A Comparison of Some Quadrature Methods for Approximating Cauchy Principal Value Integrals

Natarajan, A.; Mohankumar, N.

doi:10.1006/jcph.1995.1034

Cited by 15 publications

(6 citation statements)

References 5 publications

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“…The Cauchy principal value is central to many calculations that occur in a wide range of physics and engineering and a number of quadrature strategies have been applied to the problem [14][15][16], such as the FFT [17,18], Euler-MacLaurin [14,[19][20][21], Gaussian [22], Chebyshev [23] and variable transformation [24] (e.g. TANH [25] and Iri-Moriguti-Takasawa [26]) methods.…”

Section: Introductionmentioning

confidence: 99%

Calculation of the Kramers-Kronig transform of X-ray spectra by a piecewise Laurent polynomial method

Watts¹

2014

Opt. Express

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An algorithm is presented for the calculation of the Kramers-Kronig transform of a spectrum via a piecewise Laurent polynomial method. This algorithm is demonstrated to be highly accurate, while also being computationally efficient. The algorithm places no requirements on data point spacing and is capable of integrating across the full spectrum (i.e. from zero to infinity). Further, we present a computer application designed to aid in calculating the Kramers-Kronig transform on near-edge experimental X-ray absorption spectra (extended with atomic scattering factor data) in order to produce the dispersive part of the X-ray refractive index, including near-edge features.

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Section: Introductionmentioning

confidence: 99%

Calculation of the Kramers-Kronig transform of X-ray spectra by a piecewise Laurent polynomial method

Watts¹

2014

Opt. Express

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“…The main reason for this interest is probably due to the fact that integral equations with Cauchy principal value integrals have been shown to be an adequate tool [1] for the modeling of many physical situations, such as acoustics, fluid mechanics, elasticity, fracture mechanics, and electromagnetic scattering problems. Numerous work has been devoted in developing efficient quadrature formulas, such as the Gaussian method [2][3][4][5][6][7][8], the Newton-Cotes methods [9][10][11][12][13], spline methods [14,15], and some other methods [16][17][18][19][20][21][22]. It is the aim of this paper to investigate the superconvergence phenomenon of trapezoidal rule and, in particular, to derive error estimates.…”

Section: Introductionmentioning

confidence: 99%

The Trapezoidal Rule for Computing Cauchy Principal Value Integral on Circle

2015

Mathematical Problems in Engineering

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The composite trapezoidal rule for the computation of Cauchy principal value integral with the singular kernelcot((x-s)/2)is discussed. Our study is based on the investigation of the pointwise superconvergence phenomenon; that is, when the singular point coincides with some a priori known point, the convergence rate of the trapezoidal rule is higher than what is globally possible. We show that the superconvergence rate of the composite trapezoidal rule occurs at middle of each subinterval and obtain the corresponding superconvergence error estimate. Some numerical examples are provided to validate the theoretical analysis.

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“…So far several numerical methods in approximating the CPV integration have been developed. These methods, which include Gaussian Integration [7], Chebysheusingn polynomial [8], SINC function [9,10], approximation via an Ostrowski type inequality [11] and variable transformation [12][13][14], have been developed. However, they differ significantly from each other with regard to quadrature scheme, complexity, computation accuracy and numerical stability.…”

Section: Introductionmentioning

confidence: 99%

“…The category of variable transformations includes IMT (Iri-Moriguti-Takasava), TANH (in principle, it is a SINC method), erf function and DE (also called Tanh-Sinh method or double exponential quadrature scheme). A comparison of different quadrature methods can be found in literature [13,14], where the authors used the same set of test problems to compare several important schemes including the Quadpack package, Chebyshev, TANH, IMT, and DE schemes. These schemes show different computation stabilities and complexities from smooth integrands to complex conjugate poles to integrable singularity at boundaries.…”

Section: Introductionmentioning

confidence: 99%

Fast finite Hilbert transform via DE quadrature scheme

Cui

Cao

et al. 2010

Journal of X-Ray Science and Technology

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Purpose: The finite Hilbert transform (FHT) or inverse finite Hilbert transform (IFHT) is recently found to have some important applications in computerized tomography (CT) arena [1-6], where they are used to filter the derivatives of backprojected data in the chord-line based CT reconstruction algorithms. In this paper, we implemented, improved and validated a fast numerical solution to the FHT via a double exponential (DE) integration scheme. A same strategy can be used to compute IFHT.Methods: To overcome the underflow of floating-point numbers, we first determined the range of variable transformation from the minimum positive value of single or double precision floating point number, the integration step can be further determined by the range of variable transformation and the integration level. Two functions with their known analytical FHTs are used to validate the implementation of the FHT via DE scheme. The surface map and 2D contour of the FHT transformation error with respect to integration level and the range of the variable transformation are used to numerically determine the optimal numbers for a fast FHT.Results: Given a specific precision, the lowest integration level and the optimal range of variable transformation, which are used to transform a signal with a certain degree of fluctuation, can be numerically determined by the surface map and 2D contour of the standard deviation of transformation error. These two numbers can then be taken to efficiently compute the FHT for other signals with the same or less degree of fluctuation.Conclusions: The FHT via DE scheme and the numerical method to determine the integration level and the range of transformation can be used for fast FHT in certain applications, such as data filtering in chord-line based CT reconstruction algorithms.

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A Comparison of Some Quadrature Methods for Approximating Cauchy Principal Value Integrals

Cited by 15 publications

References 5 publications

Calculation of the Kramers-Kronig transform of X-ray spectra by a piecewise Laurent polynomial method

Calculation of the Kramers-Kronig transform of X-ray spectra by a piecewise Laurent polynomial method

The Trapezoidal Rule for Computing Cauchy Principal Value Integral on Circle

Fast finite Hilbert transform via DE quadrature scheme

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