Fast computation of incomplete elliptic integral of first kind by half argument transformation | Numerische Mathematik Skip to main content
Springer Nature Link
Log in

Fast computation of incomplete elliptic integral of first kind by half argument transformation

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

We developed a new method to calculate the incomplete elliptic integral of the first kind, \({F(\varphi|m)}\) , by using the half argument formulas of Jacobian elliptic functions. The method reduces the magnitude of \({\varphi}\) by repeated usage of the formulas while fixing m. The method is sufficiently precise in the sense that the maximum relative error is 3–5 machine epsilons at most. Thanks to the simplicity of the half argument formulas, the new procedure is significantly faster than the existing procedures. For example, it runs 20–60% faster than Bulirsch’ function, el1, and 1.9–2.2 times faster than the method using Carlson’s function, R F .

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
¥17,985 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (Japan)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abramowitz, M. Stegun, I.A. (eds.): Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Chapter 17. National Bureau of Standards, Washington (1964)

  2. Bulirsch R.: Numerical computation of elliptic integrals and elliptic functions. Numer. Math. 7, 78–90 (1965a)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bulirsch R.: Numerical computation of elliptic integrals and elliptic functions II. Numer. Math. 7, 353–354 (1965b)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bulirsch R.: An extension of the Bartky-transformation to incomplete elliptic integrals of the third kind. Numer. Math. 13, 266–284 (1969a)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bulirsch R.: Numerical computation of elliptic integrals and elliptic functions III. Numer. Math. 13, 305–315 (1969b)

    Article  MATH  MathSciNet  Google Scholar 

  6. Byrd P.F., Friedman M.D.: Handbook on Elliptic Integrals for Engineers and Physicists, 2nd edn. Springer, Berlin (1971)

    Google Scholar 

  7. Carlson B.C.: On computing elliptic integrals and functions. J. Math. Phys. 44, 332–345 (1965)

    Google Scholar 

  8. Carlson B.C.: Elliptic integrals of the first kind. SIAM J. Math. Anal. 8, 231–242 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  9. Carlson B.C.: Short proofs of three theorems on elliptic integrals. SIAM J. Math. Anal. 9, 524–528 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  10. Carlson B.C.: Computing elliptic integrals by duplication. Numer. Math. 33, 1–16 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  11. Carlson B.C., Notis E.M.: Algorithm 577. Algorithms for incomplete elliptic integrals. ACM Trans. Math. Software 7, 398–403 (1981)

    Article  MATH  Google Scholar 

  12. Cody W.J.: Chebyshev approximations for the complete elliptic integrals K and E. Math. Comp. 19, 105–112 (1965a)

    MATH  MathSciNet  Google Scholar 

  13. Cody W.J.: Chebyshev polynomial expansions of complete elliptic integrals K and E. Math. Comp. 19, 249–259 (1965b)

    MATH  MathSciNet  Google Scholar 

  14. Cody W.J.: Corrigenda: Chebyshev approximations for the complete elliptic integrals K and E. Math. Comp. 20, 207 (1966)

    Google Scholar 

  15. Didonato A.R., Hershey A.V.: New formulas for computing incomplete elliptic integrals of the first and second kind. J. Assoc. Comput. Mach. 6, 515–526 (1959)

    MATH  MathSciNet  Google Scholar 

  16. Fukushima T.: Gaussian element formulation of short-axis-mode rotation of a rigid body. Astron. J. 136, 649–653 (2008a)

    Article  Google Scholar 

  17. Fukushima T.: Canonical and universal elements of rotational motion of triaxial rigid body. Astron. J. 136, 1728–1735 (2008b)

    Article  Google Scholar 

  18. Fukushima T.: Fast computation of Jacobian elliptic functions and incomplete elliptic integrals for constant values of elliptic parameter and elliptic characteristic. Celest. Mech. Dyn. Astron. 105, 245–260 (2009a)

    Article  MathSciNet  Google Scholar 

  19. Fukushima T.: Efficient solution of initial-value problem of torque-free rotation. Astron. J. 137, 210–218 (2009b)

    Article  Google Scholar 

  20. Fukushima T.: Fast computation of complete elliptic integrals and Jacobian elliptic functions. Celest. Mech. Dyn. Astron. 105, 305–328 (2009c)

    Article  MathSciNet  Google Scholar 

  21. Fukushima T., Ishizaki H.: Numerical computation of incomplete elliptic integrals of a general form. Celest. Mech. Dyn. Astron. 59, 237–251 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  22. Glynn, E.F.: efg’s Computer Lab and Reference Library. http://www.efg2.com/Lab/Library/mathematics.htm (2009)

  23. Hastings C. Jr: Approximations for Digital Computers. Princeton University Press, Princeton (1955)

    MATH  Google Scholar 

  24. Hofsommer D.J., van de Riet R.P.: On the Numerical Calculation of Elliptic Integrals of the First and Second Kind and Elliptic Functions of Jacobi. Numer. Math. 5, 291–302 (1963)

    Article  MATH  MathSciNet  Google Scholar 

  25. Morris, A.H. Jr.: NSWC Library of Mathematics Subroutines, Tech. Rep. NSWCDD/TR-92/425, 107-110. Naval Surface Warfare Center, Dahlgren (1993)

  26. Press W.H., Flannery B.P., Teukolsky S.A., Vetterling W.T.: Numerical Recipes: the Art of Scientific Computing. Cambridge University Press, Cambridge (1986)

    Google Scholar 

  27. Press W.H., Teukolsky S.A., Vetterling W.T., Flannery B.P.: Numerical Recipes: the Art of Scientific Computing, 3rd edn. Cambridge University Press, Cambridge (2007)

    MATH  Google Scholar 

  28. Vande Vel H.: On the series expansion method for computing incomplete elliptic integrals of the first and second kinds. Math. Comp. 23, 61–69 (1969)

    MathSciNet  Google Scholar 

  29. Wolfram S.: The Mathematica Book, 5th edn. Wolfram Research Inc./Cambridge University Press, Cambridge (2003)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. National Astronomical Observatory of Japan, 2-21-1, Ohsawa, Mitaka, Tokyo, 181-8588, Japan

    Toshio Fukushima

Authors
  1. Toshio Fukushima
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Toshio Fukushima.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fukushima, T. Fast computation of incomplete elliptic integral of first kind by half argument transformation. Numer. Math. 116, 687–719 (2010). https://doi.org/10.1007/s00211-010-0321-8

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-010-0321-8

Mathematics Subject Classification (2000)