Legendre Laguerre and Hermite - Gauss Quadrature



19 Feb 2010 (Updated )

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
% geert@vandamme-iliano.be
% 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;

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