Convergence Acceleration Via Combined Nonlinear-Condensation Transformations
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.
U. D. Jentschura et al., "Convergence Acceleration Via Combined Nonlinear-Condensation Transformations," Computer Physics Communications Package, vol. 116, no. 1, pp. 28-54, Elsevier, Jan 1999.
The definitive version is available at https://doi.org/10.1016/S0010-4655(98)00111-8
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)
Article - Journal
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