Eigenvectors of the Successive Over-Relaxation Process, and its Combination with Chebyshev Semi-Iteration

Abstract

The eigenvector decomposition of the errors of the S.O.R. process is examined, and proofs are given for conjectures which have been published concerning the elementary divisors of the process. The results of numerical experiments with a Chebyshev semi-iterative procedure based on S.O.R. are interpreted in the light of this analysis, and it is concluded that the structure of the eigenvectors of the S.O.R. process makes the process unsuitable for use in Chebyshev semi-iteration. It is demonstrated that a knowledge of the maximum eigenvalue of an interative process is not always adequate for specifying the convergence of such a process—the structure of the eigenvectors can have a profound influence upon the convergence.