mirror of https://github.com/CGAL/cgal
1st revision
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Efi Fogel
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namespace CGAL {
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/*! \ingroup PkgArrangementOnSurface2TraitsClasses
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*
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* `CORE_algebraic_number_traits` is a traits class for CORE's algebraic
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* number types.
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*
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* \sa `Arr_conic_traits_2<RatKernel, AlgKernel, NtTraits>`
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*/
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class CORE_algebraic_number_traits {
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public:
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/// \name Types
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/// @{
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//! The integer number type.
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typedef CORE::BigInt Integer;
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//! The rational number type.
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typedef CORE::BigRat Rational;
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//! The polynomial type.
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typedef CORE::Polynomial<Integer> Polynomial;
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//! The algebraic number type.
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typedef CORE::Expr Algebraic;
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/// @}
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/// \name Utility Functions
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/// @{
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/*! Obtain the numerator of a rational number.
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* \param q A rational number.
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* \return The numerator of q.
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*/
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Integer numerator(const Rational& q) const;
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/*! Obtain the denominator of a rational number.
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* \param q A rational number.
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* \return The denominator of q.
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*/
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Integer denominator(const Rational& q) const;
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/*! Convert an integer to an algebraic number.
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* \param z An integer.
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* \return The algebraic number equivalent to z.
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*/
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Algebraic convert(const Integer& z) const;
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/*! Convert a rational number to an algebraic number.
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* \param q A rational number.
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* \return The algebraic number equivalent to q.
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*/
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Algebraic convert(const Rational& q) const;
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/*! Construct a rational number that lies strictly between two algebraic
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* values.
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* \param x1 The first algebraic value.
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* \param x2 The second algebraic value.
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* \pre The two values are not equal.
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* \return A rational number that lies in the open interval (x1, x2).
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*/
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Rational rational_in_interval(const Algebraic& x1, const Algebraic& x2) const;
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/*! Obtain a range of double-precision floats that contains the given
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* algebraic number.
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* \param x The given number.
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* \return A pair <x_lo, x_hi> that contain x.
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*/
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std::pair<double, double> double_interval(const Algebraic& x) const;
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/*! Convert a sequence of rational coefficients to an equivalent sequence
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* of integer coefficients. If the input coefficients are q(1), ..., q(k),
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* where q(i) = n(i)/d(i) then the output coefficients will be of the
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* form:
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* n(i) * lcm {d(1), ... , d(k)}
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* a(i) = -------------------------------
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* d(i) * gcd {n(1), ... , n(k)}
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*
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* \param q_begin The begin iterator of the rational sequence.
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* \param q_end The past-the-end iterator of the rational sequence.
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* \param zoi An output iterator for the integer coefficients.
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* \return A past-the-end iterator for the output container.
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* \pre The value type of q_begin and q_end is `Rational`, and
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* the value type of zoi is `Integer`.
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*/
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template <typename InputIterator, typename OutputIterator>
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OutputIterator convert_coefficients(InputIterator q_begin,
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InputIterator q_end,
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OutputIterator zoi) const;
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/*! Compute the square root of an algebraic number.
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* \param x The number.
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* \return The square root of x.
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* \pre x is non-negative.
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*/
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Algebraic sqrt(const Algebraic& x) const;
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/*! Compute the roots of a quadratic equations \f$a*x^2+ b*x + c = 0\f$
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* with integer coefficients.
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* \param a The coefficient of \f$x^2\f$
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* \param b The coefficient of \f$x\f$
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* \param c The free term.
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* \param oi An output iterator for the real-valued solutions of the
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* quadratic equation.
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* \return A past-the-end iterator for the output container.
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* \pre The value type of oi is Algebraic.
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*/
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template <typename NT, typename OutputIterator>
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OutputIterator solve_quadratic_equation(const NT& a, const NT& b, const NT& c,
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OutputIterator oi) const;
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/*! Construct a polynomial with integer coefficients.
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* \param coeffs The coefficients of the input polynomial.
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* \param degree The degree of the input polynomial.
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* \return The polynomial.
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*/
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Polynomial construct_polynomial(const Integer* coeffs,
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unsigned int degree) const;
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/*! Construct a polynomial with integer coefficients given rational
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* coefficients.
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* \param coeffs The coefficients of the input polynomial.
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* \param degree The degree of the input polynomial.
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* \param poly Output: The resulting polynomial with integer coefficients.
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* \param poly_denom Output: The denominator for the polynomial.
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* \return Whether this polynomial is non-zero (false if the polynomial is
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* zero).
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*/
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bool construct_polynomial(const Rational *coeffs,
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unsigned int degree,
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Polynomial& poly, Integer& poly_denom) const;
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/*! Construct two polynomials with integer coefficients such that P(x)/Q(x)
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* is a rational function equivalent to the one represented by the two
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* given vectors of rational coefficients. It is guaranteed that the GCD
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* of P(x) and Q(x) is trivial.
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* \param p_coeffs The coefficients of the input numerator polynomial.
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* \param p_degree The degree of the input numerator polynomial.
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* \param q_coeffs The coefficients of the input denominator polynomial.
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* \param q_degree The degree of the input denominator polynomial.
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* \param p_poly Output: The resulting numerator polynomial with integer
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* coefficients.
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* \param q_poly Output: The resulting denominator polynomial with integer
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* coefficients.
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* \return (true) on success; (false) if the denominator is 0.
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*/
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bool construct_polynomials(const Rational* p_coeffs,
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unsigned int p_degree,
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const Rational* q_coeffs,
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unsigned int q_degree,
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Polynomial& p_poly, Polynomial& q_poly) const;
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/*! Compute the degree of a polynomial.
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*/
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int degree(const Polynomial& poly) const;
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/*! Evaluate a polynomial at a given x-value.
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* \param poly A polynomial.
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* \param x The value to evaluate at.
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* \return The value of the polynomial at x.
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*/
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template <typename NT>
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NT evaluate_at(const Polynomial& poly, NT& x) const;
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/*! Compute the derivative of the given polynomial.
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* \param poly The polynomial p(x).
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* \return The derivative p'(x).
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*/
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Polynomial derive(const Polynomial& poly) const;
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/*! Multiply a polynomial by some scalar coefficient.
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* \param poly The polynomial P(x).
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* \param a The scalar value.
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* \return The scalar multiplication a*P(x).
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*/
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Polynomial scale(const Polynomial& poly, const Integer& a) const;
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/*! Perform "long division" of two polynomials: Given A(x) and B(x) compute
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* two polynomials Q(x) and R(x) such that: A(x) = Q(x)*B(x) + R(x) and
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* R(x) has minimal degree.
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* \param polyA The first polynomial A(x).
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* \param polyB The second polynomial A(x).
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* \param rem Output: The remainder polynomial R(x).
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* \return The quontient polynomial Q(x).
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*/
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Polynomial divide(const Polynomial& polyA,
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const Polynomial& polyB,
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Polynomial& rem) const;
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/*! Compute the real-valued roots of a polynomial with integer coefficients,
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* sorted in ascending order.
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* \param poly The input polynomial.
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* \param oi An output iterator for the real-valued root of the polynomial.
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* \return A past-the-end iterator for the output container.
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* \pre The value type of oi is Algebraic.
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*/
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template <typename OutputIterator>
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OutputIterator compute_polynomial_roots(const Polynomial& poly,
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OutputIterator oi) const;
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/*! Compute the real-valued roots of a polynomial with integer coefficients,
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* within a given interval. The roots are sorted in ascending order.
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* \param poly The input polynomial.
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* \param x_min The left bound of the interval.
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* \param x_max The right bound of the interval.
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* \param oi An output iterator for the real-valued root of the polynomial.
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* \return A past-the-end iterator for the output container.
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* \pre The value type of oi is Algebraic.
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*/
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template <typename OutputIterator>
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OutputIterator compute_polynomial_roots(const Polynomial& poly,
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double x_min, double x_max,
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OutputIterator oi) const;
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/// @}
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};
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} /* end namespace CGAL */
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