function [Q,R] = mgs(A)
% usage: [Q,R] = mgs(A)
% description: this function calculates a reduced QR 
% factorizaton of full column rank matrix A (this is
% assumed!) by the modified Gram-Schmidt algorithm.

% local variables:
% ii,jj: index variables

[m,n] = size(A);
Q = A;
R = zeros(n,n);
for ii = 1:n
  R(ii,ii) = norm(Q(:,ii));
  Q(:,ii) = Q(:,ii)/R(ii,ii);
  for jj = (ii+1):n
    R(ii,jj) = Q(:,ii)'*Q(:,jj);
    Q(:,jj) = Q(:,jj) - R(ii,jj)*Q(:,ii);
  end
end