Enhanced and Generalized One–step Neville Algorithm: Fractional Powers and Access to the Convergence Rate

Author

Ulrich D. Jentschura, Missouri University of Science and TechnologyFollow
Ludovico T. Giorgini

Abstract

The recursive Neville algorithm allows one to calculate interpolating functions recursively. Upon a judicious choice of the abscissas used for the interpolation (and extrapolation), this algorithm leads to a method for convergence acceleration. For example, one can use the Neville algorithm in order to successively eliminate inverse powers of the upper limit of the summation from the partial sums of a given, slowly convergent input series. Here, we show that, for a particular choice of the abscissas used for the extrapolation, one can replace the recursive Neville scheme by a simple one-step transformation, while also obtaining access to subleading terms for the transformed series after convergence acceleration. The matrix-based, unified formulas allow one to estimate the rate of convergence of the partial sums of the input series to their limit. In particular, Bethe logarithms for hydrogen are calculated to 100 decimal digits. Generalizations of the method to series whose remainder terms can be expanded in terms of inverse factorial series, or series with half-integer powers, are also discussed.

Recommended Citation

U. D. Jentschura and L. T. Giorgini, "Enhanced and Generalized One–step Neville Algorithm: Fractional Powers and Access to the Convergence Rate," Computer Physics Communications, vol. 303, article no. 109280, Elsevier, Oct 2024.

The definitive version is available at https://doi.org/10.1016/j.cpc.2024.109280

Department(s)

Physics

Comments

Vetenskapsrådet, Grant 638-2013-9243

International Standard Serial Number (ISSN)

0010-4655

Document Type

Article - Journal

Document Version

Citation

File Type

text

Language(s)

English

Rights

© 2024 Elsevier, All rights reserved.

Publication Date

01 Oct 2024