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function X = cs_qleft (V, Beta, p, Y)
%CS_QLEFT apply Householder vectors on the left.
% X = cs_qleft(V,Beta,p,Y) computes X = Hn*...*H2*H1*P*Y = Q'*Y where Q is
% represented by the Householder vectors V, coefficients Beta, and
% permutation p. p can be [], which denotes the identity permutation.
%
% Example:
% Prob = ssget ('HB/well1033') ; A = Prob.A ; [m n] = size (A) ;
% b = rand (m,1) ;
% [V,beta,p,R] = cs_qr (A) ; % QR factorization of A(p,:)
% b1 = cs_qleft (V, beta, p, b) ;
% x1 = R (1:n,1:n) \ b1 (1:n) ;
% x2 = A\b ;
% norm (x1-x2)
%
% See also CS_QR, CS_QRIGHT.
% CXSparse, Copyright (c) 2006-2022, Timothy A. Davis. All Rights Reserved.
% SPDX-License-Identifier: LGPL-2.1+
[m2 n] = size (V) ;
[m ny] = size (Y) ;
X = Y ;
if (m2 > m)
if (issparse (Y))
X = [X ; sparse(m2-m,ny)] ;
else
X = [X ; zeros(m2-m,ny)] ;
end
end
if (~isempty (p))
X = X (p,:) ;
end
for k = 1:n
X = X - V (:,k) * (Beta (k) * (V (:,k)' * X)) ;
end
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