Approximating the zeros of analytic functions by the exclusion algorithm
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.
scientific article; zbMATH DE number 519505