% QR decomposition of R + u*v^T
function [Q,R]=QR_Ruv(R,u,v)
n = length(u);
Q = eye(n);
% zero out u(2:n) using Givens rotations; update R
for i=n:-1:2,
    rot=Givens(u(i-1:i));
    u(i-1:i)=rot*u(i-1:i);
    R(i-1:i,i-1:n) = rot*R(i-1:i,i-1:n);
    Q(i-1:n,i-1:i) = Q(i-1:n,i-1:i)*rot';
end
% add u(1)*v' to the first row of R
R(1,:) = R(1,:) + u(1)*v';
% return R to upper triangular form
for i=2:n
    rot=Givens(R(i-1:i,i-1));
    R(i-1:i,i-1:n) = rot*R(i-1:i,i-1:n);
    Q(:,i-1:i) = Q(:,i-1:i)*rot';
end