function v = haar_step_1d ( u )
%*****************************************************************************80
%
%% haar_step_1d() computes the Haar transform of a vector.
%
% Discussion:
%
% Typically, given a vector of length N, one computes the
% Haar transforms on V(1:N), then on V(1:N/2), then on V(1:N/4),
% and so on, producing a sequence of compressed representations
% of the data.
%
% The length of the vector U does not have to be a power of 2,
% nor does it have to be even.
%
% Licensing:
%
% This code is distributed under the MIT license.
%
% Modified:
%
% 04 March 2011
%
% Author:
%
% John Burkardt
%
% Input:
%
% real U(N,1), the vector to be transformed.
%
% Output:
%
% real V(N,1), the transformed vector.
%
%
% Force U to be a dadblasted column vector of dimensions N by 1.
%
u = u(:);
n = length ( u );
if ( mod ( n, 2 ) == 1 )
vx = u(n);
else
vx = [];
end
nhalf = floor ( n / 2 );
A = reshape ( u(1:2*nhalf), 2, nhalf );
lp = A' * [ 1/2; 1/2 ];
hp = A' * [ -1/2; 1/2 ];
v = [ lp; vx; hp ];
return
end