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_chol_dense
```classdef factorization_chol_dense < factorization
%FACTORIZATION_CHOL_DENSE A = R'*R where A is full and symmetric pos. def.
% Adds an extra method, cholupdate, which acts just like the builtin cholupdate.

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

methods

function F = factorization_chol_dense (A)
%FACTORIZATION_CHOL_DENSE : A = R'*R
[f.R, g] = chol (A) ;
if (g ~= 0)
error ('MATLAB:posdef', 'Matrix must be positive definite.') ;
end
F.A = A ;
F.Factors = f ;
F.A_rank = size (A,1) ;
F.kind = 'dense Cholesky factorization: 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.A - f.R'*f.R, 1) / norm (F.A, 1) ;
end

function x = mldivide_subclass (F,b)
%MLDIVIDE_SUBCLASS x=A\b using a dense Cholesky factorization
% x = R \ (R' \ b)
f = F.Factors ;
opU.UT = true ;
opUT.UT = true ;
opUT.TRANSA = true ;
y = b ;
if (issparse (y))
y = full (y) ;
end
x = linsolve (f.R, linsolve (f.R, y, opUT), opU) ;
end

function x = mrdivide_subclass (b,F)
%MRDIVIDE_SUBCLASS x = b/A using dense Cholesky
% x = (R \ (R' \ b')'
x = (mldivide_subclass (F, b'))' ;
end

function F = cholupdate (F,w,updown)
%CHOLUPDATE update/downdate a dense Cholesky factorization
%
% Example
%   % F becomes the Cholesky factorization of A+w*w'
%   F = factorize (A) ;
%   w = rand (size (A,1),1) ;
%   G = cholupdate (F,w) ;
%   x = G\b ;               % computes x = (A+w*w')\b
%   G = cholupdate (F,w,'-') ;
%   x = G\b ;               % computes x = (A-w*w')\b
%
if (nargin < 3)
updown = '+' ;
end
switch updown
case '+'
F.Factors.R = cholupdate (F.Factors.R, w, '+') ;
F.A = F.A + w*w' ;
case '-'
F.Factors.R = cholupdate (F.Factors.R, w, '-') ;
F.A = F.A - w*w' ;
otherwise
error ('MATLAB:cholupdate:incorrectThirdInputArgument', ...
'Third argument must be ''+'' or ''-''.') ;
end
end
end
end
```