Overview

As we will see, iterative methods are not only great fun to play with and interesting objects for analysis, but they are really useful in many situations. For truly large problems they may sometimes offer the only way towards a solution.

Henk A. van der Vorst. Iterative Krylov Methods for Large Linear Systems . Cambridge University Press (2003)

A similar work [on the fast multipole method] was done in 3D by Rokhlin. As in 2D, the multistep algorithm was not properly explained.

Eric Darve. Fast Multipole Method, preprint, Paris, France (1997)

An iterative method for linear system Ax = b finds an infinite sequence of approximate solutions x ^{( j )} to the exact answer x*, each ideally with a decreased error, by using A repeatedly and without modifying it. The saving from using an iterative method lies in a hopefully early termination of the sequence as most practical applications are only interested in finding a solution x close enough to x*. Therefore, it almost goes without saying that the essence of an iterative method is fast convergence or at least convergence. When this is not possible for (1.1), we shall consider (1.2) with a suitable M.

This chapter will review a selection of iterative methods for later use as building blocks for preconditioner designs and testing. No attempt is made to exhaust all the iterative methods as one can find them in many books and surveys (e.g. [41,464,416])...