Approximating the zeros of analytic functions by the exclusion algorithm
type: Item
instance of: scholarly article
title: Approximating the zeros of analytic functions by the exclusion algorithm
zbMATH Open document ID: 0790.65038
DOI: 10.1007/BF02149763
author: Jean-Claude Yakoubsohn
published in: Numerical Algorithms
publication date: 1994-04-10T00:00:00Z
review text: The author establishes an iterative algorithm for approximating those zeros of a given function \(f\) holomorphic on \(\mathbb{C}\) lying within a prescribed compact set \(E_ 0\subset \mathbb{C}\). The main tool is an exclusion function \(m\) satisfying \(m(z_ 0) = 0\) iff \(f(z_ 0) = 0\) and s.t. \(f(z_ 0) \neq 0\) implies \(f(z) \neq 0\) for every \(z \in B(z_ 0,m(z_ 0))\). The boundary of \(E_ 0\) is supposed to be a Jordan curve. After choosing a point \(z_ 0 \in E_ 0\) satisfying \(m(z_ 0) \neq 0\) the set \(E_ 0\) is replaced by \(E_ 1 := E_ 0\setminus B(z_ 0,m(z_ 0))\). \(E_ 1\) is a compact subset of \(E_ 0\) but might split into several components each having a Jordan curve as its boundary. The author describes an algorithm for determining the components of \(E_ 1\) and explains how to choose a point \(z_ 1\) lying on the boundary of \(E_ 1\) which ought to replace \(z_ 0\). The algorithm stops after a finite number of steps and the according set \(E_ n\) splits into some components each of diameter less than a given bound and containing a zero of \(f\). The paper is completed by estimates of the optimality and a couple of examples.
Mathematics Subject Classification ID: 65H05
Mathematics Subject Classification ID: 65E05
Mathematics Subject Classification ID: 26C10
zbMATH DE Number: 519505
zbMATH Keywords: analytic function
zbMATH Keywords: polynomial
zbMATH Keywords: iterative algorithm
zbMATH Keywords: zeros
zbMATH Keywords: exclusion function
zbMATH Keywords: optimality
reviewed by: Hartje Kriete
MaRDI profile type: Publication
cites work: Q3241303
cites work: Q4074972
cites work: Localization of an algebraic hypersurface by the exclusion algorithm
cites work: Computing the real roots of a polynomial by the exclusion algorithm
cites work: Q5285697
description: scientific article; zbMATH DE number 519505