sign_at(low); CGAL_assertion_code(CGAL::Sign sign_p_high = p.sign_at(high)); CGAL_precondition(sign_p_low != CGAL::ZERO); CGAL_precondition(sign_p_high != CGAL::ZERO); CGAL_precondition(sign_p_high != sign_p_low); Polynomial gcd_pq; // computed below if necessary for (;;) { int sv = internal::descartes(q,low,high); CGAL_assertion(sv >= 0); if (sv == 0) { // q has no zero in ]low,high[ return false; } else if (sv == 1) { if (CGAL::degree(gcd_pq) < 0) { if (may_have_common_factor(p, q)) { gcd_pq = gcd_utcf(p, q); } else { gcd_pq = Polynomial(1); } } std::cout << CGAL::to_double(low) << " " << CGAL::to_double(high) << " " << CGAL::degree(gcd_pq) << " " << gcd_pq << std::endl; if (CGAL::degree(gcd_pq) > 0 // constant poly cannot change sign && gcd_pq.sign_at(low) != gcd_pq.sign_at(high)) { // q has exactly one zero in ]low,high[ // and it's equal to that of p return true; } } // q may have a zero in ]low,high[ not equal to that of p Field mid = (low+high)/Field(2); CGAL::Sign s = p.sign_at(mid); if (s == CGAL::ZERO) { mid = (low+mid)/Field(2); simplify(mid); s = p.sign_at(mid); } CGAL_postcondition(s != CGAL::ZERO); if (s == sign_p_low) { low = mid; sign_p_low = s; } else { CGAL_assertion(s == sign_p_high); high = mid; CGAL_assertion_code(sign_p_high = s); } } } // Uses refine_zero_against first and refines the interval further, if any // of the interval boarders has sign zero. template < class Polynomial, class Field > static bool strong_refine_zero_against(Field& low, Field& high, Polynomial p, Polynomial q){ std::cout << "comp has_common_root" << std::endl; bool has_common_root = refine_zero_against(low,high,p,q); std::cout << "done, " << has_common_root << std::endl; CGAL::Sign sign_p_low = p.sign_at(low); CGAL_assertion_code(CGAL::Sign sign_p_high = p.sign_at(high)); Field mid; CGAL::Sign s; while ((q.sign_at(low)==CGAL::ZERO)||(q.sign_at(high)==CGAL::ZERO)) { mid = (low+high)/Field(2); simplify(mid); s = p.sign_at(mid); if (s == CGAL::ZERO) { mid = (low+mid)/Field(2); simplify(mid); s = p.sign_at(mid); } CGAL_postcondition(s != CGAL::ZERO); if (s == sign_p_low) { low = mid; sign_p_low = s; //bogus? } else { CGAL_assertion(s == sign_p_high); high = mid; CGAL_assertion_code(sign_p_high = s); //bogus? } } return has_common_root; } } //namespace internal } //namespace CGAL #endif //CGAL_REFINE_ZERO_AGAINST_H