Remark on “Algorithm 916: Computing the Faddeyeva and Voigt Functions”

Abstract

This remark describes efficiency improvements to Algorithm 916 [Zaghloul and Ali 2011]. It is shown that the execution time required by the algorithm, when run at its highest accuracy, may be improved by more than a factor of two. A better accuracy vs efficiency trade off scheme is also implemented; this requires the user to supply the number of significant figures desired in the computed values as an extra input argument to the function. Using this trade-off, it is shown that the efficiency of the algorithm may be further improved significantly while maintaining reasonably accurate and safe results that are free of the pitfalls and complete loss of accuracy seen in other competitive techniques.The current version of the code is provided in Matlab and Scilab in addition to a Fortran translation prepared to meet the needs of real-world problems where very large numbers of function evaluations would require the use of a compiled language. To fulfill this last requirement, a recently proposed reformed version of Humlíček's w4 routine, shown to maintain the claimed accuracy of the algorithm over a wide and fine grid is implemented in the present Fortran translation for the case of 4 significant figures. This latter modification assures the reliability of the code to be employed in the solution of practical problems requiring numerous evaluation of the function for applications tolerating low accuracy computations (<10 -4 ).

Stable Distributions as Noise Models for Molecular Communication

“…These functions can be numerically calculated using efficient algorithms such as [28]. Moreover, for the case of general stable distributions with any parameters µ, c, α, and β it is possible to calculate the PDFs and the cumulative distribution functions (CDF)s numerically using the fast Fourier transform or by numerically solving definite integrals [29].…”

“…3 we plot the tail probability P (X > x) and the approximate tail probabilities from (27) and (28). In the plot the circles indicate the approximate values.…”

A Rational Approximation for Efficient Computation of the Voigt Function in Quantitative Spectroscopy

“…(x, y) is the reference. The highly accurate reference values can be generated, for example, by using the Algorithm 680 [18,30] or recently published Algorithm 916 [31]. Figures 2a and 2b show the logarithm log 10 ∆ of the relative error of the series approximation (12) at m max = 16.…”

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

“…To generate reference values we can use, for example, the Matlab code in recently published Algorithm 916 (Zaghloul & Ali, 2011). Alternatively, these values can be generated by latest versions of Mathematica computational software that supports error function of complex argument.…”

“…At |x + iy| > ∼ 1000, the high-accuracy can be obtained by using very simple rational approximations (Zaghloul & Ali, 2011):…”