Concepts

Classes
class	Polynomial_d
	A model of Polynomial_d is representing a multivariate polynomial in \( d \geq 1\) variables over some basic ring \( R\). This type is denoted as the innermost coefficient. A model of Polynomial_d must be accompanied by a traits class CGAL::Polynomial_traits_d<Polynomial_d>, which is a model of PolynomialTraits_d. Please have a look at the concept PolynomialTraits_d, since nearly all functionality related to polynomials is provided by the traits. More...
class	PolynomialTraits_d::Canonicalize
	For a given polynomial \( p\) this AdaptableUnaryFunction computes the unique representative of the set. More...
class	PolynomialTraits_d::Compare
	This AdaptableBinaryFunction compares two polynomials, with respect to the lexicographic order with preference to the outermost variable. More...
class	PolynomialTraits_d::ConstructCoefficientConstIteratorRange
	This AdaptableUnaryFunction returns a const iterator range over the coefficients of the given polynomial, with respect to the outermost variable, \( x_{d-1}\). The range starts with the coefficient for \( x_{d-1}^0\). More...
class	PolynomialTraits_d::ConstructInnermostCoefficientConstIteratorRange
	This AdaptableUnaryFunction returns a const iterator range over all innermost coefficients of the given polynomial. More...
class	PolynomialTraits_d::ConstructPolynomial
	This AdaptableFunctor provides several operators to construct objects of type PolynomialTraits_d::Polynomial_d. More...
class	PolynomialTraits_d::Degree
	This AdaptableUnaryFunction computes the degree of a PolynomialTraits_d::Polynomial_d with respect to a certain variable. More...
class	PolynomialTraits_d::DegreeVector
	For a given PolynomialTraits_d::Polynomial_d \( p\) this AdaptableUnaryFunction returns the degree vector, that is, it returns the exponent vector of the monomial of highest order in \( p\), where the monomial order is the lexicographic order giving outer variables a higher priority. In particular, this is the monomial that belongs to the innermost leading coefficient of \( p\). More...
class	PolynomialTraits_d::Differentiate
	This AdaptableUnaryFunction computes the derivative of a PolynomialTraits_d::Polynomial_d with respect to one variable. More...
class	PolynomialTraits_d::Evaluate
	This AdaptableBinaryFunction evaluates PolynomialTraits_d::Polynomial_d with respect to one variable. More...
class	PolynomialTraits_d::EvaluateHomogeneous
	This AdaptableFunctor provides evaluation of a PolynomialTraits_d::Polynomial_d interpreted as a homogeneous polynomial in one variable. More...
class	PolynomialTraits_d::GcdUpToConstantFactor
	This AdaptableBinaryFunction computes the \( gcd\) up to a constant factor (utcf) of two polynomials of type PolynomialTraits_d::Polynomial_d. More...
class	PolynomialTraits_d::GetCoefficient
	This AdaptableBinaryFunction provides access to coefficients of a PolynomialTraits_d::Polynomial_d. More...
class	PolynomialTraits_d::GetInnermostCoefficient
	For the given PolynomialTraits_d::Polynomial_d this AdaptableBinaryFunction returns the coefficient of the (multivariate) monomial specified by the given CGAL::Exponent_vector. More...
class	PolynomialTraits_d::InnermostLeadingCoefficient
	This AdaptableUnaryFunction computes the innermost leading coefficient of a PolynomialTraits_d::Polynomial_d \( p\). The innermost leading coefficient is recursively defined as the innermost leading coefficient of the leading coefficient of \( p\). In case \( p\) is univariate it coincides with the leading coefficient. More...
class	PolynomialTraits_d::IntegralDivisionUpToConstantFactor
	This AdaptableBinaryFunction computes the integral division of two polynomials of type PolynomialTraits_d::Polynomial_d up to a constant factor (utcf) . More...
class	PolynomialTraits_d::Invert
	This AdaptableUnaryFunction inverts one variable in a given PolynomialTraits_d::Polynomial_d, that is, for a given polynomial \( p\) it computes \( x^{degree(p)}p(1/x)\). More...
class	PolynomialTraits_d::IsSquareFree
	This AdaptableUnaryFunction computes whether the given a polynomial of type PolynomialTraits_d::Polynomial_d is square free. More...
class	PolynomialTraits_d::IsZeroAt
	This AdaptableFunctor returns whether a PolynomialTraits_d::Polynomial_d \( p\) is zero at a given Cartesian point, which is represented as an iterator range. More...
class	PolynomialTraits_d::IsZeroAtHomogeneous
	This AdaptableFunctor returns whether a PolynomialTraits_d::Polynomial_d \( p\) is zero at a given homogeneous point, which is given by an iterator range. More...
class	PolynomialTraits_d::LeadingCoefficient
	This AdaptableUnaryFunction computes the leading coefficient of a PolynomialTraits_d::Polynomial_d. More...
class	PolynomialTraits_d::MakeSquareFree
	This AdaptableUnaryFunction computes the square-free part of a polynomial of type PolynomialTraits_d::Polynomial_d up to a constant factor. More...
class	PolynomialTraits_d::MonomialRepresentation
	This Functor outputs the monomial representation of the given polynomial, that is, it writes all non zero terms of the polynomial as std::pair<CGAL::Exponent_vector, PolynomialTraits_d::Innermost_coefficient_type> into the given output iterator. More...
class	PolynomialTraits_d::Move
	This AdaptableFunctor moves a variable at position \( i\) to a new position \( j\). The relative order of the other variables is preserved, that is, the variables between \( x_i\) and \( x_j\) (including \( x_j\)) are moved by one position while \( x_i\) is moved to the former position of \( x_j\). More...
class	PolynomialTraits_d::MultivariateContent
	This AdaptableUnaryFunction computes the content of a PolynomialTraits_d::Polynomial_d with respect to the symmetric view on the polynomial, that is, it computes the gcd of all innermost coefficients. More...
class	PolynomialTraits_d::Negate
	This AdaptableUnaryFunction computes \( p(-x)\) for a given polynomial \( p\). More...
class	PolynomialTraits_d::Permute
	This AdaptableFunctor permutes the variables of the given polynomial with respect to a permutation \( \sigma\), that is, each monomial \( \prod x_i^{e_i}\) will be mapped to the monomial \( \prod x_{\sigma(i)}^{e_i}\). The permutation \( \sigma\) is given by the iterator range of length PolynomialTraits_d::d, which is supposed to contain the second row of the permutation. More...
class	PolynomialTraits_d::PolynomialSubresultants
	Computes the polynomial subresultant of two polynomials \( p\) and \( q\) of type PolynomialTraits_d::Polynomial_d with respect to outermost variable. Let \( p=\ccSum{i=0,\ldots,n}{} p_i x^i\) and \( q=\ccSum{i=0,\ldots,m}{} q_i x^i\), where \( x\) is the outermost variable. The \( i\)-th subresultant (with \( i=0,\ldots,\min\{n,m\}\)) is defined by. More...
class	PolynomialTraits_d::PolynomialSubresultantsWithCofactors
	Computes the polynomial subresultant of two polynomials \( p\) and \( q\) of degree \( n\) and \( m\), respectively, as defined in the documentation of PolynomialTraits_d::PolynomialSubresultants. Moreover, for \( \mathrm{Sres}_i(p,q)\), polynomials \( u_i\) and \( v_i\) with \( \deg u_i\leq m-i-1\) and \( \deg v_i\leq n-i-1\) are computed such that \( \mathrm{Sres}_i(p,q)=u_i p + v_i q\). \( u_i\) and \( v_i\) are called the cofactors of \( \mathrm{Sres}_i(p,q)\). More...
class	PolynomialTraits_d::PrincipalSturmHabichtSequence
	Computes the principal leading coefficients of the Sturm-Habicht sequence of a polynomials \( f\) of type PolynomialTraits_d::Polynomial_d with respect a certain variable \( x_i\). This means that for the \( j\)-th Sturm-Habicht polynomial, this methods returns the coefficient of \( x_i^j\). More...
class	PolynomialTraits_d::PrincipalSubresultants
	Computes the principal subresultant of two polynomials \( p\) and \( q\) of type PolynomialTraits_d::Coefficient_type with respect to the outermost variable. The \( i\)-th principal subresultant, \( \mathrm{sres}_i(p,q)\), is defined as the coefficient at \( t^i\) of the \( i\)-th polynomial subresultant \( \mathrm{Sres}_i(p,q)\). Thus, it is either the leading coefficient of \( \mathrm{Sres}_i\), or zero in the case where its degree is below \( i\). More...
class	PolynomialTraits_d::PseudoDivision
	This AdaptableFunctor computes the pseudo division of two polynomials \( f\) and \( g\). More...
class	PolynomialTraits_d::PseudoDivisionQuotient
	This AdaptableBinaryFunction computes the quotient of the pseudo division of two polynomials \( f\) and \( g\). More...
class	PolynomialTraits_d::PseudoDivisionRemainder
	This AdaptableBinaryFunction computes the remainder of the pseudo division of two polynomials \( f\) and \( g\). More...
class	PolynomialTraits_d::Resultant
	This AdaptableBinaryFunction computes the resultant of two polynomials \( f\) and \( g\) of type PolynomialTraits_d::Polynomial_d with respect to a certain variable. More...
class	PolynomialTraits_d::Scale
	Given a constant \( c\) this AdaptableBinaryFunction scales a PolynomialTraits_d::Polynomial_d \( p\) with respect to one variable, that is, it computes \( p(c\cdot x)\). More...
class	PolynomialTraits_d::ScaleHomogeneous
	Given a numerator \( a\) and a denominator \( b\) this AdaptableFunctor scales a PolynomialTraits_d::Polynomial_d \( p\) with respect to one variable, that is, it computes \( b^{degree(p)}\cdot p(a/b\cdot x)\). More...
class	PolynomialTraits_d::Shift
	This AdaptableBinaryFunction multiplies a PolynomialTraits_d::Polynomial_d by the given power of the specified variable. More...
class	PolynomialTraits_d::SignAt
	This AdaptableFunctor returns the sign of a PolynomialTraits_d::Polynomial_d \( p\) at given Cartesian point represented as an iterator range. More...
class	PolynomialTraits_d::SignAtHomogeneous
	This AdaptableFunctor returns the sign of a PolynomialTraits_d::Polynomial_d \( p\) at a given homogeneous point, which is given by an iterator range. More...
class	PolynomialTraits_d::SquareFreeFactorize
	This Functor computes a square-free factorization of a PolynomialTraits_d::Polynomial_d. More...
class	PolynomialTraits_d::SquareFreeFactorizeUpToConstantFactor
	This AdaptableFunctor computes a square-free factorization up to a constant factor (utcf) of a PolynomialTraits_d::Polynomial_d. More...
class	PolynomialTraits_d::SturmHabichtSequence
	Computes the Sturm-Habicht sequence (aka the signed subresultant sequence) of a polynomial \( f\) of type PolynomialTraits_d::Polynomial_d with respect to a certain variable \( x_i\). The Sturm-Habicht sequence is similar to the polynomial subresultant sequence of \( f\) and its derivative \( f':=\frac{\partial f}{\partial x_i}\) with respect to \( x_i\). The implementation is based on the following definition: More...
class	PolynomialTraits_d::SturmHabichtSequenceWithCofactors
	Computes the Sturm-Habicht polynomials of a polynomial \( f\) of degree \( n\), as defined in the documentation of PolynomialTraits_d::SturmHabichtSequence. Moreover, for \( \mathrm{Stha}_i(f)\), polynomials \( u_i\) and \( v_i\) with \( \deg u_i\leq n-i-2\) and \( \deg v_i\leq n-i-1\) are computed such that \( \mathrm{Sres}_i(p,q)=u_i f + v_i f'\). \( u_i\) and \( v_i\) are called the cofactors of \( \mathrm{Stha}_i(f)\). More...
class	PolynomialTraits_d::Substitute
	This Functor substitutes all variables of a given multivariate PolynomialTraits_d::Polynomial_d by the values given in the iterator range, where begin refers the value for the innermost variable. More...
class	PolynomialTraits_d::SubstituteHomogeneous
	This Functor substitutes all variables of a given multivariate PolynomialTraits_d::Polynomial_d \( p\) by the values given in the iterator range, where begin refers the value for the innermost variable. In contrast to PolynomialTraits_d::Substitute the given polynomial \( p\) is interpreted as a homogeneous polynomial. Hence the iterator range is required to be of length PolynomialTraits_d::d+1. More...
class	PolynomialTraits_d::Swap
	This AdaptableFunctor swaps two variables of a multivariate polynomial. More...
class	PolynomialTraits_d::TotalDegree
	This AdaptableUnaryFunction computes the total degree of a PolynomialTraits_d::Polynomial_d. More...
class	PolynomialTraits_d::Translate
	This AdaptableBinaryFunction translates a PolynomialTraits_d::Polynomial_d with respect to one variable, that is, for a given polynomial \( p\) and constant \( c\) it computes \( p(x+c)\). More...
class	PolynomialTraits_d::TranslateHomogeneous
	Given numerator \( a\) and denominator \( b\) this AdaptableFunctor translates a PolynomialTraits_d::Polynomial_d \( p\) with respect to one variable by \( a/b\), that is, it computes \( b^{degree(p)}\cdot p(x+a/b)\). More...
class	PolynomialTraits_d::UnivariateContent
	This AdaptableUnaryFunction computes the content of a PolynomialTraits_d::Polynomial_d with respect to the univariate (recursive) view on the polynomial, that is, it computes the gcd of all coefficients with respect to one variable. More...
class	PolynomialTraits_d::UnivariateContentUpToConstantFactor
	This AdaptableUnaryFunction computes the content of a PolynomialTraits_d::Polynomial_d with respect to the univariate (recursive) view on the polynomial up to a constant factor (utcf), that is, it computes the \( \mathrm{gcd\_utcf}\) of all coefficients with respect to one variable. More...
class	PolynomialTraits_d
	A model of PolynomialTraits_d is associated with a type Polynomial_d. The type Polynomial_d represents a multivariate polynomial. The number of variables is denoted as the dimension \( d\) of the polynomial, it is arbitrary but fixed for a certain model of this concept. Note that univariate polynomials are not excluded by this concept. In this case \( d\) is just set to one. More...