% Straight-forward implementation of the 
% QR-factorization using modified Gram-Schmidt:
% A*R = Q

function [Q,R] = MGS_qr(V)

Q = V;
[rows,cols]=size(V);m=cols;
R = eye(m);
for k=1:m
    WR = eye(m);
    vec_coef=zeros(1,m);
    n(k,k) = norm(Q(:,k)); ni(k,k)=1/n(k,k);
    vec_coef(1,k) = ni(k,k);
    Q(:,k)=Q(:,k)/n(k,k);
    for l=k+1:m
        nn(k,l)= Q(:,k)'*Q(:,l);
        Q(:,l) = Q(:,l) - nn(k,l)*Q(:,k);
	vec_coef(1,l) = - nn(k,l)*ni(k,k);
    end
    WR(k,1:m)=vec_coef;
    R = R*WR;
end
        
%