A general extrapolation procedure revisited

The E{algorithm is the most general extrapolation algorithm actually known. The aim of this paper is to provide a new approach to this algorithm. This approach gives a deeper insight into the E{algorithm, allows to obtain new properties and to relate it to other algorithms. Some extensions of the procedure are discussed. The E{algorithm is the most general extrapolation algorithm actually known. It was rst obtained by an interpolation procedure by Schneider in 1975 21] and then by Meinardus and Taylor 14] in the context of best uniform approximation. However, its present formulation as an extrapolation procedure was only given some years later by H avie 12] and Brezinski 2]. H avie's approach was by elimination while Brezinski used Sylvester's determinantal identity. The quantities computed by the E{algorithm can be expressed as ratios of determinants and they also satisfy a recursive scheme which generalizes Richardson extrapolation process. All these approaches are reviewed and explained in 4]. The E{algorithm contains most of the extrapolation algorithms found in the literature (see 7] for a review) and it has interesting acceleration properties 2, 13, 20, 24]. The aim of this paper is to give a new derivation of the E{algorithm. This derivation is based on the concepts of annihilation diierence operator and remainder estimate introduced by Weniger 25] and developed by Brezinski and Matos 6]. This new approach will also leads us to new results about the E{algorithm and it clariies its properties. Some extensions will be discussed. 1. The mechanism of extrapolation Let u = (u n) be an arbitrary sequence. We shall denote by 1 the sequence whose terms are all equal to 1. We shall deene the sequence (N k) of diierence operators by N (n) 0 (u) = u n N (n) k+1 (u) = 0 @ N (n) k (u) N (n) k (g k+1) 1 A where N (n) k (u) denotes to n th term of the sequence (N k (u)) and where g i = (g i (n)) are given auxiliary sequences for i = 1; 2; : : :.