function a = routh_matrix ( n, x )

%*****************************************************************************80
%
%% routh_matrix returns the ROUTH matrix.
%
%  Formula:
%
%    A is tridiagonal.
%    A(1,1)   =          X(1).
%    A(I-1,I) =   sqrt ( X(I) ), for I = 2 to N.
%    A(I,I-1) = - sqrt ( X(I) ), for I = 2 to N.
%
%  Example:
%
%    N = 5, X = ( 1, 4, 9, 16, 25 )
%
%    1 -2  0  0  0
%    2  0 -3  0  0
%    0  3  0 -4  0
%    0  0  4  0 -5
%    0  0  0  5  0
%
%  Properties:
%
%    A is generally not symmetric: A' /= A.
%
%    A is tridiagonal.
%
%    Because A is tridiagonal, it has property A (bipartite).
%
%    A is banded, with bandwidth 3.
%
%    A is integral, therefore det ( A ) is integral, and 
%    det ( A ) * inverse ( A ) is integral.
%
%    det ( A ) = product ( X(N) * X(N-2) * X(N-4) * ... * X(N+1-2*(N/2)) )
%
%    The family of matrices is nested as a function of N.
%
%    A is not diagonally dominant.
%
%  Licensing:
%
%    This code is distributed under the MIT license.
%
%  Modified:
%
%    20 October 2007
%
%  Author:
%
%    John Burkardt
%
%  Input:
%
%    integer N, the order of A.
%
%    real X(N), the data that defines the matrix.
%
%  Output:
%
%    real A(N,N), the matrix.
%
  a = zeros ( n, n );

  for i = 1 : n
    for j = 1 : n

      if ( i == 1 && j == 1 )
        a(i,j) = abs ( x(1) );
      elseif ( i == j + 1 )
        a(i,j) = sqrt ( abs ( x(i) ) );
      elseif ( i == j - 1 )
        a(i,j) = - sqrt ( abs ( x(i+1) ) );
      else
        a(i,j) = 0.0;
      end

    end
  end

  return
end
