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Summation of divergent series: Order-dependent mapping

APPLIED NUMERICAL MATHEMATICS 2010年第12期60卷 1454-1464页

作者： Zinn-Justin, Jean CEA IRFU F-91191 Gif Sur Yvette France Ctr Saclay Inst Phys Theor F-91191 Gif Sur Yvette France

Summation methods play a very important role in quantum field theory because all perturbation series are divergent and the expansion parameter is not always small. A number of methods have been tried in this context, most notably Pade approximants, Borel-Pade summation, Borel transformation with mapping, which we briefly describe and one on which we concentrate here, Order-Dependent Mapping (ODM). We recall the basis of the method, for a class of series we give intuitive arguments to explain its convergence and illustrate its properties by several simple examples. Since the method was proposed, some rigorous convergence proofs were given. The method has also found a number of applications and we shall list a few. (C) 2010 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： divergent series Summation methods Borel transformation Quantum mechanics

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On the ambiguity of functions represented by divergent power series

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APPLIED NUMERICAL MATHEMATICS 2010年第12期60卷 1264-1272页

作者： Caprini, Irinel Fischer, Jan Vrkoc, Ivo Acad Sci Czech Republ Inst Phys CZ-18221 Prague 8 Czech Republic Natl Inst Phys & Nucl Engn R-077125 Bucharest Romania Acad Sci Czech Republ Math Inst CZ-11567 Prague 1 Czech Republic

Assuming the asymptotic character of divergent perturbation series, we address the problem of ambiguity of a function determined by an asymptotic power expansion. We consider functions represented by an integral of the Laplace-Borel type, with a curvilinear integration contour. This paper is a continuation of results recently obtained by us in a previous work. Our new result contained in Lemma 3 of the present paper represents a further extension of the class of contours of integration (and, by this, of the class of functions possessing a given asymptotic expansion), allowing the curves to intersect themselves or return back, closer to the origin. Estimates on the remainders are obtained for different types of contours. Methods of reducing the ambiguity by additional inputs are discussed using the particular case of the Adler function in QCD. (C) 2010 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： divergent series Perturbative QCD

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Survey of numerical stability issues in convergence acceleration

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APPLIED NUMERICAL MATHEMATICS 2010年第12期60卷 1395-1410页

作者： Sidi, Avram Technion Israel Inst Technol Dept Comp Sci IL-32000 Haifa Israel

An important issue that arises in application of convergence acceleration (extrapolation) methods is that of stability in the presence of floating-point arithmetic. This issue turns out to be critical because numerical instability is inherent, even built in, when convergence acceleration methods are applied to certain types of sequences that occur commonly in practice. If methods are applied without taking this issue into account, the attainable accuracy is limited, and eventually destroyed completely, as more terms are added in the process. Therefore, it is important to understand the origin of the problem and to propose practical ways to solve it effectively. In this work, we present a general discussion of the issue of stability within the context of a generalization of the Richardson extrapolation process, and review some of the recent developments that have taken place in the theoretical study of many of the known acceleration methods. We discuss approaches that have been proposed to cope with built-in instabilities when applying various methods, and illustrate the effectiveness of these strategies with some numerical examples. (C) 2010 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： Acceleration of convergence extrapolation Generalized Richardson extrapolation process GREP((m)) Levin transformation Levin Sidi d((m)) transformation Logarithmic sequences Linear sequences General linear sequences Numerical stability Floating-point arithmetic

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Summation of divergent power series by means of factorial series

Summation of divergent power series by means of factorial se...

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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

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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Asymptotic and factorial expansions of Euler series truncation errors via exponential polynomials

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APPLIED NUMERICAL MATHEMATICS 2010年第12期60卷 1242-1250页

作者： Borghi, Riccardo Univ Roma Tre Dipartimento Elettron Applicata Rome Italy

A detailed analysis of the remainder obtained by truncating the Euler series up to the nth-order term is presented. In particular, by using an approach recently proposed by Weniger, asymptotic expansions of the remainder, both in inverse powers and in inverse rising factorials of n, are obtained. It is found that the corresponding expanding coefficients are expressed, in closed form, in terms of exponential polynomials, well known in combinatorics, and in terms of associated Laguerre polynomials, respectively. A study of the divergence and/or of the convergence of the above expansions is also carried out for positive values of the Euler series argument. (C) 2010 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： Euler series Exponential polynomials Asymptotics Factorial expansions

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Efficient algorithm for summation of some slowly convergent series

Efficient algorithm for summation of some slowly convergent ...

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

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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Summation of divergent series: Order-dependent mapping

Summation of divergent series: Order-dependent mapping

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

作者： Zinn-Justin, Jean CEA IRFU F-91191 Gif Sur Yvette France Ctr Saclay Inst Phys Theor F-91191 Gif Sur Yvette France

关键词： divergent series Summation methods Borel transformation Quantum mechanics

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Extensions of Drummond's process for convergence acceleration

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APPLIED NUMERICAL MATHEMATICS 2010年第12期60卷 1231-1241页

作者： Brezinski, C. Redivo-Zaglia, M. Univ Padua Dipartimento Matemat Pura & Applicata I-35121 Padua Italy Univ Sci & Technol Lille CNRS UFR Math Pures & Appliquees Lab Paul PainleveUMR 8524 F-59655 Villeneuve Dascq France

In this paper, we give some new algebraic properties of a sequence transformation due to Drummond for accelerating the convergence. In the particular case where the terms of the sequence to be transformed are the partial sums of a formal power series, this transformation leads to Fade-type approximants. Then, using various operators, some extensions of Drummond's process are proposed, and their properties are given. A new expression for the E-transformation, using generalized divided differences, is obtained, and it is also generalized. The recurrence relation for these differences can be used for its implementation, and the Ford-Sidi algorithm is also related to them. Lubkin's T and W-transformations, which could be related to Drummond's, are discussed in Appendix A. (C) 2010 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： extrapolation Convergence acceleration Sequence transformation Pade approximants

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Asymptotic and factorial expansions of Euler series truncation errors via exponential polynomials

Asymptotic and factorial expansions of Euler series truncati...

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

作者： Borghi, Riccardo Univ Roma Tre Dipartimento Elettron Applicata Rome Italy

关键词： Euler series Exponential polynomials Asymptotics Factorial expansions

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Survey of numerical stability issues in convergence acceleration

Survey of numerical stability issues in convergence accelera...

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

作者： Sidi, Avram Technion Israel Inst Technol Dept Comp Sci IL-32000 Haifa Israel

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