Convergence Acceleration Via Combined Nonlinear-Condensation Transformations

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.

Department(s)

Physics

Keywords and Phrases

Approximation Theory; Atomic Physics; Computational Methods; Electrodynamics; Electron Energy Levels; Functions; Mathematical Transformations; Molecular Physics; Numerical Analysis; Quantum Theory; Wave Equations; Bessel Functions; Nonlinear-condensation Transformations; Partial Wave Expansions; Quantum Electrodynamics (QED); Convergence Of Numerical Methods

International Standard Serial Number (ISSN)

0010-4655

Document Type

Article - Journal

Document Version

Citation

File Type

text

Language(s)

English

Rights

© 1999 Elsevier, All rights reserved.

Publication Date

01 Jan 1999

Link to Full Text

Share

 
COinS