The epsilon algorithm and related topics

Graves‐Morris, P. R.; Roberts, David E.; Salam, A.

doi:10.1016/s0377-0427(00)00355-1

Cited by 54 publications

(49 citation statements)

References 42 publications

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“…The Padé approximation can then be evaluated in O(ν 2 ) operations directly from the corresponding Taylor expansion [14], [31]. However, there is no built-in version in MATLAB of the types of recursions involved, and tests showed that a 'for loop'-based implementation is about 10 times slower for the present problem sizes (when taking into account that separate evaluations are needed for u(z + h) and u ′ (z + h), whereas the Toeplitz approach can compute the derivative by the quotient rule of calculus).…”

Section: Accuracy and Time Comparisonsmentioning

confidence: 99%

A numerical methodology for the Painlevé equations

Fornberg

Weideman

2011

Journal of Computational Physics

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Abstract. The six Painlevé transcendents P I -P V I have both applications and analytic properties that make them stand out from most other classes of special functions. Although they have been the subject of extensive theoretical investigations for about a century, they still have a reputation for being numerically challenging. In particular, their extensive pole fields in the complex plane have often been perceived as 'numerical mine fields'. In the present work, we note that the Painlevé property in fact provides the opportunity for very fast and accurate numerical solutions throughout such fields. When combining a Taylor/Padé-based ODE initial value solver for the pole fields with a boundary value solver for smooth regions, numerical solutions become available across the full complex plane. We focus here on the numerical methodology, and illustrate it for the P I equation. In later studies, we will concentrate on mathematical aspects of both the P I and the higher Painlevé transcendents.Key words. Painlevé transcendents, P I equation, Taylor series method, Padé approximation, Chebyshev collocation method.1. Introduction. The six Painlevé transcendents were discovered about 100 years ago [4], [13], [23], [24], and have since received extensive attention. They satisfy nonlinear second order ODEs, cannot be reduced to other types special functions, and obey the Painlevé property of all its solutions being free from movable branch points (however, the solutions may have movable poles or movable isolated essential singularities) [18], [21]. They arise in many branches of physics, including classical and quantum spin models, scattering theory, self-similar solutions to nonlinear dispersive wave PDEs, string theory and random matrix theory. For a summary of their applications, with extensive references, we refer to Sections 32.13 -32.16 in the NIST Handbook of Mathematical Functions [21].The attention that the Painlevé functions have received over the last century has primarily been limited to their analytic properties and applications. The numerical computation of these functions, by contrast, has received comparatively little attention. With the exceptions of [8], [20], [22] (discussed later), numerical studies have so far mostly focused on pole-free intervals along the real line [17], [21], [25] or on a polefree region in the complex plane [11]. A possible explanation for this state of affairs is that the large pole fields in the complex plane associated with the Painlevé ODEs may be thought of as a challenge to standard numerical methods for the integration of ODEs.We argue in the present work that these pole fields provide, in fact, especially advantageous integration opportunities. A 'pole friendly' algorithm based on Padé approximation is devised that is capable of integrating accurately and stably through pole fields for distances in the 10 5 − 10 6 range. This allows numerical studies not only of 'near-fields', but also of transition processes towards 'far fields', for which an extensive body of asymptotic...

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Section: Accuracy and Time Comparisonsmentioning

confidence: 99%

A numerical methodology for the Painlevé equations

Fornberg

Weideman

2011

Journal of Computational Physics

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“…Two well-known techniques are extrapolation or polynomial methods and vectorial -algorithms; see the papers [7,8] where a large bibliography can be found. A speciÿc sub-class of polynomial extrapolation, called the modiÿed minimal polynomial extrapolation (MMPE) has been evaluated recently [9].…”

Section: Introductionmentioning

confidence: 99%

Mathematical and numerical connections between polynomial extrapolation and Padé approximants: applications in structural mechanics

Damil

Cadou

Potier‐Ferry³

2004

Commun. Numer. Meth. Engng.

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“…The elements ε (n) 2k+1 are only auxiliary quantities which diverge if the whole transformation process converges and shouldn't be used for convergence tests or output. A recent review of the applications of the epsilon algorithm can be found in [60].…”

Section: How To Compute the Padé Approximantsmentioning

confidence: 99%

Basic Methods for Computing Special Functions

Gil

Segura

Temme

2011

Recent Advances in Computational and Applied Mathematics

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This paper gives an overview of methods for the numerical evaluation of special functions, that is, the functions that arise in many problems from mathematical physics, engineering, probability theory, and other applied sciences. We consider in detail a selection of basic methods which are frequently used in the numerical evaluation of special functions: converging and asymptotic series, including Chebyshev expansions, linear recurrence relations, and numerical quadrature. Several other methods are available and some of these will be discussed in less detail. We give examples of recent software for special functions where these methods are used. We mention a list of new publications on computational aspects of special functions available on our website.2000 Mathematics Subject Classification: 65D20, 41A60, 33C05, 33C10, 33C15.

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The epsilon algorithm and related topics

Cited by 54 publications

References 42 publications

A numerical methodology for the Painlevé equations

A numerical methodology for the Painlevé equations

Mathematical and numerical connections between polynomial extrapolation and Padé approximants: applications in structural mechanics

Basic Methods for Computing Special Functions

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