PolynomialTraits_d::PolynomialSubresultants Concept Reference

Definition

Computes the polynomial subresultant of two polynomials $p$ and $q$ of type PolynomialTraits_d::Polynomial_d with respect to outermost variable. Let $p=\ccSum{i=0,\ldots,n}{} p_i x^i$ and $q=\ccSum{i=0,\ldots,m}{} q_i x^i$, where $x$ is the outermost variable. The $i$-th subresultant (with $i=0,\ldots,\min\{n,m\}$) is defined by.

$$\mathrm{Sres}_{i}(p,q) = \det \begin{pmatrix} p_{n} & \dots & & \dots & p_{2i-m+2} & x^{m-i-1}p \\ & \ddots & & & \vdots & \vdots\\ & & p_{n} & \dots & p_{i+1} & p \\ q_{m} & \dots & & \dots & q_{2i-n+2} & x^{n-i-1}q \\ & \ddots & & & \vdots & \vdots\\ & & q_{m} & \dots & q_{i+1} & q \end{pmatrix}$$

where $p_i$ and $q_i$ are set to zero if $i<0$. In the case that $n=m$, $\mathrm{Sres_n}$ is set to $q$.

The result is written in an output range, starting with the $0$-th subresultant $\mathrm{Sres}_0(p,q)$ (aka as the resultant of $p$ and $q$).

Note

This functor is optional.

Refines

AdaptableBinaryFunction

CopyConstructible

DefaultConstructible

See also

Polynomial_d

PolynomialTraits_d

PolynomialTraits_d::Resultant

PolynomialTraits_d::PrincipalSubresultants

PolynomialTraits_d::PolynomialSubresultantsWithCofactors

PolynomialTraits_d::SturmHabichtSequence

Operations
template<typename OutputIterator >
OutputIterator	operator() (Polynomial_d p, Polynomial_d q, OutputIterator out)
	computes the polynomial subresultants of $p$ and $q$, with respect to the outermost variable.
template<typename OutputIterator >
OutputIterator	operator() (Polynomial_d p, Polynomial_d q, OutputIterator out, int i)
	computes the polynomial subresultants of $p$ and $q$, with respect to the variable $x_i$.

Member Function Documentation

operator()()

template<typename OutputIterator >

OutputIterator PolynomialTraits_d::PolynomialSubresultants::operator()	(	Polynomial_d	p,
		Polynomial_d	q,
		OutputIterator	out
	)

computes the polynomial subresultants of $p$ and $q$, with respect to the outermost variable.

Each element is of type PolynomialTraits_d::Polynomial_d.