Legendre Laguerre and Hermite - Gauss Quadrature
19 Feb 2010
(Updated 21 Feb 2010)
Nodes and weights for Legendre Laguerre and Hermite - Gauss Quadrature
function [x, w] = GaussHermite_2(n)
% This function determines the abscisas (x) and weights (w) for the
% Gauss-Hermite quadrature of order n>1, on the interval [-INF, +INF].
% This function is valid for any degree n>=2, as the companion matrix
% (of the n'th degree Hermite polynomial) is constructed as a
% symmetrical matrix, guaranteeing that all the eigenvalues (roots)
% will be real.
% Geert Van Damme
% February 21, 2010
% Building the companion matrix CM
% CM is such that det(xI-CM)=L_n(x), with L_n the Hermite polynomial
% under consideration. Moreover, CM will be constructed in such a way
% that it is symmetrical.
i = 1:n-1;
a = sqrt(i/2);
CM = diag(a,1) + diag(a,-1);
% Determining the abscissas (x) and weights (w)
% - since det(xI-CM)=L_n(x), the abscissas are the roots of the
% characteristic polynomial, i.d. the eigenvalues of CM;
% - the weights can be derived from the corresponding eigenvectors.
[V L] = eig(CM);
[x ind] = sort(diag(L));
V = V(:,ind)';
w = sqrt(pi) * V(:,1).^2;