Abstract
We describe an algorithm for the numerical evaluation of the generalized exponential integral E ν (x) for positive values of ν and x. A detailed description of the numerical methods used in the algorithm is provided, including error bounds. Different approaches from earlier algorithms are also summarised. The performance and accuracy of the resulting algorithm is analysed and compared with open-source software packages. This analysis shows that our implementation is competitive and more robust than other state-of-the-art codes. Finally, a brief study of the implementation of E ν (x) in arbitrary-precision arithmetic is discussed.
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Altaç, Z.: Integrals involving Bickley and Bessel functions in radiative transfer, and generalized exponential integral functions. J. Heat Transfer 118(3), 789–792 (1996)
Berend, D., Tassa, T.: Improved bounds on bell numbers and on moments of sums of random variables. Probab. Math. Stat. 30(2), 185–205 (2010)
Boost. Boost C++ Libraries. http://www.boost.org/, 2016. Last accessed 2016-12-29
Borwein, D., Borwein, J.M., Crandall, R.E.: Effective laguerre asymptotics. SIAM J. Numer. Anal. 46(6), 3285–3312 (2008)
de Bruijn, N.G.: Asymptotic Methods in Analysis. Bibliotheca mathematica Dover Publications (1970)
Chandrasekhar, S.: Radiative transfer. Dover books on intermediate and advanced mathematics. Dover Publications (1960)
Chiccoli, C., Lorenzutta, S., Maino, G.: A numerical method for generalized exponential integrals. Comput. Math. Appl. 14(4), 261–268 (1987)
Chiccoli, C., Lorenzutta, S., Maino, G.: An algorithm for exponential integrals of real order. Computing 45(3), 269–276 (1990)
Chiccoli, C., Lorenzutta, S., Maino, G.: Recent results for generalized exponential integrals. Comput. Math. Appl. 19(5), 21–29 (1990)
Chiccoli, C., Lorenzutta, S., Maino, G.: Concerning some integrals of the generalized exponential-integral function. Comput. Math. Appl. 23(11), 13–21 (1992)
Cody, W.J., Thacher, H.C. Jr.: Chebyshev approximations for the exponential integral E i(x). Math. Comp. 23(106), 289–303 (1969)
Cuyt, A., Petersen, V., Verdonk, B., Waadeland, H., Jones, W.B., Bonan-Hamada, C.: Handbook of continued fractions for special functions. Kluwer Academic Publishers Group, Dordrecht (2007)
Dekker, T.J.: A floating-point technique for extending the available precision. Numer. Math. 18(3), 224–242 (1971)
NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/, Release 1.0.14 of 2016-12-21. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller and B. V. Saunders (eds.)
Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91(2), 201–213 (2002)
Galassi, et al.: Gnu scientific library: reference manual, 3rd edn. Network Theory Ltd. (2003)
Gautschi, W.: Exponential integral \({\int }_{1}^{\infty }e^{-xt} t^{-n}dt\) for large values of n. J. Res. Nat. Bur. Standards 62, 123–125 (1959)
Gil, A., Ruiz-antolín, D., Segura, J., Temme, N.M.: Algorithm 969: computation of the incomplete gamma function for negative values of the argument. ACM Trans. Math. Softw. 43(3), 26,1–26,9 (2016)
Gil, A., Segura, J., Temme, N.M.: Numerical methods for special functions. SIAM (2007)
Gil, A., Segura, J., Temme, N.M.: Efficient and accurate algorithms for the computation and inversion of the incomplete gamma function ratios. SIAM J. Sci. Comput. 34(6), (2012)
Henrici, P.: Applied and computational complex analysis. Vol. 2: special functions integral transforms asymptotics continued fractions. Wiley-Interscience [John Wiley & Sons], New York (1977). Reprinted in 1991
Hida, Y., Li, X.S., Bailey, D.H.: Algorithms for quad-double precision floating point arithmetic. In: Proceedings 15th IEEE Symposium on Computer Arithmetic. ARITH-15 2001, pp. 155–162 (2001)
Johansson, F.: Arb: A c library for ball arithmetic. ACM Commun. Comput. Algebra 47(3/4), 166–169 (2014)
Johansson, F.: Computing hypergeometric functions rigorously. Working paper or preprint (2016)
Johansson , F., et al.: mpmath: a Python library for arbitrary-precision floating-point arithmetic (version 0.19). http://mpmath.org/ (2014)
Kečlić, J.D., Vasić, P.M.: Some inequalities for the gamma function. Publ. Inst. Math. (Beograd) (N.S.) 11, 107–114 (1971)
Knuth, D.E.: The art of computer programming, volume 2 (3rd edn.): seminumerical algorithms. Addison-Wesley Longman Publishing Co., Inc., Boston (1997)
LeCaine, J.: A table of integrals involving the functions En(x). N.R.C. (National Research Council). National Research Council Canada, Division of Atomic Energy (1949)
Moshier, S.L.: Cephes mathematical function library (2000)
Olde Daalhuis, A.B.: Hyperasymptotic expansions of confluent hypergeometric functions. IMA J. Appl. Math. 49, 203–216 (1992)
Ooura, T., Mori, M.: The double exponential formula for oscillatory functions over the half infinite interval. J. Comput. Appl. Math. 38(1), 353–360 (1991)
Watson, G.N.: The harmonic functions associated with the parabolic cylinder. Proc. London Math. Soc. s2-17(1), 116–148 (1918)
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The author acknowledges Javier Segura for reading the preliminary version of the manuscript and providing useful remarks. The author also thanks the anonymous reviewer for indicating errors and suggesting improvements.
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Navas-Palencia, G. Fast and accurate algorithm for the generalized exponential integral E ν (x) for positive real order. Numer Algor 77, 603–630 (2018). https://doi.org/10.1007/s11075-017-0331-z
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DOI: https://doi.org/10.1007/s11075-017-0331-z