The generalized Richardson extrapolation process GREP (1) and computation of derivatives of limits of sequences with applications to the d(1)-transformation

Abstract

Let { S m } be an infinite sequence whose limit or antilimit S can be approximated very efficiently by applying a suitable extrapolation method E 0 to { S m }. Assume that the S m and hence also S are differentiable functions of some parameter ξ, ( d/ dξ)S being the limit or antilimit of {(d/d ξ) S m }, and that we need to approximate (d/d ξ) S. A direct way of achieving this would be by applying again a suitable extrapolation method E 1 to the sequence {(d/d ξ) S m }, and this approach has often been used efficiently in various problems of practical importance. Unfortunately, as has been observed at least in some important cases, when (d/d ξ) S m and S m have essentially different asymptotic behaviors as m→∞, the approximations to (d/d ξ) S produced by this approach, despite the fact that they are good, do not converge as quickly as those obtained for S, and this is puzzling. In a recent paper (A. Sidi, Extrapolation methods and derivatives of limits of sequences, Math. Comp., 69 (2000) 305–323) we gave a rigorous mathematical explanation of this phenomenon for the cases in which E 0 is the Richardson extrapolation process and E 1 is a generalization of it, and we showed that the phenomenon has nothing to do with numerics. Following that we proposed a very effective procedure to overcome this problem that amounts to first applying the extrapolation method E 0 to { S m } and then differentiating the resulting approximations to S. As a practical means of implementing this procedure we also proposed the direct differentiation of the recursion relations of the extrapolation method E 0 used in approximating S. We additionally provided a thorough convergence and stability analysis in conjunction with the Richardson extrapolation process from which we deduced that the new procedure for (d/d ξ) S has practically the same convergence properties as E 0 for S. Finally, we presented an application to the computation of integrals with algebraic/logarithmic endpoint singularities via the Romberg integration. In this paper we continue this research by treating Sidi's generalized Richardson extrapolation process GREP (1) in detail. We then apply the new procedure to various infinite series of logarithmic type (whether convergent or divergent) in conjunction with the d (1)-transformation of Levin and Sidi. Both the theory and the numerical results of this paper too indicate that this approach is the preferred one for computing derivatives of limits of infinite sequences and series.