function [Q,R]=prob_4_4_27(A)
% returns the QR factorization of the matrix A
%
% Written by:
% -- 
% John L. Weatherwax                2006-10-23
% 
% email: wax@alum.mit.edu
% 
% Please send comments and especially bug reports to the
% above email address.
% 
%-----

[m,n]=size(A); Q=zeros(m,m); R=zeros(m,n); 
for k=1:n,
  R(k,k) = sqrt(sum(A(:,k).*A(:,k)));
  for i=1:m,
    Q(i,k) = A(i,k)/R(k,k);
  end
  for j=k+1:n,
    R(k,j) = sum(Q(:,k).*A(:,j)); 
  end
  for i=1:m,
    A(i,j)=A(i,j)-Q(i,k)*R(k,j);
  end
end