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Multistep $ \varepsilon$-algorithm, Shanks' transformation, and the Lotka-Volterra system by Hirota's method


Authors: Claude Brezinski, Yi He, Xing-Biao Hu, Michela Redivo-Zaglia and Jian-Qing Sun
Journal: Math. Comp. 81 (2012), 1527-1549
MSC (2010): Primary 65B05, 39A14, 37K10
DOI: https://doi.org/10.1090/S0025-5718-2011-02554-8
Published electronically: October 19, 2011
MathSciNet review: 2904589
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Abstract: In this paper, we propose a multistep extension of the Shanks sequence transformation. It is defined as a ratio of determinants. Then, we show that this transformation can be recursively implemented by a multistep extension of the $ \varepsilon $-algorithm of Wynn. Some of their properties are specified. Thereafter, the multistep $ \varepsilon $-algorithm and the multistep Shanks transformation are proved to be related to an extended discrete Lotka-Volterra system. These results are obtained by using Hirota's bilinear method, a procedure quite useful in the solution of nonlinear partial differential and difference equations.


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Additional Information

Claude Brezinski
Affiliation: Laboratoire Paul Painlevé, UMR CNRS 8524, UFR de Mathématiques Pures et Appliquées, Université des Sciences et Technologies de Lille, France
Email: Claude.Brezinski@univ-lille1.fr

Yi He
Affiliation: LSEC, Institute of Computational Mathematics and Scientific Engineering Computing, AMSS, Chinese Academy of Sciences, and Graduate School of the Chinese Academy of Sciences, Beijing, People’s Republic of China
Email: heyi@lsec.cc.ac.cn

Xing-Biao Hu
Affiliation: LSEC, Institute of Computational Mathematics and Scientific Engineering Computing, AMSS, Chinese Academy of Sciences, Beijing, People’s Republic of China
Email: hxb@lsec.cc.ac.cn

Michela Redivo-Zaglia
Affiliation: Università degli Studi di Padova, Dipartimento di Matematica Pura ed Applicata, Italy
Email: Michela.RedivoZaglia@unipd.it

Jian-Qing Sun
Affiliation: LSEC, Institute of Computational Mathematics and Scientific Engineering Computing, AMSS, Chinese Academy of Sciences, and Graduate School of the Chinese Academy of Sciences, Beijing, People’s Republic of China
Email: sunjq@lsec.cc.ac.cn

DOI: https://doi.org/10.1090/S0025-5718-2011-02554-8
Keywords: Convergence acceleration algorithm, $𝜖$–algorithm, Shanks’ transformation, Lotka–Volterra system
Received by editor(s): December 21, 2010
Received by editor(s) in revised form: March 14, 2011
Published electronically: October 19, 2011
Article copyright: © Copyright 2011 American Mathematical Society

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