A global Newton method for the zeros of cylinder functions
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.
scientific article; zbMATH DE number 1268409