A User-Friendly Extrapolation Method for Oscillatory Infinite Integrals

Sidi, Avram

doi:10.2307/2008589

Cited by 12 publications

(12 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Numerical confirmation to 600 and to 80 digits respectively is detailed in [2]. The difference in our level of numerical confirmation is explained by the already mentioned failure of Sidi's method [46] for n = 6.…”

Section: Multiple Mahler Measuresmentioning

confidence: 93%

See 1 more Smart Citation

Mahler measures, short walks and log-sine integrals

Borwein

2012

Proceedings of the 2011 International Workshop on Symbolic-Numeric Computation

View full text Add to dashboard Cite

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.

show abstract

Section: Multiple Mahler Measuresmentioning

confidence: 93%

“…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

View full text Add to dashboard Cite

show abstract

“…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

View full text Add to dashboard Cite

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.

show abstract

“…Among various options, in cases where the information about the asymptotic behavior of the terms is not available or is difficult to extract, the following four choices of the numerically derived remainder estimates have been suggested as optimal in the literature [10,26,27]:…”

Section: Extrapolation Techniquesmentioning

confidence: 99%

Acceleration of Slowly Convergent Series via the Generalized Weighted-Averages Method

Polimeridis¹,

Niciforovi²,

Mosig³

2010

PIER M

View full text Add to dashboard Cite

Abstract-A generalized version of the weighted-averages method is presented for the acceleration of convergence of sequences and series over a wide range of test problems, including linearly and logarithmically convergent series as well as monotone and alternating series.This method was originally developed in a partitionextrapolation procedure for accelerating the convergence of semiinfinite range integrals with Bessel function kernels (Sommerfeld-type integrals), which arise in computational electromagnetics problems involving scattering/radiation in planar stratified media. In this paper, the generalized weighted-averages method is obtained by incorporating the optimal remainder estimates already available in the literature. Numerical results certify its comparable and in many cases superior performance against not only the traditional weighted-averages method but also against the most proven extrapolation methods often used to speed up the computation of slowly convergent series.

show abstract

A User-Friendly Extrapolation Method for Oscillatory Infinite Integrals

Cited by 12 publications

References 5 publications

Mahler measures, short walks and log-sine integrals

Mahler measures, short walks and log-sine integrals

High-Precision Arithmetic in Mathematical Physics

Acceleration of Slowly Convergent Series via the Generalized Weighted-Averages Method

Contact Info

Product

Resources

About