Computing clustered close-roots of univariate polynomials

Abstract

Given a univariate polynomial having well-separated clusters of close roots, we give a method of computing close roots in a cluster simultaneously, without computing other roots. We first determine the position and the size of the cluster, as well as the number of close roots contained. Then, we move the origin to a near center of the cluster and perform the scale transformation so that the cluster is enlarged to be of size O(1). These operations transform the polynomial to a very characteristic one. We modify Durand-Kerner's method so as to compute only the close roots in the cluster. The method is very efficient because we can discard most terms of the transformed polynomial. We also give a formula of quite tight error bound. We show high efficiency of our method by empirical experiments.