Code covered by the BSD License

# Don't let that INV go past your eyes; to solve that system, FACTORIZE!

14 May 2009 (Updated )

A simple-to-use object-oriented method for solving linear systems and least-squares problems.

### Editor's Notes:

This file was selected as MATLAB Central Pick of the Week

factorization_qr_sparse
```classdef factorization_qr_sparse < factorization
%FACTORIZATION_QR_SPARSE (A*P)'*(A*P) = R'*R where A is sparse.

% Copyright 2011-2012, Timothy A. Davis, http://www.suitesparse.com

methods

function F = factorization_qr_sparse (A, fail_if_singular)
%FACTORIZATION_QR_SPARSE economy sparse QR: (A*P)'*(A*P) = R'*R
if (~isa (A, 'double'))
error ('FACTORIZE:wrongtype', 'A must be double') ;
end
[m, n] = size (A) ;
if (m < n)
error ('FACTORIZE:wrongdim', 'QR of A requires m >= n.') ;
end
[~, f.R, p] = qr (A, sparse (m,0), 0) ;
f.P = sparse (p, 1:n, 1) ;
F.A_condest = cheap_condest (get_diag (f.R), fail_if_singular) ;
F.A = A ;
F.Factors = f ;
F.A_rank = rank_est (f.R, m, n) ;
F.kind = 'sparse QR factorization of A: (A*P)''*A*P = R''*R' ;
end

function e = error_check (F)
%ERROR_CHECK : return relative 1-norm of error in factorization
% meant for testing only
f = F.Factors ;
e = norm ((F.A*f.P)'*(F.A*f.P) - f.R'*f.R, 1) / norm (F.A'*F.A, 1) ;
end

function x = mldivide_subclass (F,b)
%MLDIVIDE_SUBLCASS x = A\b using economy sparse QR of A
% least-squares solution of an overdetermined problem
A = F.A ;
f = F.Factors ;
x = f.P * (f.R \ (f.R' \ (f.P' * (A' * b)))) ;
e = f.P * (f.R \ (f.R' \ (f.P' * (A' * (b - A * x))))) ;
x = x + e ;
end

function x = mrdivide_subclass (b,F)
%MRDIVIDE_SUBCLASS x = b/A using economy sparse QR of A
% minimum 2-norm solution of an underdetermined problem
bT = b' ;
A = F.A ;
f = F.Factors ;
x = A * (f.P * (f.R \ (f.R' \ (f.P' * bT)))) ;
e = A * (f.P * (f.R \ (f.R' \ (f.P' * (bT - A' * x))))) ;
x = (x + e)' ;
end
end
end
```