gaussian_quadrature.hpp File Reference

implementation of Gaussian quadratures More...

#include "quadrature.hpp"
#include "../library/lib_domain.hpp"
#include <stdexcept>

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Classes
class	NiHu::gaussian_quadrature< Domain >
struct	NiHu::quadrature_traits< gaussian_quadrature< Domain > >
	traits of a Gaussian quadrature More...
class	NiHu::gaussian_quadrature< line_domain >
	Gaussian quadrature over a line domain. More...
class	NiHu::gaussian_quadrature< quad_domain >
	Gaussian quadrature over a quad domain. More...
class	NiHu::gaussian_quadrature< tria_domain >
	specialisation of gauss_quadrature for a triangle domain More...
struct	NiHu::gauss_family_tag
	tag for the family of Gaussian quadratures More...
struct	NiHu::quadrature_type< gauss_family_tag, Domain >
	specialisation of quadrature_type to Gaussian family on line More...
struct	NiHu::quadrature_traits< log_gaussian_quadrature >
	traits of a Log-Gaussian quadrature More...
class	NiHu::log_gaussian_quadrature
	Log-Gaussian quadrature over a line domain. More...

Functions
template<class scalar_t >
Eigen::Matrix< scalar_t, Eigen::Dynamic, 2 >	NiHu::gauss_impl (size_t N)
	return 1D N-point Guassian quadrature More...

Detailed Description

implementation of Gaussian quadratures

Definition in file gaussian_quadrature.hpp.

Function Documentation

gauss_impl()

template<class scalar_t >

Eigen::Matrix<scalar_t, Eigen::Dynamic, 2> NiHu::gauss_impl

(

size_t

N

)

return 1D N-point Guassian quadrature

Template Parameters

scalar_t

the scalar type

Parameters

[in]

N

number of Gaussian points

Returns

matrix containing the Gaussian locations and weights

The Gaussian locations are roots of the Legendre polynomials \(\varphi_n(x)\). The polynomials obey the recurrence relation

\(\varphi_{0}(x) = 1\)

\(\varphi_{1}(x) = x\)

\(\varphi_{n+1}(x) = x\varphi_{n}(x) - b_n \varphi_{n-1}(x)\), where \(b_n = \frac{n^2}{4n^2-1}\)

so the Legendre polynomials are the characteristic polynomial of the tridiagonal Jacobi matrix

\(J = \begin{pmatrix} 0 & \sqrt{b_1} \\ \sqrt{b_1} & 0 & \sqrt{b_2} \\ & \ddots & & \ddots \\ & & \sqrt{b_{N-2}} & 0 & \sqrt{b_{N-1}} \\ & & & \sqrt{b_{N-1}} & 0 \end{pmatrix}\)

and therefore the quadrature locations are the eigenvalues of \(J\)

Definition at line 62 of file gaussian_quadrature.hpp.