A global Newton method for the zeros of cylinder functions
type: Item
instance of: scholarly article
title: A global Newton method for the zeros of cylinder functions
zbMATH Open document ID: 0921.65037
DOI: 10.1023/A:1019125616736
author: Javier Segura
published in: Numerical Algorithms
publication date: 1999-03-22T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 65H05
Mathematics Subject Classification ID: 65D20
Mathematics Subject Classification ID: 33C10
zbMATH DE Number: 1268409
zbMATH Keywords: Newton-Raphson iterative method
zbMATH Keywords: cylinder function
zbMATH Keywords: convergence
zbMATH Keywords: positive zeros
zbMATH Keywords: algorithm
zbMATH Keywords: Bessel function
reviewed by: Nácere Hayek
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1268409