1function [V,beta,p,R,q] = cs_qr (A) %#ok 2%CS_QR sparse QR factorization (Householder-based). 3% [V,beta,p,R] = cs_qr(A) computes the QR factorization of A(p,:). 4% [V,beta,p,R,q] = cs_qr(A) computes the QR factorization of A(p,q). 5% The V, beta, and p terms represent the Householder vectors and coefficients. 6% The fill-reducing ordering q is found via q = cs_amd(A,3). 7% The orthogonal factor Q can be obtained via 8% Q = cs_qright(V,beta,p,speye(size(V,1))), in which case Q*R=A(:,q) is the 9% resulting factorization (the permutation p is folded into Q). A must be 10% m-by-n with m >= n. If A is structurally rank deficient, additional empty 11% rows may have been added to V and R. Note that V is typically much sparser 12% than Q. 13% 14% Example: 15% 16% Prob = ssget ('HB/well1033') ; A = Prob.A ; [m n] = size (A) ; 17% b = rand (m,1) ; 18% [V,beta,p,R,q] = cs_qr (A) ; % QR factorization of A(p,q) 19% b1 = cs_qleft (V, beta, p, b) ; 20% x1 = R (1:n,1:n) \ b1 (1:n) ; 21% x1 (q) = x1 ; 22% x2 = A\b ; 23% norm (x1-x2) 24% Q = cs_qright(V,beta,p,speye(size(V,1))) ; % Note: p accounted for in Q 25% norm (Q*R-A(:,q),1) 26% fprintf ('nnz(R) %d, nnz(V) %d, nnz(Q) %d\n', nnz(R), nnz(V), nnz(Q)) ; 27% 28% See also CS_AMD, CS_QRIGHT, CS_QR, CS_DMPERM, QR, COLAMD. 29 30% Copyright 2006-2012, Timothy A. Davis, http://www.suitesparse.com 31 32error ('cs_qr mexFunction not found') ; 33