function Q = orthog(n, k)
%ORTHOG Orthogonal and nearly orthogonal matrices.
%       Q = ORTHOG(N, K) selects the K'th type of matrix of order N.
%       K > 0 for exactly orthogonal matrices, K < 0 for diagonal scalings of
%       orthogonal matrices.
%       Available types: (K = 1 is the default)
%       K = 1:  Q(i,j) = SQRT(2/(n+1)) * SIN( i*j*PI/(n+1) )
%               Symmetric eigenvector matrix for second difference matrix.
%       K = 2:  Q(i,j) = 2/SQRT(2*n+1)) * SIN( 2*i*j*PI/(2*n+1) )
%               Symmetric.
%       K = 3:  Q(r,s) = EXP(2*PI*i*(r-1)*(s-1)/n) / SQRT(n)  (i=SQRT(-1))
%               Unitary, the Fourier matrix.  Q^4 is the identity.
%               This is essentially the same matrix as FFT(EYE(N))/SQRT(N)!
%       K = 4:  Helmert matrix: a permutation of a lower Hessenberg matrix,
%               whose first row is ONES(1:N)/SQRT(N).
%       K = 5:  Q(i,j) = SIN( 2*PI*(i-1)*(j-1)/n ) + COS( 2*PI*(i-1)*(j-1)/n ).
%               Symmetric matrix arising in the Hartley transform.
%       K = -1: Q(i,j) = COS( (i-1)*(j-1)*PI/(n-1) )
%               Chebyshev Vandermonde-like matrix, based on extrema of T(n-1).
%       K = -2: Q(i,j) = COS( (i-1)*(j-1/2)*PI/n) )
%               Chebyshev Vandermonde-like matrix, based on zeros of T(n).

%       References:
%       N.J. Higham and D.J. Higham, Large growth factors in Gaussian
%            elimination with pivoting, SIAM J. Matrix Analysis and  Appl.,
%            10 (1989), pp. 155-164.
%       P. Morton, On the eigenvectors of Schur's matrix, J. Number Theory,
%            12 (1980), pp. 122-127. (Re. ORTHOG(N, 3))
%       H.O. Lancaster, The Helmert Matrices, Amer. Math. Monthly, 72 (1965),
%            pp. 4-12.
%       D. Bini and P. Favati, On a matrix algebra related to the discrete
%            Hartley transform, SIAM J. Matrix Anal. Appl., 14 (1993),
%            pp. 500-507.

if nargin == 1, k = 1; end

if k == 1
                                       % E'vectors second difference matrix
   m = (1:n)'*(1:n) * (pi/(n+1));
   Q = sin(m) * sqrt(2/(n+1));

elseif k == 2

   m = (1:n)'*(1:n) * (2*pi/(2*n+1));
   Q = sin(m) * (2/sqrt(2*n+1));

elseif k == 3                          %  Vandermonde based on roots of unity

   m = 0:n-1;
   Q = exp(m'*m*2*pi*sqrt(-1)/n) / sqrt(n);

elseif k == 4                          %  Helmert matrix

   Q = tril(ones(n));
   Q(1,2:n) = ones(1,n-1);
   for i=2:n
       Q(i,i) = -(i-1);
   end
   Q = diag( sqrt( [n 1:n-1] .* [1:n] ) ) \ Q;

elseif k == 5                          %  Hartley matrix

   m = (0:n-1)'*(0:n-1) * (2*pi/n);
   Q = (cos(m) + sin(m))/sqrt(n);

elseif k == -1
                                       %  extrema of T(n-1)
   m = (0:n-1)'*(0:n-1) * (pi/(n-1));
   Q = cos(m);

elseif k == -2
                                       % zeros of T(n)
   m = (0:n-1)'*(.5:n-.5) * (pi/n);
   Q = cos(m);

else

   error('Illegal value for second parameter.')

end