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The paper describes, discusses, and numerically illustrates the method for obtaining a priori estimates of the optimum relaxation factor in the SOR iteration method. The computational strategy of this method uses the so-called Sigma-SOR algorithm based on the theoretical result proven in the paper. The method presented is especially efficient for problems with slowly convergent iteration process and in this case is strongly competitive with adaptive procedures used for determining dynamically the optimum relaxation factor during the course of the SOR solution.

It is always a pleasant event when apparently disparate areas of mathematics intersect: when a theorem in one discipline translates into, proves, generalizes, simplifies or "explains" a theorem in another. This phenomenon occurs frequently between probability theory and various aspects of classical analysis. Examples are so abundant that it is perilous even to begin a list. This note presents another such interaction. Iteration of functions produces fixed points, at least under suitable conditions. We are going to present a modification of the usual iteration procedure which has the pernicious property of retarding or even destroying the convergence to the fixed point and ascertain (purely analytically) just when this convergence is destroyed. Our modification comes about not purely by malice, but turns out to correspond to picking a non-optimal strategy in a certain iterative game of chance. The analytic criterion then reappears as an immediate corollary to an elementary pointwise convergence theorem. Here is the analytic set-up. G(w) is a strictly convex, continuously differentiable function on the interval 0 6 w 0, G(1) = 1, and 0 < G'(w) < 1 for 0 < w < 1. These conditions are more than enough to ensure that if mqnl1 = G(qn) (io = 0), then qn -* 1, the fixed point of G. Let wn be a sequence with 0 < wn < 1. Modify the in to get a sequence (n by defining 40 = 0 and

A large part of the theory of the iteration of functions of several variables is illustrated by the relatively simple case of functions of one variable, and it is to the study of this case that the present paper is devoted. Several distinct classes of topics immediately suggest themselves in this connection. One of these classes comprises subjects of a purely formal nature, and is concerned only with expansions in the neighborhood of certain significant points. Another class of topics deals with the iteration of a real function within an interval. Still a third treats of the total behavior of the transforming function B (x), when the given function A1 (x) is analytic and defined for all values of x. Except for certain simple cases the function B(x) cannot be one-valued. The present paper deals almost exclusively with the first two of the three classes of subjects just mentioned. In the formal problem, certain new points of view, and new results and formulke are obtained, although a large proportion of this part is a systematization of results already known but largely unrelated. The second part of this paper deals with the iteration of a real function, a question which seems to have been neglected up to date.* The third topic mentioned above, has not here been touched upon, except in the case of certain simple rational functions. A few known theorems concerning the iteration of functions of a single variable, have been omitted, in view of the fact that they seem to have little significance in connection with the subject of the iteration of functions of several variables. Only those subjects have here been discussed which appear as natural introductory material to the more general case. For references to those parts of the formal problem which have already been treated in the literature, one may consult the following: Pincherle, on " Functional Equations and Operations," in the Encyklopadie d. Math. Wiss., II, A 11, and Encyclopedie d. Sci. Math., II, 26. E. Schroeder, Math. Ann., 2 (1870), p. 317, and 3 (1871), p. 296. J. Farkas, Journ. de Math. (3), 10 (1884), p. 102. G. Koenigs, Bull. Sc. Math. (2), 7 (1883), p. 340, and (3) 1 (1884), suppl., p. 14.