A NEW APPROACH TO CONVERGENCE ACCELERATION METHODS Claude BREZINSKI Laboratoire d'Analyse Numerique et d'Optimisation UFR IEEA M3 Universite de Lille 1 59655 Villeneuve d'Ascq - Cedex FRANCE ABSTRACT. A new notion, the perfect estimation of the error of a sequen- ce, is introduced. This approach explains and relates many concepts, ideas and algorithmic procedures used for accelerating the convergence of sequences, which were indepently developed. It thus provides a more syn- thetic and profound view of the entire field. 1 - INTRODUCTION. Let (Sn) be a sequence converging to S. All the methods to accelerate the convergence of (Sn) consist in transforming (Sn) into another sequen- ce (Tn) with the hope that : T -S = o(S -S) n n (n + 00). Such a method is called a sequence transformation. There are two ways for obtaining sequence transformations , 1°) Let N be a (S ) + (T ) is n n limit of (S ). n set of sequences. The sequence built such that V(S ) E N, then n transformation T Vn, Tn=S where S is the The set N is called the kernel of the transformation T. Many transformations have been obtained by this procedure and quite often the numbers T are defined as ratios of determinants. Thus the first duty of a analyst is to derive a recursive algorithm to calcu- late the T 's without computing the determinants involved in their defi- nition singe a numerical analyst dont't know how to compute determinants. Then one has to look for the convergence and acceleration properties of the transformation T that is to find classes of sequences for which the transformed sequence (T ) converges to the same limit as (S ) and faster. See [7] for an expositi8n of this kind of approach and [lO,n 29] for a very general algorithm of this type. 2°) The second way consists in gLVLng directly an algorithm for computing the new sequence (Tn)' This was, for example, the method followed for the 373 A. Cuyt (ed.), Nonlinear Numerical Methods and Rational Approximation, © 1988 by D. Reidel Publishing Company.