function A = chow(n, alpha, delta)
%CHOW    Chow matrix - a singular Toeplitz lower Hessenberg matrix.
%        A = CHOW(N, ALPHA, DELTA) is a Toeplitz lower Hessenberg matrix
%        A = H(ALPHA) + DELTA*EYE, where H(i,j) = ALPHA^(i-j+1).
%        H(ALPHA) has p = FLOOR(N/2) zero eigenvalues, the rest being
%        4*ALPHA*COS( k*PI/(N+2) )^2, k=1:N-p.
%        Defaults: ALPHA = 1, DELTA = 0.

%        References:
%        T.S. Chow, A class of Hessenberg matrices with known
%           eigenvalues and inverses, SIAM Review, 11 (1969), pp. 391-395.
%        G. Fairweather, On the eigenvalues and eigenvectors of a class of
%           Hessenberg matrices, SIAM Review, 13 (1971), pp. 220-221.
%        I. Singh, G. Poole and T. Boullion, A class of Hessenberg matrices
%           with known pseudoinverse and Drazin inverse, Math. Comp.,
%           29 (1975), pp. 615-619.

if nargin < 3, delta = 0; end
if nargin < 2, alpha = 1; end

A = toeplitz( alpha.^(1:n), [alpha 1 zeros(1,n-2)] ) + delta*eye(n);