Numerical Evaluation of Special Functions

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

4.3. Exponential Integrals, Logarithmic Integral, Sine and Cosine Integrals .

4.3.1. Exponential Integrals of Real Argument.

Algorithms:

[ Cle62] , [ CT69] , [ Luk69b] , [ Luk76] .

Software Packages:

[ Amo80a, Fortran] , [ CMW63, Algol] , [ Gau73, Algol] , [ Pac70, Fortran] , [ SZ76, Fortran] .

Intermediate Libraries:

[ Bak92] , [ ULI90] .

Comprehensive Libraries:

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

4.3.2. Logarithmic Integral of Real Argument.

Intermediate Libraries:

[ Bak92] , [ ULI90] .

Comprehensive Libraries:

IMSL, Scientific Desk, SLATEC.

4.3.3. Sine and Cosine Integrals and Hyperbolic Sine and Cosine Integrals of Real Argument.

Algorithms:

[ Luk69b] .

Software Packages:

[ Bul67, Algol] .

Intermediate Libraries:

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

Comprehensive Libraries:

IMSL, NAG, Numerical Recipes, Scientific Desk.

4.3.4. Complex Arguments.

Algorithms:

[ Luk69b] .

Software Packages:

[ Amo90a, Fortran] .

Intermediate Libraries:

[ Bak92] .

Interactive Systems:

Maple, Mathematica.

4.3.5. Articles.

[ Amo80b] , [ Amo90b] , [ CT68] , [ TM68] , [ vdLT84] .

References

Amo80a: D. E. Amos, Algorithm 556. Exponential integrals, ACM Trans. Math. Software 6 (1980), 420--428, for remark see same journal v. 9 (1983), p. 525.
Amo80b: D. E. Amos, Computation of exponential integrals, ACM Trans. Math. Software 6 (1980), 365--377.
Amo90a: D. E. Amos, Algorithm 683. A portable Fortran subroutine for exponential integrals of a complex argument, ACM Trans. Math. Software 16 (1990), 178--182.
Amo90b: D. E. Amos, Computation of exponential integrals of a complex argument, ACM Trans. Math. Software 16 (1990), 169--177.
Bak92: L. Baker, C mathematical function handbook, McGraw-Hill, Inc., New York, 1992, includes diskette.
Bul67: R. Bulirsch, Numerical calculation of the sine, cosine and Fresnel integrals, Numer. Math. 9 (1967), 380--385.
Cle62: C. W. Clenshaw, Chebyshev series for mathematical functions, National Physical Laboratory Mathematical Tables, vol. 5, Her Majesty's Stationery Office, London, 1962.
CMW63: C. W. Clenshaw, G. F. Miller, and M. Woodger, Algorithms for special functions I, Numer. Math. 4 (1963), 403--419.
CT68: W. J. Cody and H. C. Thacher, Jr., Rational Chebyshev approximations for the exponential integral , Math. Comp. 22 (1968), 641--649.
CT69: W. J. Cody and H. C. Thacher, Jr., Chebyshev approximations for the exponential integral , Math. Comp. 23 (1969), 289--303.
Gau73: W. Gautschi, Algorithm 471. Exponential integrals, Comm. ACM 16 (1973), 761--763.
Luk69: Y. L. Luke, The special functions and their approximations, vol. 2, Academic Press, New York, 1969.
Luk76: Y. L. Luke, On the expansion of exponential type integrals in series of Chebyshev polynomials, Theory of Approximation with Applications (A. G. Law and B. N. Sahney, eds.), Academic Press, Inc., New York, 1976, pp. 180--199.
Mos89: S. L. B. Moshier, Methods and programs for mathematical functions, Ellis Horwood Limited, Chichester, 1989, separate diskette.
Pac70: K. A. Paciorek, Algorithm 385. Exponential integral , Comm. ACM 13 (1970), 446--447, for certification and remarks see same journal v. 13 (1970), pp. 448--449 and p. 750; v. 15 (1972), p. 1074.
SZ76: I. A. Stegun and R. Zucker, Automatic computing methods for special functions. Part III. The sine, cosine, exponential integrals, and related functions, J. Res. Nat. Bur. Standards 80B (1976), 291--311.
TM68: R. F. Tooper and J. Mark, Simplified calculation of for positive arguments, and a short table of , Math. Comp. 22 (1968), 448--449.
ULI90: Mathematical function library for Microsoft--C, United Laboratories, Inc., John Wiley & Sons, 1990, includes diskettes. Edition also exists in Fortran (1989).
vdLT84: 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.

Abstract:

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: dlozier@nist.gov

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

E-mail address: olver@bessel.umd.edu

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 13:50:43 EDT 1995