Konvergenzbeschleunigung bei Einstufigen Iterationsverfahren durch Summierungsmethoden

Abstract

Given a one-step iteration method (with iteration matrix T) for the solution of a system of linear equations we consider an “averaging formula”, i.e. a k-step formula resulting from linear combinations of the last k iterates. It is shown that this iteration scheme can be obtained as a recurrence formula for the partial sums of a series which results from applying a summation method to the NEUMANN series of T. Using this relationship one gets results concerning the “domain of convergence” and the “domain of acceleration” of the k-step scheme we consider. Both domains may reach far beyond the unit circle, the domain of convergence of the given one-step method. For k = 1 and k = 2 an explicit description of these domains is given.