\( \newcommand{\E}{\mathrm{E}} \) \( \newcommand{\A}{\mathrm{A}} \) \( \newcommand{\R}{\mathrm{R}} \) \( \newcommand{\N}{\mathrm{N}} \) \( \newcommand{\Q}{\mathrm{Q}} \) \( \newcommand{\Z}{\mathrm{Z}} \) \( \def\ccSum #1#2#3{ \sum_{#1}^{#2}{#3} } \def\ccProd #1#2#3{ \sum_{#1}^{#2}{#3} }\)
CGAL 5.0.2 - Polynomial
PolynomialTraits_d::SturmHabichtSequence Concept Reference

Definition

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:

Let \( n:=\deg f\) and \( \delta_k:=(-1)^{k(k+1)/2}\). For \( k\in\{0,\ldots,n\}\), the \( k\)-th Sturm-Habicht polynomial of \( f\) is defined as:

sturm_habicht_def.png

where \( \mathrm{Sres}_k(f,f')\) is defined as in the concept PolynomialTraits_d::PolynomialSubresultants.

The result is written in an output range, starting with the \( 0\)-th Sturm-Habicht polynomial (which is equal to the discriminant of \( f\) up to a multiple of the leading coefficient).

Note
This functor is optional.
Refines:

AdaptableBinaryFunction

CopyConstructible

DefaultConstructible

See also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::Resultant
PolynomialTraits_d::PrincipalSturmHabichtSequence
PolynomialTraits_d::SturmHabichtSequenceWithCofactors
PolynomialTraits_d::PolynomialSubresultants

Operations

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

Member Function Documentation

◆ operator()()

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

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

Each element is of type PolynomialTraits_d::Polynomial_d.