function x = ring_adj_null_left ( m, n )

%*****************************************************************************80
%
%% RING_ADJ_NULL_LEFT returns a left null vector of the RING_ADJ matrix.
%
%  Licensing:
%
%    This code is distributed under the MIT license.
%
%  Modified:
%
%    12 March 2015
%
%  Author:
%
%    John Burkardt
%
%  Input:
%
%    integer M, N, the order of the matrix.
%
%  Output:
%
%    real X(M,1), the null vector.
%
  if ( mod ( m, 4 ) ~= 0 )
    fprintf ( 1, '\n' );
    fprintf ( 1, 'RING_ADJ_NULL_LEFT - Fatal error!\n' );
    fprintf ( 1, '  M must be a multiple of 4.\n' );
    error ( 'RING_ADJ_NULL_LEFT - Fatal error!' );
  end

  x = zeros ( m, 1 );

  x(1:4:m,1) = + 1.0;
  x(2:4:m,1) = + 1.0;
  x(3:4:m,1) = - 1.0;
  x(4:4:m,1) = - 1.0;

  return
end
