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:

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.