TABLE OF CONTENTS For Poisson's equation, there will be a short list of numerical methods that are clearly superior to all the others we discuss. But for other linear systems it is not always clear which method is best (which is why we talk about so many!). To help users select the best method for solving their linear systems among the many available, on-line help is available at NETLIB/templates. This directory contains a short book [24] and software for most of the iterative methods discussed in this chapter. The book is available in both PostScript (NETLIB/templates/templates.ps) and Hypertext Markup Language (NETLIB/templates/Templates.html). The software is available in Matlab, Fortran, and C++.
The word template is used to describe this book and the software, because the implementations separate the details of matrix representations from the algorithm itself. In particular, the Krylov subspace methods (see section 6.6) require only the ability to multiply the matrix A by an arbitrary vector z. The best way to do this depends on how A is represented but does not otherwise affect the organization of the algorithm. In other words, matrix-vector multiplication is a "black-box" called by the template. It is the user's responsibility to supply an implementation of this black-box.
An analogous templates project for eigenvalue problems is underway. Other recent textbooks on iterative methods are [15, 136, 214].
For the most challenging practical problems arising from differential equations more challenging than our model problem, the linear system
