Optimized implementations of rational approximations—a case study on the Voigt and complex error function

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Abstract

Rational functions are frequently used as efficient yet accurate numerical approximations for real and complex valued special functions. For the complex error function w(x+iy), whose real part is the Voigt function K(x,y), the rational approximation developed by Hui, Armstrong, and Wray [Rapid computation of the Voigt and complex error functions, J. Quant. Spectrosc. Radiat. Transfer 19 (1978) 509–516] is investigated. Various optimizations for the algorithm are discussed. In many applications, where these functions have to be calculated for a large x grid with constant y, an implementation using real arithmetic and factorization of invariant terms is especially efficient.

Introduction

The convolution of a Lorentz and a Gauss function, commonly known as the Voigt function, is important in many branches of physics, e.g., atomic and molecular spectroscopy, atmospheric radiative transfer, plasma physics, astrophysics, etc. [1]. The Voigt function is identical to the real part of the complex error function (or complex probability function), and is also closely related to Dawson's function [2]. Unfortunately, none of these functions can be evaluated in closed analytical form and a large number of numerical algorithms have been developed in the past. Most modern algorithms for the Voigt function employ approximations for the complex error function. Actually this approach has further advantages, in particular it simultaneously provides derivatives of these functions required for, e.g., sensitivity analysis or optimization.

Rational approximations [3] have been proven to be an efficient approach for a wide variety of functions, and have been used successfully also for evaluation of the complex error function, e.g., Hui et al. [4], Humlicek [5], [6], Weideman [7]. Whereas the Humlicek algorithms are based on different approximations in different regions of the arguments, an attractive and unique feature of the Hui–Armstrong–Wray and Weideman algorithms is their applicability in the entire domain.

In this paper we present optimized implementations of the Hui, Armstrong, and Wray [4] algorithm. Definition and fundamental properties of the Voigt and complex error function are given in Section 2. Various implementations of the rational approximation are developed in Section 3 and performance tests are described in Section 4. A summary and conclusions are given in Section 5. Listing of constants are given in Appendix A.

Section snippets

Basic definitions and fundamental properties

It is convenient to define the Voigt function K(x,y) normalized to π,K(x,y)=yπet2(xt)2+y2dt, where the dimensionless variables x, y are a measure for the distance from the peak center and for the ratio of Lorentz to Gauss width, respectively. The Voigt function represents the real part of the complex functionW(z)K(x,y)+iL(x,y)=iπet2ztdtwith z=x+iy, that, for y>0, is identical to the complex error function (probability function) defined by [2]w(z)=ez2(1+2iπ0zet2dt)=ez2(1erf(iz))

Original implementation: complex polynomials

Approximation of an arbitrary function by a rational function, i.e., the quotient PM/QN of two polynomials of degree M and N is generally superior to polynomial approximations [3]. Because of the asymptotic behavior (6) of the complex error function, the degree of the nominator and denominator polynomials are constrained by N=M+1. For M=6 Hui et al. [4] have developed a complex rational approximation with a relative accuracy of 6 digits in the entire x, y plane,w(z)=P(z˜)Q(z˜)=m=0Mamz˜mn=0M+1b

Line-by-line modeling of cross sections

As an example we will discuss the evaluation of molecular absorption cross sections for high resolution “line-by-line” atmospheric radiative transfer modeling in the microwave and infrared spectral range. Accurate cross sections are required for the simulation and analysis of atmospheric remote sensing data. Especially for the operational data processing of spaceborne atmospheric sounders highly optimized algorithms are mandatory. In case of temperature sounding instruments derivatives of the

Conclusions

The potential of optimization of rational function approximations has been investigated using the Hui, Armstrong, and Wray [4] algorithm for the complex error function w(x+iy). Except for the continued fraction implementation most optimizations exploit the fact that in many applications w has to be computed for a series of y values and a moderate to large array of grid points x. Utilizing factorizations of terms independent of the grid point x and real number arithmetic results in a significant

Acknowledgement

Financial support by the World Data Center for Remote Sensing of the Atmosphere (WDC-RSAT) hosted by DLR is greatly appreciated.

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