function A = prolate(n, w)
%PROLATE Prolate matrix (symmetric, ill-conditioned Toeplitz matrix).
%   A = GALLERY('PROLATE',N,W) is the N-by-N prolate matrix with 
%   parameter W. It is a symmetric Toeplitz matrix.
%   If 0 < W < 0.5 then
%     1. A is positive definite
%     2. The eigenvalues of A are distinct, lie in (0, 1), and tend to 
%        cluster around 0 and 1.
%   W defaults to 0.25.

%   Reference:
%   [1] J.M. Varah. The Prolate matrix. Linear Algebra and Appl.,
%       187:269--278, 1993.
%
%   Copyright 1984-2002 The MathWorks, Inc. 
%   $Revision: 1.9 $  $Date: 2002/04/15 03:43:17 $

if nargin == 1, w = 0.25; end

a = zeros(n,1);
a(1) = 2*w;
a(2:n) = sin( 2*pi*w*(1:n-1) ) ./ ( pi*(1:n-1) );

A = toeplitz(a);