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Numerical Evaluation of Special Functions
D. W. Lozier and F. W. J. Olver
5.3. Elliptic Integrals and Functions
.
An important recent change in the old subject of elliptic integrals is
a renormalization of the definitions of the integrals. This is due to
B. C. Carlson: references will be found in § 5.3.5.
5.3.1. Complete Elliptic Integrals.
Algorithms:
[ Bel88]
,
[ Cod65a]
,
[ Cod65b]
,
[ Luk69b]
.
Software Packages:
[ Bul65a, Algol]
,
[ Bul65b, Algol]
,
[ Bul69b, Algol]
,
[ MH73, Algol]
.
Intermediate Libraries:
[ Bak92]
,
[ Mos89]
,
[ ULI90]
.
Comprehensive Libraries:
IMSL,
Numerical Recipes.
Interactive Systems:
Mathematica.
5.3.2. Incomplete Elliptic Integrals.
Algorithms:
[ Luk69b]
.
Software Packages:
[ Bul65a, Algol]
,
[ Bul69b, Algol]
,
[ Car87, Fortran]
,
[ Car88, Fortran]
,
[ CN81, Fortran]
,
[ PT90, Fortran]
.
Intermediate Libraries:
[ Bak92]
,
[ Mos89]
,
[ ULI90]
.
Comprehensive Libraries:
IMSL,
NAG,
Numerical Recipes,
Scientific Desk,
SLATEC.
Interactive Systems:
Mathematica.
5.3.3. Jacobi's Elliptic Functions.
This subsection includes the theta functions.
Software Packages:
[ Bul65a, Algol]
.
Intermediate Libraries:
[ Bak92]
,
[ Mos89]
,
[ ULI90]
.
Comprehensive Libraries:
IMSL,
NAG,
Numerical Recipes.
Interactive Systems:
Mathematica (includes inverse functions).
5.3.4. Weierstrass' Elliptic Functions.
Algorithms:
[ Eck76]
,
[ Eck77]
.
Software Packages:
[ Eck80, Fortran]
.
Intermediate Libraries:
[ Bak92]
,
[ ULI90]
.
Comprehensive Libraries:
IMSL.
Interactive Systems:
Mathematica.
5.3.5. Articles.
[ ACJP85, includes survey]
,
[ Bul69a]
,
[ Car65]
,
[ Car77a]
,
[ Car77b]
,
[ Car79]
,
[ Car87]
,
[ Car88]
,
[ Car89]
,
[ Car91]
,
[ Car92]
,
[ CGL90]
,
[ Cri89]
,
[ FGG82]
,
[ FL67]
,
[ Lee90]
,
[ Lee92]
,
[ Luk68]
,
[ Luk70b]
,
[ LY88]
,
[ Mid75]
,
[ NC66]
,
[ Sal89]
,
[ War60]
.
References
- ACJP85
-
J. Arazy, T. Claesson, S. Janson, and J. Peetre, Means and their
iterations, Proceedings of the Nineteenth Nordic Congress of Mathematicians,
Reykjavik 1984, Icelandic Mathematical Society, Reykjavik, 1985,
pp. 191--212.
- Bak92
-
L. Baker, C mathematical function handbook, McGraw-Hill, Inc., New
York, 1992, includes diskette.
- Bel88
-
V. N. Belykh, Calculation on a computer of the complete elliptic integrals
and , Boundary value problems for partial differential
equations, Akad. Nauk SSSR Sibirsk. Otdel., Inst. Mat., Novosibirsk, 1988,
pp. 3--15 and 137 (Russian).
- Bul65a
-
R. Bulirsch, Numerical calculation of elliptic integrals and elliptic
functions, Numer. Math. 7 (1965), 78--90.
- Bul65b
-
R. Bulirsch, Numerical calculation of elliptic integrals and elliptic
functions. II, Numer. Math. 7 (1965), 353--354.
- Bul69a
-
R. Bulirsch, An extension of the Bartky-transformation to incomplete
elliptic integrals of the third kind, Numer. Math. 13 (1969),
266--284.
- Bul69b
-
R. Bulirsch, Numerical calculation of elliptic integrals and elliptic
functions. III, Numer. Math. 13 (1969), 305--315.
- Car65
-
B. C. Carlson, On computing elliptic integrals and functions, J. Math.
and Phys. 44 (1965), 36--51.
- Car77a
-
B. C. Carlson, Elliptic integrals of the first kind, SIAM J. Math. Anal.
8 (1977), 231--242.
- Car77b
-
B. C. Carlson, Special functions of applied mathematics, Academic Press,
New York, 1977.
- Car79
-
B. C. Carlson, Computing elliptic integrals by duplication, Numer. Math.
33 (1979), 1--16.
- Car87
-
B. C. Carlson, A table of elliptic integrals of the second kind, Math.
Comp. 49 (1987), 595--606 and S13--S17.
- Car88
-
B. C. Carlson, A table of elliptic integrals of the third kind, Math.
Comp. 51 (1988), 267--280 and S1--S5.
- Car89
-
B. C. Carlson, A table of elliptic integrals : Cubic cases, Math.
Comp. 53 (1989), 327--333.
- Car91
-
B. C. Carlson, A table of elliptic integrals : One quadratic
factor, Math. Comp. 56 (1991), 267--280.
- Car92
-
B. C. Carlson, A table of elliptic integrals : Two quadratic
factors, Math. Comp. 59 (1992), 165--180.
- CGL90
-
R. Coquereaux, A. Grossmann, and B. E. Lautrup, Iterative method for
calculation of the Weierstrass elliptic function, IMA J. Numer. Anal.
10 (1990), 119--128.
- CN81
-
B. C. Carlson and E. M. Notis, Algorithm 577. Algorithms for incomplete
elliptic integrals, ACM Trans. Math. Software 7 (1981), 398--403.
- Cod65a
-
W. J. Cody, Chebyshev approximations for the complete elliptic integrals
K and E, Math. Comp. 19 (1965), 105--112, for corrigenda see
same journal v. 20 (1966), p. 207.
- Cod65b
-
W. J. Cody, Chebyshev polynomial expansions of complete elliptic
integrals, Math. Comp. 19 (1965), 249--259.
- Cri89
-
C. L. Critchfield, Computation of elliptic functions, J. Math. Phys.
30 (1989), 295--297.
- Eck76
-
U. Eckhardt, A rational approximation to Weierstrass'
--function, Math. Comp. 30 (1976), 818--826.
- Eck77
-
U. Eckhardt, A rational approximation to Weierstrass'
--function. II. The lemniscatic case, Computing 18
(1977), 341--349.
- Eck80
-
U. Eckhardt, Algorithm 549. Weierstrass' elliptic functions, ACM Trans.
Math. Software 4 (1980), 112--120.
- FGG82
-
J. D. Fenton and R. S. Gardiner-Garden, Rapidly-convergent methods for
evaluating elliptic integrals and theta and elliptic functions, J. Austral.
Math. Soc. Ser. B 24 (1982), 47--58.
- FL67
-
W. G. Fair and Y. L. Luke, Rational approximations to the incomplete
elliptic integrals of the first and second kinds, Math. Comp. 21
(1967), 418--422.
- Lee90
-
D. K. Lee, Application of theta functions for numerical evaluation of
complete elliptic integrals of the first and second kinds, Comput. Phys.
Comm. 60 (1990), 319--327.
- Lee92
-
D. K. Lee, Calculation of coefficients in a power-series expansion of the
nome , Comput. Phys. Comm. 70 (1992),
292--296.
- Luk68
-
Y. L. Luke, Approximations for elliptic integrals, Math. Comp. 22
(1968), 627--634.
- Luk69
-
Y. L. Luke, The special functions and their approximations, vol. 2,
Academic Press, New York, 1969.
- Luk70
-
Y. L. Luke, Further approximations for elliptic integrals, Math. Comp.
24 (1970), 191--198.
- LY88
-
T. Y. Lemczyk and M. M. Yovanovich, Efficient evaluation of incomplete
elliptic integrals and functions, Comput. Math. Appl. 16 (1988),
747--757.
- MH73
-
T. Morita and T. Horiguchi, Convergence of arithmetic-geometric mean
procedure for the complex variables and the calculation of the complete
elliptic integrals with complex modulus, Numer. Math. 20 (1973),
425--430, for correction see same journal v. 29 (1978), pp. 233--236.
- Mid75
-
P. Midy, An improved calculation of the general elliptic integral of the
second kind in the neighbourhood of x=0, Numer. Math. 25 (1975),
99--101.
- Mos89
-
S. L. B. Moshier, Methods and programs for mathematical functions, Ellis
Horwood Limited, Chichester, 1989, separate diskette.
- NC66
-
W. J. Nellis and B. C. Carlson, Reduction and evaluation of elliptic
integrals, Math. Comp. 20 (1966), 223--231.
- PT90
-
W. H. Press and S. A. Teukolsky, Elliptic integrals, Computers in Physics
4 (1990), 92--96.
- Sal89
-
K. L. Sala, Transformations of the Jacobian amplitude function and its
calculation via the arithmetic-geometric mean, SIAM J. Math. Anal. 20
(1989), 1514--1528.
- ULI90
-
Mathematical function library for Microsoft--C, United Laboratories,
Inc., John Wiley & Sons, 1990, includes diskettes. Edition also
exists in Fortran (1989).
- War60
-
M. Ward, The calculation of the complete elliptic integral of the third
kind, Amer. Math. Monthly 67 (1960), 205--213.
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 14:09:20 EDT 1995