Polynomials module

This module provides a QR based polynomial solver.

To use this module, add

#include <unsupported/Eigen/Polynomials>

at the start of your source file.


class	Eigen::PolynomialSolver< _Scalar, _Deg >
	A polynomial solver. More...

class	Eigen::PolynomialSolverBase< _Scalar, _Deg >
	Defined to be inherited by polynomial solvers: it provides convenient methods such as. More...


template<typename Polynomial >
NumTraits< typename Polynomial::Scalar >::Real	Eigen::cauchy_max_bound (const Polynomial &poly)

template<typename Polynomial >
NumTraits< typename Polynomial::Scalar >::Real	Eigen::cauchy_min_bound (const Polynomial &poly)

template<typename Polynomials , typename T >
T	Eigen::poly_eval (const Polynomials &poly, const T &x)

template<typename Polynomials , typename T >
T	Eigen::poly_eval_horner (const Polynomials &poly, const T &x)

template<typename RootVector , typename Polynomial >
void	Eigen::roots_to_monicPolynomial (const RootVector &rv, Polynomial &poly)

cauchy_max_bound()

template<typename Polynomial >

NumTraits<typename Polynomial::Scalar>::Real Eigen::cauchy_max_bound

(

const Polynomial &

poly

)

inline

Returns: a maximum bound for the absolute value of any root of the polynomial.

Parameters

[in]

poly

: the vector of coefficients of the polynomial ordered by degrees i.e. poly[i] is the coefficient of degree i of the polynomial e.g. \( 1 + 3x^2 \) is stored as a vector \( [ 1, 0, 3 ] \).

Precondition: the leading coefficient of the input polynomial poly must be non zero

cauchy_min_bound()

template<typename Polynomial >

NumTraits<typename Polynomial::Scalar>::Real Eigen::cauchy_min_bound

(

const Polynomial &

poly

)

inline

Returns: a minimum bound for the absolute value of any non zero root of the polynomial.

Parameters

[in]

poly

: the vector of coefficients of the polynomial ordered by degrees i.e. poly[i] is the coefficient of degree i of the polynomial e.g. \( 1 + 3x^2 \) is stored as a vector \( [ 1, 0, 3 ] \).

poly_eval()

template<typename Polynomials , typename T >

T Eigen::poly_eval	(	const Polynomials &	poly,
		const T &	x
	)

inline

Returns: the evaluation of the polynomial at x using stabilized Horner algorithm.

Parameters

[in]	poly	: the vector of coefficients of the polynomial ordered by degrees i.e. poly[i] is the coefficient of degree i of the polynomial e.g. \( 1 + 3x^2 \) is stored as a vector \( [ 1, 0, 3 ] \).
[in]	x	: the value to evaluate the polynomial at.

poly_eval_horner()

template<typename Polynomials , typename T >

T Eigen::poly_eval_horner	(	const Polynomials &	poly,
		const T &	x
	)

inline

Returns: the evaluation of the polynomial at x using Horner algorithm.

Parameters

[in]	poly	: the vector of coefficients of the polynomial ordered by degrees i.e. poly[i] is the coefficient of degree i of the polynomial e.g. \( 1 + 3x^2 \) is stored as a vector \( [ 1, 0, 3 ] \).
[in]	x	: the value to evaluate the polynomial at.

Note: for stability: \( |x| \le 1 \)

roots_to_monicPolynomial()

template<typename RootVector , typename Polynomial >

void Eigen::roots_to_monicPolynomial	(	const RootVector &	rv,
		Polynomial &	poly
	)

Given the roots of a polynomial compute the coefficients in the monomial basis of the monic polynomial with same roots and minimal degree. If RootVector is a vector of complexes, Polynomial should also be a vector of complexes.

Parameters

[in]	rv	: a vector containing the roots of a polynomial.
[out]	poly	: the vector of coefficients of the polynomial ordered by degrees i.e. poly[i] is the coefficient of degree i of the polynomial e.g. \( 3 + x^2 \) is stored as a vector \( [ 3, 0, 1 ] \).