function C = chebspec(n, k)
%CHEBSPEC  Chebyshev spectral differentiation matrix.
%          C = CHEBSPEC(N, K) is a Chebyshev spectral differentiation
%          matrix of order N.  K = 0 (the default) or 1.
%          For K = 0 (`no boundary conditions'), C is nilpotent, with
%              C^N = 0 and it has the null vector ONES(N,1).
%              C is similar to a Jordan block of size N with eigenvalue zero.
%          For K = 1, C is nonsingular and well-conditioned, and its eigenvalues
%              have negative real parts.
%          For both K, the computed eigenvector matrix X from EIG is
%              ill-conditioned (MESH(REAL(X)) is interesting).

%          References:
%          C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang, Spectral
%             Methods in Fluid Dynamics, Springer-Verlag, Berlin, 1988; p. 69.
%          L.N. Trefethen and M.R. Trummer, An instability phenomenon in
%             spectral methods, SIAM J. Numer. Anal., 24 (1987), pp. 1008-1023.
%          D. Funaro, Computing the inverse of the Chebyshev collocation
%             derivative, SIAM J. Sci. Stat. Comput., 9 (1988), pp. 1050-1057.

if nargin == 1, k = 0; end

% k = 1 case obtained from k = 0 case with one bigger n.
if k == 1, n = n + 1; end

n = n-1;
C = zeros(n+1);

one = ones(n+1,1);
x = cos( (0:n)' * (pi/n) );
d = ones(n+1,1); d(1) = 2; d(n+1) = 2;

% eye(size(C)) on next line avoids div by zero.
C = (d * (one./d)') ./ (x*one'-one*x' + eye(size(C)));

%  Now fix diagonal and signs.

C(1,1) = (2*n^2+1)/6;
for i=2:n+1
    if rem(i,2) == 0
       C(:,i) = -C(:,i);
       C(i,:) = -C(i,:);
    end
    if i < n+1
       C(i,i) = -x(i)/(2*(1-x(i)^2));
    else
       C(n+1,n+1) = -C(1,1);
    end
end

if k == 1
   C = C(2:n+1,2:n+1);
end