Function classes for double exponential integration formulas

Tanaka, Ken’ichiro; Sugihara, Masaaki; Murota, Kazuo; Mori, Miwako

doi:10.1007/s00211-008-0195-1

Cited by 51 publications

(44 citation statements)

References 14 publications

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“…This is the class of functions defined on (−1, 1) that can be extended analytically into a domain defined by the image of the strip |Im(x)| < a under the transformation = tanh( 1 2 π sinh(x)), which is an infinitely sheeted Riemann surface that wraps around ξ = ±1. A proof can be found in [165], where related theorems for other single and double exponential transformations are also given. Further variations are considered in [1,141].…”

Section: Exponential and Double Exponentialmentioning

confidence: 99%

The Exponentially Convergent Trapezoidal Rule

Trefethen¹,

Weideman²

2014

SIAM Rev.

469

362

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Abstract.It is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed, and it is shown that far from being a curiosity, it is linked with computational methods all across scientific computing, including algorithms related to inverse Laplace transforms, special functions, complex analysis, rational approximation, integral equations, and the computation of functions and eigenvalues of matrices and operators.

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Section: Exponential and Double Exponentialmentioning

confidence: 99%

The Exponentially Convergent Trapezoidal Rule

Trefethen¹,

Weideman²

2014

SIAM Rev.

469

362

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“…Actually, those are theoretically supported in the literature [4,5]. Furthermore, in the case of algebraic singularity (3), explicit (computable) error bounds of those rules have been recently given [6], and the results were utilized to construct a verified numerical integration library [7] in that case.…”

Section: Introductionmentioning

confidence: 87%

Explicit error bound for the tanh rule and the DE formula for integrals with logarithmic singularity

Okayama

2014

JSIAM Letters

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The tanh rule and the double-exponential (DE) formula are known empirically and theoretically as quite efficient quadrature formulas, especially for integrals with endpoint singularity, including algebraic singularity and logarithmic singularity. Furthermore, in the case of integrals with algebraic singularity, explicit error bounds have been given for those formulas, which enables us to guarantee their approximation accuracy. In the case of integrals with logarithmic singularity, however, such explicit error bounds have not ever given thus far, although those formulas should work accurately in this case as well. This paper presents the desired theoretical explicit error bounds, with numerical experiments.

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“…In (4.6), h is not set based on a theoretical criterion but it is determined experimentally in reference to the settings in the DE formulas for definite integration (Tanaka et al, 2009). All computations are performed through MATLAB R2013a programs with double precision floating point arithmetic on a PC with a 3.0 GHz CPU and 2.0 GB RAM.…”

Section: Numerical Examplesmentioning

confidence: 99%

A fast and accurate numerical method for symmetric Lévy processes based on the Fourier transform and sinc-Gauss sampling formula

Tanaka¹

2015

IMA J Numer Anal

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In this paper, we propose a fast and accurate numerical method based on Fourier transform to solve Kolmogorov forward equations of symmetric scalar Lévy processes. The method is based on the accurate numerical formulas for Fourier transform proposed by Ooura. These formulas are combined with nonuniform fast Fourier transform (FFT) and fractional FFT to speed up the numerical computations. Moreover, we propose a formula for numerical indefinite integration on equispaced grids as a component of the method. The proposed integration formula is based on the sinc-Gauss sampling formula, which is a function approximation formula. This integration formula is also combined with the FFT. Therefore, all steps of the proposed method are executed using the FFT and its variants. The proposed method allows us to be free from some special treatments for a non-smooth initial condition and numerical time integration. The numerical solutions obtained by the proposed method appeared to be exponentially convergent on the interval if the corresponding exact solutions do not have sharp cusps. Furthermore, the real computational times are approximately consistent with the theoretical estimates.

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Function classes for double exponential integration formulas

Cited by 51 publications

References 14 publications

The Exponentially Convergent Trapezoidal Rule

The Exponentially Convergent Trapezoidal Rule

Explicit error bound for the tanh rule and the DE formula for integrals with logarithmic singularity

A fast and accurate numerical method for symmetric Lévy processes based on the Fourier transform and sinc-Gauss sampling formula

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