Convergence Acceleration Techniques

TL;DR

This work tackles the problem of slowly convergent and divergent series across statistics, physics, and mathematics by introducing the COMBINED NONLINEAR-CONDENSATION TRANSFORMATION (CNCT). The method first applies a Van Wijngaarden transformation to convert a nonalternating series into an alternating one, then uses a delta transformation to dramatically accelerate convergence, as illustrated by the Li_3(0.99999) example achieving $10^{-15}$ accuracy after 12 transforms. The authors demonstrate substantial gains across domains: accelerated evaluation of Lerch-related distributions in biophysics, highly precise QED calculations including the Bethe logarithm to nine orders of magnitude improvement, and rapid, high-precision sums in mathematics (e.g., the identity $\sum_{k=1}^{\infty} (1+1/2+\cdots+1/k)^2 k^{-2}=17\pi^4/360$ to 200 digits). The results indicate CNCT’s broad applicability and potential to markedly reduce computational time in scientific computing.

Abstract

This work describes numerical methods that are useful in many areas: examples include statistical modelling (bioinformatics, computational biology), theoretical physics, and even pure mathematics. The methods are primarily useful for the acceleration of slowly convergent and the summation of divergent series that are ubiquitous in relevant applications. The computing time is reduced in many cases by orders of magnitude.