A robust double exponential formula for Fourier-type integrals

Ooura, Takuya; Mori, Miwako

doi:10.1016/s0377-0427(99)00223-x

Cited by 71 publications

(70 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Lastly, we applied the formula to the computations of integrals of the Hankel transform type. In relation to integrals of the Hankel transform type, we also remark that Ooura and Mori have presented DE-type formulae for oscillatory functions with slow decay [5,6,7] in addition to their first DE-type formula for Fourier integrals [8].…”

Section: Discussionmentioning

confidence: 99%

“…A remedy for this weakness is to adopt a DE-type transform such that the singularities of the transformed integrand function does not approach to the real axis as h → 0 as the one used in the robust DE formula for Fourier transformation type integrals by Ooura and Mori [7]. However, it is difficult to find the explicit form of a transform satisfying the above property.…”

Section: Application To Integrals Of the Hankel Transformation Typementioning

confidence: 99%

See 1 more Smart Citation

A Numerical Integration Formula Based on the Bessel Functions

Ogata¹

2005

Publ. Res. Inst. Math. Sci.

View full text Add to dashboard Cite

In this paper, we discuss the properties of a quadrature formula with the zeros of the Bessel functions as nodes for integralswhere ν is a real constant greater than −1 and f (x) is a function analytic on the real axis (−∞, +∞). We show from theoretical error analysis that (i) the quadrature formula converges exponentially, (ii) it is as accurate as the trapezoidal formula over (−∞, +∞) and (iii) the accuracy of the quadrature formula doubles that of an interpolation formula with the same nodes. Numerical examples support the above theoretical results. We also apply the quadrature formula to the numerical integration of integral involving the Bessel function. §1. IntroductionIn this paper, we investigate the quadrature formula with the zeros of the Bessel functions as nodes, namely,, k = ±1, ±2, . . . , (1.2)

show abstract

Section: Discussionmentioning

confidence: 99%

Section: Application To Integrals Of the Hankel Transformation Typementioning

confidence: 99%

A Numerical Integration Formula Based on the Bessel Functions

Ogata¹

2005

Publ. Res. Inst. Math. Sci.

View full text Add to dashboard Cite

show abstract

“…A recently exploited additional advantage of DE is the possible adjustement of the parameter η in (26). Although, as mentioned, the historical choice has been η = π/2 and this choice has given consistently excellent results in onedimensional integrals, nothing prevents to try to optimize this parameter for multidimensional integrals.…”

Section: De and Multidimensional Singular Integralsmentioning

confidence: 99%

“…A final development of paramount relevance for this paper is the transformation, obtained by T. Ooura, of the DE quadrature into a form that can be applied to Fourier transforms of slowly decaying functions [26], [27]. This modified DE version can be easily applied to the evaluation of Sommerfeld integral tails [10], [28], [29].…”

Section: Historical Introductionmentioning

confidence: 99%

Weighted Averages and Double Exponential Algorithms

Mosig

2013

IEICE Trans. Commun.

View full text Add to dashboard Cite

SUMMARYThis paper reviews two simple numerical algorithms particularly useful in Computational ElectroMagnetics (CEM): the Weighted Averages (WA) algorithm and the Double Exponential (DE) quadrature. After a short historical introduction and an elementary description of the mathematical procedures underlying both techniques, they are applied to the evaluation of Sommerfeld integrals, where WA and DE combine together to provide a numerical tool of unprecedented quality. It is also shown that both algorithms have a much wider range of applications. A generalization of the WA algorithm, able to cope with integrands including products of Bessel and similar oscillatory functions, is described. Similarly, the original DE algorithm is adapted with exceptional results to the evaluation of the multidimensional singular integrals arising in the discretization of Integral-Equation based CEM formulations. The new possibilities of WA and DE algorithms are demonstrated through several practical numerical examples.

show abstract

“…Unfortunately, the original formula was not efficient in computing slowly decaying oscillatory functions over . Hence, to overcome this weakness, Ooura and Mori proposed a robust DE formula for Fourier-type integrals [24]. The key idea of the new transformation was slightly different: the nodes of the new quadrature approach rapidly (double exponentially) the zeros of sine/cosine function, thus allowing computation of the Fouriertype integrals with a small number of function evaluations.…”

Section: Introductionmentioning

confidence: 99%

Efficient Algorithms for Computing Sommerfeld Integral Tails

Golubovic

Polimeridis

Mosig

2012

IEEE Trans. Antennas Propagat.

View full text Add to dashboard Cite

Abstract-Sommerfeld-integrals (SIs) are ubiquitous in the analysis of problems involving antennas and scatterers embedded in planar multilayered media. It is well known that the oscillating and slowly decaying nature of their integrands makes the numerical evaluation of the SI real-axis tail segment a very time consuming and computationally expensive task. Therefore, SI tails have to be specially treated. In this paper we compare two recently developed techniques for their efficient numerical evaluation. First, a partition-extrapolation method, in which the integration-then-summation procedure is combined with a new version of the weighted averages (WA) extrapolation technique, is summarized. The previous variants of WA technique are also discussed. Then, a review of double-exponential (DE) quadrature formulas for direct integration of the SI tails is presented. The efficient way of implementing the algorithms, their pros and cons, as well as comparisons of their performance are discussed in detail.

show abstract

A robust double exponential formula for Fourier-type integrals

Cited by 71 publications

References 8 publications

A Numerical Integration Formula Based on the Bessel Functions

A Numerical Integration Formula Based on the Bessel Functions

Weighted Averages and Double Exponential Algorithms

Efficient Algorithms for Computing Sommerfeld Integral Tails

Contact Info

Product

Resources

About