Nonlinear Sequence Transformations: Computational Tools for the Acceleration of Convergence and the Summation of Divergent Series

Ernst Joachim Weniger

Nonlinear Sequence Transformations: Computational Tools for the Acceleration of Convergence and the Summation of Divergent Series

Classical Analysis and ODEs 2007-05-23 v1

Authors: Ernst Joachim Weniger

Abstract

Convergence problems occur abundantly in all branches of mathematics or in the mathematical treatment of the sciences. Sequence transformations are principal tools to overcome convergence problems of the kind. They accomplish this by converting a slowly converging or diverging input sequence $\{s_n \}_{n=0}^{\infty}$ into another sequence $\{s^{\prime}_n \}_{n=0}^{\infty}$ with hopefully better numerical properties. Pad\'{e} approximants, which convert the partial sums of a power series to a doubly indexed sequence of rational functions, are the best known sequence transformations, but the emphasis of the review will be on alternative sequence transformations which for some problems provide better results than Pad\'{e} approximants.

Keywords

sequence transformations convergence acceleration divergent series summation pade approximants nonlinear transformations

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@article{arxiv.math/0107080,
  title  = {Nonlinear Sequence Transformations: Computational Tools for the Acceleration of Convergence and the Summation of Divergent Series},
  author = {Ernst Joachim Weniger},
  journal= {arXiv preprint arXiv:math/0107080},
  year   = {2007}
}

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29 pages, LaTeX, 0 figures

Nonlinear Sequence Transformations: Computational Tools for the Acceleration of Convergence and the Summation of Divergent Series

Abstract

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