Complex gamma function with error control

ABSTRACT

An algorithm to compute the gamma function and the loggamma function of a complex variable is presented. The standard algorithm is modified in several respects to insure the continuity of the function value and to reduce accumulation of round-off errors. In addition to computation of function values, this algorithm includes an object-time estimation of round-off errors. Experimental data with regard to the effectiveness of this error control are presented. A Fortran program for the algorithm appears in the algorithms section of this issue.

REFERENCES

• 1: Abramowitz, M., and Stegun, I.A. (Eds.) Handbook of Mathematical Functions. National Bureau of Standards, A.M.S. 55, Washington, D.C., 1964, pp. 255-259.
• 2: Kuki, H. Mathematical function subprograms for basic system libraries: objectives, constraints, and trade-off. Proc. Syrup. on Mathematical Software, Lafayette, Ind., Apr. 1970, Academic Press, New York, 1971.
• 3: Fan, T., and Kuki, H. Review of SDA 3230, SHARE Secretary Distribution 164, Pt. II, U-67, Apr. 1967.
• 4: John F. Hart, Computer Approximations, Krieger Publishing Co., Inc., Melbourne, FL, 1978
• 5: Luke, Y.L. Evaluation of the Gamma function by means of Pad6 approximations. SIAM J. Math. Anal. 1, 2 (May 1970), 266-281.
• 6: Hirondo Kuki, Algorithm 421: complex gamma function with error control [S14], Communications of the ACM, v.15 n.4, p.271-272, April 1972 [doi>10.1145/361284.361298]

INDEX TERMS

Keywords:
complex gamma function, complex loggamma function, error estimates, gamma function, inherent error control, loggamma function, round-off error control, run-time error estimates