TABLE OF CONTENTS To obtain improved convergence rates for the methods of successive displacement we require the coefficient matrix to have a P-condition number as small as possible. If this criterion is not satisfied, then it is advisable to prepare the system or precondition it beforehand.
D. J. Evans . Journal of the Institute of Mathematics and Applications , (1968)
In devising a preconditioner, we are faced with a choice between finding a matrix M that approximates A, and for which solving a system is easier than solving one with A, or finding a matrix M that approximates A ?1, so that only multiplication by M is needed.
R. Barrett, et al. The Templates book . SIAM Publications (1993)
This book is concerned with designing an effective matrix, the so-called preconditioner, in order to obtain a numerical solution with more accuracy or in less time. Denote a large-scale linear system of n equations, with A ? 
n n, b ? n, by
| (1.1) |  |
and one simple preconditioned system takes the following form
| (1.2) |  |
(Our primary concern is the real case; the complex case is addressed later.) We shall present various techniques of constructing such a preconditioner M that the preconditioned matrix A 1 = MA has better matrix properties than A. As we see, preconditioning can strike the balance of success and failure of a numerical method.
This chapter will review these introductory...
