Polynomials Namespace Reference

Namespaces
namespace	internal

Classes
class	HermiteInterpolation
class	HermiteLikeInterpolation
class	Hierarchical
class	LagrangeEquidistant
class	Legendre
class	Lobatto
class	Monomial
class	PiecewisePolynomial
class	Polynomial
class	PolynomialsHermite

Functions
std::vector< Polynomial< double > >	generate_complete_Lagrange_basis (const std::vector< Point< 1 > > &points)
template<typename Number >
Number	jacobi_polynomial_value (const unsigned int degree, const int alpha, const int beta, const Number x, const bool rescale_to_dealii_unit_interval=true)
template<typename Number >
Number	jacobi_polynomial_derivative (const unsigned int degree, const int alpha, const int beta, const Number x, const bool rescale_to_dealii_unit_interval)
template<typename Number >
Number	jacobi_polynomial_kth_derivative (const unsigned int derivative_order, const unsigned int degree, const int alpha, const int beta, const Number x, const bool rescale_to_dealii_unit_interval)
template<typename Number >
std::vector< Number >	jacobi_polynomial_roots (const unsigned int degree, const int alpha, const int beta)
template<typename Number >
Number	jacobi_polynomial_homogenized_value (const unsigned int degree, const int alpha, const int beta, const Number x, const Number s)
template<typename Number >
Number	jacobi_polynomial_homogenized_derivative (const unsigned int order_x, const unsigned int order_s, const unsigned int degree, const int alpha, const int beta, const Number x, const Number s)
std::vector< PiecewisePolynomial< double > >	generate_complete_Lagrange_basis_on_subdivisions (const unsigned int n_subdivisions, const unsigned int base_degree)
std::vector< PiecewisePolynomial< double > >	generate_complete_linear_basis_on_subdivisions (const std::vector< Point< 1 > > &points)

Detailed Description

A namespace in which classes relating to the description of 1d polynomial spaces are declared.

Function Documentation

generate_complete_Lagrange_basis()

std::vector< Polynomial< double > > Polynomials::generate_complete_Lagrange_basis

(

const std::vector< Point< 1 > > &

points

)

Given a set of points along the real axis, this function returns all Lagrange polynomials for interpolation of these points. The number of polynomials is equal to the number of points and the maximum degree is one less.

Definition at line 692 of file polynomial.cc.

jacobi_polynomial_value()

template<typename Number >

Number Polynomials::jacobi_polynomial_value	(	const unsigned int	degree,
		const int	alpha,
		const int	beta,
		const Number	x,
		const bool	rescale_to_dealii_unit_interval = true
	)

Definition at line 1154 of file polynomial.h.

jacobi_polynomial_derivative()

template<typename Number >

Number Polynomials::jacobi_polynomial_derivative	(	const unsigned int	degree,
		const int	alpha,
		const int	beta,
		const Number	x,
		const bool	rescale_to_dealii_unit_interval
	)

Definition at line 1197 of file polynomial.h.

jacobi_polynomial_kth_derivative()

template<typename Number >

Number Polynomials::jacobi_polynomial_kth_derivative	(	const unsigned int	derivative_order,
		const unsigned int	degree,
		const int	alpha,
		const int	beta,
		const Number	x,
		const bool	rescale_to_dealii_unit_interval
	)

Definition at line 1211 of file polynomial.h.

jacobi_polynomial_roots()

template<typename Number >

std::vector< Number > Polynomials::jacobi_polynomial_roots	(	const unsigned int	degree,
		const int	alpha,
		const int	beta
	)

Compute the roots of the Jacobi polynomials on the unit interval \([0, 1]\) of the given degree. These roots are used in several places inside the deal.II library, such as the Gauss-Lobatto quadrature formula or for the Hermite-like interpolation.

The algorithm uses a Newton algorithm, using the zeros of the Chebyshev polynomials as an initial guess. This code has been tested for alpha and beta equal to zero (Legendre case), one (Gauss-Lobatto case) as well as two, so be careful when using it for other values as the Newton iteration might or might not converge.

For the definition of the Jacobi polynomials, see https://en.wikipedia.org/wiki/Gauss%E2%80%93Jacobi_quadrature, though that page uses the interval \([-1,1]\) instead, and so the weights associated with \(\beta\) are there given as \((1+x)^\beta\) whereas in our implementation the weights are \((1-x)^\alpha\) and \(x^\beta\).

Definition at line 1254 of file polynomial.h.

jacobi_polynomial_homogenized_value()

template<typename Number >

Number Polynomials::jacobi_polynomial_homogenized_value	(	const unsigned int	degree,
		const int	alpha,
		const int	beta,
		const Number	x,
		const Number	s
	)

Definition at line 1333 of file polynomial.h.

jacobi_polynomial_homogenized_derivative()

template<typename Number >

Number Polynomials::jacobi_polynomial_homogenized_derivative	(	const unsigned int	order_x,
		const unsigned int	order_s,
		const unsigned int	degree,
		const int	alpha,
		const int	beta,
		const Number	x,
		const Number	s
	)

Definition at line 1375 of file polynomial.h.

generate_complete_Lagrange_basis_on_subdivisions()

std::vector< PiecewisePolynomial< double > > Polynomials::generate_complete_Lagrange_basis_on_subdivisions	(	const unsigned int	n_subdivisions,
		const unsigned int	base_degree
	)

Generates a complete Lagrange basis on a subdivision of the unit interval in smaller intervals for a given degree on the subintervals and number of intervals.

Definition at line 211 of file polynomials_piecewise.cc.

generate_complete_linear_basis_on_subdivisions()

std::vector< PiecewisePolynomial< double > > Polynomials::generate_complete_linear_basis_on_subdivisions

(

const std::vector< Point< 1 > > &

points

)

Generates a complete linear basis on a subdivision of the unit interval in smaller intervals for a given vector of points.

Definition at line 236 of file polynomials_piecewise.cc.