TABLE OF CONTENTS The convergence of a matrix iteration depends on the properties of the matrix-the eigenvalues, the singular values, or sometimes other information. One of the developments that made it possible for these methods to take off in the 1970s and 1980s was the discovery that in many cases, the problem of interest can be transformed so that the properties of the matrix are improved drastically. This process of "preconditioning" is essential to most successful applications of iterative methods.
In the abstract, the idea of preconditioning a system of equations is elementary. Suppose we wish to solve an m m nonsingular system
For any nonsingular m m matrix M, the system
has the same solution. If we solve (40.2) iteratively, however, the convergence will depend on the properties of M ?1 A instead of those of A. If this preconditioner M is well chosen, (40.2) may be solved much more rapidly than (40.1).
For this idea to be useful, of course, it must be possible to compute the operation represented by the product M ?1 A efficiently. As usual in numerical linear algebra, this will not mean an explicit construction of the inverse M ?1, but the solution of systems of equations of the form
Two extreme cases come quickly to mind. If M = A, then (40.3) is the same as (40.1), so applying the preconditioner is as hard as solving...
