mirror of https://github.com/CGAL/cgal
Updated doc
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Michael Kerber
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\begin{ccRefConcept}{PolynomialTraits_d::PolynomialSubresultants}
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\textbf{Note:} This functor is optional!
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\ccDefinition
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Computes the polynomial subresultant of two polynomials $p$ and $q$ of
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\begin{ccRefConcept}{PolynomialTraits_d::PolynomialSubresultantsWithCofactors}
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\textbf{Note:} This functor is optional!
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\ccDefinition
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Computes the polynomial subresultant of two polynomials $p$ and $q$ of degree
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@ -14,7 +17,7 @@ starting with the $0$th subresultant and the corresponding cofactors.
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A default implementation for the case that
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\ccc{Polynomial_traits_d::Coefficient_type}
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is a model of \ccc{CGAL::Integral_division}
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is a model of \ccc{CGAL::IntegralDomain}
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is provided by the function \ccc{CGAL::polynomial_subresultants_with_cofactors}.
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\ccCreationVariable{fo}
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@ -1,4 +1,7 @@
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\begin{ccRefConcept}{PolynomialTraits_d::PrincipalSturmHabichtSequence}
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\textbf{Note:} This functor is optional!
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\ccDefinition
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Computes the principal leading coefficients of the Sturm-Habicht sequence
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@ -17,7 +20,7 @@ the polynomial~$f$.
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A default implementation for the case that
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\ccc{Polynomial_traits_d::Coefficient_type}
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is a model of \ccc{CGAL::Integral_domain_without_division}
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is a model of \ccc{CGAL::IntegralDomainWithoutDivision}
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is provided by the function \ccc{CGAL::principal_sturm_habicht_sequence}.
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\ccOperations
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\begin{ccRefConcept}{PolynomialTraits_d::PrincipalSubresultants}
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\textbf{Note:} This functor is optional!
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\ccDefinition
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Computes the principal subresultant of two polynomials $p$ and $q$ of
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@ -17,7 +20,7 @@ $\mathrm{sres}_0(p,q)$
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A default implementation for the case that
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\ccc{Polynomial_traits_d::Coefficient_type}
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is a model of \ccc{CGAL::Integral_domain_without_division}
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is a model of \ccc{CGAL::IntegralDomainWithoutDivision}
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is provided by the function \ccc{CGAL::principal_subresultants}.
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\ccOperations
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@ -1,4 +1,7 @@
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\begin{ccRefConcept}{PolynomialTraits_d::SturmHabichtSequence}
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\textbf{Note:} This functor is optional!
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\ccDefinition
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Computes the Sturm-Habicht sequence
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@ -26,7 +29,7 @@ the discriminant of $f$ up to a multiple of the leading coefficient)
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A default implementation for the case that
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\ccc{Polynomial_traits_d::Coefficient_type}
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is a model of \ccc{CGAL::Integral_domain_without_division}
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is a model of \ccc{CGAL::IntegralDomainWithoutDivision}
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is provided by the function \ccc{CGAL::sturm_habicht_sequence}.
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\ccCreationVariable{fo}
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\begin{ccRefConcept}{PolynomialTraits_d::SturmHabichtSequenceWithCofactors}
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\textbf{Note:} This functor is optional!
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\ccDefinition
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Computes the Sturm-Habicht polynomials of a polynomial $f$ of degree $n$,
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@ -14,7 +17,7 @@ and the corresponding cofactors.
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A default implementation for the case that
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\ccc{Polynomial_traits_d::Coefficient_type}
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is a model of \ccc{CGAL::Integral_division}
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is a model of \ccc{CGAL::IntegralDomain}
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is provided by the function \ccc{CGAL::sturm_habicht_sequence_with_cofactors}.
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Computes the Sturm-Habicht sequence of a polynomials $f$ of type
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@ -811,15 +811,15 @@ Computes the polynomial subresultants
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of two polynomials, defined as in the concept
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\ccc{Polynomial_traits_d::PolynomialSubresultants},
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if \ccc{PolynomialTraits_d::Coefficient_type}
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is at least a model of \ccc{CGAL::Integral_domain_without_division}.
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is at least a model of \ccc{CGAL::IntegralDomainWithoutDivision}.
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The computation method depends on the algebraic category of
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\ccc{PolynomialTraits_d::Coefficient_type}.
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if it models \ccc{CGAL::Integral_division},
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if it models \ccc{CGAL::IntegralDomain},
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the subresultants are computed
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by a pseudo-division approach~\cite{ducos-optimizations}.
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If they only model a
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\ccc{CGAL::Integral_domain_without_division}, the subresultants
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\ccc{CGAL::IntegralDomainWithoutDivision}, the subresultants
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are computed by evaluating the determinants directly, avoiding divisions
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completely~\cite{kerber-division}.
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@ -846,7 +846,7 @@ Computes the polynomial subresultants
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of two polynomials and their cofactors, defined as in the concept
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\ccc{Polynomial_traits_d::PolynomialSubresultantsWithCofactors},
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if \ccc{PolynomialTraits_d::Coefficient_type}
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is at least a model of \ccc{CGAL::Integral_division}.
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is at least a model of \ccc{CGAL::IntegralDomain}.
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\ccOperations
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\ccFunction{template<typename Polynomial_traits_d,
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@ -879,7 +879,7 @@ Computes the principal subresultants
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of two polynomials, defined as in the concept
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\ccc{Polynomial_traits_d::PrincipalSubresultants},
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if \ccc{PolynomialTraits_d::Coefficient_type}
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is at least a model of \ccc{CGAL::Integral_domain_without_division}.
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is at least a model of \ccc{CGAL::IntegralDomainWithoutDivision}.
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\ccOperations
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\ccFunction{template<typename Polynomial_traits_d,typename OutputIterator>
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@ -904,7 +904,7 @@ Computes the Sturm-Habicht-polynomials
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of a polynomial, defined as in the concept
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\ccc{Polynomial_traits_d::SturmHabichtSequence},
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if \ccc{PolynomialTraits_d::Coefficient_type}
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is at least a model of \ccc{CGAL::Integral_domain_without_division}.
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is at least a model of \ccc{CGAL::IntegralDomainWithoutDivision}.
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The computation works simply by calling \ccc{CGAL::polynomial_subresultants},
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and adjusting the signs.
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@ -931,7 +931,7 @@ Computes the Sturm-Habicht-sequence of a polynomial
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and its cofactors, defined as in the concept
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\ccc{Polynomial_traits_d::SturmHabichtSequenceWithCofactors},
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if \ccc{PolynomialTraits_d::Coefficient_type}
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is at least a model of \ccc{CGAL::Integral_division}.
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is at least a model of \ccc{CGAL::IntegralDomain}.
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\ccOperations
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\ccFunction{template<typename Polynomial_traits_d,
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@ -964,7 +964,7 @@ Computes the principal Sturm-Habicht coefficients
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of a polynomial, defined as in the concept
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\ccc{Polynomial_traits_d::PrincipalSturmHabichtSequence},
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if \ccc{PolynomialTraits_d::Coefficient_type}
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is at least a model of \ccc{CGAL::Integral_domain_without_division}.
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is at least a model of \ccc{CGAL::IntegralDomainWithoutDivision}.
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\ccOperations
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\ccFunction{ template <typename Polynomial_traits_d,typename OutputIterator> inline
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