A simpletouse objectoriented method for solving linear systems and leastsquares problems.
Editor's Note: This file was selected as MATLAB Central Pick of the Week
What's the best way to solve a linear system? Backslash is simple, but the factorization it computes internally can't be reused to solve multiple systems (x=A\b then y=A\c). You might be tempted to use inv(A), as S=inv(A) ; x=S*b ; y=S*c, but's that's slow and inaccurate. The best solution is to use a matrix factorization (LU, CHOL, or QR) followed by forward/backsolve, but figuring out the correct method to use is not easy.
In textbooks, linear algebraic expressions often use the inverse. For example, the Schur complement is written as S=AB*inv(D)*C. It is never meant to be used that way. S=AB*(D\C) or S=A(B/D)*C are better, but the syntax doesn't match the formulas in your textbook.
The solution is to use the FACTORIZE object. It overloads the backslash and mtimes operators (among others) so that solving two linear systems A*x=b and A*y=c can be done with this:
F=factorize(A) ;
x=F\b ;
y=F\c ;
An INVERSE method is provided which does not actually compute the inverse itself (unless it is explicitly requested). The statements above can be done equivalently with this:
S=inverse(A) ;
x=S*b ;
y=S*c ;
That looks like a use of INV(A), but what happens is that S is a factorized representation of the inverse of A, and multiplying S times another matrix is implemented with a forward/backsolve.
Wiith the INVERSE function, your Schur complement computation becomes S = A  B*inverse(D)*C which looks just like the equation in your book ... except that it computes a factorization and then does a forward/backsolve, WITHOUT computing the inverse at all.
An extensive demo is included, as well as a simple and easytoread version (FACTORIZE1) with fewer features, as an example of how to create objects in MATLAB.
If the system is overdetermined, QR factorization is used to solve the leastsquares problem. If the system is underdetermined, the transpose of A is factorized via QR, and the minimum 2norm solution is found. In MATLAB, x=A\b finds a "basic" solution to an underdetermined system.
You can compute a subset of the entries of the inverse just by referencing them. For example, S=inverse(A), S(n,n) computes the (n,n) entry of the inverse, without computing all of them.
In the RARE case that you need the entire inverse (or pseduoinverse of a fullrank rectangular matrix) you can force the object to become "double", with S = double (inverse (A)). This works for both full and sparse matrices (pinv(A) only works for full matrices).
And remember ... don't ever multiply a matrix by inv(A).
This code is published as "Algorithm 930: FACTORIZE: an objectoriented linear system solver for MATLAB", T. A. Davis, ACM Transactions on Mathematical Software, Vol 39, Issue 4, pp. 28:1  28:18, 2013.
1.8  Converted to a toolbox. Updated to the version that appears as Algorithm 930 in the Collected Algorithms of the ACM. 

1.7  Major update. Added SVD and COD (complete orthogonal decomposition) and many methods, such as the uminus requested in a comment below. 

1.5  Substantial update. The factorization class is now a superclass, with 8 specific subclasses beneath it (one for each kind of matrix factorization). Minor changes to the demo. No change to the user interface. 

1.4  Added link to LINFACTOR file. 

1.3  Added "(SetAccess = protected)" to class properties. 

1.2  Bug fix so that inverse(A)\b does the same thing as A*b (a peculiar case, of course...) 

1.1  New overload for plus and minus (cholupdate). Improved performance. More extensive demo. 
Inspired by: LINFACTOR: uses LU or CHOL to factorize a matrix, or previously computed factors to solve Ax=b
Philipp Glira (view profile)
Yernar Nukezhanov (view profile)
MANIK BANSAL (view profile)
Milind (view profile)
inv() is certainly slower than \ unless you have multiple right hand side vectors to solve for. But inv() is NOT inaccurate. The link elaborates further : http://arxiv.org/abs/1201.6035
>Several widelyused textbooks lead the >reader to believe that solving a >linear system of equations Ax = b by >multiplying the vector b by a computed >inverse inv(A) is inaccurate. >Virtually all other textbooks on >numerical analysis and numerical >linear algebra advise against using >computed inverses without stating >whether this is accurate or not. In >fact, under reasonable assumptions on >how the inverse is computed, x = >inv(A)*b is as accurate as the >solution computed by the best
>backwardstable solvers.
Eric Schols (view profile)
Works as documented in the demo: like a charm!
One question: can this package also be used for eigendecomposition? For example:
A = rand(5);
[S,D] = eig(A);
norm(AS*D*inverse(S))
But this still uses EIG, does this package contain a direct way to compute the eigendecomposition?
Daniel (view profile)
Hamza (view profile)
Michael Völker (view profile)
Tim Davis (view profile)
Comment from the author: I have addressed Ben's comment about uminus in this update (Sept 2011). You can now do inverse(A)*b, or 2*inverse(A)*b, which you could not do in the previous version.
Tim Davis (view profile)
inverse(A)*b can also be done as (inverse(A)*b), which is more natural than inverse(A)*(b).
Tim Davis (view profile)
Comment from the author: implementing the uminus function would be rather painful. I would need to return an object that is unchanged from the previous one, except with a flag stating that the result of all other functions must negate their result. This would be rather ugly and would needlessly complicate the code, just to add a rather minor feature.
Thanks for the suggestion ... I agree that uminus "ought" to work. It seems simple from the perspective of an enduser. But implementing it is far more trouble than it's worth, in my opinion.
Benjamin (view profile)
Very good, it is even faster when I solve a single linear system.
Bruno Luong (view profile)
What mathworks are waiting to incorporate this package into Matlab?
Ben (view profile)
All in all this package is a great idea and it looks like a first class implementation. As such, I'd like to help improve it by pointing out an issue. I am trying to solve Ax=b, so I type in x=inverse(A)*b, but I get the following error:
??? Undefined function or method 'uminus' for input arguments of type 'factorization_dense_lu'.
The work around is simple: x=inverse(A)*(b). However, these are mathematically equivalent, so both should work.
Miao LI (view profile)
Petter (view profile)
Nice with an update to modern MATLAB OOP.
Petter (view profile)
This is an excellent piece of software. Makes great use of Matlab's OOP.
Sebastien PARIS (view profile)
Great !!!
John D'Errico (view profile)
Absolutely splendid! Many thanks to Tim for writing this. A great title too.
us (view profile)
a truly professional and INValuable package written with a twinkle in the (author's) eye and a must lookat for people dealing with linear systems...
one pedestrian thought: why not create a standalone object given ML's new and powerful OOPstrength...
us