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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/Algebraic_real_rep.h $ // $Id: include/CGAL/Algebraic_kernel_d/Algebraic_real_rep.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_ALGEBRAIC_KERNEL_D_ALGEBRAIC_REAL_REP_H #define CGAL_ALGEBRAIC_KERNEL_D_ALGEBRAIC_REAL_REP_H #include #include #include #include #include namespace CGAL { namespace internal { // definition of the Algebraic_real_rep x: //IS_GENERAL: // low_ lower bound of x // high_ upper bound of x // polynomial_ a square free polynomial // sign_at_low_ = polynomial_.evaluate(low_) // x is the only root of polynomial_ in the open interval ]low_,high_[ // low_ != x != high // ******************* EXCEPTION ******************* // x is rational: in this case low=high=x template< class Coefficient_, class Rational_> class Algebraic_real_rep{ public: typedef Coefficient_ Coefficient; typedef Rational_ Bound; typedef typename CGAL::Polynomial_type_generator::Type Polynomial_1; private: typedef Rational_ Rational; typedef typename CGAL::Polynomial_type_generator::Type Poly; typedef CGAL::Sign TRI_BOOL; typedef Algebraic_real_rep Self; public: typedef std::optional< std::pair > Interval_option; mutable Poly polynomial_; //!< square free polynomial mutable Rational low_; //!< lower endpoint of interval mutable Rational high_; //!< upper endpoint of interval mutable CGAL::Sign sign_at_low_; //!< sign of polynomial a \a low_ mutable Interval_option interval_option; mutable bool is_rational_; mutable const Self *next; mutable const Self *prev; private: void copy_all_members(const Algebraic_real_rep& y) const { polynomial_ = y.polynomial_; low_ = y.low_; high_ = y.high_; sign_at_low_ = y.sign_at_low_; interval_option = y.interval_option; is_rational_ = y.is_rational_; } protected: virtual CGAL::Sign sign_of_polynomial_at( const Rational& f ) const { return polynomial().sign_at( f ); } protected: // joins the two lists of related algebraic reals void introduce(const Algebraic_real_rep& y) const{ y.prev->next=next; next->prev=y.prev; next=&y; y.prev=this; } protected: // x leaves the list of related algebraic reals void erase_from_list() const { next->prev=prev; prev->next=next; next=prev=this; } protected: virtual void set_explicit_rep(const Rational& m) const { typename Fraction_traits::Decompose decomp; typename Fraction_traits::Numerator_type num; typename Fraction_traits::Denominator_type den; decomp(m, num, den); polynomial_ = Poly(-Coefficient(num),Coefficient(den)); is_rational_ = true; low_ = m; high_ = m; sign_at_low_= CGAL::ZERO; interval_option = Interval_option(CGAL::to_interval(m)); } protected: virtual void set_implicit_rep(const Poly & P, const Rational& LOW, const Rational& HIGH, bool use_expensive_sign = false ) const { CGAL_precondition(LOW < HIGH); CGAL_precondition(P.sign_at(LOW) != CGAL::ZERO); CGAL_precondition(P.sign_at(HIGH) != CGAL::ZERO); CGAL_precondition(P.sign_at(HIGH) != P.sign_at(LOW)); CGAL_precondition_msg(typename CGAL::Polynomial_traits_d< Poly >::Is_square_free()(P), "Polynomial not square-free."); polynomial_ = P; low_=LOW; high_=HIGH; sign_at_low_=(use_expensive_sign ? P.sign_at(LOW) : sign_of_polynomial_at(LOW) ); is_rational_=false; // interval_option left out // tries to set rational if degree is 1 typedef typename CGAL::Coercion_traits< Coefficient, Rational >::Type RET; set_rational(RET()); } private: template < class NTX > void set_rational(NTX) const { } void set_rational(Rational) const { if (CGAL::degree(polynomial()) == 1) { low_ = high_ = -Rational(polynomial()[0])/Rational(polynomial()[1]); is_rational_ = true; interval_option = Interval_option(CGAL::to_interval(rational())); } } public: //! creates the algebraic real from int \a i. explicit Algebraic_real_rep(int i = 0){ next=prev=this; set_explicit_rep(i); } //! creates the algebraic real from Rational \a m. explicit Algebraic_real_rep(const Rational& m){ next=prev=this; set_explicit_rep(m); } /*! \brief creates the algebraic real as the unique root of \a P in the open interval ]low,high[. \pre the polynomial \a P is square free \pre P(low)!=0 \pre P(high)<0 \pre x is the one and only root in the open interval of \a P. */ Algebraic_real_rep(const Poly& P, Rational LOW, Rational HIGH) { if (LOW == HIGH) { CGAL_precondition(P.sign_at(LOW) == CGAL::ZERO); set_explicit_rep(LOW); } else { CGAL_precondition(LOW < HIGH); CGAL_precondition(P.sign_at(LOW) != CGAL::ZERO); CGAL_precondition(P.sign_at(HIGH) != CGAL::ZERO); CGAL_precondition(P.sign_at(HIGH) != P.sign_at(LOW)); CGAL_precondition_msg(typename CGAL::Polynomial_traits_d::Is_square_free()(P), "Polynomial not square-free."); set_implicit_rep(P,LOW,HIGH); } next=prev=this; } //! copy constructor Algebraic_real_rep(const Self& y){ next=prev=this; copy_all_members(y); introduce(y); } //! destructor virtual ~Algebraic_real_rep(){ erase_from_list(); } // assignment Algebraic_real_rep& operator=(const Self& y) { if ( this != & y) { erase_from_list(); copy_all_members(y); introduce(y); } CGAL_expensive_postcondition(self_test()); return *this; } public: const Rational& low() const {return low_;} const Rational& high() const {return high_;} const Poly& polynomial() const {return polynomial_;} const TRI_BOOL& sign_at_low() const {return sign_at_low_;} /*const std::pair< double, double>& interval() const { if( interval_option ) { return *interval_option; } else { typedef typename Get_arithmetic_kernel< Coefficient >::Arithmetic_kernel::Bigfloat BF; //typedef typename LEDA_arithmetic_kernel::Bigfloat BF; long old_precision = get_precision( BF() ); set_precision( BF(), 53 ); interval_option = Interval_option( internal::to_interval( convert_to_bfi( (*this) ) ) ); set_precision( BF(), old_precision ); return *interval_option; } }*/ /*const std::pair& interval() const { if(interval_option){ return *interval_option; }else{ interval_option = Interval_option( compute_interval_approximation(polynomial(),low(),high())); return *interval_option; } }*/ bool is_rational() const {return is_rational_;} const Rational& rational() const { CGAL_precondition(is_rational()); CGAL_precondition(low()==high()); CGAL_precondition(polynomial().sign_at(low())==CGAL::ZERO); return low_; } /*! \brief convert algebraic real to type double */ /*double to_double() const { return (to_interval().first +to_interval().second)/2; }*/ /*! \brief convert algebraic real to type Interval */ /*std::pair to_interval() const { CGAL_precondition(compare(Rational(interval().first)) != CGAL::SMALLER); CGAL_precondition(compare(Rational(interval().second)) != CGAL::LARGER); return interval(); }*/ private: bool self_test() const{ if(!is_rational()){ CGAL_precondition(low() void strong_refine(const NTX& m) const{ if(is_rational()) return; if( NTX(low()) <= m && m <= NTX(high()) ){ CGAL_precondition(polynomial().sign_at(m)!=CGAL::ZERO); refine(); while( NTX(low()) <= m && m <= NTX(high())) refine(); } } public: virtual void refine_at(const Rational& m) const{ if(is_rational()) return; if( m <= low() || high() <= m ) return; // now: low < m < high CGAL::Sign s = sign_of_polynomial_at(m); if(s == CGAL::ZERO) learn_from(m); else (s == sign_at_low())? low_ = m : high_=m; } public: CGAL::Comparison_result compare(const Rational& y, bool are_distinct = false) const { if(is_rational()) return CGAL::compare(rational(),y); if(y <= low()) { strong_refine(y); return CGAL::LARGER; } if(high() <= y) { strong_refine(y); return CGAL::SMALLER; } // now: low < y < high if(!are_distinct){ if(sign_of_polynomial_at(y)==CGAL::ZERO){ learn_from(y); return CGAL::EQUAL ; } }else{ CGAL_precondition(polynomial().sign_at(y)!=CGAL::ZERO); } strong_refine(y); CGAL_postcondition(y < low() || high() < y ); if(y < low()) return CGAL::LARGER; else return CGAL::SMALLER; } CGAL::Comparison_result compare (const Algebraic_real_rep& y, bool are_distinct = false) const{ if( is_rational()) return -y.compare( rational()); if(y.is_rational()) return compare(y.rational()); // type of both x and y IS_GENERAL if ( high() <= y.low() ) return CGAL::SMALLER; if ( low() >= y.high()) return CGAL::LARGER; // intersection isolating intervals is ]L,R[ Rational L = (low() > y.low() ) ? low() : y.low() ; Rational R = (high() < y.high()) ? high() : y.high() ; // refine to smaller intervals at intersection interval boundaries // this can change type() only to IS_RATIONAL refine_at(L); refine_at(R); y.refine_at(L); y.refine_at(R); if ( is_rational()) return -y.compare( rational()); if (y.is_rational()) return compare(y.rational()); // type of both x and y still IS_GENERAL if ( high() <= y.low() ) return CGAL::SMALLER; if ( y.high() <= low() ) return CGAL::LARGER; // filter 2: probabilistically check coprimality if (!are_distinct) { are_distinct = !(may_have_common_factor(polynomial(), y.polynomial())); } if (!are_distinct) { // OK, filters failed. So we have to do the actual work // and compute the gcd of the defining polynomials. // we have ]low(), high()[ == ]y.low(),y.high()[ == ]L,R[ // and let both numbers decide for the gcd or its complement Poly F1,F2,G; G = CGAL::gcd_up_to_constant_factor(polynomial(),y.polynomial()); F1 = integral_division_up_to_constant_factor(polynomial(),G); CGAL_postcondition(CGAL::degree(F1)== CGAL::degree(polynomial())-CGAL::degree(G)); F2 = integral_division_up_to_constant_factor(y.polynomial(),G); CGAL_postcondition(CGAL::degree(F2)== CGAL::degree(y.polynomial())-CGAL::degree(G)); learn_from(G,F1); y.learn_from(G,F2); // this may simplify them due to degree loss if (y.is_rational()) return compare(y.rational()); if ( is_rational()) return -y.compare( rational()); // type of x and y is still IS_GENERAL // check for equality if (G.sign_at(L)!=G.sign_at(R)){ introduce(y); return CGAL::EQUAL; } } // if we are here, we know the numbers to be distinct // refine to disjointness for (;;) { y.refine(); refine(); if (y.is_rational()) return compare(y.rational()); if ( is_rational()) return -y.compare( rational()); if ( high() <= y.low()) return CGAL::SMALLER; if (y.high() <= low()) return CGAL::LARGER; } } protected: void learn_from(const Rational& m) const { if(is_rational()){ CGAL_precondition(rational()==m); return; } CGAL_precondition(polynomial().sign_at(m)==CGAL::ZERO); typename Fraction_traits::Decompose decomp; typename Fraction_traits::Numerator_type num; typename Fraction_traits::Denominator_type den; decomp(m, num, den); Poly G = Poly(-Coefficient(num),Coefficient(den)); Poly F1= integral_division_up_to_constant_factor(polynomial(),G); CGAL_postcondition(CGAL::degree(F1)== CGAL::degree(polynomial())-CGAL::degree(G)); learn_from(G,F1); if(!is_rational()) set_explicit_rep(m); } // splits the list of related algebraic reals according to the two // factors of the polynomial void learn_from(const Poly& pfactor1, const Poly& pfactor2) const { if(CGAL::degree(pfactor1)==0) return; if(CGAL::degree(pfactor2)==0) return; CGAL_precondition(may_have_common_factor(polynomial(),pfactor1)); CGAL_precondition(may_have_common_factor(polynomial(),pfactor2)); CGAL_precondition(CGAL::degree(pfactor1)+CGAL::degree(pfactor2) == CGAL::degree(polynomial())); Self dummy1, dummy2; Self const *last=prev; Self const *current=this; Self const *next; while(true){ next=current->next; if(pfactor1.sign_at(current->low())!= pfactor1.sign_at(current->high())){ current->next=&dummy1; current->prev=dummy1.prev; dummy1.prev->next=current; dummy1.prev=current; current->set_implicit_rep(pfactor1, current->low(), current->high(), true ); }else{ current->next=&dummy2; current->prev=dummy2.prev; dummy2.prev->next=current; dummy2.prev=current; current->set_implicit_rep(pfactor2, current->low(), current->high(), true ); } if(current==last) break; current=next; } } public: bool is_root_of(const Poly& Q) const { if (CGAL::degree(Q) == 0) return Q.is_zero(); if (is_rational() ) return CGAL::ZERO == Q.sign_at(rational()); if ( may_have_common_factor(polynomial(), Q) ) { Poly G = CGAL::gcd_up_to_constant_factor(polynomial(),Q); if(CGAL::degree(G)!=0){ Poly F1 = integral_division_up_to_constant_factor( polynomial(),G ); CGAL_postcondition(CGAL::degree(F1)== CGAL::degree(polynomial())-CGAL::degree(G)); learn_from(G,F1); return (G.sign_at(low()) != G.sign_at(high())); } } return false; } public: virtual void simplify() const{ if(is_rational()){ }else{ CGAL::simplify(low_); CGAL::simplify(high_); polynomial_ = typename Polynomial_traits_d::Canonicalize()(polynomial_); } } }; } // namespace internal } //namespace CGAL #include #endif // CGAL_ALGEBRAIC_KERNEL_D_ALGEBRAIC_REAL_REP_H