function [D, x] = lp_cheb(N)
% LP_CHEB  Chebyshev differentiation matrix and nodes on [-1, 1].
%
% x = cos(pi*(0:N)/N), ordered from 1 down to -1, and D is the (N+1)-by-(N+1)
% matrix with (D*v)(i) = p'(x_i) for the polynomial p of degree <= N through
% the data v.  This is Trefethen's cheb.m (Spectral Methods in MATLAB, 2000).
x = cos(pi*(0:N)'/N);
c = [2; ones(N-1, 1); 2] .* (-1).^(0:N)';
X = repmat(x, 1, N+1);
dX = X - X';
D = (c * (1 ./ c)') ./ (dX + eye(N+1));   % off-diagonal entries
D = D - diag(sum(D, 2));                  % diagonal, by the zero-row-sum rule
end