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Numerical Evaluation of Special Functions

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

4.2. Error Functions, Dawson's Integral, Fresnel Integrals .

4.2.1. Error Functions of Real Argument.

Algorithms:

[ Cle62] , [ Cod69] , [ Luk69b] , [ Luk75] , [ Sch78] , [ SL81] .

Software Packages:

[ Ada69, Algol] , [ CMW63, Algol] , [ Cod90a, Fortran] , [ CT85, Fortran] , [ Hil73, Fortran] , [ SZ70, Fortran] , [ Tem94, Pascal] .

Intermediate Libraries:

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

Comprehensive Libraries:

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

4.2.2. Inverse Error Functions of Real Argument.

Algorithms:

[ BEJ76] .

Software Packages:

[ Cun69, Fortran] , [ HD73, Algol] .

Intermediate Libraries:

[ Mos89] , [ ULI90] .

Comprehensive Libraries:

IMSL, NAG, Scientific Desk.

4.2.3. Integrals of the Error Function.

Algorithms:

[ Woo67] .

Software Packages:

[ Gau77a, Fortran] .

Intermediate Libraries:

[ Bak92] .

4.2.4. Dawson's Integral of Real Argument.

Algorithms:

[ CPT70] , [ Hum64] .

Software Packages:

[ Ryb89, Fortran] .

Intermediate Libraries:

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

Comprehensive Libraries:

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

4.2.5. Fresnel Integrals of Real Argument.

Algorithms:

[ Cod68] , [ Hea85] , [ Luk69b] , [ Luk75] .

Software Packages:

[ Bul67, Algol] , [ LG64, Algol] , [ Sny93, Fortran] .

Intermediate Libraries:

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

Comprehensive Libraries:

IMSL, NAG, Numerical Recipes.

Interactive Systems:

Maple.

4.2.6. Complex Arguments.

Algorithms:

[ Luk69b] .

Software Packages:

[ Gau69a, Algol] , [ Lyn93, Fortran] , [ PW90a, Fortran] , [ SZ81, Fortran] .

Intermediate Libraries:

[ Bak92] .

Comprehensive Libraries:

IMSL, NAG.

Interactive Systems:

Maple, Mathematica.

4.2.7. Articles.

[ BR71] , [ Cod90b, includes survey] , [ Col87a] , [ Fle68] , [ Gau70] , [ Gau77b] , [ Hen79] , [ HR72] , [ LW90] , [ LW91] , [ McC74] , [ Mor83] , [ MR71] , [ PW90b] , [ Str68] , [ vdLT84] , [ Wei94a, includes survey] , [ Wei94b] .

References

Ada69

A. G. Adams, Algorithm 39. Areas under the normal curve, Comput. J. 12 (1969), 197--198.

Bak92

L. Baker, C mathematical function handbook, McGraw-Hill, Inc., New York, 1992, includes diskette.

BEJ76

J. M. Blair, C. A. Edwards, and J. H. Johnson, Rational Chebyshev approximations for the inverse of the error function, Math. Comp. 30 (1976), 827--830.

BR71

R. D. Bardo and K. Ruedenberg, Numerical analysis and evaluation of normalized repeated integrals of the error function and related functions, J. Comput. Phys. 8 (1971), 167--174.

Bul67

R. Bulirsch, Numerical calculation of the sine, cosine and Fresnel integrals, Numer. Math. 9 (1967), 380--385.

Cle62

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

CMW63

C. W. Clenshaw, G. F. Miller, and M. Woodger, Algorithms for special functions I, Numer. Math. 4 (1963), 403--419.

Cod68

W. J. Cody, Chebyshev approximations for the Fresnel integrals, Math. Comp. 22 (1968), 450--453, with Microfiche Supplement.

Cod69

W. J. Cody, Rational Chebyshev approximations for the error function, Math. Comp. 23 (1969), 631--637.

Cod90a

W. J. Cody, The normal integral, Tech. Report MCS--89--1090, Mathematics and Computer Science Division, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, Illinois 60439--4801, 1990.

Cod90b

W. J. Cody, Performance evaluation of programs for the error and complementary error functions, ACM Trans. Math. Software 16 (1990), 29--37.

Col87

J. P. Coleman, Complex polynomial approximation by the Lanczos --method : Dawson's integral, J. Comput. Appl. Math. 20 (1987), 137--151.

CPT70

W. J. Cody, K. A. Paciorek, and H. C. Thacher, Jr., Chebyshev approximations for Dawson's integral, Math. Comp. 24 (1970), 171--178.

CT85

M. Carmignani and A. Tortorici Macaluso, Calcolo delle funzioni speciali , , , , alle alte precisioni, Atti Accad. Sci. Lett. Arti Palermo (5) 2 (1981--82) (1985), no. 1, 7--25.

Cun69

S. W. Cunningham, Algorithm AS 24. From normal integral to deviate, Appl. Statist. 18 (1969), 290--293.

Fle68

O. L. Fleckner, A method for the computation of the Fresnel integrals and related functions, Math. Comp. 22 (1968), 635--640.

Gau69

W. Gautschi, Algorithm 363. Complex error function, Comm. ACM 12 (1969), 635, for certification see same journal v. 15 (1972), pp. 465--466.

Gau70

W. Gautschi, Efficient computation of the complex error function, SIAM J. Numer. Anal. 7 (1970), 187--198.

Gau77a

W. Gautschi, Algorithm 521. Repeated integrals of the coerror function, ACM Trans. Math. Software 3 (1977), 301--302.

Gau77b

W. Gautschi, Evaluation of the repeated integrals of the coerror function, ACM Trans. Math. Software 3 (1977), 240--252.

HD73

G. W. Hill and A. W. Davis, Algorithm 442. Normal deviate, Comm. ACM 16 (1973), 51--52.

Hea85

M. A. Heald, Rational approximations for the Fresnel integrals, Math. Comp. 44 (1985), 459--461.

Hen79

P. Henrici, Zur numerischen Berechnung der Fresnelschen Integrale, Z. Angew. Math. Phys. 30 (1979), 209--219.

Hil73

I. D. Hill, Algorithm AS 66. The normal integral, Appl. Statist. 22 (1973), 424--427.

HR72

D. B. Hunter and T. Regan, A note on the evaluation of the complementary error function, Math. Comp. 26 (1972), 539--541.

Hum64

D. G. Hummer, Expansion of Dawson's function in a series of Chebyshev polynomials, Math. Comp. 18 (1964), 317--319.

LG64

H. Lotsch and M. Gray, Algorithm 244. Fresnel integrals, Comm. ACM 7 (1964), 660--661.

Luk69

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.

LW90

F. G. Lether and P. R. Wenston, An algorithm for the numerical computation of the Voigt function, Appl. Math. Comput. 35 (1990), 277--289.

LW91

F. G. Lether and P. R. Wenston, The numerical computation of the Voigt function by a corrected midpoint quadrature rule for , J. Comput. Appl. Math. 34 (1991), 75--92.

Lyn93

A. E. Lynas-Gray, VOIGTL---A fast subroutine for Voigt function evaluation on vector processors, Comput. Phys. Comm. 75 (1993), 135--142.

McC74

J. H. McCabe, A continued fraction expansion, with a truncation error estimate, for Dawson's integral, Math. Comp. 28 (1974), 811--816.

Mor83

M. Mori, A method for evaluation of the error function of real and complex variable with high relative accuracy, Publ. Res. Inst. Math. Sci. 19 (1983), 1081--1094.

Mos89

S. L. B. Moshier, Methods and programs for mathematical functions, Ellis Horwood Limited, Chichester, 1989, separate diskette.

MR71

F. Matta and A. Reichel, Uniform computation of the error function and other related functions, Math. Comp. 25 (1971), 339--344.

PW90a

G. P. M. Poppe and C. M. J. Wijers, Algorithm 680. Evaluation of the complex error function, ACM Trans. Math. Software 16 (1990), 47.

PW90b

G. P. M. Poppe and C. M. J. Wijers, More efficient computation of the complex error function, ACM Trans. Math. Software 16 (1990), 38--46.

Ryb89

G. B. Rybicki, Dawson's integral and the sampling theorem, Computers in Physics 3 (1989), 85--87.

Sch78

J. L. Schonfelder, Chebyshev expansions for the error and related functions, Math. Comp. 32 (1978), 1232--1240.

SL81

M. M. Shepherd and J. G. Laframboise, Chebyshev approximation of
in , Math. Comp. 36 (1981), 249--253.

Sny93

W. V. Snyder, Algorithm 723. Fresnel integrals, ACM Trans. Math. Software 19 (1993), 452--456.

Str68

A. J. Strecok, On the calculation of the inverse of the error function, Math. Comp. 22 (1968), 144--158.

SZ70

I. A. Stegun and R. Zucker, Automatic computing methods for special functions, J. Res. Nat. Bur. Standards 74B (1970), 211--224.

SZ81

I. A. Stegun and R. Zucker, Automatic computing methods for special functions. Part IV. Complex error function, Fresnel integrals, and other related functions, J. Res. Nat. Bur. Standards 86 (1981), 661--686.

Tem94

N. M. Temme, A set of algorithms for the incomplete gamma functions, Probab. Engrg. Inform. Sci. (1994), in press.

ULI90

Mathematical function library for Microsoft--C, United Laboratories, Inc., John Wiley & Sons, 1990, includes diskettes. Edition also exists in Fortran (1989).

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.

Wei94a

J. A. C. Weideman, Computation of the complex error function, SIAM J. Numer. Anal. (1994), in press.

Wei94b

J. A. C. Weideman, Computing integrals of the complex error function, These proceedings, 1994.

Woo67

V. E. Wood, Chebyshev expansions for integrals of the error function, Math. Comp. 21 (1967), 494--496.

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.