A reliable argument principle algorithm to find the number of zeros of an analytic function in a bounded domain
type: Item
instance of: scholarly article
title: A reliable argument principle algorithm to find the number of zeros of an analytic function in a bounded domain
zbMATH Open document ID: 0628.65015
DOI: 10.1007/BF01395882
author: Xingren Ying
author: I. N. Katz
published in: Numerische Mathematik
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/133269
review text: The argument principle is a natural and simple method to determine the number of zeros of an analytic function f(z) in a bounded domain D. N, the number of zeros (counting multiplicities) of f(z), is 1/2\(\pi\) times the change in Arg f(z) as z moves along the closed contour \(\partial D\). Since the range of Arg f(z) is (-\(\pi\),\(\pi\) ] a critical point in the computational procedure is to assure that the discretization of \(\partial D\), \(\{z_ i,i=1,...,M\}\), is such that \(| \Delta _{[z_ i,z_{i+1}]}Arg f(z)| \leq \pi.\) Discretization control which may violate this inequality can lead to an unreliable algorithm. Mathematical theorems derived for the discretization of \(\partial D\) lead to a completely reliable algorithm to compute N. This algorithm also treats in an elementary way the case when a zero is on or near the contour \(\partial D\). Numerical examples are given for the reliable algorithm formulated here and it is pointed out in these examples how inadequate discretization control can lead to failure of other algorithms.
Mathematics Subject Classification ID: 65E05
Mathematics Subject Classification ID: 30C15
zbMATH DE Number: 4022465
zbMATH Keywords: argument principle
zbMATH Keywords: number of zeros of an analytic function
zbMATH Keywords: reliable algorithm
zbMATH Keywords: numerical examples
zbMATH Keywords: discretization control
zbMATH Keywords: global information
zbMATH Keywords: nonlinear equation solver
zbMATH Keywords: solver for all roots
MaRDI profile type: Publication
cites work: A Numerical Method for Locating the Zeros of an Analytic Function
cites work: Q4072022
cites work: On Locating All Zeros of an Analytic Function within a Bounded Domain by a Revised Delves/Lyness Method
cites work: A simple reliable solver for all the roots of a nonlinear function in a given domain
cites work: A uniform formulation for the calculation of stress singularities in the plane elasticity of a wedge composed of multiple isotropic materials
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description: scientific article; zbMATH DE number 4022465