Prediction properties of Aitkenʹs iterated Δ2 process, of Wynnʹs epsilon algorithm, and of Brezinskiʹs iterated theta algorithm

The prediction properties of Aitkenʹs iterated Δ2 process, Wynnʹs epsilon algorithm, and Brezinskiʹs 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.