Matrix and Vector Sequence Transformations Revisitedc

Sequence transformations are extrapolation methods. They are used for the purpose of convergence acceleration. In the scalar case, such algorithms can be obtained by two diierent approaches which are equivalent. The rst one is an elimination approach based on the solution of a system of linear equations and it makes use of determinants. The second approach is based on the notion of annihilation diierence operators. In this paper, these two approaches are generalized to the matrix and the vector cases. There exist many algorithms for transforming a sequence of numbers, or a sequence of vectors, or a sequence of matrices into a new sequence of objects of the same type. Such sequence transformations are used for accelerating the convergence of the initial sequence. They are often much useful, and even essential, since many sequences and many iterative processes used in numerical analysis and in applied mathematics converge so slowly that their practical use is very limited. Sequence transformations are based on the notion of extrapolation as explained, for example, in 5, 9, 21] and their theory is now fully understood. Quite often, the elements of the new sequence obtained by an extrapolation method are expressed as a ratio of two determinants since the elements of the new sequence are, in fact, expressed as the solutions of systems of linear equations. Of course, since a numerical analyst is unable to compute the value of a determinant (because there are too many arithmetical operations and then too many rounding errors), it is necessary to have recursive algorithms at one's disposal for computing these ratios of determinants. The equivalence between ratios of determinants and triangular recursive schemes was studied in 8]. As showed in 4] and in 7], acceleration algorithms can also be derived by means of diierence operators, an approach rst introduced by Weniger 20]. In this paper, we shall extend these two procedures to the vector and the matrix cases. In the rst section, we shall begin by recalling the main points of these two approaches. 1. The scalar case Let (S n) be a sequence of complex numbers such that, 8n S n ? S = aD n (1) where S and a are unknown numbers and (D n) a known sequence. We shall try to compute the value of S. Writing (1) for the index n and for the index n + 1 and subtracting the rst equation from …