Iterative procedures related to relaxation methods for eigenvalue problems

Abstract

Three iterative procedures for approximating to the solutions of linear eigenvalue problems for systems with a finite number of degrees of freedom are discussed. Two of the procedures are closely related to known iterative procedures while the third is original. The procedures are shown to possess quadratic, geometric and cubic convergence. All three procedures lie within the framework of the relaxation method, each representing a particular manner of fixing the freedom of choice existent in the relaxation method. The study was made to investigate the convergence and behaviour in the large of the relaxation method and to provide guiding principles for the relaxation computer. One particular result of importance is that orthogonalization of trial modes is not essential to the success of the relaxation method.