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This paper presents uniform error bounds for fast calculation of approximate Voigt profiles that can be useful with the computationally demanding broadband line-by-line analysis of radiative transfer in the atmosphere. Formal proofs are given and rigorous criteria are provided to determine the domain on which the Voigt profile can be approximated by the Lorentz profile within any required accuracy. The most accurate Voigt-implementation to date can then be used to determine the required threshold parameters. Since most of the broadband radiative transfer calculations in the atmosphere will pertain to far wing computations, the potential saving in time is almost the same as by replacing the Voigt computation for the Lorentzian altogether completely. The error bounds can furthermore be used to derive a simple and efficient subband adaptive line selection strategy which can be used to rigorously exclude lines that will contribute to the resulting absorption coefficient less than any given threshold.

Considerable research efforts have been directed at implementing the Faddeeva function w ( z ) and its derivatives with respect to z , but these did not consider the key computing issue of a possible dependence of z on some variable t . The general case is to differentiate the compound function w ( z ( t ) ) = w ∘ z ( t ) with respect to t by applying the chain rule for a first order derivative, or Faa di Bruno’s formula for higher-order ones. Higher-order automatic differentiation (HOAD) is an efficient and accurate technique for derivative calculation along scientific computing codes. Although codes are available for w ( z ) , a special symbolic HOAD is required to compute accurate higher-order derivatives for w ∘ z ( t ) in an efficient manner. A thorough evaluation is carried out considering a nontrivial case study in optics to support this assertion. Program summary Program title: HOAD_MathFun Catalogue identifier: AFAG_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AFAG_v1_0.html Program obtainable from: CPC Program Library, Queen’s University, Belfast, N. Ireland Licensing provisions: yes No. of lines in distributed program, including test data, etc.: 4737 No. of bytes in distributed program, including test data, etc.: 103450 Distribution format: tar.gz Programming language: Fortran 90. Computer: Non-specific. Operating system: Non-specific. RAM: 1 megabyte Classification: 4.12. External routines: Algorithm 680 [8], Mathematical functions (included) Nature of problem: General optimized higher-order automatic differentiation of mathematical functions. Complex refractive index as a case study. Solution method: Higher-order differentiation for the general second order ordinary equation defining the mathematical functions. Optimized operator overloading recurrence formula. Unusual features: Automatic differentiation, Quadratic complexity. Running time: 1 second.

We 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.

A simple interpolating algorithm for rapid calculation of the Voigt function with high accuracy is presented. It can utilize any existing rapid algorithm as a subprogram for the pre-computation of the selected points. Such approach is particularly efficient for massive input arrays and, compared to the Wells algorithm, accelerates the computation, improves the accuracy, and increases the input vector size by factors greater than 35, 7, and 10, respectively. The calculated results were compared with exponential and arctangent expansion approximations of the Voigt function. It is found that the relative error in the Wells algorithm slightly exceeds the supposed accuracy of 10 −5 .

The numerical solution of Eq. (1) relating the Voigt, Lorentzian, and Gaussian full-widths at half-maximum, as shown in Fig. 2 of the article A new procedure for obtaining the Voigt function dependent upon the complex error function by Luque, Calzada and Saez (JQSRT 2005; 94:151–161), is not valid in some parts of the computational domain. The problem resides within a physical domain that is of current spectroscopic interest for the diagnostics of low temperature (∼300 K), high pressure (∼1 atm) dielectric barrier discharge plasmas. The purpose here is to identify the deficiency and to demonstrate a well-conditioned numerical approach for this domain of interest.

Existing algorithms for the calculation of the Voigt/Faddeeva function are adapted to produce a method which is more efficient for atmospheric line-by-line calculations. The approach allows greater computational speed to be obtained if the maximum relative error criteria can be relaxed. A supplementary algorithm to calculate the partial derivatives of the Voigt function simultaneously with the Faddeeva function is also specified.