Numer. Math. 46, 311-321 (1985) Numerische Mathematik 9 Springer-Verlag 1985 Composite Sequence Transformations* Claude Brezinski Laboratoire d'Analyse Num6rique et d'Optimisation, UER IEEA-M3, Universit6 de Lille l, F-59655 Villeneuve d'Ascq Cedex, France Summary. The aim of this work is to introduce the new concept of com- posite sequence transformations and to show, by very simple examples and theorems, that it can be useful in accelerating the convergence of sequences. Generalizations of classical transformations and results are obtained. Subject Classifications: AMS(MOS): 65 B I0; CR: G 1.m. 1. Introduction Let (S,) be the sequence to be accelerated and let S be its limit. We consider p sequence transformations t k ---~'4nh .. t k: (S,) t~k J k=l, .,p. The transformations t k are assumed to be strongly semi-regular that is [3 n, t~ "~= -k '~"+ 1)] ~ L~krt('0 ----lim S~]. p~ o~ We define the transformation T where, Vn T: (S.)-*(T.) P Zn ~ Z z~(n)t(n) ~k ~k" k=l T is called a composite sequence transformation and p is its rank. When the coefficients a~ ") do not depend on the terms of the sequences (S,) or (t~ ")) then T is a linear composite sequence transformation. Let us give some examples. * Work performed under the Nato Research Grant 027.81. Presented at the International Conference on Numerical Analysis, Munich, March 19-21, 1984