Numerical Evaluation of Special Functions

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

6. Testing and Library Construction

In this section we list articles and books that provide general observations on the testing of software and/or the construction of software libraries for the special functions. For information on individual libraries see § 3.

[ Cod74] , [ Cod76] , [ Cod82] , [ Cod84b] , [ Cod85] , [ CS91] , [ Eva74, especially pp. 275--301 and 357--435] , [ Ful77] , [ Gaf88] , [ Kuk71] , [ LMS73a] , [ LMS73b] , [ Mos89] , [ PTVF92, example books] , [ Ric83] , [ Sch76] , [ SL73] .


W. J. Cody, The construction of numerical subroutine libraries, SIAM Rev. 16 (1974), 36--46.

W. J. Cody, An overview of software development for special functions, Lecture Notes in Mathematics 506: Numerical Analysis Dundee, 1975 (G. A. Watson, ed.), Springer-Verlag, Berlin, 1976, pp. 38--48.

W. J. Cody, Implementation and testing of function software, Lecture Notes in Computer Science No. 142. Problems and Methodologies in Mathematical Software Production (P. C. Messina and A. Murli, eds.), Springer-Verlag, Berlin, 1982, pp. 24--47.

W. J. Cody, Observations on the mathematical software effort, Sources and Development of Mathematical Software (W. R. Cowell, ed.), Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1984, pp. 1--19.

W. J. Cody, Software for special functions, Rend. Sem. Mat. Univ. Politec. Torino Fascicolo Speciale. Special Functions: Theory and Computation (1985), 91--116.

W. J. Cody and L. Stoltz, The use of Taylor series to test accuracy of function programs, ACM Trans. Math. Software 17 (1991), 55--63.

D. J. Evans (ed.), Software for numerical mathematics, Proceedings of the Loughboro University of Technology Conference of the IMA held in April 1973, Academic Press, London, 1974.

L. W. Fullerton, Portable special function routines, Lecture Notes in Computer Science 57: Portability of Numerical Software (Oak Brook 1976) (W. R. Cowell, ed.), Springer-Verlag, Berlin, 1977, pp. 452--483.

P. W. Gaffney, When things go wrong , Pitman Research Notes in Mathematics Series (D. F. Griffiths and G. A. Watson, eds.), vol. 170, Longman Scientific and Technical, Harlow, Essex, U. K., 1988, pp. 67--114.

H. Kuki, Mathematical function subprograms for basic system libraries--- objectives, constraints and trade-off, Mathematical Software (J. R. Rice, ed.), Academic Press, New York, 1971, pp. 187--199.

D. W. Lozier, L. C. Maximon, and W. L. Sadowski, A bit comparison program for algorithm testing, Comput. J. 16 (1973), 111--117.

D. W. Lozier, L. C. Maximon, and W. L. Sadowski, Performance testing of a Fortran library of mathematical function routines---A case study in the application of testing techniques, J. Res. Nat. Bur. Standards 77B (1973), 101--110.

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

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical recipes. The art of scientific computing, second ed., Cambridge University Press, 1992, diskettes and example books available. Editions exist in Basic (1991), C (1992), Fortran (1992), Macintosh Fortran (1988) and Pascal (1989).

J. R. Rice, Numerical methods, software and analysis : IMSL reference edition, McGraw-Hill Book Company, New York, 1983.

J. L. Schonfelder, The production of special function routines for a multi-machine library, Software---Practice and Experience 6 (1976), 71--82.

W. L. Sadowski and D. W. Lozier, A unified standards approach to algorithm testing, Program Test Methods (W. C. Hetzel, ed.), Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1973, pp. 277--290.


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 14:27:57 EDT 1995