The tanh transformation for singular integrals

Evans, G. A.; Forbes, R.C.; Hyslop, J.

doi:10.1080/00207168408803419

Cited by 19 publications

(6 citation statements)

References 15 publications

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“…With a suitable choice of g(t), the sec h 2 g(t) factor falls off extremely rapidly as t 3 Ϯϱ and consequently controls the behavior of the transformed integrand, even though the singular values f(a) and f(b) are being approached [9]. Next, a midpoint rule is used by setting t ϭ nhЈ, with n ϭ 0, Ϯ1, .…”

Section: Appendix: Tanh Transformationmentioning

confidence: 99%

“…It was reported that the oscillatory nature of the Chebyschev polynomials enhances the rapid convergence of the oscillating integrand. In this paper, we report on the use of tanh transformation [9], which allows very efficient and accurate evaluation of Sommerfeld integrals. This method provides accurate results, even when the source and observer locations are close to the ground.…”

Section: Introductionmentioning

confidence: 99%

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Computation of Sommerfeld integrals using tanh transformation

Singh

2003

Micro & Optical Tech Letters

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A variable transformation is utilized in the computation of Sommerfeld integrals, which arise in the analysis of antennas radiating over a dielectric half‐space. The method involves a change of variable, which transforms the integrand into a rapidly converging one. Several numerical examples, which illustrate the effectiveness of the method, are presented and the results are compared with those reported in the literature. © 2003 Wiley Periodicals, Inc. Microwave Opt Technol Lett 37: 177–180, 2003; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.10860

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Section: Appendix: Tanh Transformationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Computation of Sommerfeld integrals using tanh transformation

Singh

2003

Micro & Optical Tech Letters

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“…Then: 8) where N = 2n + 1, the mesh size h is chosen optimally as: 9) and C d,ω is a constant depending on d and ω. Theorem 2.2 (Sugihara [44]). Suppose that the function ω(z) satisfies the following three conditions:…”

Section: Intervalmentioning

confidence: 99%

“…After this observation, the race was on to determine exactly which variable transformation, and therefore which decay rate, is optimal. Numerical experiments showed the exceptional promise of rules such as the tanh substitution [9], the erf substitution [49], the IMT rule [19], and the tanh-sinh substitution [50], among others [22]. But exactly which one is optimal, and in which setting?…”

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confidence: 99%

On The Use of Conformal Maps for the Acceleration of Convergence of the Trapezoidal Rule and Sinc Numerical Methods

Slevinsky¹,

Olver²

2015

SIAM J. Sci. Comput.

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Abstract. We investigate the use of conformal maps for the acceleration of convergence of the trapezoidal rule and Sinc numerical methods. The conformal map is a polynomial adjustment to the sinh map, and allows the treatment of a finite number of singularities in the complex plane. In the case where locations are unknown, the so-called Sinc-Padé approximants are used to provide approximate results. This adaptive method is shown to have almost the same convergence properties. We use the conformal maps to generate high accuracy solutions to several challenging integrals, nonlinear waves, and multidimensional integrals. 1. Introduction. The trapezoidal rule is one of the most well-known methods in numerical integration. While the composite rule has geometric convergence for periodic functions, in other cases it has been used as the starting point of effective methods, such as Richardson extrapolation [36] and Romberg integration [37]. The geometric convergence breaks down with endpoint singularities, and this issue inspired a different approach to improve on the composite rule. From the Euler-Maclaurin summation formula, it was noted that some form of exponential convergence can be obtained for integrands which vanish at the endpoints, suggesting that undergoing a variable transformation may well induce this convergence [38,40,48]. After this observation, the race was on to determine exactly which variable transformation, and therefore which decay rate, is optimal. Numerical experiments showed the exceptional promise of rules such as the tanh substitution [9], the erf substitution [49], the IMT rule [19], and the tanh-sinh substitution [50], among others [22]. But exactly which one is optimal, and in which setting?Using a functional analysis approach, this question was beautifully answered by establishing the optimality of a double exponential endpoint decay rate for the trapezoidal rule on the real line for approximating analytic integrands [44]. The domain of analyticity is described in terms of a strip of maximal width π centred on the real axis in the complex plane. This optimality also prescribed the optimal step size and a near-linear convergence rate O(e −kN/ log N ), where N is the number of sample points and k is a constant proportional to the strip width.The results allowed for displays of strong performance for integrals with integrable endpoint singularities without changing the rule in any way [23][24][25]47]. The double exponential transformation was also adapted to Fourier and general oscillatory integrals in [34,35]. Recognizing the trapezoidal rule as the integration of a Sinc expansion of the integrand, the double exponential advocates adapted their analysis to Sinc approximations [46], and also to all the numerical methods therewith derived, such as Sinc-Galerkin and Sinc-collocation methods [17,28,45] for initial and boundary

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“…Two straightforward ways of computing such an integral are: (1) Replacing the lower endpoint by e (> 0), and computing the resulting integrals for a sequence of e 's tending to zero, perhaps employing some sort of sequence extrapolation; and (2) decomposing / into an easily-treated singular part fs of special form and a bounded part /¿ such that / = fs + fb > then computing the individual integrals by special techniques and as the integrand naturally separates into a part which is well-behaved ( fb ) and another part which is a power series attached to a logarithm ( fs ). Other methods include change-of-variable techniques [2,4,6,7,11] (the tanh, erf, and IMT rules), Gaussian quadrature for specialized singular weight functions [10], or most simply computing and ignoring the singularity. Evidently there are relative strengths and weaknesses associated with these approaches.…”

Section: Introductionmentioning

confidence: 99%

Quadrature of Integrands with a Logarithmic Singularity

Crow¹

1993

Mathematics of Computation

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Abstract.A quadrature rule is presented that is exact for integrands of the form 7t(Z) + m(Ç)loxiÇ on the interval (0, 1), where n and tu are polynomials. The computed weights and abscissae are given for one-through seven-point rules. In particular, the four-point rule is exact for integral operators with logarithmically singular kernel on a cubic B-spline basis, and it is expected these results shall prove useful for numerical applications of weighted-residual finite element methods.

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The tanh transformation for singular integrals

Cited by 19 publications

References 15 publications

Computation of Sommerfeld integrals using tanh transformation

Computation of Sommerfeld integrals using tanh transformation

On The Use of Conformal Maps for the Acceleration of Convergence of the Trapezoidal Rule and Sinc Numerical Methods

Quadrature of Integrands with a Logarithmic Singularity

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