[Paper Review] Convergence Acceleration Techniques

U. D. Jentschura, Sergey Aksenov+3 more|ArXiv.org|Feb 1, 2002

Embedded Systems Design Techniques3 citations

TL;DR

This paper introduces a combined nonlinear-condensation transformation (CNCT) that accelerates convergence of slowly convergent or divergent series by first applying a Van Wijngaarden transformation to convert nonalternating series into alternating ones, followed by delta transformation for further acceleration. The method achieves up to 200-order-of-magnitude speedup, enabling 200-digit precision in hours using only ~84,000 terms instead of 10^205 terms for a challenging zeta-series sum.

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.

Motivation & Objective

• To develop a robust numerical method for accelerating slowly convergent or divergent series common in physics, biology, and mathematics.
• To overcome numerical instabilities in standard convergence acceleration for nonalternating series.
• To enable high-precision evaluation of special functions like polylogarithms, zeta functions, and Lerch transcendent with minimal computational cost.
• To provide a general-purpose tool applicable across scientific computing domains, including quantum physics and bioinformatics.
• To demonstrate orders-of-magnitude reduction in computational effort for series requiring extreme accuracy.

Proposed method

• The method applies a Van Wijngaarden transformation to convert a nonalternating series into an alternating one, reducing the truncation error's impact.
• It then applies a delta transformation to the transformed series to further accelerate convergence.
• The combined transformation is termed the Combined Nonlinear–Condensation Transformation (CNCT), mapping partial sums $ s_n $ to transformed values $ \\_n}^{(0)}(1,\mathbf{S}_n) $.
• The CNCT uses hidden information in trailing digits of partial sums to model and eliminate truncation error via nonlinear extrapolation.
• The algorithm is implemented in C, Fortran, and Mathematica, with sample code available for download.
• The method is particularly effective for series with $ |z| \approx 1 $, such as polylogarithms and Lerch transcendent, where standard methods fail.

Experimental results

Research questions

• RQ1 Can convergence acceleration techniques be generalized to handle nonalternating, slowly convergent series with high numerical stability?
• RQ2 To what extent can the CNCT reduce computational cost in evaluating special functions like the polylogarithm and zeta functions?
• RQ3 How accurately can the CNCT compute series requiring 10^205 terms when only 84,000 terms are used?
• RQ4 Can the CNCT achieve precision levels of 10^-15 or higher in quantum electrodynamic calculations?
• RQ5 What is the performance gain of CNCT over direct summation for series like $ \sum_{k=1}^\infty \left( H_k \right)^2 k^{-2} $?

Key findings

• The CNCT achieved 15-digit relative accuracy in computing $ 10^{-1} \mathrm{Li}_3(0.99999) $ using only 12 transformation orders, equivalent to summing ~100 million terms directly.
• For the series $ \sum_{k=1}^\infty \left( H_k \right)^2 k^{-2} $, the CNCT computed 200-digit precision in hours using only ~84,000 terms, reducing computational cost by approximately 200 orders of magnitude.
• The method enabled the calculation of the Bethe logarithm for the 4P state of hydrogen to 29-digit precision, 9 orders of magnitude more accurate than the prior best result.
• The CNCT achieved convergence for $ \Phi(z,s,v) $ with $ |z| \approx 1 $, where standard methods fail due to slow convergence.
• The transformation successfully stabilized and accelerated convergence in quantum electrodynamic bound state calculations, supporting measurements at 10^-14 relative accuracy.
• The method demonstrated robustness and high precision in evaluating special functions across mathematics, physics, and computational biology, including zeta, polylogarithm, and Lerch transcendent functions.