PolynomialTraits_d::SturmHabichtSequence Class Reference

#include <Concepts/PolynomialTraits_d--SturmHabichtSequence.h>

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:

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 \( x_i\). More...

Member Function Documentation

operator()() [1/2]

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.

operator()() [2/2]

template<typename OutputIterator >

OutputIterator PolynomialTraits_d::SturmHabichtSequence::operator()	(	Polynomial_d	f,
		OutputIterator	out,
		int	i
	)

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