AlgebraicKernel_d_1 Concept Reference

Definition

A model of the AlgebraicKernel_d_1 concept is meant to provide the algebraic functionalities on univariate polynomials of general degree \( d\).

Refines:

CopyConstructible

Assignable

Has Models:

CGAL::Algebraic_kernel_rs_gmpz_d_1

CGAL::Algebraic_kernel_rs_gmpq_d_1

See Also

AlgebraicKernel_d_2

A model of AlgebraicKernel_d_1 must provide:

Concepts
concept	ApproximateAbsolute_1
	A model of AlgebraicKernel_d_1::ApproximateAbsolute_1 is an AdaptableBinaryFunction that computes an approximation of an AlgebraicKernel_d_1::Algebraic_real_1 value with respect to a given absolute precision. More...
concept	ApproximateRelative_1
	A model of AlgebraicKernel_d_1::ApproximateRelative_1 is an AdaptableBinaryFunction that computes an approximation of an AlgebraicKernel_d_1::Algebraic_real_1 value with respect to a given relative precision. More...
concept	BoundBetween_1
	Computes a number of type AlgebraicKernel_d_1::Bound in-between two AlgebraicKernel_d_1::Algebraic_real_1 values. More...
concept	Compare_1
	Compares AlgebraicKernel_d_1::Algebraic_real_1 values. More...
concept	ComputePolynomial_1
	Computes a square free univariate polynomial \( p\), such that the given AlgebraicKernel_d_1::Algebraic_real_1 is a root of \( p\). More...
concept	ConstructAlgebraicReal_1
	Constructs AlgebraicKernel_d_1::Algebraic_real_1. More...
concept	IsCoprime_1
	Determines whether a given pair of univariate polynomials \( p_1, p_2\) is coprime, namely if \( \deg({\rm gcd}(p_1 ,p_2)) = 0\). More...
concept	Isolate_1
	Computes an open isolating interval for an AlgebraicKernel_d_1::Algebraic_real_1 with respect to the real roots of a given univariate polynomial. More...
concept	IsSquareFree_1
	Computes whether the given univariate polynomial is square free. More...
concept	IsZeroAt_1
	Computes whether an AlgebraicKernel_d_1::Polynomial_1 is zero at a given AlgebraicKernel_d_1::Algebraic_real_1. More...
concept	MakeCoprime_1
	Computes for a given pair of univariate polynomials \( p_1\), \( p_2\) their common part \( g\) up to a constant factor and coprime parts \( q_1\), \( q_2\) respectively. More...
concept	MakeSquareFree_1
	Returns a square free part of a univariate polynomial. More...
concept	NumberOfSolutions_1
	Computes the number of real solutions of the given univariate polynomial. More...
concept	SignAt_1
	Computes the sign of a univariate polynomial AlgebraicKernel_d_1::Polynomial_1 at a real value of type AlgebraicKernel_d_1::Algebraic_real_1. More...
concept	Solve_1
	Computes the real roots of a univariate polynomial. More...
concept	SquareFreeFactorize_1
	Computes a square free factorization of an AlgebraicKernel_d_1::Polynomial_1. More...

Types
typedef unspecified_type	Coefficient
	A model of IntegralDomain and RealEmbeddable. More...
typedef unspecified_type	Polynomial_1
	A univariate polynomial that is a model of Polynomial_d, where CGAL::Polynomial_traits_d<Polynomial_1>::Innermost_coefficient_type is AlgebraicKernel_d_1::Coefficient.
typedef unspecified_type	Algebraic_real_1
	A type that is used to represent real roots of univariate polynomials. More...
typedef unspecified_type	Bound
	A type to represent upper and lower bounds of AlgebraicKernel_d_1::Algebraic_real_1. More...
typedef unspecified_type	size_type
	Size type (unsigned integral type).
typedef unspecified_type	Multiplicity_type
	Multiplicity type (unsigned integral type).

Functors
typedef unspecified_type	Construct_algebraic_real_1
	A model of AlgebraicKernel_d_1::ConstructAlgebraicReal_1.
typedef unspecified_type	Compute_polynomial_1
	A model of AlgebraicKernel_d_1::ComputePolynomial_1.
typedef unspecified_type	Isolate_1
	A model of AlgebraicKernel_d_1::Isolate_1.
typedef unspecified_type	Is_square_free_1
	A model of AlgebraicKernel_d_1::IsSquareFree_1.
typedef unspecified_type	Make_square_free_1
	A model of AlgebraicKernel_d_1::MakeSquareFree_1.
typedef unspecified_type	Square_free_factorize_1
	A model of AlgebraicKernel_d_1::SquareFreeFactorize_1.
typedef unspecified_type	Is_coprime_1
	A model of AlgebraicKernel_d_1::IsCoprime_1.
typedef unspecified_type	Make_coprime_1
	A model of AlgebraicKernel_d_1::MakeCoprime_1.
typedef unspecified_type	Solve_1
	A model of AlgebraicKernel_d_1::Solve_1.
typedef unspecified_type	Number_of_solutions_1
	A model of AlgebraicKernel_d_1::NumberOfSolutions_1.
typedef unspecified_type	Sign_at_1
	A model of AlgebraicKernel_d_1::SignAt_1.
typedef unspecified_type	Compare_1
	A model of AlgebraicKernel_d_1::Compare_1.
typedef unspecified_type	Bound_between_1
	A model of AlgebraicKernel_d_1::BoundBetween_1.
typedef unspecified_type	Approximate_absolute_1
	A model of AlgebraicKernel_d_1::ApproximateAbsolute_1.
typedef unspecified_type	Approximate_relative_1
	A model of AlgebraicKernel_d_1::ApproximateRelative_1.

Operations
For each of the function objects above, there must exist a member function that requires no arguments and returns an instance of that function object. The name of the member function is the uncapitalized name of the type returned with the suffix _object appended. For example, for the function object AlgebraicKernel_d_1::Bound_between_1 the following member function must exist:
AlgebraicKernel_d_1::Bound_between_1	bound_between_1_object () const

Member Typedef Documentation

typedef unspecified_type AlgebraicKernel_d_1::Algebraic_real_1

A type that is used to represent real roots of univariate polynomials.

The type must be a model of DefaultConstructible, CopyConstructible, Assignable and RealEmbeddable.

typedef unspecified_type AlgebraicKernel_d_1::Bound

A type to represent upper and lower bounds of AlgebraicKernel_d_1::Algebraic_real_1.

The type is ExplicitInteroperable with AlgebraicKernel_d_1::Coefficient and must be a model IntegralDomain, RealEmbeddable and dense in \( \mathbb{R}\).

typedef unspecified_type AlgebraicKernel_d_1::Coefficient

A model of IntegralDomain and RealEmbeddable.

ExplicitInteroperable with AlgebraicKernel_d_1::Bound.