Convergence Acceleration via Combined Nonlinear-Condensation Transformations

TL;DR

The paper presents the combined nonlinear-condensation transformation (CNCT), a two-step method to accelerate slowly convergent monotone series: first transform the monotone series into a strictly alternating series via Van Wijngaarden condensation, then apply powerful nonlinear sequence transformations (Levin or Weniger) to the alternating series using explicit remainder estimates. This approach yields high-precision results even near the circle of convergence or singularities and extends to divergent alternating forms for analytic continuation (e.g., the Riemann zeta function). The authors demonstrate broad applicability across Dirichlet, Lerch, and generalized hypergeometric series, as well as partial-wave sums in quantum electrodynamics, achieving substantial computational savings and robust numerical stability. CNCT thus provides a versatile, stable tool for evaluating slowly convergent sums and special functions with near-boundary behavior of their arguments, enabling efficient and accurate computations in applied mathematics and theoretical physics.

Abstract

A method of numerically evaluating slowly convergent monotone series is described. First, we apply a condensation transformation due to Van Wijngaarden to the original series. This transforms the original monotone series into an alternating series. In the second step, the convergence of the transformed series is accelerated with the help of suitable nonlinear sequence transformations that are known to be particularly powerful for alternating series. Some theoretical aspects of our approach are discussed. The efficiency, numerical stability, and wide applicability of the combined nonlinear-condensation transformation is illustrated by a number of examples. We discuss the evaluation of special functions close to or on the boundary of the circle 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 state calculations.