Borghi, Riccardo and Weniger, Ernst Joachim (2015) Convergence analysis of the summation of the factorially divergent Euler series by Pade approximants and the delta transformation. APPLIED NUMERICAL MATHEMATICS, 94. pp. 149-178. ISSN 0168-9274, 1873-5460
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Sequence transformations are valuable numerical tools that have been used with considerable success for the acceleration of convergence and the summation of diverging series. However, our understanding of their theoretical properties is far from satisfactory. The Euler series xi(z) similar to Sigma(infinity)(n-0)(-1)(n)nlz(n) is a very important model for the ubiquitous factorially divergent perturbation expansions in theoretical physics and for the divergent asymptotic expansions for special functions. In this article, we analyze the summation of the Euler series by Pade approximants and by the delta transformation, which is a powerful nonlinear Levin-type transformation that works very well in the case of strictly alternating convergent or divergent series. Our analysis is based on a very recent factorial series representation of the truncation error of the Euler series. We derive explicit expressions for the transformation errors of Pade approximants and of the delta transformation. A subsequent asymptotic analysis proves rigorously the convergence of both Pade and delta. Our asymptotic estimates clearly show the superiority of the delta transformation over Pade. This is in agreement with previous numerical results. (C) 2015 IMACS. Published by Elsevier B.V. All rights reserved.
| Item Type: | Article |
|---|---|
| Uncontrolled Keywords: | GROUND-STATE ENERGY; PERTURBATION-SERIES; SEQUENCE TRANSFORMATIONS; DIFFRACTION CATASTROPHES; HYPERGEOMETRIC-SERIES; ANHARMONIC-OSCILLATOR; NUMERICAL EVALUATION; EPSILON ALGORITHM; ACCELERATION; EXPANSIONS; Euler series; Summation of factorial divergence; Weniger's delta transformation; Fade approximants; Convergence proofs |
| Subjects: | 500 Science > 540 Chemistry & allied sciences |
| Divisions: | Chemistry and Pharmacy > Institut für Physikalische und Theoretische Chemie |
| Depositing User: | Dr. Gernot Deinzer |
| Date Deposited: | 03 Jul 2019 13:11 |
| Last Modified: | 03 Jul 2019 13:11 |
| URI: | https://pred.uni-regensburg.de/id/eprint/5166 |
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