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Numerical Evaluation of Special Functions
D. W. Lozier and F. W. J. Olver
2. Mathematical Developments
Comprehensive compendia of mathematical properties of the
special functions are provided by the National Bureau of
Standards' Handbook of Mathematical Functions [ AS64]
,
published originally in 1964,
and the 3-volume set that resulted from the Bateman
Manuscript Project [ EMOT53a,EMOT53b,EMOT55]
, published in 1953 and 1955.
These references
employ the same notation for the special functions, and we shall
follow them. The NBS Handbook has been reprinted many times by
the U. S. Government Printing Office and has also been issued in
whole, or in part, by other publishers including Dover
Publications, Moscow Nauka, Verlag Harri Deutsch and Wiley-Interscience.
The forerunner of [ AS64]
is the book of Jahnke and Emde
[ JE45]
, published originally in 1909, and still in
print. It continues to be
useful, especially for its collection of graphs. Other useful
compendia include those of Magnus, Oberhettinger and Soni
[ MOS66]
, and (from the standpoint of hypergeometric
functions) Luke [ Luk69a]
. For an introductory
compendium, see the recent ``atlas'' of Spanier and Oldham
[ SO87]
.
Books and articles that include descriptions or surveys of
general methods for computing special functions include
[ Bre78b,DKK81,Gau75,HCL+68,Luk69b,Luk77b,PT84,PTVF92,Tem78,vdLT84]
.
Other books and articles that provide indepth coverage of
pertinent topics include:
[ Ask89, survey of compendia]
,
[ BG81a,BG81b, Padé approximations]
,
[ BH75, asymptotic approximations]
,
[ Bre91, continued fractions, Padé approximations]
,
[ BvI93, Padé approximations]
,
[ Cod70, polynomial and rational approximations]
,
[ Fik68, polynomial and rational approximations]
,
[ FP68, Chebyshev polynomials]
,
[ JT80, continued fractions]
,
[ Kar91, power series]
,
[ KG80, statistical computations]
,
[ Luk75, supplement to AS64---especially for functions of hypergeometric type]
,
[ Mor80, power series]
,
[ Olv74, asymptotic approximations]
,
[ Riv90, Chebyshev polynomials]
,
[ Tem77, integral representations]
,
[ Tem85, asymptotic approximations]
,
[ Wim84, recurrence relations]
,
[ Won89, asymptotic approximations]
.
References
- AS64
-
M. Abramowitz and I. A. Stegun (eds.), Handbook of mathematical functions
with formulas, graphs and mathematical tables, National Bureau of Standards
Applied Mathematics Series, vol. 55, U. S. Government Printing Office,
Washington, D. C., 1964.
- Ask89
-
R. A. Askey, Handbooks of special functions, A Century of Mathematics in
America, Part III, Hist. Math., vol. 3, American Mathematical Society,
Providence, Rhode Island, 1989, pp. 369--391.
- BG81a
-
G. A. Baker, Jr. and P. Graves-Morris, Padé approximants, part
I, Encyclopedia of Mathematics and its Applications, vol. 13,
Addison-Wesley Publishing Company, Reading, Massachusetts, 1981.
- BG81b
-
G. A. Baker, Jr. and P. Graves-Morris, Padé approximants, part
II, Encyclopedia of Mathematics and its Applications, vol. 14,
Addison-Wesley Publishing Company, Reading, Massachusetts, 1981.
- BH75
-
N. Bleistein and R. A. Handelsman, Asymptotic expansions of integrals,
Holt, Rinehart and Winston, New York, 1975.
- Bre78
-
R. P. Brent, A Fortran multiple-precision arithmetic package, ACM
Trans. Math. Software 4 (1978), 57--70.
- Bre91
-
C. Brezinski, History of continued fractions and Padé approximants,
Springer Series in Computational Mathematics, vol. 12, Springer-Verlag,
Berlin, 1991.
- BvI93
-
C. Brezinski and J. van Iseghem, Padé approximations, Handbook of
Numerical Analysis (P. G. Ciarlet and J. L. Lions, eds.), vol. 3,
North-Holland, Amsterdam, 1993, in press.
- Cod70
-
W. J. Cody, A survey of practical rational and polynomial approximation of
functions, SIAM Rev. 12 (1970), 400--423.
- DKK81
-
V. A. Ditkin, K. A. Karpov, and M. K. Kerimov, The computation of special
functions, U.S.S.R. Comput. Math. and Math. Phys. 20 (1981), no. 5,
3--12.
- EMOT53a
-
A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher
transcendental functions, vol. 1, McGraw-Hill, New York, 1953, reprinted and
published in 1981 by Krieger Publishing Company, Melbourne, Florida.
- EMOT53b
-
A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher
transcendental functions, vol. 2, McGraw-Hill, New York, 1953, reprinted and
published in 1981 by Krieger Publishing Company, Melbourne, Florida.
- EMOT55
-
A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher
transcendental functions, vol. 3, McGraw-Hill, New York, 1955, reprinted and
published in 1981 by Krieger Publishing Company, Melbourne, Florida.
- Fik68
-
C. T. Fike, Computer evaluation of mathematical functions,
Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1968.
- FP68
-
L. Fox and I. B. Parker, Chebyshev polynomials in numerical analysis,
Oxford University Press, London, 1968.
- Gau75
-
W. Gautschi, Computational methods in special functions---a survey,
Theory and application of special functions, Proc. Advanced Sem., Math. Res.
Center, Univ. Wisconsin, Madison, Wis., Academic Press, New York, 1975,
pp. 1--98.
- HCL68
-
J. F. Hart, E. W. Cheney, C. L. Lawson, H. J. Maehly, C. K. Mesztenyi, J. R.
Rice, H. C. Thacher, Jr., and C. Witzgall, Computer approximations,
John Wiley and Sons, Inc., New York, 1968.
- JE45
-
E. Jahnke and F. Emde, Tables of functions with formulae and curves,
fourth ed., Dover Publications, Inc., New York, 1945.
- JEL60
-
E. Jahnke, F. Emde, and F. Lösch, Tables of higher functions, sixth
ed., McGraw-Hill, New York, 1960.
- JT80
-
W. B. Jones and W. J. Thron, Continued fractions : analytic theory
and applications, Encyclopedia of Mathematics and its Applications, vol. 11,
Addison-Wesley Publishing Company, Reading, Massachusetts, 1980.
- Kar91
-
E. A. Karatsuba, Fast evaluation of transcendental functions, Problems
Inform. Transmission 27 (1991), 339--360.
- KG80
-
W. J. Kennedy, Jr. and J. E. Gentle, Statistical computing, Marcel
Dekker, New York, 1980.
- Luk69a
-
Y. L. Luke, The special functions and their approximations, vol. 1,
Academic Press, New York, 1969.
- Luk69b
-
Y. L. Luke, The special functions and their approximations, vol. 2,
Academic Press, New York, 1969.
- Luk75
-
Y. L. Luke, Mathematical functions and their approximations, Academic
Press, New York, 1975.
- Luk77
-
Y. L. Luke, Algorithms for the computation of mathematical functions,
Academic Press, New York, 1977.
- Mor80
-
M. Mori, Analytic representations suitable for numerical computation of
some special functions, Numer. Math. 35 (1980), 163--174.
- MOS66
-
W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and theorems for the
special functions of mathematical physics, Die Grundlehren der
mathematischen Wissenschaften in Einzeldarstellungen, vol. 52,
Springer-Verlag, New York, 1966.
- Olv74
-
F. W. J. Olver, Asymptotics and special functions, Academic Press, New
York, 1974.
- PT84
-
B. A. Popov and G. S. Tesler, Computation of functions on electronic
computers---handbook, Naukova Dumka, Kiev, 1984 (Russian), see review by K.
S. Kölbig in Math. Comp. v. 55 (1990), pp. 395--397.
- PTVF92
-
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).
- Riv90
-
T. J. Rivlin, Chebyshev polynomials. From approximation theory to
algebra and number theory, second ed., John Wiley and Sons, Inc., New York,
1990.
- SO87
-
J. Spanier and K. B. Oldham, An atlas of functions, Hemisphere Publishing
Corporation, Washington, D. C., 1987.
- Tem77
-
N. M. Temme, The numerical computation of special functions by use of
quadrature rules for saddle point integrals. I. Trapezoidal integration
rules, Tech. Report TW 164/77, Mathematisch Centrum, Amsterdam, 1977.
- Tem78
-
N. M. Temme, Some aspects of applied analysis : asymptotics, special
functions and their numerical computation, Mathematisch Centrum, Amsterdam,
1978.
- Tem85
-
N. M. Temme, Special functions as approximants in uniform asymptotic
expansions of integrals ; A survey, Rend. Sem. Mat. Univ. Politec.
Torino Fascicolo Speciale. Special Functions: Theory and Computation
(1985), 289--317.
- 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.
- Wim84
-
J. Wimp, Computation with recurrence relations, Pitman, London, 1984.
- Won89
-
R. Wong, Asymptotic approximations of integrals, Academic Press, New
York, 1989.
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.
- ...print
- A more recent edition, with F. Lösch added as
author [ JEL60]
, is no longer in print.
Daniel W Lozier
Fri Apr 7 13:32:38 EDT 1995