This book is targeted primarily toward engineers and engineering students of advanced standing (sophomores, seniors and graduate students). Familiarity with a computer language is required; knowledge of basic engineering mechanics is useful, but not essential.

The text attempts to place emphasis on numerical methods, not programming. Most engineers are not programmers, but problem solvers. They want to know what methods can be applied to a given problem, what are their strengths and pitfalls and how to implement them. Engineers are not expected to write computer code for basic tasks from scratch; they are more likely to utilize functions and subroutines that have been already written and tested. Thus programming by engineers is largely confined to assembling existing pieces of code into a coherent package that solves the problem at hand.

The “piece” of code is usually a function that implements a specific task. For the user the details of the code are unimportant. What matters is the interface (what goes in and what comes out) and an understanding of the method on which the algorithm is based. Since no numerical algorithm is infallible, the importance of understanding the underlying method cannot be overemphasized; it is, in fact, the rationale behind learning numerical methods.

This book attempts to conform to the views outlined above. Each numerical method is explained in detail and its shortcomings are pointed out.

To be able to read or work on matrix computing, a reader must have completed a course on linear algebra. The inclusion of this Appendix A is to review some selected topics from basic linear algebra for reference purposes.

Starting from this chapter, we shall first describe various preconditioning techniques that are based on manipulation of a given matrix. These are classified into four categories: direct matrix extraction (or operator splitting type), inverse approximation (or inverse operator splitting type), multilevel Schur complements and multi-level operator splitting (multilevel methods).

The acoustic scattering modelling provides a typical example of utilizing a boundary element method to derive a dense matrix application as shown in Chapter 1. Such a physical problem is only a simple model of the full wave equations or the Maxell equations from electromagnetism. The challenges are:

1. (i) the underlying system is dense and non-Hermitian;

2. (ii) the kernel of a boundary integral operator is highly oscillatory for high wavenumbers, implying that a large linear system must be solved. The oscillation means that the fast multipole method and the fast wavelet methods are not immediately applicable.

This chapter reviews the recent work on using preconditioned iterative solvers for such linear systems arising from acoustic scattering modelling and points out the various challenges for future research work. We consider the following.

Matrix computing arises in the solution of almost all linear and nonlinear systems of equations. As the computer power upsurges and high resolution simulations are attempted, a method can reach its applicability limits quickly and hence there is a constant demand for new and fast matrix solvers. Preconditioning is the key to a successful iterative solver. It is the intention of this book to present a comprehensive exposition of the many useful preconditioning techniques.

Preconditioning equations mainly serve for an iterative method and are often solved by a direct solver (occasionally by another iterative solver). Therefore it is inevitable to address direct solution techniques for both sparse and dense matrices. While fast solvers are frequently associated with iterative solvers, for special problems, a direct solver can be competitive. Moreover, there are situations where preconditioning is also needed for a direct solution method. This clearly demonstrates the close relationship between a direct and an iterative method.

Parallel computing represents a major research direction for the future and offers the best and often the only solution to large-scale computational problems in today's technology. A book on fast solvers is incomplete without a discussion of this important topic. However, any incomplete description of the subject is of no use and there are already too many books available. Nevertheless, the author believes that too much emphasis has been put from a computer scientist's view (on parallelization) so that a beginner may feel either confused with various warnings and jargons of new phrases or intimated by the complexity of some published programs (algorithms) of well-known methods. Hence we choose to give complete details for a few selected examples that fall into the category of ‘embarrassingly parallelizable’ methods.

Therefore the purpose of this chapter is to convey two simple messages.