Finding a cluster of zeros of univariate polynomials

A method to compute an accurate approximation For a zero cluster of a complex univariate polynomial is presented. The theoretical background on which this method is based deals with homotopy, Newton's method, and Rouche's theorem. First the homotopy method provides a point close to the zero cluster. Next the analysis of the behaviour of thr Newton method in the neighbourhood of a zero cluster gives the number of zeros in this cluster. in this case, it is sufficient to know three points of the Newton sequence in order to generate an open disk susceptible to contain all the zeros of the cluster, Finally, an inclusion test based on a punctual version of the Rouche theorem validates the previous step. A specific implementation of this algorithm is given. Numerical experiments illustrate how this method works and some figures are displayed. (C) 2000 Academic Press.