function X = cs_qright (V, Beta, p, Y)
%CS_QRIGHT apply Householder vectors on the right.
%   X = cs_qright(V,Beta,p,Y) computes X = Y*P'*H1*H2*...*Hn = Y*Q where Q is
%   represented by the Householder vectors V, coefficients Beta, and
%   permutation p.  p can be [], which denotes the identity permutation.
%   To obtain Q itself, use Q = cs_qright(V,Beta,p,speye(size(V,1))).
%
%   Example:
%       load west0479 ; q = colamd (west0479) ; A = west0479 (:,q) ;
%       [Q,R] = qr (A) ; norm (Q*R-A, 1)
%       [V,beta,p,R2] = cs_qr (A) ;
%       Q2 = cs_qright (V, beta, p, speye(size(V,1))) ; norm (Q2*R2-A, 1)
%
%   See also CS_QR, CS_QLEFT.

%   Copyright 2006-2007, Timothy A. Davis.
%   http://www.cise.ufl.edu/research/sparse

[m n] = size (V) ;
X = Y ;
if (~isempty (p))
    X = X (:,p) ;
end
for k = 1:n
    X = X - (X * (Beta (k) * V (:,k))) * V (:,k)' ;
end
