Convergence acceleration via combined nonlinear-condensation transformations

A method of numerically evaluating slowly convergent monotone series is des cribed. First, we apply a condensation transformation due to Van Wijngaarde n to the original series. This transforms the original monotone series into an alternating series. In the second step, the convergence of the transfor med series is accelerated with the help of suitable nonlinear sequence tran sformations that are known to be particularly powerful for alternating seri es. Some theoretical aspects of our approach are discussed. The efficiency, numerical stability, and wide applicability of the combined nonlinear-cond ensation transformation is illustrated by a number of examples. We discuss the evaluation of special functions close to or on the boundary of the circ le of convergence, even in the vicinity of singularities. We also consider a series of products of spherical Bessel functions, which serves as a model for partial wave expansions occurring in quantum electrodynamic bound stat e calculations. (C) 1999 Elsevier Science B.V.