Algorithm 838

Algorithm 838: Airy functions. We present a Fortran 90 module, which computes the solutions and their derivatives of Airy’s differential equation, both on the real line and in the complex plane. The module also computes the zeros and associated values of the solutions and their derivatives, and the modulus and phase functions on the negative real axis. The computational methods are numerical integration of the differential equation and summation of asymptotic expansions for large argument. These methods were chosen because they are simple, adaptable to any precision, and amenable to rigorous error analysis. The module can be used to validate other codes or as a component in programs that require Airy functions (Source:

This software is also peer reviewed by journal TOMS.

References in zbMATH (referenced in 15 articles , 1 standard article )

Showing results 1 to 15 of 15.
Sorted by year (citations)

  1. Dunster, T. M.; Gil, A.; Segura, J.: Simplified error bounds for turning point expansions (2021)
  2. Nemes, Gergö: Proofs of two conjectures on the real zeros of the cylinder and Airy functions (2021)
  3. Seydaoğlu, Muaz; Koçak, Hüseyin; Erdoğan, Utku: An efficient numerical treatment for the asymptotic behaviour of the nonlinear Airy-type problems (2020)
  4. Dunster, T. M.; Gil, A.; Segura, J.: Computation of asymptotic expansions of turning point problems via Cauchy’s integral formula: Bessel functions (2017)
  5. Venutelli, Maurizio: Third-order dispersion in shallow water waves (2014)
  6. Jentschura, U. D.; Lötstedt, E.: Numerical calculation of Bessel, Hankel and Airy functions (2012)
  7. Kudenatti, Ramesh B.; Bujurke, N. M.; Pedley, T. J.: Stability of two-dimensional collapsible-channel flow at high Reynolds number (2012)
  8. Gil, Amparo; Segura, Javier; Temme, Nico M.: Basic methods for computing special functions (2011)
  9. Gil, Amparo; Segura, Javier; Temme, Nico M.: Fast and accurate computation of the Weber parabolic cylinder function (W(a, x)) (2011)
  10. Kodama, Masao: Algorithm 912: A module for calculating cylindrical functions of complex order and complex argument (2011)
  11. Kodama, Masao: Algorithm 877: A subroutine package for cylindrical functions of complex order and nonnegative argument. (2008)
  12. Gil, Amparo; Segura, Javier; Temme, Nico M.: Computing the real parabolic cylinder functions (U(a, x)), (V(a, x)). (2006)
  13. Ledoux, V.; Rizea, M.; Ixaru, L.; Vanden Berghe, G.; Van Daele, M.: Solution of the Schrödinger equation by a high order perturbation method based on a linear reference potential (2006)
  14. Fabijonas, B. R.: Algorithm 838: Airy functions (2004)
  15. Fabijonas, B. R.; Lozier, D. W.; Olver, F. W. J.: Computation of complex Airy functions and their zeros using asymptotics and the differential equation (2004)