HURRY: An Acceleration Algorithm for Scalar Sequences and Series

Fessler, T. E.; Ford, William F.; Smith, David A.

doi:10.1145/356044.356051

Cited by 49 publications

(26 citation statements)

References 7 publications

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“…[31,84] concerning sequence transformations, including the ǫ and the ρ algorithms, and in Ref. [25] concerning a number of rather sophisticated algorithms, including COMPOS.…”

Section: Quick Startmentioning

confidence: 99%

From useful algorithms for slowly convergent series to physical predictions based on divergent perturbative expansions

et al. 2007

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This review is focused on the borderline region of theoretical physics and mathematics. First, we describe numerical methods for the acceleration of the convergence of series. These provide a useful toolbox for theoretical physics which has hitherto not received the attention it actually deserves. The unifying concept for convergence acceleration methods is that in many cases, one can reach much faster convergence than by adding a particular series term by term. In some cases, it is even possible to use a divergent input series, together with a suitable sequence transformation, for the construction of numerical methods that can be applied to the calculation of special functions. This review both aims to provide some practical guidance as well as a groundwork for the study of specialized literature. As a second topic, we review some recent developments in the field of Borel resummation, which is generally recognized as one of the most versatile methods for the summation of factorially divergent (perturbation) series. Here, the focus is on algorithms which make optimal use of all information contained in a finite set of perturbative coefficients. The unifying concept for the various aspects of the Borel method investigated here is given by the singularities of the Borel transform, which introduce ambiguities from a mathematical point of view and lead to different possible physical interpretations. The two most important cases are: (i) the residues at the singularities correspond to the decay width of a resonance, and (ii) the presence of the singularities indicates the existence of nonperturbative contributions which cannot be accounted for on the basis of a Borel resummation and require generalizations toward resurgent expansions. Both of these cases are illustrated by examples.

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“…[31,84] concerning sequence transformations, including the ǫ and the ρ algorithms, and in Ref. [25] concerning a number of rather sophisticated algorithms, including COMPOS.…”

Section: Quick Startmentioning

confidence: 99%

From useful algorithms for slowly convergent series to physical predictions based on divergent perturbative expansions

et al. 2007

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“…(7.1-6) Fessler, Ford, and Smith [57] could derive a recursive scheme which allows a convenient computation of both the numerator and the denominator of the general Levin transformation, eq. (7.1-7).…”

Section: Recursive Computation Of the Levin Transformationmentioning

confidence: 99%

“…However, Longman's derivation of the recurrence formula (7.2-6) is based upon Sister Celine's technique [59] and not on properties of the difference operator ∆ as the derivation by Fessler, Ford, and Smith [57].…”

Section: Recursive Computation Of the Levin Transformationmentioning

confidence: 99%

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Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series

Weniger

1989

Computer Physics Reports

414

535

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Slowly convergent series and sequences as well as divergent series occur quite frequently in the mathematical treatment of scientific problems. In this report, a large number of mainly nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series are discussed. Some of the sequence transformations of this report as for instance Wynn's ǫ algorithm or Levin's sequence transformation are well established in the literature on convergence acceleration, but the majority of them is new. Efficient algorithms for the evaluation of these transformations are derived. The theoretical properties of the sequence transformations in convergence acceleration and summation processes are analyzed. Finally, the performance of the sequence transformations of this report are tested by applying them to certain slowly convergent and divergent series, which are hopefully realistic models for a large part of the slowly convergent or divergent series that can occur in scientific problems and in applied mathematics.

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“…Because of this property, the Levin «-transform has been used in a general-purpose computer program for acceleration of convergence [5]. The ö-algorithm and the Levin transforms are among several procedures for acceleration of convergence tested in [16] and [17].…”

Section: Introductionmentioning

confidence: 99%

Acceleration of convergence of a family of logarithmically convergent sequences

Tuyl¹

1994

Math. Comp.

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Abstract. The asymptotic behavior of several sequence transformations is investigated as n -► oo when applied to a certain family of logarithmically convergent sequences. The transformations considered are the iterations of the transformations e^\An) of Shanks and Wn of Lubkin, the ^-algorithm of Brezinski, the Levin u-and u-transforms, and generalizations of the palgorithm and the Neville table. Computational results are given for both real and complex sequences.

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HURRY: An Acceleration Algorithm for Scalar Sequences and Series

Cited by 49 publications

References 7 publications

From useful algorithms for slowly convergent series to physical predictions based on divergent perturbative expansions

From useful algorithms for slowly convergent series to physical predictions based on divergent perturbative expansions

Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series

Acceleration of convergence of a family of logarithmically convergent sequences

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