function [x, w] = plate_gauss_legendre(n)
% PLATE_GAUSS_LEGENDRE  Nodes and weights of n-point Gauss-Legendre quadrature on [-1, 1].
%
%   [x, w] = plate_gauss_legendre(n) returns row vectors, nodes ascending,
%   by the Golub-Welsch method: the nodes are the eigenvalues of the
%   symmetric tridiagonal Jacobi matrix of the Legendre recurrence and the
%   weights are twice the squared first components of its eigenvectors.

if n == 1
    x = 0; w = 2;
    return
end
k = 1:n - 1;
b = k ./ sqrt(4 * k.^2 - 1);
J = diag(b, 1) + diag(b, -1);
[V, D] = eig(J);
[x, order] = sort(diag(D)');
w = 2 * V(1, order).^2;
end