A Note on a Method of Computing the Gamma Function

ABSTRACT

Numerous formulas are available for the computation of the Gamma function [1, 2]. The purpose of this note is to indicate the value of a well-known method that is easily extended for higher accuracy requirements. Using the recursion formula for the Gamma function, &Ggr;(x + 1) = x&Ggr;(x), (1) and Stirling's asymptotic expansion for ln &Ggr;(x) [3], we have ln &Ggr;(x) ∼ (x - 1/2) ln x - x + 1/2 ln 2&pgr; + ∑Nr=1 Cr/x2r-1. (2) It follows that, if k and N are appropriately selected positive integers, &Ggr;(x + 1) can be represented by &Ggr;(x + 1) ∼ √2&pgr; exp (x + k - 1/2) ln (x + k) - (x + k) exp ∑Nr=1 Cr/(x + k)2r-1/(x + 1)(x + 2) ··· (x + k - 1) (3) where Cr = (- 1)r-1 Br/(2r - 1)(2r), Br being the Bernoulli numbers [4]. These coefficients have been published by Uhler [5]. Requiring the range 0 ≦ x ≦ 1 is no restriction since, if necessary, &Ggr;(x + 1) can be generated for other arguments using (1). For a given N, the error in (2) can be estimated from |&egr;| < |CN+1|/x2N+1. (4) The curves of Figure 1 show contours of constant error bound as a function of N and x. These curves represent single and double-precision floating-arithmetic requirements of &egr; < 5·10-9 and &egr; < 5·10-17. For a given N, k is defined as the minimum integral x greater than or equal to those on the curves. Then N and k can be chosen to minimize round-off and computing time. For N and k equal to 4, formula (3) yields &Ggr;(x + 1) ∼ &radic2&pgr; exp (x + 4 - 1/2) ln (x + 4) - (x + 4) exp ∑4r=1Cr/(x + 4)2r-1/(x + 1)(x + 2)(x + 3). (5) A similar expression suitable for double precision results for N = 8 and k = 9. The exponents in (5) are split to reduce roundoff. Various algebraic manipulations might result in a further reduction of roundoff.

REFERENCES

• 1: Cecil Hastings, Approximations for Digital Computers, Princeton University Press, Princeton, NJ, 1955
• 2: C. LANCZOS, Trigonometric interpolation of empirical and analytic functions J Math Phys 17 (1938), 123-199
• 3: M. E SHERRY AND S FULDA, Calculation of Gamma functions to high accuracy, Math Tables Azds Compz~t 13 (1959), 314-315
• 4: F T WHITTAKER AND G N. WATSON, A Course. of Modern Analysis, p 125 Cambridge University Press, Cambridge, Englund.
• 5: HORACE S UHLER, The coefficients of Stirling's serms for log F(x), Proc Nat. Acad Scz. 28 (1942), 59-62

INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.2 DISCRETE MATHEMATICS

Additional Classification:
  F. Theory of Computation
  F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY
      F.2.2 Nonnumerical Algorithms and Problems
          Subjects: Computations on discrete structures

General Terms:
Measurement, Performance, Reliability, Theory