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A global Newton method for the zeros of cylinder functions. (English) Zbl 0921.65037
The author constructs a Newton-Raphson iterative (NRI) method based on the monotonic function $f_\nu(x)= x^{2\nu-1}H_\nu(x)$; where $H_\nu(x)= C_\nu(x)/C_{\nu-1}(x)$ and $C_\nu(x)= \cos\alpha J_\nu(x)- \sin\alpha Y_\nu(x)$ is a general cylinder function. He then proves a theorem asserting the convergence of the NRI method in order to evaluate the positive zeros of any $C_\nu(x)$ for all real $\nu$ and $\alpha$ and for any starting value $x_0> 0$. Some applications of the method are discussed and a very simple algorithm to compute the zeros of the Bessel function $J_\nu(x)$ (for all real $\nu)$ is explicitly given.

65H05Single nonlinear equations (numerical methods)
65D20Computation of special functions, construction of tables
33C10Bessel and Airy functions, cylinder functions, ${}_0F_1$
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