%
% Matlab function mgs.m
%
% Purpose: This function uses the modified Gram-Schmidt
%          orthonormalization algorithm to compute the
%          reduced QR factorization of the matrix A. If
%          the matrix does not possess linearly independent
%          columns, then this algorithm will trap the error
%          and return to the calling environment. Also, the
%          function will print a diagnostic that measures the
%          orthonormality of the Q matrix.
%
% Author:
%   Copyright (c)2001 by John S. Tamaresis.
%   All rights reserved.
%
% Date:
%   27 February 2001
%
% Inputs:
%   A = m-by-n matrix with real entries (m >= n)
%
% Outputs:
%   Q = m-by-n matrix with orthonormal columns
%   R = n-by-n upper triangular matrix
%
function [Q,R] = mgs(A)

% Determine the size and condition number of the matrix.
   [mrows,ncols] = size(A);
   if mrows < ncols
      error('invalid dimension for A')
   end

% Initialize the factor matrices.
   Q = zeros(mrows,ncols);
   R = zeros(ncols);

% Begin the factorization.
   for ic=1:ncols,
      R(ic,ic) = norm(A(:,ic));
% Test the current diagonal entry to determine
% whether the matrix A is rank deficient.
      if R(ic,ic) == 0.
         error('The original matrix is rank deficient.')
      end
      Q(:,ic) = (1./R(ic,ic)).*A(:,ic);
      for ir=ic+1:ncols,
         R(ic,ir) = (Q(:,ic)')*A(:,ir);
         A(:,ir) = A(:,ir) - R(ic,ir)*Q(:,ic);
      end
   end

% End of function