function [Q,R]=mgs(X)
%
%  function [Q,R]=mgs(X)
%   Modified Gram-Schmidt algorithm for QR factorization
%
[m,n]=size(X);
R=[];
Q=[];
p=min(m,n);
for k=1:p-1
  rkk=norm(X(:,k));
  qk=X(:,k)/rkk;
  v=X(:,k+1:p)'*qk;
  X(:,k+1:p)=X(:,k+1:p)-qk*v';
  Q=[Q qk ];
  R=[R  ; zeros(1,k-1) rkk v'];
end;
rkk=norm(X(:,p));
qk=X(:,p)/rkk;
Q=[Q qk];
R=[R ; zeros(1,p-1) rkk ];