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] = GaussLaguerre_2(n, alpha)
% This function determines the abscisas (x) and weights (w) for the
% Gauss-Laguerre quadrature of order n>1, on the interval [0, +infinity].
% Unlike the function 'GaussLaguerre', this function is valid for
% n>=34. This is due to the fact that the companion matrix (of the n'th
% degree Laguerre polynomial) is now 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 Laguerree polynomial
% under consideration. Moreover, CM will be constructed in such a way
% that it is symmetrical.
i = 1:n;
a = (2*i-1) + alpha;
b = sqrt( i(1:n-1) .* ((1:n-1) + alpha) );
CM = diag(a) + diag(b,1) + diag(b,-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 = gamma(alpha+1) .* V(:,1).^2;