Gil, Amparo; Segura, Javier; Temme, Nico M. Basic methods for computing special functions. (English) Zbl 1216.65034 Simos, Theodore E. (ed.), Recent advances in computational and applied mathematics. Dordrecht: Springer (ISBN 978-90-481-9980-8/hbk; 978-90-481-9981-5/ebook). 67-121 (2011). Summary: This paper gives an overview of methods for the numerical evaluation of special functions, that is, functions that arise in many problems from mathematical physics, engineering, probability theory, and other applied sciences. We consider in detail a selection of basic methods which are frequently used in the numerical evaluation of special functions: converging and asymptotic series, including Chebyshev expansions, linear recurrence relations, and numerical quadrature. Several other methods are available and some of these will be discussed in less detail. We give examples of recent software for special functions where these methods are used. We mention a list of new publications on computational aspects of special functions available on our website.For the entire collection see [Zbl 1201.65004]. Cited in 2 Documents MSC: 65D20 Computation of special functions and constants, construction of tables 33C05 Classical hypergeometric functions, \({}_2F_1\) 33C10 Bessel and Airy functions, cylinder functions, \({}_0F_1\) 33C15 Confluent hypergeometric functions, Whittaker functions, \({}_1F_1\) 33F05 Numerical approximation and evaluation of special functions 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.) Keywords:numerical evaluation of special functions; Chebyshev expansions; quadrature methods; transformation of series; continued fractions; asymptotic analysis; numerical examples; linear recurrence relations Software:AIZ; Algorithm 804; FGH; Algorithm 814; BIZ; F1; Algorithm 858; Algorithm 850; NUMER; Algorithm 644; Algorithm 682; DTORH3; NSWC; Algorithm 885; Algorithm 877; Algorithm 838; Algorithm 831; REMES2; Algorithm 822; Algorithm 855; Algorithm 861; NumSBT; Hypexp PDFBibTeX XMLCite \textit{A. Gil} et al., in: Recent advances in computational and applied mathematics. Dordrecht: Springer. 67--121 (2011; Zbl 1216.65034) Full Text: DOI Link