function a = integration_inverse ( alpha, n )

%*****************************************************************************80
%
%% INTEGRATION_INVERSE returns the inverse of the INTEGRATION matrix.
%
%  Formula:
%
%    if ( I = J )
%      A(I,J) = 1
%    else if ( I < J )
%      A(I,J) = (-ALPHA)^(J-I) / (I*...*J-1)
%    else
%      A(I,J) = 0
%
%  Example:
%
%    ALPHA = 2, N = 5
%
%    1 -2   2  -4/3   2/3
%    0  1  -1   2/3  -1/3
%    0  0   1  -2/3   1/3
%    0  0   0    1   -1/2
%    0  0   0    0     1
%
%  Properties:
%
%    A is generally not symmetric: A' /= A.
%
%    A is unit upper triangular.
%
%    A is nonsingular.
%
%    det ( A ) = 1.
%
%    A is unimodular.
%
%    LAMBDA(1:N) = 1.
%
%  Licensing:
%
%    This code is distributed under the GNU LGPL license.
%
%  Modified:
%
%    12 October 2007
%
%  Author:
%
%    John Burkardt
%
%  Parameters:
%
%    Input, real ALPHA, the scalar which defines the first
%    superdiagonal of the matrix.
%
%    Input, integer N, the number of rows and columns of A.
%
%    Output, real A(N,N), the matrix.
%
  a = zeros ( n, n );

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

      if ( j == i )
        a(i,j) = 1.0;
      elseif ( i < j )
        a(i,j) = ( -alpha ) ^ ( j - i ) / i4_rise ( i, j - i );
      else
        a(i,j) = 0.0;
      end

    end
  end

  return
end
