Some applications of nonlinear convergence accelerators

The efficient and reliable evaluation of infinite series is a frequently occurring problem. In many cases one is confronted with series expansions which converge so slowly that an evaluation by a direct summation of their terms would not be feasible, or even with series expansions which diverge. Consequently, alternative methods for their evaluation have to be used. In this article we want to report our experiences with two nonlinear convergence accelerators, the Shanks transformation and the u transformation of Levin. The Shanks transformation was designed to accelerate linear convergence whereas the u transformation is especially powerful for slowly converging monotonic series and for some converging or diverging alternating series. The numerical examples presented are drawn from molecular multicenter integrals and from special function theory.