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Numerical Evaluation of Special Functions

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

5.1. Bessel Functions .

All of the following subsections apply to the ordinary Bessel functions (J and Y) and the modified Bessel functions (I and K).

5.1.1. Orders 0 and 1, Real Arguments.

Interactive Systems:

[ Bla74] , [ Cle62] , [ Luk69b] , [ Luk75] , [ WBR82] .

Software Packages:

[ BS92, Fortran] , [ Hil81, Fortran] .

Intermediate Libraries:

[ Bak92] , [ Mos89] .

Comprehensive Libraries:

IMSL, NAG, Numerical Recipes, Scientific Desk, SLATEC.

5.1.2. Integer or Half-Integer Orders, Real Arguments.

This subsection includes spherical Bessel functions.

Interactive Systems:

[ AM61] , [ MM90] , [ PB82] .

Software Packages:

[ AM78, Fortran] , [ Col80, Fortran] , [ Hil81, Fortran] , [ RF93, Fortran] .

Intermediate Libraries:

[ Bak92] , [ Mos89] , [ ULI90] .

Comprehensive Libraries:

IMSL, Numerical Recipes.

5.1.3. Real Orders, Real Arguments.

Interactive Systems:

[ CP66] , [ Luk69b] , [ Luk71a] , [ Luk71b] , [ Luk72a] , [ Luk75] .

Software Packages:

[ ADW77a, Fortran] , [ Bar82b, Fortran] , [ Cam79, Fortran] , [ Cod83, Fortran] , [ Mat93b, Fortran] , [ Pie84b, Fortran] , [ Tem75, Algol] , [ Tem76, Algol] .

Intermediate Libraries:

[ Mos89] , [ ULI90] .

Comprehensive Libraries:

IMSL, Numerical Recipes, Scientific Desk, SLATEC.

Interactive Systems:

Maple.

5.1.4. Integer or Half-Integer Orders, Complex Arguments.

This subsection includes Kelvin functions.

Interactive Systems:

[ Bur63] , [ CM83] .

Software Packages:

[ BKN88a, Fortran] , [ BKN88b, Fortran] ,

Intermediate Libraries:

[ Bak92] , [ ULI90] .

Comprehensive Libraries:

IMSL, NAG.

5.1.5. Real Orders, Complex Arguments.

Interactive Systems:

[ Luk69b] , [ Luk75] .

Software Packages:

[ Amo86, Fortran] , [ Cam81, Fortran] , [ TB87, Fortran] . IMSL,

Comprehensive Libraries:

NAG, Scientific Desk, SLATEC.

5.1.6. Complex Orders, Complex Arguments.

Software Packages:

[ TB85, Fortran] .

Interactive Systems:

Mathematica.

5.1.7. Integrals of Bessel Functions.

Interactive Systems:

[ BEJ78] , [ GP64] .

Software Packages:

[ Amo83a, Fortran] , [ And82a, Fortran] , [ Cha83, Fortran] , [ Feu91, Fortran] , [ PB84, Fortran] , [ Pie82, Fortran] , [ SZ79, Fortran] , [ Tal83, Fortran] .

Intermediate Libraries:

[ Bak92] .

Comprehensive Libraries:

SLATEC.

5.1.8. Zeros of Bessel Functions.

Interactive Systems:

[ Pie84a] .

Software Packages:

[ Cam84, Fortran] , [ Pie90, Fortran] , [ Tem79, Algol] .

Intermediate Libraries:

[ Bak92] .

5.1.9. Articles---Functions.

[ Ach86] , [ ADW77b] , [ Amo74] , [ Bar81a] , [ BGV93] , [ Cam80] , [ CF87] , [ CMF77] , [ Cod80, includes survey] , [ Col87b] , [ CS89, includes survey] , [ Gau91b] , [ GB87] , [ GS78] , [ Hit68] , [ KS84b] , [ Luk72b] , [ Luk77b] , [ Nes84] , [ OS72] , [ Rem73] , [ TB86] , [ VGK+91] , [ Wal84] , [ WC90] , [ YN74] , [ Yos92] .

5.1.10. Articles---Integrals.

[ Amo83c] , [ And82b] , [ BFST86] , [ BGV93] , [ Can81] , [ Chr90] , [ Cof91] , [ Cor72] , [ DK90] , [ Gab79] , [ Gab80] , [ GM81] , [ Han85] , [ Joh75] , [ Lew91] , [ Lin72] , [ LK73] , [ LPM81] , [ Lun85] , [ MDS92] , [ Moo83] , [ OFM78] , [ PB82] , [ PB83] , [ PB85, includes survey] , [ Puo88] , [ SBK92] , [ Sie77] .

5.1.11. Articles---Zeros.

[ CH70a] , [ IKF91] , [ KS84a] , [ KS84c] , [ KS85a] , [ KS85b] , [ KS85c] , [ KS87] , [ MF86] , [ Sko85] .

References

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J.-J. Achenbach, Numerik. Implementierung von Zylinderfunktionen, Friedr. Vieweg & Sohn, Braunschweig/Wiesbaden, 1986.

ADW77a

D. E. Amos, S. L. Daniel, and M. K. Weston, Algorithm 511. CDC 6600 subroutines IBESS and JBESS for Bessel functions and , , , ACM Trans. Math. Software 3 (1977), 93--95, for erratum see same journal v. 4 (1978), p. 411.

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D. E. Amos, S. L. Daniel, and M. K. Weston, CDC 6600 subroutines IBESS and JBESS for Bessel functions and , , , ACM Trans. Math. Software 3 (1977), 76--92.

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A. M. Arthurs and R. McCarroll, Expansion of spherical Bessel functions in a series of Chebyshev polynomials, Math. Comp. 15 (1961), 159--162.

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R. W. B. Ardill and K. J. M. Moriarty, Spherical Bessel functions and of integer order and real argument, Comput. Phys. Comm. 14 (1978), 261--265.

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D. E. Amos, Computation of modified Bessel functions and their ratios, Math. Comp. 28 (1974), 239--251.

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D. E. Amos, Algorithm 609. A portable Fortran subroutine for the Bickley functions , ACM Trans. Math. Software 9 (1983), 480--493.

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D. E. Amos, Uniform asymptotic expansions for exponential integrals and Bickley functions , ACM Trans. Math. Software 9 (1983), 467--479.

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D. E. Amos, Algorithm 644. A portable package for Bessel functions of a complex argument and nonnegative order, ACM Trans. Math. Software 12 (1986), 265--273, for remark see same journal v. 16 (1990), p. 404.

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W. L. Anderson, Algorithm 588. Fast Hankel transforms using related and lagged convolutions, ACM Trans. Math. Software 8 (1982), 369--370.

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W. L. Anderson, Fast Hankel transforms using related and lagged convolutions, ACM Trans. Math. Software 8 (1982), 344--368.

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A. R. Barnett, COULFG : Coulomb and Bessel functions and their derivatives, for real arguments, by Steed's method, Comput. Phys. Comm. 27 (1982), 147--166.

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BFST86

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Bla74

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BS92

R. F. Boisvert and B. V. Saunders, Portable vectorized software for Bessel function evaluation, ACM Trans. Math. Software 18 (1992), 456--469, for corrigendum see same journal v. 19 (1993), p. 131.

Bur63

F. D. Burgoyne, Approximations to Kelvin functions, Math. Comp. 17 (1963), 295--298.

Cam79

J. B. Campbell, Bessel functions and of real order and real argument, Comput. Phys. Comm. 18 (1979), 133--142.

Cam80

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Cam81

J. B. Campbell, Bessel functions and of real order and complex argument, Comput. Phys. Comm. 24 (1981), 97--105, for erratum see same journal v. 25 (1982), p. 207.

Cam84

J. B. Campbell, Determination of --zeros of Hankel functions, Comput. Phys. Comm. 32 (1984), 333--339.

Can81

S. M. Candel, An algorithm for the Fourier-Bessel transform, Comput. Phys. Comm. 23 (1981), 343--353.

CF87

Ll. Closas and J. Fernández Rubio, Calculo rapido de las funciones de Bessel modificadas e y sus derivadas, Stochastica 11 (1987), no. 1, 53--61.

CH70

J. A. Cochran and J. N. Hoffspiegel, Numerical techniques for finding --zeros of Hankel functions, Math. Comp. 24 (1970), 413--422.

Cha83

A. D. Chave, Numerical integration of related Hankel transforms by quadrature and continued fraction expansion, Geophysics 48 (1983), 1671--1686.

Chr90

N. B. Christensen, Optimized fast Hankel transform filters, Geophysical Prospecting 38 (1990), 545--568, for comment and reply, see same journal v. 39 (1991), pp. 445--447 and 449--450.

Cle62

C. W. Clenshaw, Chebyshev series for mathematical functions, National Physical Laboratory Mathematical Tables, vol. 5, Her Majesty's Stationery Office, London, 1962.

CM83

J. P. Coleman and A. J. Monaghan, Chebyshev expansions for the Bessel function in the complex plane, Math. Comp. 40 (1983), 343--366.

CMF77

W. J. Cody, R. M. Motley, and L. W. Fullerton, The computation of real fractional order Bessel functions of the second kind, ACM Trans. Math. Software 3 (1977), 232--239.

Cod80

W. J. Cody, Preliminary report on software for the modified Bessel functions of the first kind, Tech. Memorandum TM--357, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, Illinois 60439--4801, 1980.

Cod83

W. J. Cody, Algorithm 597. Sequence of modified Bessel functions of the first kind, ACM Trans. Math. Software 9 (1983), 242--245.

Cof91

M. W. Coffey, Calculation of generalized Lommel integrals for modified Bessel functions, J. Phys. A 24 (1991), 23--33.

Col80

J. P. Coleman, A Fortran subroutine for the Bessel function of order 0 to 10, Comput. Phys. Comm. 21 (1980), 109--118.

Col87

J. P. Coleman, Polynomial approximations in the complex plane, J. Comput. Appl. Math. 18 (1987), 193--211.

Cor72

P. Cornille, Computation of Hankel transforms, SIAM Rev. 14 (1972), 278--285.

CP66

C. W. Clenshaw and S. M. Picken, Chebyshev series for Bessel functions of fractional order, National Physical Laboratory Mathematical Tables, vol. 8, Her Majesty's Stationery Office, London, 1966.

CS89

W. J. Cody and L. Stoltz, Performance evaluation of programs for certain Bessel functions, ACM Trans. Math. Software 15 (1989), 41--48.

DK90

S. L. Dvorak and E. F. Kuester, Numerical computation of the incomplete Lipschitz-Hankel integral , J. Comput. Phys. 87 (1990), 301--327.

Feu91

F. Feuillebois, Numerical calculation of singular integrals related to Hankel transform, Comput. Math. Appl. 21 (1991), no. 2-3, 87--94.

Gab79

B. Gabutti, On high precision methods for computing integrals involving Bessel functions, Math. Comp. 33 (1979), 1049--1057.

Gab80

B. Gabutti, On the generalization of a method for computing Bessel function integrals, J. Comput. Appl. Math. 6 (1980), 167--168.

Gau91

W. Gautschi, On the paper ``A continued fraction approximation of the modified Bessel function '' by P. R. Parthasarathy and N. Balakrishnan, Appl. Math. Lett. 4 (1991), no. 5, 47--51.

GB87

A. Ganguli and R. Baskaran, Generation of Bessel functions with complex arguments and integer orders, Internat. J. Comput. Math. 21 (1987), 43--64.

GM81

B. Gabutti and B. Minetti, A new application of the discrete Laguerre polynomials in the numerical evaluation of the Hankel transform of a strongly decreasing even function, J. Comput. Phys. 42 (1981), 277--287.

GP64

I. Gargantini and T. Pomentale, Rational Chebyshev approximations to the Bessel function integrals , Comm. ACM 7 (1964), 727--730.

GS78

W. Gautschi and J. Slavik, On the computation of modified Bessel function ratios, Math. Comp. 32 (1978), 865--875.

Han85

E. W. Hansen, Fast Hankel transform algorithm, IEEE Trans. Acoust. Speech Signal Process. 33 (1985), 666--671.

Hil81

G. W. Hill, Algorithm 571. Statistics for von Mises' and Fisher's distributions of directions : and their inverses, ACM Trans. Math. Software 7 (1981), 233--238.

Hit68

S. Hitotumatu, On the numerical computation of Bessel functions through continued fraction, Comment. Math. Univ. St. Paul. 16 (1967/68), 89--113.

IKF91

Y. Ikebe, Y. Kikuchi, and I. Fujishiro, Computing zeros and orders of Bessel functions, J. Comput. Appl. Math. 38 (1991), 169--184.

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H. K. Johansen, An interactive computer/graphic-display-terminal system for interpretation of resistivity soundings, Geophysical Prospecting 23 (1975), 449--458.

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KS84b

M. K. Kerimov and S. L. Skorokhodov, Calculation of the complex zeros of the modified Bessel function of the second kind and its derivatives, U.S.S.R. Comput. Math. and Math. Phys. 24 (1984), no. 4, 115--123.

KS84c

M. K. Kerimov and S. L. Skorokhodov, Evaluation of complex zeros of Bessel functions and and their derivatives, U.S.S.R. Comput. Math. and Math. Phys. 24 (1984), no. 5, 131--141.

KS85a

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KS85b

M. K. Kerimov and S. L. Skorokhodov, Calculation of the complex zeros of Hankel functions and their derivatives, U.S.S.R. Comput. Math. and Math. Phys. 25 (1985), no. 6, 26--36.

KS85c

M. K. Kerimov and S. L. Skorokhodov, Calculation of the multiple zeros of the derivatives of the cylindrical Bessel functions and , U.S.S.R. Comput. Math. and Math. Phys. 25 (1985), no. 6, 101--107.

KS87

M. K. Kerimov and S. L. Skorokhodov, On the calculation of the multiple complex roots of the derivatives of cylindrical Bessel functions, U.S.S.R. Comput. Math. and Math. Phys. 27 (1987), no. 6, 18--25.

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T. Yoshida, Error analysis of the recurrence technique for calculation of Bessel functions , J. Inform. Process. 15 (1992), 213--221.

Abstract:

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.