function [Q,R] = mgs2(A);
% MGS2 computes QR reduced factors by modified Gram-Schmidt, but it completes
%      Q in the rank-deficient case, and it estimates rank(A)

% demonstration/testing:
%   >> m=17; n=7; A=randn(m,n); A(:,n)=A(:,2); A(:,n-1)=A(:,n-2)+A(:,n-3);
%   >> [Q,R]=mgs2(A); norm(A-Q*R,1), norm(Q'*Q-eye(n,n),1), rank(A)

[m n] = size(A);
rank = n; % reduces this if it finds very small vectors
R = zeros(n,n);
scal = max(max(abs(A)));
tol = (max([m n]) + 10) * eps;
if scal == 0
    Q = eye(m,n);
    return, end

Q = A;
for i = 1:n
    r = norm(Q(:,i),2);
    R(i,i) = r;
    if abs(r)/scal < tol
        warning(['column ' num2str(i) ' gives nearly zero internal vector'])
        rank = rank - 1;
        w = randn(m,1);
        for j = 1:i-1
          w = w - (Q(:,j)' * w) * Q(:,j);
          end
        w = w / norm(w,2);
    else
        w = Q(:,i)/r; end
    Q(:,i) = w;
    for j = i+1:n
        r = w'*Q(:,j);
        R(i,j) = r;
        Q(:,j) = Q(:,j)-r*w;
    end
end

if rank < n
    warning(['input essentially rank-deficient: rank = ' num2str(rank) '?'])
end