The numerical evaluation of very oscillatory infinite integrals by extrapolation

Sidi, Avram

doi:10.1090/s0025-5718-1982-0645667-5

Cited by 82 publications

(49 citation statements)

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“…These integrand functions, which are highly oscillatory, are very challenging to evaluate to high precision. We employed tanh-sinh quadrature and Gaussian quadrature, together with the Sidi mW extrapolation algorithm, as described in a 1994 paper by Lucas and Stone [78], which, in turn, is based on two earlier papers by Sidi [79,80]. While the computations were relatively expensive, we were able to compute 1000-digit values of these integrals for odd n up to 17 [76].…”

Section: Ramble Integralsmentioning

confidence: 99%

High-Precision Arithmetic in Mathematical Physics

Bailey

Borwein

2015

Mathematics

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For many scientific calculations, particularly those involving empirical data, IEEE 32-bit floating-point arithmetic produces results of sufficient accuracy, while for other applications IEEE 64-bit floating-point is more appropriate. But for some very demanding applications, even higher levels of precision are often required. This article discusses the challenge of high-precision computation, in the context of mathematical physics, and highlights what facilities are required to support future computation, in light of emerging developments in computer architecture.

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Section: Ramble Integralsmentioning

confidence: 99%

High-Precision Arithmetic in Mathematical Physics

Bailey

Borwein

2015

Mathematics

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“…This work and various more recent papers, especially [2], highlight the continuing difficulty of computing the underlying Bessel integrals. Interestingly, Sidi's methods [45,46] proved useful only for a product of an odd number of Bessel functions. Subsequent to the work in [2], Sidi has however proposed an enhanced method [47].…”

Section: The Density Pmentioning

confidence: 99%

Mahler measures, short walks and log-sine integrals

Borwein

2012

Proceedings of the 2011 International Workshop on Symbolic-Numeric Computation

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The Mahler measure of a polynomial in several variables has been a subject of much study over the past thirty years -very few closed forms are proven but more are conjectured. In the case of multiple Mahler measures more tractable but interesting families exist. Using values of log-sine integrals we provide systematic evaluations of various higher and multiple Mahler measures.The evaluations in terms of log-sine integrals become particularly useful in light of the fact that log-sine integrals may be automatically reexpressed as polylogarithmic values. We present this correspondence along with related generating functions for log-sine integrals.Our initial interest in considering Mahler measures stems from a study of uniform random walks in the plane as first introduced by Pearson. The main results on the moments of the distance traveled by an n-step walk, as well as the corresponding probability density functions, are reviewed. It is the derivative values of the moments that are Mahler measures.This work would be impossible without very extensive symbolic and numeric computations. It also makes frequent use of the new NIST Handbook of Mathematical Functions and similar tools. Our intention is to show off the interplay between numeric and symbolic computing while exploring the three mathematical topics in the title.

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“…In [4] the author developed an extrapolation method, the Wtransformation, by which a large class of convergent infinite oscillatory integrals can be computed very efficiently. The approach of [4] was later extended to divergent infinite oscillatory integrals in [6], and it was shown that the VK-transformation, with no modifications, can be applied to such integrals with the same efficiency.…”

Section: Introductionmentioning

confidence: 99%

“…The approach of [4] was later extended to divergent infinite oscillatory integrals in [6], and it was shown that the VK-transformation, with no modifications, can be applied to such integrals with the same efficiency. The use of the ^-transformation involves some asymptotic analysis of the integrand as the variable of integration tends to infinity.…”

Section: Introductionmentioning

confidence: 99%

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A User-Friendly Extrapolation Method for Oscillatory Infinite Integrals

Sidi¹

1988

Mathematics of Computation

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Abstract.In a recent publication [4] the author developed an extrapolation method, the ^-transformation, for the accurate computation of convergent oscillatory infinite integrals. In yet another publication [6] this method was shown to be applicable to divergent oscillatory infinite integrals that are defined in the sense of summability. The application of the VK-transformation involves some asymptotic analysis of the integrand as the variable of integration tends to infinity. In the present work the W-transformation is modified so as to keep this asymptotic analysis to a minimum, involving only the phase of oscillations. This modified version, which turns out to be as efficient as the original IV-transformation, can also be applied to convergent or divergent oscillatory infinite integrals other than those dealt with in

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The numerical evaluation of very oscillatory infinite integrals by extrapolation

Cited by 82 publications

References 10 publications

High-Precision Arithmetic in Mathematical Physics

High-Precision Arithmetic in Mathematical Physics

Mahler measures, short walks and log-sine integrals

A User-Friendly Extrapolation Method for Oscillatory Infinite Integrals

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