Numerical Algorithms

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

A global Newton method for the zeros of cylinder functions

  • Javier Segura


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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Copyright information

© Kluwer Academic Publishers 1998

Authors and Affiliations

  • Javier Segura

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