mirror of https://github.com/CGAL/cgal
60 lines
2.6 KiB
TeX
60 lines
2.6 KiB
TeX
\begin{ccRefConcept}{PolynomialTraits_d::SturmHabichtSequenceWithCofactors}
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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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as defined in the documentation of \ccc{PolynomialTraits_d::SturmHabichtSequence}.
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Moreover, for $\mathrm{Stha}_i(f)$, polynomials $u_i$ and $v_i$
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with $\deg u_i\leq n-i-2$ and $\deg v_i\leq n-i-1$ are computed
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such that $\mathrm{Sres}_i(p,q)=u_i f + v_i f'$. $u_i$ and $v_i$ are called
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the \emph{cofactors} of $\mathrm{Stha}_i(f)$.
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The result is written in three output ranges, each of length $\min\{n,m\}+1$,
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starting with the $0$th Sturm-Habicht polynomial $\mathrm{Stha_0(f)}$
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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 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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\ccc{PolynomialTraits_d::Polynomial_d} with respect a certain variable $x_i$.
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Additionally, it computes two ranges of cofactors, {\tt co\_f} and {\tt co\_fx}
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with the property that {\tt stha[i] == co\_f[i] f + co\_fx[i] f'}.
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\ccCreationVariable{fo}
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\ccOperations
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\ccMethod{template< typename OutputIterator1,
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typename OutputIterator2,
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typename OutputIterator3 >
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OutputIterator1 operator()(Polynomial_d f,
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OutputIterator1 stha,
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OutputIterator2 co_f,
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OutputIterator3 co_fx);}
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{ computes the Sturm-Habicht sequence of $f$, and the cofactors,
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with respect to the outermost variable. Each element is of type
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\ccc{PolynomialTraits_d::Polynomial_d}.}
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\ccMethod{template< typename OutputIterator1,
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typename OutputIterator2,
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typename OutputIterator3 >
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OutputIterator1 operator()(Polynomial_d f,
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OutputIterator1 stha,
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OutputIterator2 co_f,
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OutputIterator3 co_fx,
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int i);}
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{ computes the Sturm-Habicht sequence of $f$, and the cofactors,
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with respect to $x_i$. Each element is of type
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\ccc{PolynomialTraits_d::Polynomial_d}.}
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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{PolynomialTraits_d::SturmHabichtSequence}\\
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\ccRefIdfierPage{CGAL::sturm_habicht_sequence_with_cofactors}\\
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\end{ccRefConcept}
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