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// Copyright (c) 1997 INRIA Sophia-Antipolis (France). // All rights reserved. // // This file is part of CGAL (www.cgal.org) // // $URL: https://github.com/CGAL/cgal/blob/v6.1/Polynomial/include/CGAL/Polynomial/square_free_factorize.h $ // $Id: include/CGAL/Polynomial/square_free_factorize.h b26b07a1242 $ // SPDX-License-Identifier: LGPL-3.0-or-later OR LicenseRef-Commercial // // // Author(s) : Michael Hemmer // // ============================================================================ #ifndef CGAL_POLYNOMIAL_SQUARE_FREE_FACTORIZATION_H #define CGAL_POLYNOMIAL_SQUARE_FREE_FACTORIZATION_H #include #include #include #include namespace CGAL { namespace internal { // square-free factorization // // the implementation uses two dispatches: // a) first look at the coefficient's algebra type // b) if the algebra type is of the concept field, try to decompose // the same holds for square-free factorization up to constant factors (utcf) // // sqff -------> algebra type ? ----field-----> decomposable ? // | A | | // UFD | no yes // | | | | // V | | // Yun's algo <---------------------- | // A | // | | | // UFD | V // | | decompose and use // sqff_utcf --> algebra_type ? ----field-- sqff_utcf with numerator // | // integral domain // | // V // Yun's algo (utcf) template inline int square_free_factorize(const IC&, OutputIterator1, OutputIterator2){ return 0; } template inline int square_free_factorize_for_regular_polynomial(const IC&, OutputIterator1, OutputIterator2){ return 0; } template inline int square_free_factorize(const Polynomial&, OutputIterator1, OutputIterator2); template inline int square_free_factorize_for_regular_polynomial(const Polynomial&, OutputIterator1, OutputIterator2); template inline int square_free_factorize_for_regular_polynomial_(const Polynomial&, OutputIterator1, OutputIterator2, CGAL::Tag_true); template inline int square_free_factorize_for_regular_polynomial_(const Polynomial&, OutputIterator1, OutputIterator2, CGAL::Tag_false); template inline int square_free_factorize_for_regular_polynomial_(const Polynomial&, OutputIterator1, OutputIterator2, Integral_domain_tag); template inline int square_free_factorize_for_regular_polynomial_(const Polynomial&, OutputIterator1, OutputIterator2, Unique_factorization_domain_tag); template inline int square_free_factorize (const Polynomial& poly, OutputIterator1 factors, OutputIterator2 multiplicities) { typedef Polynomial POLY; typedef Polynomial_traits_d< POLY > PT; typedef typename PT::Construct_polynomial Construct_polynomial; typedef typename PT::Univariate_content_up_to_constant_factor Ucont_utcf; typedef typename PT::Integral_division_up_to_constant_factor Idiv_utcf; if (typename PT::Total_degree()(poly) == 0){return 0;} Coeff ucont_utcf = Ucont_utcf()(poly); POLY regular_poly = Idiv_utcf()(poly,Construct_polynomial()(ucont_utcf)); int result = square_free_factorize_for_regular_polynomial( regular_poly, factors, multiplicities); if (CGAL::total_degree(ucont_utcf) > 0){ typedef std::vector< Coeff > Factors_uc; typedef std::vector< int > Multiplicities_uc; Factors_uc factors_uc; Multiplicities_uc multiplicities_uc; result += square_free_factorize( ucont_utcf, std::back_inserter(factors_uc), std::back_inserter(multiplicities_uc) ); for( typename Factors_uc::iterator it = factors_uc.begin(); it != factors_uc.end(); ++it ){ *factors++ = Construct_polynomial()(*it); } for( Multiplicities_uc::iterator it = multiplicities_uc.begin(); it != multiplicities_uc.end(); ++it ){ *multiplicities++ = (*it); } } return result; } template inline int square_free_factorize_for_regular_polynomial (const Polynomial& p, OutputIterator1 factors, OutputIterator2 multiplicities){ typedef Polynomial POLY; typedef typename CGAL::Fraction_traits::Is_fraction Is_fraction; return square_free_factorize_for_regular_polynomial_(p,factors,multiplicities,Is_fraction()); } template inline int square_free_factorize_for_regular_polynomial_ (const Polynomial& p, OutputIterator1 factors, OutputIterator2 multiplicities, CGAL::Tag_true){ typedef Polynomial POLY; typedef Polynomial_traits_d< POLY > PT; typedef Fraction_traits FT; typename FT::Numerator_type num; typename FT::Denominator_type denom; typename FT::Decompose()(p,num,denom); std::vector ifacs; int result = square_free_factorize_for_regular_polynomial(num,std::back_inserter(ifacs),multiplicities); typedef typename std::vector::iterator Iterator; denom = typename FT::Denominator_type(1); for ( Iterator it = ifacs.begin(); it != ifacs.end(); ++it) { POLY q = typename FT::Compose()(*it, denom); *factors++ = typename PT::Canonicalize()(q); } return result; } template inline int square_free_factorize_for_regular_polynomial_ (const Polynomial& p, OutputIterator1 factors, OutputIterator2 multiplicities, CGAL::Tag_false){ typedef Polynomial POLY; typedef typename Algebraic_structure_traits::Algebraic_category Algebraic_category; return square_free_factorize_for_regular_polynomial_(p,factors,multiplicities,Algebraic_category()); } template inline int square_free_factorize_for_regular_polynomial_ (const Polynomial& p, OutputIterator1 factors, OutputIterator2 multiplicities, Integral_domain_tag){ // Yun's Square-Free Factorization // see [Geddes et al, 1992], Algorithm 8.2 typedef Polynomial POLY; typedef Polynomial_traits_d PT; typedef typename PT::Innermost_coefficient_type IC; typename PT::Innermost_leading_coefficient ilcoeff; //typename PT::Innermost_coefficient_to_polynomial ictp; typename PT::Construct_innermost_coefficient_const_iterator_range range; typename Algebraic_extension_traits::Denominator_for_algebraic_integers dfai; typename Algebraic_extension_traits::Normalization_factor nfac; typename Scalar_factor_traits::Scalar_factor sfac; typename Scalar_factor_traits::Scalar_div sdiv; typedef typename Scalar_factor_traits::Scalar Scalar; if (typename PT::Total_degree()(p) == 0) return 0; POLY a = CGAL::canonicalize(p); POLY b = CGAL::differentiate(a); POLY c = CGAL::internal::gcd_utcf_(a, b); if (c == Coeff(1)) { *factors = a; *multiplicities = 1; return 1; } int i = 1, n = 0; // extending both polynomials a and b by the denominator for algebraic // integers, which comes out from c=gcd(a,b), such that a and b are // divisible by c IC lcoeff = ilcoeff(c); IC denom = dfai(range(c).first, range(c).second); lcoeff *= denom * nfac(denom); POLY w = (a * POLY(lcoeff)) / c; POLY y = (b * POLY(lcoeff)) / c; // extracting a common scalar factor out of w=a/c and y=b/c simultaneously, // such that the parts in z=y-w' are canceled out as they should Scalar sfactor = sfac(y,sfac(w)); sdiv(w, sfactor); sdiv(y, sfactor); POLY z = y - CGAL::differentiate(w); POLY g; while (!z.is_zero()) { g = CGAL::internal::gcd_utcf_(w, z); if (g.degree() > 0) { *factors++ = g; *multiplicities++ = i; n++; } i++; lcoeff = ilcoeff(g); // same as above denom =dfai(range(c).first, range(c).second); lcoeff *= denom * nfac(denom); w = (w * POLY(lcoeff)) / g; y = (z * POLY(lcoeff)) / g; Scalar sfactor = sfac(y,sfac(w)); sdiv(w, sfactor); sdiv(y, sfactor); z = y - CGAL::differentiate(w); } *factors = w; *multiplicities++ = i; n++; return n; } template inline int square_free_factorize_for_regular_polynomial_ (const Polynomial& p, OutputIterator1 factors, OutputIterator2 multiplicities, Unique_factorization_domain_tag){ // Yun's Square-Free Factorization // see [Geddes et al, 1992], Algorithm 8.2 /* @inproceedings{y-osfda-76, author = {David Y.Y. Yun}, title = {On square-free decomposition algorithms}, booktitle = {SYMSAC '76: Proceedings of the third ACM symposium on Symbolic and algebraic computation}, year = {1976}, pages = {26--35}, location = {Yorktown Heights, New York, United States}, doi = {https://dl.acm.org/citation.cfm?doid=800205.806320}, publisher = {ACM Press}, address = {New York, NY, USA}, } */ typedef Polynomial POLY; typedef Polynomial_traits_d PT; if (typename PT::Total_degree()(p) == 0) return 0; POLY a = CGAL::canonicalize(p); POLY b = CGAL::differentiate(a); POLY c = CGAL::gcd(a, b); if (c == Coeff(1)) { *factors = a; *multiplicities = 1; return 1; } int i = 1, n = 0; POLY w = a/c, y = b/c, z = y - CGAL::differentiate(w), g; while (!z.is_zero()) { g = CGAL::gcd(w, z); if (g.degree() > 0) { *factors++ = g; *multiplicities++ = i; n++; } i++; w /= g; y = z/g; z = y - CGAL::differentiate(w); } *factors = w; *multiplicities++ = i; n++; return n; } // square-free factorization utcf template inline int square_free_factorize_utcf (const Polynomial& p, OutputIterator1 factors, OutputIterator2 multiplicities) { return square_free_factorize(p,factors,multiplicities); } template inline int square_free_factorize_utcf_for_regular_polynomial (const Polynomial& p, OutputIterator1 factors, OutputIterator2 multiplicities) { return square_free_factorize_for_regular_polynomial(p,factors,multiplicities); } // filtered square-free factorization // ### filtered versions #### /*! \brief As NiX::square_free_factorize, but filtered by * NiX::may_have_multiple_root * * Use this function if the polynomial might be square free. */ template inline int filtered_square_free_factorize( Polynomial p, OutputIterator1 factors, OutputIterator2 multiplicities) { if(CGAL::internal::may_have_multiple_factor(p)){ return internal::square_free_factorize(p, factors, multiplicities); }else{ *factors++ = CGAL::canonicalize(p); *multiplicities++ = 1; return 1; } } /*! \brief As NiX::square_free_factorize_utcf, but filtered by * NiX::may_have_multiple_root * * Use this function if the polynomial might be square free. */ template inline int filtered_square_free_factorize_utcf( const Polynomial& p, OutputIterator1 factors, OutputIterator2 multiplicities) { return filtered_square_free_factorize(p,factors,multiplicities); } } // namespace internal } //namespace CGAL #endif // CGAL_POLYNOMIAL_SQUARE_FREE_FACTORIZATION_H