Accelerating Innnite Products Deenition 1.1 -the Class A

Slowly convergent innnite products Q 1 n=1 b n are considered, where fb n g is a sequence of numbers, or a sequence of linear operators. Using an asymptotic expansion for the 'remainder' of the innnite product a method for convergence acceleration is suggested. The method is in the spirit of the d-transformation for series. It is very simple and eecient for some classes of sequences fb n g. For complicated sequences fb n g it involves the solution of some linear systems, but it is still eeective. x1. Introduction In this work we investigate convergence acceleration of the innnite product P = 1 Y n=1 b n = b 1 b 2 : : : b n : : : ; (1:1) i.e., evaluating P using a small number of its partial products P m = m Y n=1 b n = b 1 b 2 : : : b m : (1:2) If the elements fb n g are numbers ('the scalar case') then a possible strategy is to transform the problem into a series summation problem, evaluating Log(P) = P 1 n=1 Log(b n). In many applications one encounters innnite products of operators, e.g., fb n g can be s s matrices. In this case the Logarithm operation is not applicable. Hence, we try in this work to act directly on the innnite product, using operations which are natural to the general linear operators, i.e., products and inverses. Our strategy is based upon the methodology of the d-transformation for accelerating the convergence of innnite series S = P 1 n=1 a n using its partial sums S N = P N n=1 a n. Hence we start by reviewing some of the deenitions and results in 7]. is the class of all functions p(x) which have a Poincar e-type asymptotic expansion of the form: p(x) x