A generalization of the continuous Euler transformation and its application to numerical quadrature

Ooura, Takuya

doi:10.1016/s0377-0427(03)00378-9

Cited by 6 publications

(6 citation statements)

References 6 publications

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“…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 Numerical Integration Formula Based on the Bessel Functions

Ogata¹

2005

Publ. Res. Inst. Math. Sci.

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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)

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Section: Discussionmentioning

confidence: 99%

A Numerical Integration Formula Based on the Bessel Functions

Ogata¹

2005

Publ. Res. Inst. Math. Sci.

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“…. (17) where in the first column and is the best approximation of , given the partial sums . In the original version of the WA method [19], the remainders are expanded into infinite series using integration by parts, which is truncated for numerical purposes (18) where (19) The weights obtained in this way, although different, are asymptotically equivalent to those in (15).…”

Section: A Partition-extrapolation Methods Involving Wa Techniquementioning

confidence: 99%

“…Therefore, the direct application of the formulas (24)- (25) using as default the maximal number of points (160, corresponding to 10 iterations with a quadrature order of 16) can be implemented in MATLAB as a simple matrix multiplication, which is very convenient and indeed leads to lower computational time than the iterative procedure proposed in [18], while resulting in the same accuracy for the final results. Under these assumptions, the DE transformation needs, approximately 5 times less computational time than the two other methods, as can be seen from weights (15)- (16) involved in the Mosig-Michalski WA technique are very easy to evaluate, but the method calls for the recursion (17). On the other hand, the evaluation of the weights (21) involved in new WA algorithm is more complex, since it calls for the computation of the binomial coefficients.…”

Section: B De Techniquementioning

confidence: 99%

Efficient Algorithms for Computing Sommerfeld Integral Tails

Golubovic

Polimeridis

Mosig

2012

IEEE Trans. Antennas Propagat.

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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.

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“…This formula, however, has some restriction for the setting of the range, and direct application of the FFT to the formula is not straightforward. On the other hand, Ooura [8] [9] proposed other useful formulas…”

Section: Introductionmentioning

confidence: 99%

Error control of a numerical formula for the Fourier transform by Ooura’s continuous Euler transform and fractional FFT

Tanaka

2014

Journal of Computational and Applied Mathematics

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In this paper, we consider a method for fast numerical computation of the Fourier transform of a slowly decaying function with given accuracy in given ranges of the frequency. In these decades, some useful formulas for the Fourier transform are proposed to recover difficulty of the computation due to the slow decay and the oscillation of the integrand. In particular, Ooura proposed formulas with continuous Euler transformation and showed their effectiveness. It is, however, also reported that errors of them become large outside some ranges of the frequency. Then, for an illustrating representative of the formulas, we choose parameters in the formula based on its error analysis to compute the Fourier transform with given accuracy in given ranges of the frequency. Furthermore, combining the formula and fractional FFT, a generalization of the fast Fourier transform (FFT), we execute the computation in the same order of computation time as the FFT.

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A generalization of the continuous Euler transformation and its application to numerical quadrature

Cited by 6 publications

References 6 publications

A Numerical Integration Formula Based on the Bessel Functions

A Numerical Integration Formula Based on the Bessel Functions

Efficient Algorithms for Computing Sommerfeld Integral Tails

Error control of a numerical formula for the Fourier transform by Ooura’s continuous Euler transform and fractional FFT

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