Legendre Laguerre and Hermite - Gauss Quadrature

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19 Feb 2010 (Updated )

Nodes and weights for Legendre Laguerre and Hermite - Gauss Quadrature

GaussLegendre.m
function [x, w] = GaussLegendre_2(n)

% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %
% This function determines the abscisas (x) and weights (w)  for the        %
% Gauss-Legendre quadrature, of order n>1, on the interval [-1, +1].        %
%   Unlike many publicly available functions, 'GaussLegendre_2' is valid    %
%   for n>=46. This is due to the fact that 'GaussLegendre_2' does not      %
%   rely on the build-in Matlab routine 'roots' to determine the roots of   %
%   the Legendre polynomial, but finds the roots by looking for the         %
%   eigenvalues of an alternative version of the companion matrix of the    %
%   n'th degree Legendre polynomial. The companion matrix is constructed    %
%   as a symmetrical matrix, guaranteeing that all the eigenvalues          %
%   (roots) will be real. On the contrary, the 'roots' function uses a      %
%   general form for the companion matrix, which becomes unstable at        %
%   higher orders n, leading to complex roots.                              %
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %


%  Geert Van Damme
% geert@vandamme-iliano.be
% February 21, 2010    



% Building the companion matrix CM
    % CM is such that det(xI-CM)=P_n(x), with P_n the Legendre polynomial
    % under consideration. Moreover, CM will be constructed in such a way
    % that it is symmetrical.
i   = 1:n-1;
a   = i./sqrt(4*i.^2-1);
CM  = diag(a,1) + diag(a,-1);

% Determining the abscissas (x) and weights (w)
    % - since det(xI-CM)=P_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       = 2 * V(:,1).^2;

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