Asymptotic expansions and acceleration of convergence for higher order iteration processes

Abstract

We analyze the convergence behavior of sequences of real numbers {x n }, which are defined through an iterative process of the formx n :=T(x n ?1), whereT is a suitable real function. It will be proved that under certain mild assumptions onT, these numbersx n possess an asymptotic (error) expansion, where the type of this expansion depends on the derivative ofT in the limit point $$\xi : = \mathop {\lim }\limits_{n \to \infty } x_n $$ ; this generalizes a result of G. Meinardus [6]. It is well-known that the convergence of sequences, which possess an asymptotic expansion, can be accelerated significantly by application of a suitable extrapolation process. We introduce two types of such processes and study their main properties in some detail. In addition, we analyze practical aspects of the extrapolation and present the results of some numerical tests. As we shall see, even the convergence of Newton's method can be accelerated using the very simple linear extrapolation process.