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ASYMPTOTIC ANALYSIS OF A GENERALIZED RICHARDSON extrapolation PROCESS ON LINEAR sequences

MATHEMATICS OF COMPUTATION 2010年第271期79卷 1681-1695页

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

In this work, we give a detailed convergence and stability analysis for the author's generalized Richardson extrapolation process GREP((m)) as this is being applied to linearly convergent or divergent infinite sequences {A(n)}, where A(n) similar to A + Sigma(m)(k=1) zeta(n)(k) Sigma(infinity)(i=0) beta(ki)n(gamma k-i) as n -> infinity, zeta(k) not equal 1 being distinct. The quantity we would like to compute is A, whether it is the limit or antilimit of {A(n)}. Such sequences arise, for example, as partial sums of power series and of Fourier series of functions that have algebraic and/or logarithmic branch singularities. Specifically, we define the GREP((m)) approximation A(n)((m,j)) to A, with n = (n(1) ,..., n(m)) and alpha > 0, via the linear systems A(l) = A(n)((m,j)) + Sigma(m)(k=1) zeta(l)(k) Sigma(nk-1)(i=0) (beta) over bar (ki) (alpha + l)(gamma k-i), j <= l <= j + Sigma(m)(k=1) n(k), where (beta) over bar (ki) are additional unknowns. We study the convergence and stability properties of A(n)((m,j)) as j -> infinity. We show, in particular, that A(n)((m,j)) - A = Sigma(m)(k=1) O(zeta(j)(k) j(gamma k-2nk)) as j -> infinity. When compared with Aj - A = Sigma(m)(k=1) O(zeta(j)(k) j(gamma k)) as j -> infinity, this result shows that GREP((m)) is a true convergence acceleration method for the sequences considered. In addition, we show that GREP((m)) is stable for the case being studied, and we also quantify its stability properties. The results of this work are the first ones pertaining to GREP((m)) with m > 1.

关键词： Acceleration of convergence generalized Richardson extrapolation process GREP((m)) power series Fourier series asymptotic expansions

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Strong coupling asymptotics of the β-function in φ⁴ theory and QED

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APPLIED NUMERICAL MATHEMATICS 2010年第12期60卷 1418-1428页

作者： Suslov, I. M. Kapitza Inst Phys Problems Moscow Russia

The well-known algorithm for summing divergent series is based on the Borel transformation in combination with the conformal mapping. A modification of this algorithm allows one to determine a strong coupling asymptotics of the sum of the series through the values of the expansion coefficients. An application of the algorithm to the beta-function of phi(4) theory leads to the asymptotics beta(g) = beta(infinity)g(alpha) at g -> infinity, where alpha approximate to 1 for space dimensions d = 2, 3, 4. The natural hypothesis arises, that the asymptotic behavior is beta(g) similar to g for all d. Consideration of the "toy" zero-dimensional model confirms the hypothesis and reveals the origin of this result: it is related to a zero of a certain functional integral. A generalization of this mechanism to the arbitrary space dimensionality leads to the linear asymptotics of beta(g) for all d. The same idea can be applied to QED and gives the asymptotics beta(g)= g, where g is the running fine structure constant. A relation to the "zero charge" problem is discussed. (C) 2010 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： divergent series Renormalization group Cell-Mann-Low function

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On the ambiguity of functions represented by divergent power series

On the ambiguity of functions represented by divergent power...

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

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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Study of a generalized Levin-Weniger convergence accelerator of numerical series: Particular cases and applications to multi-electron integrals in atomic physics

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APPLIED NUMERICAL MATHEMATICS 2010年第12期60卷 1371-1381页

作者： Porras, Ignacio Cordobes-Aguilar, Francisco Univ Granada Fac Ciencias Dept Fis Atom Mol & Nucl E-18071 Granada Spain Univ London Dept Phys Egham TW20 0EX Surrey England

In this work the general Levin transformation for a new remainder estimate is studied and compared to the u-transformation for some particular test cases. Also, the Weniger transformation and a generalization of both are studied specifically for the three-electron problem in atomic physics. In addition, we study the problem of the instability of these convergence accelerators and discuss attempts to reduce them. (C) 2010 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： Convergence acceleration Levin transformation Weniger transformation Few-electron atoms

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A recursive algorithm for the G transformation and accurate computation of incomplete Bessel functions

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APPLIED NUMERICAL MATHEMATICS 2010年第12期60卷 1411-1417页

作者： Slevinsky, Richard M. Safouhi, Hassan Univ Alberta Math Sect Edmonton AB T6C 4G9 Canada

In the present contribution, we develop an efficient algorithm for the recursive computation of the G(n)((1)) transformation for the approximation of infinite-range integrals. Previous to this contribution, the theoretically powerful G(n)((1)) transformation was handicapped by the lack of an algorithmic implementation. Our proposed algorithm removes this handicap by introducing a recursive computation of the successive G(n)((1)) transformations with respect to the order n. This recursion, however, introduces the (x(2) d/dx) operator applied to the integrand. Consequently, we employ the Slevinsky-Safouhi formula I for the analytical and numerical developments of these required successive derivatives. Incomplete Bessel functions, which pose as a numerical challenge, are computed to high pre-determined accuracies using the developed algorithm. The numerical results obtained show the high efficiency of the new method, which does not resort to any numerical integration in the computation. (C) 2010 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： The Slevinsky-Safouhi formulae extrapolation methods Nonlinear transformations Incomplete Bessel functions

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Strong coupling asymptotics of the β-function in φ⁴ theory and QED

Strong coupling asymptotics of the β-function in φ<SUP>4</...

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

作者： Suslov, I. M. Kapitza Inst Phys Problems Moscow Russia

关键词： divergent series Renormalization group Cell-Mann-Low function

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A recursive algorithm for the G transformation and accurate computation of incomplete Bessel functions

A recursive algorithm for the <i>G</i> transformation and ac...

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

作者： Slevinsky, Richard M. Safouhi, Hassan Univ Alberta Math Sect Edmonton AB T6C 4G9 Canada

关键词： The Slevinsky-Safouhi formulae extrapolation methods Nonlinear transformations Incomplete Bessel functions

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Study of a generalized Levin-Weniger convergence accelerator of numerical series: Particular cases and applications to multi-electron integrals in atomic physics

Study of a generalized Levin-Weniger convergence accelerator...

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

作者： Porras, Ignacio Cordobes-Aguilar, Francisco Univ Granada Fac Ciencias Dept Fis Atom Mol & Nucl E-18071 Granada Spain Univ London Dept Phys Egham TW20 0EX Surrey England

关键词： Convergence acceleration Levin transformation Weniger transformation Few-electron atoms

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Extensions of Drummond's process for convergence acceleration

Extensions of Drummond's process for convergence acceleratio...

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

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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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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