Efficient algorithmic implementation of the Voigt/complex error function based on exponential series approximation

Abstract

a b s t r a c tWe show that a Fourier expansion of the exponential multiplier yields an exponential series that can compute high-accuracy values of the complex error function in a rapid algorithm. Numerical error analysis and computational test reveal that with essentially higher accuracy it is as fast as FFT-based Weideman's algorithm at a regular size of the input array and considerably faster at an extended size of the input array. As this exponential series approximation is based only on elementary functions, the algorithm can be implemented utilizing freely available functions from the standard libraries of most programming languages. Due to its simplicity, rapidness, high-accuracy and coverage of the entire complex plane, the algorithm is efficient and practically convenient in numerical methods related to the spectral line broadening and other applications requiring errorfunction evaluation over extended input arrays.

Spectral curve fitting of dielectric constants

“…The errors are due to the K-K inconsistency of the real and imaginary parts, 18 and to lesser extent, the errors in the evaluation of the Feddeeva function. 38 Therefore, the Voigt line shape may be applicable only to species that exhibit narrow spectral bands.…”

“…5,18,20 The Voigt function cannot be expressed analytically in closed form, however, it can be calculated using various approximations. 38,39 Specifically, the Voigt function, K(x,y), is the real part of Faddeeva function, w(z): 38…”

“…38 The complex optical constants can be described by a slightly modified version of the Faddeeva function 18 that satisfies the causality rule. In this work we evaluate the Faddeeva function using the exponential series approximation as employed by Abrarov.…”

Master-Slave Algorithm for Highly Accurate and Rapid Computation of the Voigt/Complex Error Function

“…Due to rational function representation, the Weideman's approximation (17) is rapid in computation and, consequently, widely applied in practice. However, as it has been shown in a recent publication, in order to sustain accuracy 10 −6 at y ≥ 10 −5 and 0 ≤ x ≤ 15 the integer N determining the number of the summation terms in Weideman's approximation must be increased up to 32 (Abrarov & Quine, 2011). Furthermore, according to Schreier et al (2014) in the range y < 10 −5 and 4 < x < 15 the Weideman's approximation (17) at N = 32 cannot provide accuracy better than 8 × 10 −5 .…”

Stable Distributions as Noise Models for Molecular Communication

“…In this work, we derive closed-form expressions for the PDF of the noise terms in our channels in terms of the complex error function and Voigt functions [22], which are used in other fields of science such as physics. We numerically compare the stable-distributed noise densities and distribution functions to the Gaussian distribution, and show that the stable distribution exhibits longer tails.…”