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Efficient multiple-precision computation of the scaled complementary error function and the Dawson integral

NUMERICAL ALGORITHMS 2024年第3期95卷 1291-1308页

作者： Zaghloul, Mofreh R. United Arab Emirates Univ Coll Sci Dept Phys Al Ain 15551 U Arab Emirates

We present algorithms to approximate the scaled complementary error function, ??????(x(2))erfc(x) , and the Dawson integral, e-x(2x)?(0)e(t2)dt , to the best accuracy in the standard single, double, and quadruple precision arithmetic. The algorithms are based on expansion in Chebyshev subinterval polynomial approximations together with expansion in terms of Taylor series and/or Laplace continued fraction. The present algorithms, implemented as Fortran elemental modules, have been bench-marked versus competitive algorithms available in the literature and versus functions built-in in modern Fortran compilers, in addition to comprehensive tables generated with variable precision computations using the Matlab (TM) symbolic tool-box. The present algorithm for calculating the scaled complementary error function showed an overall significant efficiency improvement (factors between 1.3 and 20 depending on the compiler and tested dataset) compared to the built-in function "Erfc_scaled" in modern Fortran compilers, whereas the algorithm for calculating the Dawson integral is exceptional in calculating the function to 32 significant digits (compared to 19 significant digits reported in the literature) while being more effi-cient than competitive algorithms as well.

关键词： Special functions scaled complementary error function Dawson integral Fortran

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Closed-form approximations to the error and complementary error functions and their applications in atmospheric science

引用

ATMOSPHERIC SCIENCE LETTERS 2007年第3期8卷 70-73页

作者： Ren, C. MacKenzie, A. R. Univ Lancaster Div Environm Sci Lancaster LA1 4YQ England

The error function, as well as related functions, occurs in theoretical aspects of many parts of atmospheric science. This note presents a closed-form approximation for the error, complementary error, and scaled complementary error functions, with maximum relative errors within 0.8%. Unlike other approximate solutions, this single equation gives answers within the stated accuracy for real variable x is an element of [0 infinity). The approximation is very useful in solving atmospheric science problems by providing analytical solutions. Examples of the utility of the approximation are: the computation of cirrus cloud physics inside a general circulation model, the cumulative distribution functions of normal and log-normal distributions, and the recurrence period for risk assessment. Copyright (C) 2007 Royal Meteorological Society

关键词： error function complementary error function scaled complementary error function normal distribution log-normal distribution cumulative distribution function recurrence interval

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