A historical overview of iterative methods

Abstract

The object of this paper is to present a historical overview of the development of iterative methods for the solution of large sparse systems of linear equations. The emphasis is on methods which are applicable to linear systems arising in the numerical solution of partial differential equations. Aspects to be covered including the methods of L.F. Richardson and of Liebmann as well as relaxation methods used by Southwell and others; the SOR method and extensions such as block SOR methods, and p-cyclic matrices; Chebyshev polynomial methods; alternating direction implicit methods; the SSOR method; approximate matrix factorization methods including the strongly implicit method (SIP) and the incomplete Cholesky method (ICC); fast direct methods; conjugate gradient methods; adaptive methods for the automatic determination of iteration parameters; multigrid methods; methods for nonsymmetric systems; and iterative software. Future developments will be discussed with emphasis on the use of vector and parallel processors.