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Zbl 0545.33001
Amos, D.E.
Algorithm 609 a portable FORTRAN subroutine for the Bickley functions $Ki\sb n(x)$. (English)
[J] ACM Trans. Math. Softw. 9, 480-493 (1983). ISSN 0098-3500

The author discribes a FORTRAN subroutine for the computation of the so- called Bickley functions $$ Ki\sb n(x)=\int\sp{\infty}\sb{x}Ki\sb{n- 1}(t)dt\quad(x\ge 0,n=1,2,...) $$ where $Ki\sb 0(x)=K\sb 0(x)$ is the modified Bessel function of the second kind. The algorithm consists of two main parts: For 0$\le x\le 2$ it uses a four-term recurrence relation with starting values for $n=0,1,2$ obtained from a power series. For $2<x<\infty$ a uniform asymptotic expansion is used for $n\ge 0$. The author also discusses the computation of the exponential integral $E\sb n(x)$, the gamma ratio $R(x)=\Gamma(x)/\Gamma(x+{1\over2}),$ and the digamma function $\psi$ (x), all used as auxiliary functions. The description of the procedures is fundamentally mathematical and a careful error analysis as well as extensive tests are presented. The author emphazises that the main feature of these subroutines is that they are portable for an accuracy up to 18 digits. He states that, by using the special initialization routines provided, the user has a choice of 15 machine environments representing more than 19 machines and operating systems.
[K.S.Kölbig]

MSC 2000:: *33-04 Machine computation, programs (special functions)
33B15 Gamma-functions, etc.
33C10 Cylinder functions, etc.
33E99 Special functions
41A30 Approximation by other special function classes

Keywords: gamma function; psi function; function evaluation; FORTRAN; Bickley functions; modified Bessel function; asymptotic expansion; exponential integral; digamma function

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