Algorithm 930: FACTORIZE: An Object-Oriented Linear System Solver for MATLAB
The MATLAB (TM) backslash (x=A\b) is an elegant and powerful interface to a suite of high-performance factorization methods for the direct solution of the linear system Ax = b and the least-squares problem min(x) parallel to b - Ax parallel to. It is a meta-algorithm that selects the best factorization method for a particular matrix, whether sparse or dense. However, the simplicity and elegance of its single-character interface prohibits the reuse of its factorization for subsequent systems. Requiring MATLAB users to find the best factorization method on their own can lead to suboptimal choices; even MATLAB experts can make the wrong choice. Furthermore, naive MATLAB users have a tendency to translate mathematical expressions from linear algebra directly into MATLAB, so that x = A(-1)b becomes the inferior yet all-to-prevalent x=inv(A)*b. To address these issues, an object-oriented FACTORIZE method is presented. Via simple-to-use operator overloading, solving two linear systems can be written as F=factorize(A); x=F\b; y=F\c, where A is factorized only once. The selection of the best factorization method (LU, Cholesky, LDLT, QR, or a complete orthogonal decomposition for rank-deficient matrices) is hidden from the user. The mathematical expression x = A(-1)b directly translates into the MATLAB expression x=inverse(A)*b, which does not compute the inverse at all, but does the right thing by factorizing A and solving the corresponding triangular systems.
algorithms linear systems least-square problems experimentation performance matrix factorization object-oriented methods