function [Q, R] = gs(A);
%
%   Gram-Schmidt. this is not a stable way to compute a
%   qr factorization
%   written by Ming Gu for Math 221, Fall 2007
%
   [m, n] = size(A);
%
%   we assume that m>= n.
%   
   V = A;
   R = zeros(n,n);
   for j=1:n
      vj = A(:,j);
      if (j > 1)
         for i = 1:j-1
            R(i,j) = Q(:,i)' * A(:,j);
            vj     = vj - R(i,j) * Q(:,i);
         end
      end
      R(j,j) = norm(vj);
      Q(:,j) = vj / R(j,j);
   end