Prediction Properties of Aitken's Iterated Delta^2 Process, of Wynn's Epsilon Algorithm, and of Brezinski's Iterated Theta Algorithm

TL;DR

This paper analyzes the prediction capabilities of Aitken's Delta^2 process, Wynn's epsilon algorithm, and Brezinski's iterated theta algorithm for formal power series, focusing on their recursive structures and accuracy predictions.

Contribution

It provides a detailed analysis of the prediction properties of these three transformation algorithms, including their recursive schemes and accuracy-through-order relationships.

Findings

Derived accuracy-through-order relationships for the transformations

Rewrote rational approximants as partial sums plus transformation terms

Produced predictions for series coefficients not used in approximants

Abstract

The prediction properties of Aitken's iterated Delta^2 process, Wynn's epsilon algorithm, and Brezinski's iterated theta algorithm for (formal) power series are analyzed. As a first step, the defining recursive schemes of these transformations are suitably rearranged in order to permit the derivation of accuracy-through-order relationships. On the basis of these relationships, the rational approximants can be rewritten as a partial sum plus an appropriate transformation term. A Taylor expansion of such a transformation term, which is a rational function and which can be computed recursively, produces the predictions for those coefficients of the (formal) power series which were not used for the computation of the corresponding rational approximant.