Development of the Overholt transformation for accelerating the convergence of sequences

Abstract

In this paper we demonstrate the development of the Overholt transformation. We consider the use of the Overholt-type transformations to accelerate the convergence of scalar sequence and their effectiveness for approximating the solution of a given power series is illustrated. In process we shall demonstrate the convergence of each of the methods considered. The approximate solution of the improved Overholt transformation and the modified Overholt transformation are found to be substantially more accurate than the classical Overholt transformation and the Wimp transformation.

Introduction

In this paper, we shall demonstrate the improvement of the Overholt transformation. The classical Overholt transformation and the new transformations are of rational form and are essentially designed for accelerating the convergence of scalar sequence. The improved Overholt transformation is actually a variant of the modified Overholt transformation. The prime motive for the development of these new transformations was to improve the Overholt transformation. Consequently, we have found that the new transformations are consistent, stable and much more accurate than the other similar transformations considered. In order to construct these new transformations the following proposition is essential.

Proposition

Let us assume that

$$s=\sum _{i=0}^{\infty }{c}_i\text{,}$$

is a slowly convergent or divergent sequence, whose elements s_n are the partial sum of an infinite series, given as

$$s_n=\sum _{i=0}^n{c}_i\text{.}$$

The basic assumption of all sequence transformation is that a sequence element s_n can be for all indices $n⩾0$ to be partitioned into a limit s and a truncation error e_n according to

$$s_n=s+e_n\text{.}$$

The conventional approach of evaluating an infinite series consists of adding up so many terms that the error e_n ultimately becomes zero. Unfortunately, this is not always possible because of obvious practical limitations. Moreover, adding up further terms does not work in the case of a divergent series. Therefore the development of the new transformations plays an important role.

The structure of this paper is as follows. In Section 2 we briefly review two well-known methods, the Overholt transformation and the Wimp transformation. In Section 3 we shall define the new transformations, namely the improved Overholt transformation and the modified Overholt transformation. Moreover, in Section 4 we examine the effectiveness of these new transformations for determining the approximate solution of four particular power series. We make four distinct comparisons of the estimates derived by the new transformations. For k equal one to five we compare estimates formed using the row sequence of the improved Overholt transformation of type $(n\text{,}k)$ with corresponding estimates derived from the modified Overholt transformation of type $(n\text{,}k)$, the Overholt transformation of type $(n\text{,}k)$ and the Wimp transformation of type $(n\text{,}k)$. The effectiveness of the new method for accelerating the convergence of a scalar sequence was investigated. The new improved Overholt transformation and the modified Overholt transformation are proved to be the most effective of the methods considered.