Numerical Algorithms 11(1996)327-337 327 An algebraic approach to the vector e-algorithm A. Salam Laboratoire de Math6matiques Appliqu~es, Universit~ du Littoral, Centre Universitaire de la Mi- Voix, Bhtiment Poincar~, 50 rue F. Buisson, B.P. 699, F-62228 Calais Cddex, France The vector e-algorithm is obtained from the scalar e-algorithm by taking the pseudo-inverse of a vector instead of the inverse of a scalar. Thus the vector e-algorithm is known only through its rules contrarily to the scalar e-algorithm and some other extrapolation algorithms. The aim of this paper is to provide an algebraic approach to the vector e-algorithm. Keywords: Designant, Clifford algebra, Clifford group, vector e-algorithm. AMS subject classification: 65B05. 1. Introduction The vector e-algorithm was first introduced by Wynn [27]. It is a quite powerful method for accelerating the convergence of vector sequences. It has many other applications: solution of systems of linear and nonlinear equations [12,5], com- putation of eigenvalues of a matrix [6]. It occurs likewise in approximation theory (vector continued fractions [27], vector interpolation and Pad~ approxi- mation [13,14], ...). Although it was the subject of intensive research, it still lacks a complete set of theoretical results. In [19,20], I showed that this algorithm realizes some extra- polation in a Clifford algebra and gave explicitly the quantities computed as a "ratio" of two designants. It was the first approach to the algorithm. Recently, in [8], a new approach of the E-algorithm was given, using the elimination system first introduced by Weniger [25]. This approach was then generalized to vector and matrix sequences in [9]. In this paper, we use this basic idea to present an algebraic approach to the vector c-algorithm. We will see that the nature of the problem encountered is different, and hence we will need some knowledge of Clifford group and algebra. 2. Clifford algebra We recall first some definitions and properties of a Clifford algebra. 9 J.C. Baltzer AG, Science Publishers