The SIAM 100-Digit Challenge

Bornemann, Folkmar; Laurie, Dirk; Wagon, Stan; Waldvogel, Jörg

doi:10.1137/1.9780898717969

Cited by 80 publications

(37 citation statements)

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“…As an example, consider the last problem of the SIAM 100-Digit Challenge [25]. This involved a particle in a 10 × 1 rectangle undergoing a two-dimensional Brownian motion.…”

Section: Laplace Transforms and Hankel Contours The Laplace Transformentioning

confidence: 99%

“…The problem can be posed as a Laplace problem for the determination of a harmonic measure; see [25,Chapter 10]. We consider a box of dimensions 2b × 2a × 2a and, making use of the symmetries, we reduce the problem to solving (16.5) (16.9) where D is the Chebyshev spectral differentiation matrix on [0, a] that incorporates homogeneous Neumann and Dirichlet conditions at the left and right endpoints, respectively.…”

Section: Laplace Transforms and Hankel Contours The Laplace Transformentioning

confidence: 99%

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The Exponentially Convergent Trapezoidal Rule

Trefethen¹,

Weideman²

2014

SIAM Rev.

436

362

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Abstract.It is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed, and it is shown that far from being a curiosity, it is linked with computational methods all across scientific computing, including algorithms related to inverse Laplace transforms, special functions, complex analysis, rational approximation, integral equations, and the computation of functions and eigenvalues of matrices and operators.

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“…As an example, consider the last problem of the SIAM 100-Digit Challenge [25]. This involved a particle in a 10 × 1 rectangle undergoing a two-dimensional Brownian motion.…”

Section: Laplace Transforms and Hankel Contours The Laplace Transformentioning

confidence: 99%

Section: Laplace Transforms and Hankel Contours The Laplace Transformentioning

confidence: 99%

The Exponentially Convergent Trapezoidal Rule

Trefethen¹,

Weideman²

2014

SIAM Rev.

436

362

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“…Nevertheless, they often accomplish spectacular results when applied to appropriate problems. Consequently, nonlinear transformations now dominate both mathematical research as well as practical applications, as documented in a large number of recent books [1,10,[17][18][19][20][21][22][23][24][25][26][27][28][29] and review articles [30][31][32][33] on this topic.…”

Section: Introduction To Convergence Acceleration and Resummationmentioning

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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“…This article was stimulated by the recent SIAM "100 Digit Challenge" of Nick Trefethen, beautifully described in [12]. Indeed, these ten numeric challenge problems are also listed in [14], where they are followed by the ten symbolic/numeric challenge problems that are discussed in this article.…”

Section: Introductionmentioning

confidence: 99%