A global Newton method for the zeros of cylinder functions
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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.
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- M. Abramowitz and I.A. Stegun, eds., Handbook of Mathematical Functions (Dover, 1972).Google Scholar
- A. Gray and G.B. Mathews, A Treatise on Bessel Functions and Their Applications to Physics (Dover, 1966).Google Scholar
- K.S. Kölbig, CERNLIB-short writeups, subroutine DBZEJY(C345), library MATHLIB; http:// wwwcn.cern.ch/shortwrupsdir/index.html.Google Scholar
- W.H. Press, S.A. Teukolski, W.T. Vetterling and B.P. Flannery, Numerical Recipes in Fortran 77 (Cambridge University Press, 1992).Google Scholar
- 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
- SLATEC public domain library; gopher://archives.math.utk.edu/11/software/multi-platform/ SLATEC.Google Scholar
- N.M. Temme, Special Functions: An Introduction to the Classical Functions of Mathematical Physics (Wiley, New York, 1996).Google Scholar
- G.N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University Press, London, 1962).Google Scholar