Incomplete Factorization Preconditioning Methods

Abstract

In Chapter 5 we presented basic iterative methods, both of stationary and of nonstationary type. The parameters in the methods were chosen to accelerate their convergence. The ability to accelerate depends, however, on the eigenvalue distribution. As we have seen, for iteration matrices with real and positive eigenvalues, or with nearly real eigenvalues but positive real parts, the parameters can be chosen so that the rate of convergence is increased by an order of magnitude, such as in the Chebyshev iteration method. Certain cases of an indefinite or more general complex spectrum can also be handled. The eigenvalue distribution depends on the matrix splitting method used. Some splitting methods, such as the SOR method, lead to a “dead-end”—i.e., an iteration matrix where no, or only minor, polynomial acceleration of convergence is possible, because the spectrum of the iteration matrix is typically located on a circle and, hence, powers of the iteration matrix are also located on a circle. The purpose of this and the following chapters is to present some practically important splitting methods of A , that is, A = C – R (where C is nonsingular), which can improve the eigenvalue distribution of the iteration matrix C −1 A in such a way that the iterative method will converge much faster with the splitting than without it.