mirror of https://github.com/CGAL/cgal
53 lines
2.0 KiB
TeX
53 lines
2.0 KiB
TeX
\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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of a polynomials $f$ of type \ccc{PolynomialTraits_d::Polynomial_d}
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with respect a certain variable $x_i$.
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This means that for the $j$th Sturm-Habicht polynomial, this methods returns
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the coefficient of $x_i^j$.
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Note that the degree of the $j$th Sturm-Habicht polynomial is at most $j$,
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but the principal coefficient might be zero, thus, this functor does not
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necessarily give the leading coefficient of the Sturm-Habicht polynomials
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In case that \ccc{PolynomialTraits_d::Coefficient_type} is \ccc{RealEmbeddable}, the function \ccc{CGAL::stha_count_number_of_real_roots} can be used
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on the resulting sequence to count the number of distinct real roots of
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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::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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\ccMethod{template<typename OutputIterator>
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OutputIterator operator()(Polynomial_d f,
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OutputIterator out);}
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{ computes the principal coefficients of the
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Sturm-Habicht sequence of $f$,
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with respect to the outermost variable. Each element is of type
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\ccc{PolynomialTraits_d::Coefficient_type}.}
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\ccMethod{template<typename OutputIterator>
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OutputIterator operator()(Polynomial_d f,
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OutputIterator out,
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int i);}
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{ computes the principal coefficients
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of the Sturm-Habicht sequence of $f$
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with respect to the variable $x_i$.}
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%\ccHasModels
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\ccSeeAlso
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\ccRefIdfierPage{Polynomial_d}\\
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\ccRefIdfierPage{PolynomialTraits_d}\\
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\ccRefIdfierPage{CGAL::principal_sturm_habicht_sequence}\\
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\ccRefIdfierPage{CGAL::stha_count_number_of_real_roots}\\
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\end{ccRefConcept}
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