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Numerical Evaluation of Special Functions

D. W. Lozier and F. W. J. Olver

4.4. Gamma, Psi, and Polygamma Functions .

4.4.1. Gamma Function of Real Argument.


[ CH67] , [ Cle62] , [ Luk69b] , [ Luk75] .

Software Packages:

[ CMW63, Algol] , [ CT85, Fortran] , [ FS67, Algol] , [ Mac89, Fortran] , [ Tem94, Pascal] .

Intermediate Libraries:

[ Bak92] , [ Mos89] , [ ULI90] .

Comprehensive Libraries:

IMSL, NAG, Numerical Recipes, Scientific Desk, SLATEC.

4.4.2. Psi and Polygamma Functions of Real Argument.


[ CST73] , [ Luk69b] , [ Luk75] .

Software Packages:

[ Amo83b, Fortran] , [ Bow84, Fortran] .

Intermediate Libraries:

[ Bak92] , [ Mos89] , [ ULI90] .

Comprehensive Libraries:

IMSL, NAG, Scientific Desk, SLATEC.

4.4.3. Complex Arguments.


[ Luk69b] .

Software Packages:

[ BD80, Fortran] , [ Kol72a, Fortran] , [ Kuk72a, Fortran] .

Intermediate Libraries:

[ Bak92] .

Comprehensive Libraries:

IMSL, Scientific Desk, SLATEC.

Interactive Systems:

Maple, Mathematica.

4.4.4. Articles.

[ AB87b] , [ Cha80] , [ Cod91, includes survey] , [ FW80] , [ Kat78] , [ Kra90] , [ Kuk72b] , [ Luk70a] , [ McC81] , [ Ng75, includes survey] , [ vdLT84] .


G. Allasia and R. Besenghi, Numerical computation of Tricomi's psi function by the trapezoidal rule, Computing 39 (1987), 271--279.

D. E. Amos, Algorithm 610. A portable Fortran subroutine for derivatives of the psi function, ACM Trans. Math. Software 9 (1983), 494--502.

L. Baker, C mathematical function handbook, McGraw-Hill, Inc., New York, 1992, includes diskette.

A. Bañuelos and R. A. Depine, A program for computing the Riemann zeta function for complex argument, Comput. Phys. Comm. 20 (1980), 441--445.

K. O. Bowman, Computation of the polygamma functions, Comm. Statist. B---Simulation Comput. 13 (1984), 409--415.

W. J. Cody and K. E. Hillstrom, Chebyshev approximations for the natural logarithm of the gamma function, Math. Comp. 21 (1967), 198--203.

B. W. Char, On Stieltjes' continued fraction for the gamma function, Math. Comp. 34 (1980), 547--551.

C. W. Clenshaw, Chebyshev series for mathematical functions, National Physical Laboratory Mathematical Tables, vol. 5, Her Majesty's Stationery Office, London, 1962.

C. W. Clenshaw, G. F. Miller, and M. Woodger, Algorithms for special functions I, Numer. Math. 4 (1963), 403--419.

W. J. Cody, Performance evaluation of programs related to the real gamma function, ACM Trans. Math. Software 17 (1991), 46--54.

W. J. Cody, A. J. Strecok, and H. C. Thacher, Jr., Chebyshev approximations for the psi function, Math. Comp. 27 (1973), 123--127.

M. Carmignani and A. Tortorici Macaluso, Calcolo delle funzioni speciali , , , , alle alte precisioni, Atti Accad. Sci. Lett. Arti Palermo (5) 2 (1981--82) (1985), no. 1, 7--25.

A. M. S. Filho and G. Schwachheim, Algorithm 309. Gamma function with arbitrary precision, Comm. ACM 10 (1967), 511--512.

A. Fransén and S. Wrigge, High-precision values of the gamma function and of some related coefficients, Math. Comp. 34 (1980), 553--566, for addendum and corrigendum see same journal v. 37 (1981), pp. 233--235.

C. R. Katholi, On the computation of values of the psi function from rapidly converging power series expansions, J. Statist. Comput. Simulation 8 (1978), 25--42.

K. S. Kölbig, Programs for computing the logarithm of the gamma function, and the digamma function, for complex argument, Comput. Phys. Comm. 4 (1972), 221--226.

W. Krämer, Berechnung der Gammafunktion für reelle Punkt- und Intervallargumente, Z. Angew. Math. Mech. 70 (1990), T581--T584.

H. Kuki, Algorithm 421. Complex gamma function with error control, Comm. ACM 15 (1972), 271--272.

H. Kuki, Complex gamma function with error control, Comm. ACM 15 (1972), 262--267.

Y. L. Luke, The special functions and their approximations, vol. 2, Academic Press, New York, 1969.

Y. L. Luke, Evaluation of the gamma function by means of Padé approximations, SIAM J. Math. Anal. 1 (1970), 266--281.

Y. L. Luke, Mathematical functions and their approximations, Academic Press, New York, 1975.

A. J. MacLeod, Algorithm AS 245. A robust and reliable algorithm for the logarithm of the gamma function, Appl. Statist. 38 (1989), 397--402.

P. McCullagh, A rapidly convergent series for computing and its derivatives, Math. Comp. 36 (1981), 247--248.

S. L. B. Moshier, Methods and programs for mathematical functions, Ellis Horwood Limited, Chichester, 1989, separate diskette.

E. W. Ng, A comparison of computational methods and algorithms for the complex gamma function, ACM Trans. Math. Software 1 (1975), 56--70.

N. M. Temme, A set of algorithms for the incomplete gamma functions, Probab. Engrg. Inform. Sci. (1994), in press.

Mathematical function library for Microsoft--C, United Laboratories, Inc., John Wiley & Sons, 1990, includes diskettes. Edition also exists in Fortran (1989).

C. G. van der Laan and N. M. Temme, Calculation of special functions : The gamma function, the exponential integrals and error-like functions, CWI Tract, vol. 10, Centrum voor Wiskunde en Informatica, Amsterdam, 1984.


This document is an excerpt from the current hypertext version of an article that appeared in Walter Gautschi (ed.), Mathematics of Computation 1943--1993: A Half-Century of Computational Mathematics, Proceedings of Symposia in Applied Mathematics 48, American Mathematical Society, Providence, RI 02940, 1994. The symposium was held at the University of British Columbia August 9--13, 1993, in honor of the fiftieth anniversary of the journal Mathematics of Computation.

The original abstract follows.

Higher transcendental functions continue to play varied and important roles in investigations by engineers, mathematicians, scientists and statisticians. The purpose of this paper is to assist in locating useful approximations and software for the numerical generation of these functions, and to offer some suggestions for future developments in this field.

Applied and Computational Mathematics Division, National Institute of Standards and Technology, Gaithersburg, Md 20899

E-mail address:

Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742

E-mail address:

The research of the second author has been supported by NSF Grant CCR 89-14933.

1991 Mathematics Subject Classification. Primary 65D20; Secondary 33-00.

Daniel W Lozier
Fri Apr 7 13:52:51 EDT 1995