Numerical comparisons of nonlinear convergence accelerators

Smith, David A.; Ford, William F.

doi:10.1090/s0025-5718-1982-0645665-1

Cited by 125 publications

(71 citation statements)

References 16 publications

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“…In this context, one might note that according to Refs. [106,302], the nonlinear sequence transformation listed in Table 1, notably the d and δ transformations discussed in Sec. 2.2.3 below, are among the most powerful and most versatile sequence transformations for alternating series that are currently known.…”

Section: Discussionmentioning

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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Section: Discussionmentioning

confidence: 99%

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

et al. 2007

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“…The problem of extrapolating such a sequence has been discussed in several reviews (Smith and Ford 1982, Barber and Hamer 1982, Guttmann 1989 Henkel and Patkos (1987), and Henkel and Schütz (1988). This algorithm involves an explicit parameter ω which can be optimized to match the leading power-law correction.…”

Section: Methodsmentioning

confidence: 99%

Finite-size scaling in the transverse Ising model on a square lattice

Hamer

2000

J. Phys. A: Math. Gen.

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“…The best-known example of such a sequence transformation is Levin's transformation [58] which is both very versatile and very powerful [6,12,[59][60][61]:…”

Section: Appendix A: Sequence Transformationsmentioning

confidence: 99%

Irregular input data in convergence acceleration and summation processes: General considerations and some special Gaussian hypergeometric series as model problems

Weniger

2001

Computer Physics Communications

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Sequence transformations accomplish an acceleration of convergence or a summation in the case of divergence by detecting and utilizing regularities of the elements of the sequence to be transformed. For sufficiently large indices, certain asymptotic regularities normally do exist, but the leading elements of a sequence may behave quite irregularly. The Gaussian hypergeometric series 2F1(a, b; c; z) is well suited to illuminate problems of that kind. Sequence transformations perform quite well for most parameters and arguments. If, however, the third parameter c of a nonterminating hypergeometric series 2F1 is a negative real number, the terms initially grow in magnitude like the terms of a mildly divergent series. The use of the leading terms of such a series as input data leads to unreliable and even completely nonsensical results. In contrast, sequence transformations produce good results if the leading irregular terms are excluded from the transformation process. Similar problems occur also in perturbation expansions. For example, summation results for the infinite coupling limit k3 of the sextic anharmonic oscillator can be improved considerably by excluding the leading terms from the transformation process. Finally, numerous new recurrence formulas for the 2F1(a, b; c; z) are derived.

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Numerical comparisons of nonlinear convergence accelerators

Cited by 125 publications

References 16 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

Finite-size scaling in the transverse Ising model on a square lattice

Irregular input data in convergence acceleration and summation processes: General considerations and some special Gaussian hypergeometric series as model problems

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