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An introduction to the topics presented at the conference "approximation and extrapolation of convergent and divergent sequences and series" CIRM Luminy: September 28, 2009-October 2, 2009

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APPLIED NUMERICAL MATHEMATICS 2010年第12期60卷 1184-1187页

作者： Weniger, Ernst Joachim Univ Regensburg Inst Phys & Theoret Chem D-93040 Regensburg Germany

The conference "approximation and extrapolation of convergent and divergent sequences and series" discusses some very old problems of mathematical analysis and modern approaches for their solution based on so-called nonlinear sequence transformations. In spite of all the advances in computer hard and software, slowly convergent or divergent sequences and series are still annoying obstacles not only in mathematics, but in particular also in other mathematically oriented disciplines. This articles tries to give a highly condensed survey of the topics treated at this conference. (C) 2010 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： Numerical approximation and evaluation approximation by rational functions Fade approximation extrapolation to the limit Summation of series

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An introduction to the topics presented at the conference "approximation and extrapolation of convergent and divergent sequences and series" CIRM Luminy: September 28, 2009-October 2, 2009

An introduction to the topics presented at the conference "A...

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conference on approximation and extrapolation of convergent and divergent sequences and series

作者： Weniger, Ernst Joachim Univ Regensburg Inst Phys & Theoret Chem D-93040 Regensburg Germany

关键词： Numerical approximation and evaluation approximation by rational functions Fade approximation extrapolation to the limit Summation of series

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Special Issue: approximation and extrapolation of convergent and divergent sequences and series (CIRM, Luminy - France, 2009) Preface

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APPLIED NUMERICAL MATHEMATICS 2010年第12期60卷 1183-1183页

作者： Brezinski, Claude Redivo-Zaglia, Michela Weniger, Ernst Joachim Univ Sci & Technol Lille CNRS Lab Paul Painleve UFR Math Pures & AppliqueesUMR 8524 F-59655 Villeneuve Dascq France Univ Padua Dipartimento Matemat Pura & Applicata I-35121 Padua Italy Univ Regensburg Inst Phys & Theoret Chem D-93040 Regensburg Germany

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Pade approximation and continued fractions

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APPLIED NUMERICAL MATHEMATICS 2010年第12期60卷 1364-1370页

作者： Lorentzen, Lisa Norwegian Univ Sci & Technol Dept Math Sci N-7491 Trondheim Norway

The purpose of this paper is to tell how continued fraction expansions of functions are derived, how they relate to Pade approximation, and how they can improve Pade approximants. (C) 2010 IMACS. Published by Elsevier... 详细信息

关键词： Pade approximation Continued fractions Convergence acceleration Special functions Modified approximants Tail sequences

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From numerical quadrature to Pade approximation

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APPLIED NUMERICAL MATHEMATICS 2010年第12期60卷 1209-1220页

作者： Brezinski, C. Univ Sci & Technol Lille CNRS Lab Paul Painleve UFR Math Pures & AppliqueesUMR 8524 F-59655 Villeneuve Dascq France

The paper reviews the relation between Pade-type approximants of a power series and interpolatory quadrature formulae with free nodes, and between Pade approximants and Gaussian quadrature methods. Then, it is shown how the Kronrod procedure and the anti-Gaussian quadrature methods could be used for estimating the error in Pade approximation. The epsilon-algorithm for accelerating the convergence of sequences, and computing recursively Parte approximants is evoked, and its error estimated by the same procedures. Finally, the case of series of functions is considered. Considerations on further research topics end the paper. (C) 2010 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： Pade approximantion Quadrature Convergence acceleration extrapolation

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Pade approximation and continued fractions

Pade approximation and continued fractions

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conference on approximation and extrapolation of convergent and divergent sequences and series

作者： Lorentzen, Lisa Norwegian Univ Sci & Technol Dept Math Sci N-7491 Trondheim Norway

关键词： Pade approximation Continued fractions Convergence acceleration Special functions Modified approximants Tail sequences

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Summation of divergent power series by means of factorial series

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APPLIED NUMERICAL MATHEMATICS 2010年第12期60卷 1429-1441页

作者： Weniger, Ernst Joachim Univ Regensburg Inst Phys & Theoret Chem D-93040 Regensburg Germany

Factorial series played a major role in Stirling's classic book Methodus Differentialis (1730), but now only a few specialists still use them. This article wants to show that this neglect is unjustified, and that factorial series are useful numerical tools for the summation of divergent (inverse) power series. This is documented by summing the divergent asymptotic expansion for the exponential integral E-1 (z) and the factorially divergent Rayleigh-Schrodinger perturbation expansion for the quartic anharmonic oscillator. Stirling numbers play a key role since they occur as coefficients in expansions of an inverse power in terms of inverse Pochhammer symbols and vice versa. It is shown that the relationships involving Stirling numbers are special cases of more general orthogonal and triangular transformations. (C) 2010 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： Factorial series divergent asymptotic (inverse) power series Stieltjes series Quartic anharmonic oscillator Stirling numbers General orthogonal and triangular transformations

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From numerical quadrature to Pade approximation

From numerical quadrature to Pade approximation

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conference on approximation and extrapolation of convergent and divergent sequences and series

作者： Brezinski, C. Univ Sci & Technol Lille CNRS Lab Paul Painleve UFR Math Pures & AppliqueesUMR 8524 F-59655 Villeneuve Dascq France

关键词： Pade approximantion Quadrature Convergence acceleration extrapolation

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Efficient algorithm for summation of some slowly convergent series

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APPLIED NUMERICAL MATHEMATICS 2010年第12期60卷 1442-1453页

作者： Wozny, Pawel Univ Wroclaw Inst Comp Sci PL-50383 Wroclaw Poland

The 2 transformation, introduced recently by Wozny and Nowak, may serve as a good tool for summation of slowly convergence series. This approach can be easily applied to the case of generalized or basic hypergeometric series, as well as some orthogonal polynomial expansions. It is closely related to the famous Wynn's epsilon algorithm. Weniger's or Homeier's transformations, and the method introduced by Cizek, Zamastil and Skala. However, it is difficult to use the algorithm proposed by Woiny and Nowak in the general case, because of its high complexity, and some other restrictions. We propose another realization of the 2 transformation, which results in obtaining a simpler and faster algorithm. Four illustrative numerical examples are given. (C) 2010 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： convergent acceleration Sequence transformations Orthogonal series

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Scalar Levin-type sequence transformations

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2000年第1-2期122卷 81-147页

作者： Homeier, HHH Univ Regensburg Inst Phys & Theoret Chem D-93040 Regensburg Germany

Sequence transformations are important tools for the convergence acceleration of slowly convergent scalar sequences or series and for the summation of divergent series. The basic idea is to construct from a given sequence {{s(n)}} a new sequence {{s(n)'}} = T({{s(n)}}) where each s(n)' depends on a finite number of elements s(n1), ..., s(nm). Often, the s(n) are the partial sums of an infinite series. The aim is to find transformations such that {{s(n)'}} converges faster than (or sums) {{s(n)}}. Transformations T{{s(n)}}, {{w(n)}}) that depend not only on the sequence elements or partial sums s(n) but also on an auxiliary sequence of the so-called remainder estimates w(n) are of Levin-type if they are linear in the s(n), and nonlinear in the w(n). Such remainder estimates provide an easy-to-use possibility to use asymptotic information on the problem sequence for the construction of highly efficient sequence transformations. As shown first by Levin, it is possible to obtain such asymptotic information easily for large classes of sequences in such a way that the w(n) are simple functions of a few sequence elements s(n). Then, nonlinear sequence transformations are obtained. Special cases of such Levin-type transformations belong to the most powerful currently known extrapolation methods for scalar sequences and series. Here, we review known Levin-type sequence transformations and put them in a common theoretical framework. It is discussed how such transformations may be constructed by either a model sequence approach or by iteration of simple transformations. As illustration, two new sequence transformations are derived. Common properties and results on convergence acceleration and stability are given. For important special cases, extensions of the general results are presented. Also, guidelines for the application of Levin-type sequence transformations are discussed, and a few numerical examples are given. (C) 2000 Elsevier Science B.V. All rights reserved. MSC: 65

关键词： convergence acceleration extrapolation summation of divergent series stability analysis hierarchical consistency iterative sequence transformation Levin-type transformations algorithm linear convergence logarithmic convergence Fourier series power series rational approximation

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