A rational approximation of the Dawson’s integral for efficient computation of the complex error function

Abrarov, S. M.; Quine, Brendan M.

doi:10.1016/j.amc.2017.10.032

Cited by 22 publications

(31 citation statements)

References 47 publications

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“…We applied the highly accurate MATLAB function fadsamp.m shown in our recent paper [34] as a subroutine for this purpose (see a brief description of this method in Appendix B). Alternatively, a MATLAB function file that is highly accurate and suitable for computation of the interpolating grid-points can be found in [35] (these two codes can also be downloaded from the Matlab Central websites [36] and [37], respectively). Highly accurate C/C++ implementation by Johnson [29] with MEX plugins for MATLAB [38] can also be used for a subroutine that can be invoked from the MATLAB environment to compute these 1587 interpolating grid-points.…”

Section: Algorithmic Implementationmentioning

confidence: 99%

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A Single-Domain Implementation of the Voigt/Complex Error Function by Vectorized Interpolation

Abrarov¹,

Quine²,

Siddiqui³

et al. 2019

ESR

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In this work we show how to perform a rapid computation of the Voigt/complex error over a single domain by vectorized interpolation. This approach enables us to cover the entire set of the parameters x, y ∈ R required for the HITRAN-based spectroscopic applications. The computational test reveals that within domains x ∈ [0, 15] ∩ y ∈ 10 −8 , 15 and x ∈ [0, 50000] ∩ y ≥ 10 −8 our algorithmic implementation is faster in computation by factors of about 8 and 3, respectively, as compared to the fastest known C/C++ code for the Voigt/complex error function. A rapid MATLAB code is presented.

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Section: Algorithmic Implementationmentioning

confidence: 99%

“…This implementation is known to be the fastest C/C++ program. It represents a modified Algorithm 680 [32,33] with inclusion of the Salzer's approximations [35] (see also [39]). As the computation complexity prevails at smaller values of the parameter y, we imply that it is close to zero, say y = 10 −5 .…”

Section: Figures 1a and 1bmentioning

confidence: 99%

A Single-Domain Implementation of the Voigt/Complex Error Function by Vectorized Interpolation

Abrarov¹,

Quine²,

Siddiqui³

et al. 2019

ESR

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“…Similar to our previous work [7] we applied only three approximations bounded inside domains as shown in Fig. 1; in fact, due to symmetric properties of the complex error function…”

Section: Approximations and Boundariesmentioning

confidence: 99%

“…Approximation (7) contains poles that can be readily found by solving the following quadratic equation…”

Section: Location Of Polesmentioning

confidence: 99%

“…Consequently, we can conclude that the error function erf (z) is just a reformulation of the complex error function w (z). Some other functions of practical importance that can be reformulated in terms of the complex error function are the Dawson's integral [3,4,5,6,7] daw (z) = e −z 2 z 0 e t 2 dt = √ π 2i w (z) − e −z 2 , the Fresnel integral [2] F (z) = z 0 e i(π/2)t 2 dt = (1 + i) 1 − e i(π/2)z 2 w √ π (1 + i) z/2 /2 and the plasma dispersion function [8] Z…”

Section: Introductionmentioning

confidence: 99%

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A sampling-based approximation of the complex error function and its implementation without poles

Abrarov

Quine

Jagpal

2018

Applied Numerical Mathematics

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Recently we developed a new sampling methodology based on incomplete cosine expansion of the sinc function and applied it in numerical integration in order to obtain a rational approximation for the complex error function w (z) = e −z 2 1 + 2i √ π z 0 e t 2 dt , where z = x + iy. As a further development, in this work we show how this sampling-based rational approximation can be transformed into alternative form for efficient computation of the complex error function w (z) at smaller values of the imaginary argument y = Im [z]. Such an approach enables us to avoid poles in implementation and to cover the entire complex plain with high accuracy in a rapid algorithm. An optimized Matlab code utilizing only three rapid approximations is presented.

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Approximating fractional derivative of Faddeeva function, Gaussian function, and Dawson's integral

Yarman

2019

Math Methods in App Sciences

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A new method for approximating fractional derivatives of Faddeeva function, Gaussian function, and Dawson's integral are presented. Unlike previous approaches, which are dominantly based on some discretization of Riemann-Liouville integral over a finite interval using polynomial or discrete Fourier basis, we take an alternative approach that is based on expressing computation of Riemann-Liouville definition of the fractional integral for the semi-infinite axis in terms of a moment problem. As a result, fractional derivatives of Faddeeva function, Gaussian function, and Dawson's integral are expressed as a weighted sum of complex scaled Gaussian and Dawson's integral. Error bounds for the approximations are provided. Another feature of the proposed method compared with the previous approaches is it can be extended to approximate partial derivatives with respect to the order of the fractional derivative that can be used in PDE constraint optimization problems.

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A rational approximation of the Dawson’s integral for efficient computation of the complex error function

Cited by 22 publications

References 47 publications

A Single-Domain Implementation of the Voigt/Complex Error Function by Vectorized Interpolation

A Single-Domain Implementation of the Voigt/Complex Error Function by Vectorized Interpolation

A sampling-based approximation of the complex error function and its implementation without poles

Approximating fractional derivative of Faddeeva function, Gaussian function, and Dawson's integral

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