A verified method for bounding clusters of zeros of analytic functions

Authors:
Xiao-Ming Niu;Tetsuya Sakurai;Hiroshi Sugiura
Affiliations:
Department of Computer Science & Media Technology, Ritsumeikan University, Kusatsu, Japan;Department of Computer Science University of Tsukuba, Tsukuba, Japan;Department of Mathematical Sciences Faculty of Mathematical Sciences and Information Engineering, Nanzan Univerisity, Seto, Japan
Venue:
Journal of Computational and Applied Mathematics - Special issue: Scientific computing, computer arithmetic, and validated numerics (SCAN 2004)
Year:
2007

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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.