Comparison of some methods for evaluating infinite range oscillatory integrals

Blakemore, Michael; Evans, G. A.; Hyslop, J.

doi:10.1016/0021-9991(76)90054-1

Cited by 51 publications

(14 citation statements)

References 30 publications

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“…Squire [10] also used this method as well as the algorithm and suggested the Levin transformation [23] as a promising alternative. Blakemore et al [3] have found Levin's transformation to be more efficient than the algorithm, albeit only marginally so. Bubenik [25] employed the algorithm.…”

Section: Introductionmentioning

confidence: 98%

“…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%

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Extrapolation methods for Sommerfeld integral tails

Michalski

1998

IEEE Trans. Antennas Propagat.

254

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: 98%

Section: Introductionmentioning

confidence: 99%

Extrapolation methods for Sommerfeld integral tails

Michalski

1998

IEEE Trans. Antennas Propagat.

254

234

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“…Some of these integrals (in the given form or in a transformed form) were investigated in other studies [3,22,26,27]. The results obtained using Programs 1 and 2 are displayed in Table 5.…”

Section: Numerical Results For Some Selected Integralsmentioning

confidence: 97%

On numerical computation of integrals with integrands of the form f(x)sin(w/xr) on [0, 1]

Hascelik

2009

Journal of Computational and Applied Mathematics

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With existing numerical integration methods and algorithms it is difficult in general to obtain accurate approximations to integrals of the formwhere f is a sufficiently smooth function on [0, 1]. Gautschi has developed software (as scripts in Matlab) for computing these integrals for the special case r = ω = 1. In this paper, an algorithm (as a Mathematica program) is developed for computing these integrals to arbitrary precision for any given values of the parameters in a certain range. Numerical examples are given of testing the performance of the algorithm/program.

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“…Now we write (3)(4)(5)(6)(7) so that k(x) = e~xlK(x), k*(x) = e~x2K*(x), \k*-k\\p+)= \\K*-K\\p-\ It follows from the assumption (2.1) that K G Lp~\ Let us now require that K* be a polynomial. Since the polynomials are dense in the space L<p) (see [13, Lemma 2]), we can choose K* so that the first two terms on the right-hand side of (3.6) are each less than e, where e > 0 is a given arbitrary number.…”

mentioning

confidence: 99%

“…A quite different method for handling oscillatory integrals over semi-infinite intervals is the acceleration method, which has been reviewed and extended by Blakemore, Evans and Hyslop [3]. That method is based on the use of accurate quadrature rules to integrate between successive zeros, combined with powerful acceleration techniques to speed the convergence as the upper limit is taken to infinity.…”

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confidence: 99%

Product integration over infinite intervals. I. Rules based on the zeros of Hermite polynomials

Smith¹,

Sloan²,

Opie³

1983

Math. Comp.

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Abstract. The paper discusses both theoretical properties and practical implementation of product integration rules of the formwhere / is continuous, k is absolutely integrable, the nodes {*",} are roots of the Hermite polynomials H"(x), and the weights {%,•} are chosen so that the rule is exact if/ is any polynomial of degree < n. Convergence of the rule to the exact integral as n -» oo is proved for a wide class of functions/and k (including singular or oscillatory functions k), and rates of convergence are estimated. The rules are shown to have the property of asymptotic positivity, and as a consequence exhibit good numerical stability. Numerical calculations for some practical cases are presented, which show the method to be computationally effective for integrands (including highly oscillatory ones) that decay suitably at infinity. Applications of the method to integration over [ 0, oo) are also discussed.

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Comparison of some methods for evaluating infinite range oscillatory integrals

Cited by 51 publications

References 30 publications

Extrapolation methods for Sommerfeld integral tails

Extrapolation methods for Sommerfeld integral tails

On numerical computation of integrals with integrands of the form f(x)sin(w/xr) on [0, 1]

Product integration over infinite intervals. I. Rules based on the zeros of Hermite polynomials

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