B. R. Fabijonas
Abstract
We describe a method by which one can compute the solutions of Airy's differential equation, and their derivatives, both on the real line and in the complex plane. The computational methods are numerical integration of the differential equation and summation of asymptotic expansions for large argument. We give details involved in obtaining all of the parameter values, and we control the truncation errors rigorously. Using the same computational methods, we describe an algorithm that computes the zeros and associated values of the Airy functions and their derivatives, and the modulus and phase functions on the negative real axis.
- Amos, D. E. 1986. Algorithm 644. A portable package for Bessel functions of a complex argument and nonnegative order. ACM Trans. Math. Softw. 12, 3, 265--273. Google Scholar
- Amos, D. E. 1990. Remark on Algorithm 644. ACM Trans. Math. Softw. 16, 4, 404. Google Scholar
- Amos, D. E. 1995. A remark on Algorithm 644. ACM Trans. Math. Softw. 21, 4, 388--393. Google Scholar
- Barnett, A. R. 1981a. An algorithm for regular and irregular Coulomb and Bessel functions of real order to machine accuracy. Comput. Phys. Commun. 21, 297--314.Google Scholar
- Barnett, A. R. 1981b. KLEIN: Coulomb functions for real λ and positive energy to high accuracy. Comput. Phys. Commun. 24, 141--159.Google Scholar
- Bleistein, N. and Handelsman, R. A. 1975. Asymptotic Expansions of Integrals. Holt, Rinehart, and Winston, New York.Google Scholar
- Bond, G. and Pitteway, M. L. V. 1967. Algorithm 301. Airy function. Commun. ACM 10, 5, 291--292. Google Scholar
- Char, B. W., Geddes, K. O., Gonnet, G. H., Leong, B., Monaghan, M., and Watt, S. 1991. MAPLE V Library Reference Manual. New York.Google Scholar
- Connor, J. N. L., Curtis, P. R., and Farrelly, D. 1983. A differential equation method for the numerical evaluation of the Airy, Pearcey, and Swallowtail canonical integrals and their derivatives. Molec. Physics 48, 6, 1305--1330.Google Scholar
- Corless, R. M., Jeffrey, D. J., and Rasmussen, H. 1992. Numerical evaluation of Airy functions with complex arguments. J. Comput. Phys. 99, 1, 106--114. Google Scholar
- Fabijonas, B. R. 2004. Algorithm 838: Airy functions. ACM Trans. Math. Softw. 30, 4, XXX--XXX. Google Scholar
- Fabijonas, B. R., Lozier, D. W., and Rappoport, J. M. 2003. Algorithms and codes for Macdonald functions: recent progress and comparisons. J. Comput. Appl. Math. 161, 179--192. Google Scholar
- Fabijonas, B. R. and Olver, F. W. J. 1999. On the reversion of an asymptotic expansion and the zeros of the Airy functions. SIAM Rev. 41, 4, 762--773. Google Scholar
- Gil, A., Segura, J., and Temme, N. M. 2002. Algorithm 819: AIZ, BIZ: Two Fortran 77 routines for the computation of complex Airy functions. ACM Trans. Math. Softw. 28, 3, 325--336. Google Scholar
- Jeffreys, H. 1942. Asymptotic solutions of linear differential equations. Philos. Mag. 7, 33, 451--456.Google Scholar
- Jeffreys, H. 1953. On approximate solutions of linear differential equations. Proc. Cambridge Philos. Soc. 49, 4, 601--611.Google Scholar
- Lozier, D. W. and Olver, F. W. J. 1993. Airy and Bessel functions by parallel integration of ODEs. In Proceedings of the Sixth SIAM Conference on Parallel Processing for Scientific Computing, R. Sincovec, D. Keyes, M. Leuze, L. Petzold, and D. Reed, Eds. Vol. 2. Society for Industrial and Applied Mathematics, 530--538.Google Scholar
- Miller, J. C. P. 1946. The Airy Integral. British Assoc. Adv. Sci. Mathematical Tables Part-Vol.B. Cambridge Univ. Press, Cambridge.Google Scholar
- Miller, J. C. P. 1950. On the choice of standard solutions for a homogeneous linear differential equation of the second order. Quart. J. Mech. Appl. Math. 3, 2, 225--235.Google Scholar
- Muldoon, M. E. 1977. Higher monotonicity properties of certain Sturm-Liouville functions V. Proc. Roy. Soc. Edinburgh Sect. A 77, 23--37.Google Scholar
- Olver, F. W. J. 1954. The asymptotic expansion of Bessel functions of large order. Philos. Trans. Roy. Soc. London Ser. A 247, 328--368.Google Scholar
- Olver, F. W. J. 1974. Asymptotics and Special Functions. Academic Press, New York. Reprinted by AK Peters, Wellesley, 1997.Google Scholar
- Olver, F. W. J. 1998. Airy Functions, Chapter AI. In Digital Library of Mathematical Functions Project. National Institute of Standards and Technology. http://dlmf.nist.gov.Google Scholar
- Prince, P. J. 1975. Algorithm 498. Airy functions using Chebyshev series approximations. ACM Trans. Math. Softw. 1, 4, 372--379. Google Scholar
- Razaz, M. and Schonfelder, J. L. 1981. Remark on Algorithm 498. ACM Trans. Math. Softw. 7, 3, 404--405. Google Scholar
- Schulten, Z., Anderson, D. G. M., and Gordon, R. G. 1979. An algorithm for the evaluation of the complex Airy functions. J. Comput. Physics 31, 1, 60--75.Google Scholar
- Sherry, M. 1959. The zeros and maxima of the Airy function and its first derivative to 25 significant figures. Tech. Rep. AFCRC-TR-59-135, ASTIA AD214568, Air Force Cambridge Research Center. April.Google Scholar
- Wong, R. 1989. Asymptotic Approximations of Integrals. Academic Press, New York. Reprinted by SIAM, Philadelphia, 2001. Google Scholar
Index Terms
- Computation of complex Airy functions and their zeros using asymptotics and the differential equation
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