Code covered by the BSD License

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

### Tim Davis (view profile)

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_qrt_sparse
```classdef factorization_qrt_sparse < factorization
%FACTORIZATION_QRT_SPARSE (P*A)*(P*A)'=R'*R where A is sparse.

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

methods

function F = factorization_qrt_sparse (A, fail_if_singular)
%FACTORIZATION_QRT_SPARSE economy sparse QR: (P*A)*(P*A)'=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 (n,0), 0) ;
f.P = sparse (1:m, p, 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'': (P*A)*(P*A)'' = 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.P*F.A)*(f.P*F.A)' - f.R'*f.R, 1) / norm (F.A*F.A', 1) ;
end

function x = mldivide_subclass (F,b)
%MLDIVIDE_SUBCLASS x = A\b using economy sparse QR of A'
% minimum 2-norm solution of an underdetermined system
A = F.A ;
f = F.Factors ;
x = A' * (f.P' * (f.R \ (f.R' \ (f.P * b)))) ;
e = A' * (f.P' * (f.R \ (f.R' \ (f.P * (b - A * x))))) ;
x = x + e ;
end

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

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