Evaluating infinite integrals involving Bessel functions of arbitrary order

Lucas, Stephen K.; Stone, Howard A.

doi:10.1016/0377-0427(95)00142-5

Cited by 63 publications

(32 citation statements)

References 13 publications

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“…Since a significant fraction of the overall computational effort is typically spent on the tail integral, it is essential that this integration be done as efficiently as possible. The proven and most popular approach is the integration then summation procedure [2]- [4] in which the integral is evaluated as a sum of a series of partial integrals over finite subintervals as follows: (2) with (3) where with and is a sequence of suitably selected interpolation points. These break points may be selected based on the asymptotic behavior of the integrand .…”

Section: Introductionmentioning

confidence: 99%

Extrapolation methods for Sommerfeld integral tails

Michalski

1998

IEEE Trans. Antennas Propagat.

253

234

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A review is presented of the extrapolation methods for accelerating the convergence of Sommerfeld-type integrals (i.e., semi-infinite range integrals with Bessel function kernels), which arise in problems involving antennas or scatterers embedded in planar multilayered media. Attention is limited to partition-extrapolation procedures in which the Sommerfeld integral is evaluated as a sum of a series of partial integrals over finite subintervals and is accelerated by an extrapolation method applied over the real-axis tail segment (a a a, 1 1 1) of the integration path, where a a a > > > 0 is selected to ensure that the integrand is well behaved. An analytical form of the asymptotic truncation error (or the remainder), which characterizes the convergence properties of the sequence of partial sums and serves as a basis for some of the most efficient extrapolation methods, is derived. Several extrapolation algorithms deemed to be the most suitable for the Sommerfeld integrals are described and their performance is compared. It is demonstrated that the performance of these methods is strongly affected by the horizontal displacement of the source and field points and by the choice of the subinterval break points. Furthermore, it is found that some well-known extrapolation techniques may fail for a number of values of and ways to remedy this are suggested. Finally, the most effective extrapolation methods for accelerating Sommerfeld integral tails are recommended.

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Section: Introductionmentioning

confidence: 99%

Extrapolation methods for Sommerfeld integral tails

Michalski

1998

IEEE Trans. Antennas Propagat.

253

234

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“…The component integrals are then evaluated by treating the areas between integrand zeros as terms in alternating series which are readily summed by the well-known accelerated ε-convergence method. A single series summation (Lucas and Stone, 1995) results when r = 0 in U 3 (r, z, p).…”

Section: Numerical Evaluationsmentioning

confidence: 99%

One- and Three-Dimensional Soil Transportation Models for Volatiles Migrating from Soils to House Interiors

Robinson

Turczynowicz

2005

Transp Porous Med

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Two models are presented for the transient migration of a volatile organic compound (VOC) from soil to the interor of a house with a crawl space. The migration in the house is taken as one-dimensional (1D) and coupled to a soil transportation model with diffusion, leaching and VOC degradation. The diffusion is either vertical, providing a 1D model, or three-dimensional (3D) axisymmetric, providing a 3D model. The initial subsurface VOC deposition is assumed to be a finite layer extending over the footprint of the house in the 1D case or as a cylinder of arbitrary radius in the 3D case. Using data for benzene as the VOC, comparisons are made for results from the two models for VOC concentrations in soil, crawl space and dwelling space as well as the cumulative dwelling space concentration used in human health assessment, for a wide range of soil parameters and subsurface sizes of the initial VOC cylinder. In all cases the values from the 1D transport model were slightly higher than those for the 3D model. The analysis uses Laplace transforms with numerical inversions. For the 3D soil transport model, different soil surface flux boundary conditions underneath and outside the house give rise to novel dual integral equations.

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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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Evaluating infinite integrals involving Bessel functions of arbitrary order

Cited by 63 publications

References 13 publications

Extrapolation methods for Sommerfeld integral tails

Extrapolation methods for Sommerfeld integral tails

One- and Three-Dimensional Soil Transportation Models for Volatiles Migrating from Soils to House Interiors

High-Precision Arithmetic in Mathematical Physics

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