A verified method for bounding clusters of zeros of analytic functions

Abstract

In this paper, we propose a verified method for bounding clusters of zeros of analytic functions. Our method gives a disk that contains a cluster of m zeros of an analytic function f ( z ) . Complex circular arithmetic is used to perform a validated computation of n-degree Taylor polynomial p ( z ) of f ( z ) . Some well known formulae for bounding zeros of a polynomial are used to compute a disk containing a cluster of zeros of p ( z ) . A validated computation of an upper bound for Taylor remainder series of f ( z ) and a lower bound of p ( z ) on a circle are performed. Based on these results, Rouché's theorem is used to verify that the disk contains the cluster of zeros of f ( z ) . This method is efficient in computation of the initial disk of a method for finding validated polynomial factor of an analytic function. Numerical examples are presented to illustrate the efficiency of the proposed method.