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Precise and fast computation of inverse Fermi-Dirac integral of order 1/2 by minimax rational function approximation. (English) Zbl 06881760
Summary: The single and double precision procedures are developed for the inverse Fermi-Dirac integral of order 1/2 by the minimax rational function approximation. The maximum error of the new approximations is one and 7 machine epsilons in the single and double precision computations, respectively. Meanwhile, the CPU time of the new approximations is so small as to be comparable to that of elementary functions. As a result, the new double precision approximation achieves the 15 digit accuracy and runs 30–84% faster than H. M. Antia’s 28 bit precision approximation [“Rational function approximations for Fermi-Dirac integrals”, Astrophys. J. Suppl. Ser. 84, 101–108 (1993; doi:10.1086/191748)]. Also, the new single precision approximation is of the 24 bit accuracy and runs 10-86% faster than Antia’s 15 bit precision approximation.

##### MSC:
 65D30 Numerical integration 68W25 Approximation algorithms 82D20 Statistical mechanical studies of solids 68W30 Symbolic computation and algebraic computation 65D20 Computation of special functions and constants, construction of tables
##### Software:
DLMF; Fermi-Dirac; Mathematica
Full Text:
##### References:
 [1] McDougall, J.; Stoner, E. C., The computation of Fermi-Dirac functions, Phil. Trans. Royal Soc. London, Ser. A., Math. Phys. Sci., 237, 67-104, (1938) · JFM 64.1500.04 [2] Ashcroft, N. W.; Mermin, N. D., Solid state physics, (1976), Holt, Rinehalt, and Winston Dumfries · Zbl 1118.82001 [3] Dingle, R., The Fermi-Dirac integrals $$\mathcal{F}_p(\eta) = (p!)^{- 1} \int_0^\infty \varepsilon^p / \left(e^{\varepsilon - \eta} + 1\right) d \varepsilon$$, Appl. Sci. Res., 6, 225-239, (1957) · Zbl 0077.23704 [4] (Olver, F. W.J.; Lozier, D. W.; Boisvert, R. F.; Clark, C. W., NIST Handbook of Mathematical Functions, (2010), Cambridge Univ. Press Cambridge), · Zbl 1198.00002 [5] Blakemore, J. S., Approximations for Fermi-Dirac integrals, especially the function $$\mathcal{F}_{1 / 2}(\eta)$$ used to describe electron density in a semiconductor, Solid-State Electron., 25, 1067-1076, (1982) [6] Sommerfeld, A., Zur elektronentheorie der metalle auf grund der fermischen statistik. I. teil: allgemeines, strömungs und austrittsvorgänge, Zeitschrift für Physik, 47, 1-32, (1929) · JFM 54.0987.06 [7] Macleod, A. J., Algorithm 779: Fermi-Dirac functions of order −1/2, 1/2, 3/2, and 5/2, ACM Trans. Math. Software, 24, 1-12, (1998) · Zbl 0917.65017 [8] Cody, W. J.; Thatcher, H. C., Rational Chebyshev approximations for Fermi-Dirac integrals of orders −1/2, 1/2 and 3/2, Math. Comp., 21, 30-40, (1967) · Zbl 0149.11905 [9] Antia, H. M., Rational function approximations for Fermi-Dirac integrals, Astrophys. J. Suppl. Ser., 84, 101-108, (1993) [10] Fukushima, T., Precise and fast computation of Fermi-Dirac integral of integer and half integer order by piecewise minimax rational approximation, Appl. Math. Comp., submitted of publication. · Zbl 06881761 [11] Ehrenberg, W., The electric conductivity of simple semiconductors, Proc. Phys. Soc. London, A63, 75-76, (1950) · Zbl 0034.28802 [12] Nilsson, N. G., An accurate approximation of the generalized Einstein relation for degenerate semiconductors, Phys. Stat. Solidi, 19, K75-K78, (1973) [13] Joyce, W. B.; Dixon, R. W., Analytic approximations for the Fermi energy of an ideal Fermi gas, Appl. Phys. Lett., 31, 354-356, (1977) [14] Joyce, W. B., Analytic approximations for the Fermi energy in (al, ga)as, Appl. Phys. Lett., 32, 680-681, (1978) [15] Nilsson, N. G., Empirical approximations for the Fermi energy in a semiconductor with parabolic bands, Appl. Phys. Lett., 33, 653-654, (1978) [16] Bednarczyk, D.; Bednarczyk, J., The approximation of the Fermi-Dirac integral $$\mathcal{F}_{1 / 2}(\eta)$$, Phys. Lett., 64, 409-410, (1978) [17] Chang, T. Y.; Izabelle, A., Full range analytic approximations for Fermi energy and Fermi-Dirac integral $$F_{- 1 / 2}$$ in terms of $$F_{1 / 2}$$, J. Appl. Phys., 65, 2162-2164, (1989) [18] Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P., Numerical recipes: the art of scientific computing, (2007), Cambridge Univ. Press Cambridge · Zbl 1132.65001 [19] T. Ooura, Numerical Integration (Quadrature) - DE Formula (Almighty Quadrature), 2006. [20] Takahashi, H.; Mori, H., Double exponential formulas for numerical integration, Publ. RIMS, Kyoto Univ., 9, 721-741, (1974) · Zbl 0293.65011 [21] Wolfram, S., The Mathematica book, (2003), Wolfram Research Inc./Cambridge Univ. Press Cambridge [22] Wolfram Research, Function Approximations Package Tutorial, Wolfram Research Inc., 2014. [23] Fukushima, T., Analytical computation of generalized Fermi-Dirac integrals by truncated Sommerfeld expansions, Appl. Math. Comm., 234, 417-433, (2014) · Zbl 1305.82058
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