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 methodology is based on homotopy. Newton's method, and Rouché's theorem. The homotopy method provides a point close to the cluster. The Newton method in the neighborhood of a zero cluster gives the number of zeros in the cluster, where it is sufficient to know three points of the Newton sequence in order to generate an open disk susceptible to contain all zeros of the cluster. An inclusion test based on a punctual version of the Rouché theorem validates the previous step. Numerical examples illustrate the effectiveness of the algorithm.
scientific article; zbMATH DE number 1558137
scientific article