CGAL 6.1.2 - Polynomial: Concepts

CGAL 6.1.2 - Polynomial

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##

## Concepts
concept | 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...
concept | 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...
concept | PolynomialTraits_d::Canonicalize
| For a given polynomial \( p\) this `AdaptableUnaryFunction` computes the unique representative of the set. More...
concept | 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...
concept | PolynomialTraits_d::ConstructPolynomial
| This `AdaptableFunctor` provides several operators to construct objects of type `PolynomialTraits_d::Polynomial_d`. More...
concept | 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...
concept | PolynomialTraits_d::Evaluate
| This `AdaptableBinaryFunction` evaluates `PolynomialTraits_d::Polynomial_d` with respect to one variable. More...
concept | PolynomialTraits_d::GcdUpToConstantFactor
| This `AdaptableBinaryFunction` computes the \( gcd\) up to a constant factor (utcf) of two polynomials of type `PolynomialTraits_d::Polynomial_d`. More...
concept | 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...
concept | 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...
concept | PolynomialTraits_d::IsSquareFree
| This `AdaptableUnaryFunction` computes whether the given a polynomial of type `PolynomialTraits_d::Polynomial_d` is square free. More...
concept | 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...
concept | 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...
concept | 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...
concept | PolynomialTraits_d::Negate
| This `AdaptableUnaryFunction` computes \( p(-x)\) for a given polynomial \( p\). More...
concept | 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...
concept | 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...
concept | PolynomialTraits_d::PseudoDivision
| This `AdaptableFunctor` computes the pseudo division of two polynomials \( f\) and \( g\). More...
concept | PolynomialTraits_d::PseudoDivisionRemainder
| This `AdaptableBinaryFunction` computes the remainder of the pseudo division of two polynomials \( f\) and \( g\). More...
concept | 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...
concept | PolynomialTraits_d::Shift
| This `AdaptableBinaryFunction` multiplies a `PolynomialTraits_d::Polynomial_d` by the given power of the specified variable. More...
concept | 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...
concept | PolynomialTraits_d::SquareFreeFactorizeUpToConstantFactor
| This `AdaptableFunctor` computes a square-free factorization up to a constant factor (utcf) of a `PolynomialTraits_d::Polynomial_d`. More...
concept | 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...
concept | 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...
concept | PolynomialTraits_d::TotalDegree
| This `AdaptableUnaryFunction` computes the total degree of a `PolynomialTraits_d::Polynomial_d`. More...
concept | 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...
concept | 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...
concept | PolynomialTraits_d::Compare
| This `AdaptableBinaryFunction` compares two polynomials, with respect to the lexicographic order with preference to the outermost variable. More...
concept | PolynomialTraits_d::Degree
| This `AdaptableUnaryFunction` computes the degree of a `PolynomialTraits_d::Polynomial_d` with respect to a certain variable. More...
concept | PolynomialTraits_d::EvaluateHomogeneous
| This `AdaptableFunctor` provides evaluation of a `PolynomialTraits_d::Polynomial_d` interpreted as a homogeneous polynomial in one variable. More...
concept | 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...
concept | 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...
concept | 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...
concept | 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...
concept | 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...
concept | 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...
concept | 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...
concept | 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...
concept | PolynomialTraits_d::Swap
| This `AdaptableFunctor` swaps two variables of a multivariate polynomial. More...
concept | 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...
concept | PolynomialTraits_d::ConstructInnermostCoefficientConstIteratorRange
| This `AdaptableUnaryFunction` returns a const iterator range over all innermost coefficients of the given polynomial. More...
concept | PolynomialTraits_d::GetCoefficient
| This `AdaptableBinaryFunction` provides access to coefficients of a `PolynomialTraits_d::Polynomial_d`. More...
concept | PolynomialTraits_d::LeadingCoefficient
| This `AdaptableUnaryFunction` computes the leading coefficient of a `PolynomialTraits_d::Polynomial_d`. More...
concept | 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...
concept | 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...
concept | 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...
concept | PolynomialTraits_d::Differentiate
| This `AdaptableUnaryFunction` computes the derivative of a `PolynomialTraits_d::Polynomial_d` with respect to one variable. More...
concept | 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...
concept | PolynomialTraits_d::SquareFreeFactorize
| This `Functor` computes a square-free factorization of a `PolynomialTraits_d::Polynomial_d`. More...
concept | 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...
concept | 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...
concept | PolynomialTraits_d::PseudoDivisionQuotient
| This `AdaptableBinaryFunction` computes the quotient of the pseudo division of two polynomials \( f\) and \( g\). More...