Numerical Algorithms

, Volume 18, Issue 3–4, pp 259–276 | Cite as

A global Newton method for the zeros of cylinder functions

  • Javier Segura
Article

Abstract

The zeros of cylinder functions Cu(x)=cos α, Ju(x) - sin α, Yu(x) coincide with those of the ratios Hu(x)=Cu(x)/Cu-1(x) except, perhaps, at x = 0. We show monotonicity properties of Hu(x) and fu(x) = x2v-1Hu(x) and their derivatives for x > 0. We then build a Newton-Raphson iterative method based on the monotonic function fu(x) which is shown to be convergent, for any real values of u and α and any starting value x0 > 0, to an sth positive root c,s of Cu(x) = 0, s being such that c,s and x0 belong to the same interval (cu-1,s', cu -1,s'+1].

We also show applications of the method. In particular, taking advantage of the fact that the ratio Hu(x) for first kind Bessel functions Ju(x) can be evaluated by using a continued fraction, a very simple algorithm is built; it becomes especially efficient for low values of u and s and it allows the evaluation of the real zeros for arbitrary orders u, positive or negative.

zeros of cylinder functions Bessel functions global Newton method primary 33C10 

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References

  1. [1]
    M. Abramowitz and I.A. Stegun, eds., Handbook of Mathematical Functions (Dover, 1972).Google Scholar
  2. [2]
    Á. Elbert, An approximation for the zeros of Bessel functions, Numer. Math. 59 (1991) 647–657.MATHMathSciNetGoogle Scholar
  3. [3]
    Á. Elbert and A. Laforgia, On the square of the zeros of Bessel functions, SIAM J. Math. Anal. 15 (1984) 206–212.MATHMathSciNetCrossRefGoogle Scholar
  4. [4]
    A. Gray and G.B. Mathews, A Treatise on Bessel Functions and Their Applications to Physics (Dover, 1966).Google Scholar
  5. [5]
    K.S. Kölbig, CERNLIB-short writeups, subroutine DBZEJY(C345), library MATHLIB; http:// wwwcn.cern.ch/shortwrupsdir/index.html.Google Scholar
  6. [6]
    R. Piessens, Chebyshev series approximations for the zeros of the Bessel functions, J. Comput. Phys. 53 (1984) 188–192.MATHMathSciNetCrossRefGoogle Scholar
  7. [7]
    W.H. Press, S.A. Teukolski, W.T. Vetterling and B.P. Flannery, Numerical Recipes in Fortran 77 (Cambridge University Press, 1992).Google Scholar
  8. [8]
    J. Segura and A. Gil, ELF and GNOME: two tiny codes to evaluate the real zeros of the Bessel functions of the first kind for real orders, submitted to Comput. Phys. Commun.Google Scholar
  9. [9]
    SLATEC public domain library; gopher://archives.math.utk.edu/11/software/multi-platform/ SLATEC.Google Scholar
  10. [10]
    N.M. Temme, An algorithm with ALGOL 60 program for the computation of the zeros of ordinary Bessel functions and those of their derivatives, J. Comput. Phys. 32 (1979) 270–279.MATHCrossRefGoogle Scholar
  11. [11]
    N.M. Temme, Special Functions: An Introduction to the Classical Functions of Mathematical Physics (Wiley, New York, 1996).Google Scholar
  12. [12]
    I.J. Thompson and A.R. Barnett, Coulomb and Bessel functions of complex arguments and order, J. Comput. Phys. 64 (1986) 490–509.MATHMathSciNetCrossRefGoogle Scholar
  13. [13]
    G.N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University Press, London, 1962).Google Scholar

Copyright information

© Kluwer Academic Publishers 1998

Authors and Affiliations

  • Javier Segura

There are no affiliations available

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