function [xls_new, Pinvsr_new ] = qr_rls( Pinvsr, xls, a, y )
% function [xls_new, Pinvsr_new ] = qr_rls( Pinvsr, xls, a, y )
%
% QR version of recursive least-squares estimator for solving y=Px.
% 'a' is a new row of matrix P, and 'y' is the associated target value.
%
% 'Pinvsr' is the square root of Pinv=(P'*P)^-1,
%    and should be LOWER triangular (ie, Pinvsr = chol(Pinv)' ).
% 'a' should be specified as a column vector.
% 'y' may be a row vector, in which case 'xls' will be a matrix.
%
% This version does not include a forgetting factor.
%
% D. Wingate 5/12/2006
%

  err = y - a'*xls;
  n = size(Pinvsr,1);

  M = [ 1, a'*Pinvsr ; zeros( n, 1 ), Pinvsr ];

%  [Q,R] = qr( M );
  R = triu( qr( M' ) );

  Pinvsr_new = R( 2:1+n, 2:1+n )';

  Kt = R( 1, 2:1+n )' / R(1,1);

  xls_new = xls + Kt * err;

return;