Asymptotic expansions and acceleration of convergence for higher order iteration processes
The paper analyzes the convergence behavior of sequences of real numbers \(\{x_ n\}\), which are defined through an iterative process of the form \(x_ n:=T(x_{n-1})\), where T is a suitable real function. It is proved that under certain mild assumptions on T, these numbers \(x_ n\) possess an asymptotic (error) expansion, where the type of this expansion depends on the derivative of T in the limit point \(\xi:=\lim_{n\to \infty}x_ n.\) 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. In the present paper two types of such processes are studied in some detail. In addition, the author analyzes practical aspects of the extrapolation and presents the results of some numerical tests. It turns out that even the convergence of Newton's method can be accelerated using the very simple linear extrapolation process.
scientific article; zbMATH DE number 4213300
scientific article