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**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**]
.

**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*, Math. Comp.*:*Cubic cases**53**(1989), 327--333. **Car91**-
B. C. Carlson,
*A table of elliptic integrals*, Math. Comp.*:*One quadratic factor**56**(1991), 267--280. **Car92**-
B. C. Carlson,
*A table of elliptic integrals*, Math. Comp.*:*Two quadratic factors**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*, Math. Comp.**K**and**E****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*, Numer. Math.**x=0****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.

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.

Fri Apr 7 14:09:20 EDT 1995