Convergence acceleration via combined nonlinear-condensation transformations

Ulrich D. Jentschura, Peter J. Mohr, Gerhard Soff, Ernst Joachim Weniger

Research output: Contribution to journalArticle

48 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)28-54
Number of pages27
JournalComputer Physics Communications
Issue number1
Publication statusPublished - Jan 1999


  • Calculations and mathematical techniques in atomic and molecular physics
  • Computational techniques
  • Numerical approximation and analysis
  • Quantum electrodynamics (specific calculations)

ASJC Scopus subject areas

  • Hardware and Architecture
  • Physics and Astronomy(all)

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