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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.
% CXSparse, Copyright (c) 2006-2022, Timothy A. Davis. All Rights Reserved.
% SPDX-License-Identifier: LGPL-2.1+
[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
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