BIT 0006-3835/98/3804-0101 $12.00 1998, Vol. 38, No. 2, pp. 101–104 c  Swets & Zeitlinger On locating clusters of zeros of analytic functions * P. KRAVANJA 1 , T. SAKURAI 2 , and M. VAN BAREL 3 † 1 Katholieke Universiteit Leuven, Department of Computer Science Celestijnenlaan 200 A, B-3001 Heverlee, Belgium. email: Peter.Kravanja@na-net.ornl.gov 2 University of Tsukuba, Institute of Information Sciences and Electronics Tsukuba 305, Japan. email: sakurai@is.tsukuba.ac.jp 3 Katholieke Universiteit Leuven, Department of Computer Science Celestijnenlaan 200 A, B-3001 Heverlee, Belgium. email: Marc.VanBarel@cs.kuleuven.ac.be Abstract. Given an analytic function f and a Jordan curve γ that does not pass through any zero of f , we consider the problem of computing all the zeros of f that lie inside γ, together with their respective multiplicities. Our principal means of obtaining infor- mation about the location of these zeros is a certain symmetric bilinear form that can be evaluated via numerical integration along γ. If f has one or several clusters of zeros, then the mapping from the ordinary moments associated with this form to the zeros and their respective multiplicities is very ill-conditioned. We present numerical methods to calculate the centre of a cluster and its weight, i.e., the arithmetic mean of the zeros that form a certain cluster and the total number of zeros in this cluster, respectively. Our approach relies on formal orthogonal polynomials and rational in- terpolation at roots of unity. Numerical examples illustrate the effectiveness of our techniques. AMS subject classification: Primary 65H05; Secondary 65E05. Key words: zeros of analytic functions, clusters of zeros, logarithmic residue integrals, formal orthogonal polynomials, rational interpolation. 1 Introduction Let W be a simply connected region in C, f : W → C analytic in W , and γ a positively oriented Jordan curve in W that does not pass through any zero of f . We consider the problem of computing all the zeros of f that lie in the interior of γ , together with their respective multiplicities. Our principal means of obtaining information about the location of these zeros is a certain symmetric * Received July 1993. Revised January 1994. † The first author was supported by a grant from the Flemish Institute for the Promotion of Scientific and Technological Research in the Industry (IWT). This work is part of the projects #G.0261.96 “Counting and Computing all Isolated Solutions of Systems of Nonlinear Equations” and #G.0278.97 “Orthogonal Systems and their Applications” funded by the Fund for Scientific Research, Flanders.