\( \newcommand{\E}{\mathrm{E}} \) \( \newcommand{\A}{\mathrm{A}} \) \( \newcommand{\R}{\mathrm{R}} \) \( \newcommand{\N}{\mathrm{N}} \) \( \newcommand{\Q}{\mathrm{Q}} \) \( \newcommand{\Z}{\mathrm{Z}} \) \( \def\ccSum #1#2#3{ \sum_{#1}^{#2}{#3} } \def\ccProd #1#2#3{ \sum_{#1}^{#2}{#3} }\)
CGAL 4.5 - Polynomial
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PolynomialTraits_d::PrincipalSturmHabichtSequence Concept Reference

Definition

Note: This functor is optional!

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 \( x_i^j\).

Note that the degree of the \( j\)-th Sturm-Habicht polynomial is at most \( j\), but the principal coefficient might be zero, thus, this functor does not necessarily give the leading coefficient of the Sturm-Habicht polynomials.

In case that PolynomialTraits_d::Coefficient_type is RealEmbeddable, the function CGAL::number_of_real_roots can be used on the resulting sequence to count the number of distinct real roots of the polynomial \( f\).

Refines:

AdaptableBinaryFunction

CopyConstructible

DefaultConstructible

See Also
Polynomial_d
PolynomialTraits_d
CGAL::number_of_real_roots()
PolynomialTraits_d::Resultant
PolynomialTraits_d::SturmHabichtSequence
PolynomialTraits_d::PrincipalSubresultants

Operations

template<typename OutputIterator >
OutputIterator operator() (Polynomial_d f, OutputIterator out)
 computes the principal coefficients of the Sturm-Habicht sequence of \( f\), with respect to the outermost variable. More...
 
template<typename OutputIterator >
OutputIterator operator() (Polynomial_d f, OutputIterator out, int i)
 computes the principal coefficients of the Sturm-Habicht sequence of \( f\) with respect to the variable \( x_i\).
 

Member Function Documentation

template<typename OutputIterator >
OutputIterator PolynomialTraits_d::PrincipalSturmHabichtSequence::operator() ( Polynomial_d  f,
OutputIterator  out 
)

computes the principal coefficients of the Sturm-Habicht sequence of \( f\), with respect to the outermost variable.

Each element is of type PolynomialTraits_d::Coefficient_type.