Numerical Evaluation of Special Functions

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

5.9. Mathieu, Lamé, and Spheroidal Wave Functions .

5.9.1. Characteristic Values of Mathieu's Equation.

Software Packages:

[ Cle69, Fortran] , [ Del73, Algol] , [ Lee79, Fortran] , [ RL80, Fortran] , [ Shi93a, Fortran] .

Intermediate Libraries:

[ Bak92] .

Comprehensive Libraries:

IMSL.

5.9.2. Mathieu Functions.

Software Packages:

[ Cle69, Fortran] , [ Del73, Algol] , [ RL80, Fortran] , [ Shi93a, Fortran] .

Intermediate Libraries:

[ Bak92] .

Comprehensive Libraries:

IMSL.

5.9.3. Spheroidal Wave Functions.

Software Packages:

[ BC83a, Fortran] , [ BC83b, Fortran] , [ KBH70, Fortran] , [ KvB70, Fortran] , [ vBBH70, Fortran] .

Intermediate Libraries:

[ Bak92] .

5.9.4. Articles.

[ ADK+84] , [ ADKL89] , [ ADKL91] , [ ATZ83] , [ Bla46] , [ Cal88] , [ Can71] , [ Egl84] , [ EP69] , [ Hod70] , [ Pal69] , [ Shi93b] , [ SM75] , [ TP83] , [ vBBHK72] .

References

ADK84: A. A. Abramov, A. L. Dyshko, N. B. Konyukhova, T. V. Pak, and B. S. Pariiskii, Evaluation of prolate spheroidal function by solving the corresponding differential equations, U.S.S.R. Comput. Math. and Math. Phys. 24 (1984), no. 1, 1--11.
ADKL89: A. A. Abramov, A. L. Dyshko, N. B. Konyukhova, and T. V. Levitina, Evaluation of Lamé angular wave functions by solving auxiliary differential equations, U.S.S.R. Comput. Math. and Math. Phys. 29 (1989), no. 3, 119--131.
ADKL91: A. A. Abramov, A. L. Dyshko, N. B. Konyukhova, and T. V. Levitina, Computation of radial wave functions for spheroids and triaxial ellipsoids by the modified phase function method, Comput. Math. and Math. Phys. 31 (1991), no. 2, 25--42.
ATZ83: F. M. Arscott, P. J. Taylor, and R. V. M. Zahar, On the numerical construction of ellipsoidal wave functions, Math. Comp. 40 (1983), 367--380.
Bak92: L. Baker, C mathematical function handbook, McGraw-Hill, Inc., New York, 1992, includes diskette.
BC83a: T. A. Beu and R. I. Câmpeanu, Prolate angular spheroidal wave functions, Comput. Phys. Comm. 30 (1983), 187--192.
BC83b: T. A. Beu and R. I. Câmpeanu, Prolate radial spheroidal wave functions, Comput. Phys. Comm. 30 (1983), 177--185.
Bla46: G. Blanch, On the computation of Mathieu functions, J. Math. and Phys. 25 (1946), 1--20.
Cal88: J. Caldwell, Computation of eigenvalues of spheroidal harmonics using relaxation, J. Phys. A 21 (1988), 3685--3693.
Can71: J. Canosa, Numerical solution of Mathieu's equation, J. Comput. Phys. 7 (1971), 255--272.
Cle69: D. S. Clemm, Algorithm 352. Characteristic values and associated solutions of Mathieu's differential equation, Comm. ACM 12 (1969), 399--407, for remarks see same journal v. 13 (1970), p. 750 and v. 15 (1972), p. 1074.
Del73: Delft Numerical Analysis Group, On the computation of Mathieu functions, J. Engrg. Math. 7 (1973), 39--61.
Egl84: A. P. Eglaya, Eigenvalues of wave spheroidal functions with a complex parameter, Latv. Mat. Ezhegodnik (1984), no. 28, 143--150 (Russian).
EP69: S. P. Erasevskaja and A. A. Pal'cev, The computation of spheroidal functions and their first derivatives on a computer. II, Vesci Akad. Navuk BSSR Ser. Fiz.-Mat. Navuk (1969), no. 4, 37--46 (Russian).
Hod70: D. B. Hodge, Eigenvalues and eigenfunctions of the spheroidal wave equation, J. Math. Phys. 11 (1970), 2308--2312.
KBH70: B. J. King, R. V. Baier, and S. Hanish, A Fortran computer program for calculating the prolate spheroidal radial functions of the first and second kind and their first derivatives, NRL Report No. 7012, Naval Res. Lab., Washington, D. C., 1970.
KvB70: B. J. King and A. L. van Buren, A Fortran computer program for calculating the prolate and oblate angle functions of the first kind and their first and second derivatives, NRL Report No. 7161, Naval Res. Lab., Washington, D. C., 1970.
Lee79: W. R. Leeb, Algorithm 537. Characteristic values of Mathieu's differential equation, ACM Trans. Math. Software 5 (1979), 112--117.
Pal69: A. A. Pal'cev, The computation of spheroidal functions and their first derivatives on a computer. I, Vesci Akad. Navuk BSSR Ser. Fiz.-Mat. Navuk (1969), no. 1, 19--25 (Russian).
RL80: S. R. Rengarajan and J. E. Lewis, Mathieu functions of integral orders and real arguments, IEEE Trans. Microwave Theory Tech. 28 (1980), 276--277.
Shi93a: R. B. Shirts, Algorithm 721. MTIEU1 and MTIEU2 : Two subroutines to compute eigenvalues and solutions to Mathieu's differential equation for noninteger and integer order, ACM Trans. Math. Software 19 (1993), 391--406.
Shi93b: R. B. Shirts, The computation of eigenvalues and solutions of Mathieu's differential equation for noninteger order, ACM Trans. Math. Software 19 (1993), 377--390.
SM75: B. P. Sinha and R. H. MacPhie, On the computation of the prolate spheroidal radial functions of the second kind, J. Math. Phys. 16 (1975), 2378--2381.
TP83: N. Tugbay and E. Panayirci, An efficient algorithm for generation of prolate spheroidal wave functions, Bull. Tech. Univ. Istanbul 36 (1983), 563--577.
vBBH70: A. L. van Buren, R. V. Baier, and S. Hanish, A Fortran computer program for calculating the oblate spheroidal radial functions of the first and second kind and their first derivatives, NRL Report No. 6959, Naval Res. Lab., Washington, D. C., 1970.
vBBHK72: A. L. van Buren, R. V. Baier, S. Hanish, and B. J. King, Calculation of spheroidal wave functions, J. Acoust. Soc. Amer. 51 (1972), 414--416.

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 14:19:39 EDT 1995