Some Properties of a Generalization of the Richardson Extrapolation Process

The Richardson extrapolation process is generalized to cover a large class of sequences. Error bounds for the approximations are obtained and some convergence theorems for two different limiting processes are given. The results are illustrated by an oscillatory infinite integral.. THE PURPOSE of this paper is to generalize the well-known extrapolation process due to Richardson and to analyse, in some detail, the convergence properties of this generalization. In view of this analysis we shall also give some simple criteria for the efficient implementation of this "generalized Richardson extrapolation process" (GREP). An illustrative numerical example will also be appended. Definition 1.1. We shall say that a function A(y), defined for °< y ~ b, for some b > 0, where y can be a discrete or continuous variable, belongs to the set F(m), for some integer m > 0, if there exist functions ¢ k(y), Ih(y), 0 ~ k ~ m-1, and a constant A, such that m-l A = A(y) + L ¢k(y)flk(y), (Ll) k=O where A = lim A(y) whenever this limit exists, in which case lim ¢k(y) = 0, y~O+ y~O+ o~ k ~ m-l, and flk(n, as functions of the continuous variable~, are continuous for o~ ~ ~ b, and for some constants r k > 0, as ~ ..... 0+, have Poincare-type asymptotic expansions of the form oc flk(~)""" L flk. (1.2) i=O If, in addition, the functions Bk(t) == flk(i 1 / rk), as functions of the continuous variable t, b rk are infinitely differentiable for 0 ~ t ~ , we shall say that A(y) belongs to the set F (m)