Complexity Analysis of Root Clustering for a Complex Polynomial

TL;DR

This paper introduces an efficient, certified subdivision algorithm for local root clustering of complex polynomials, extending classical root isolation with complexity analysis based on root cluster geometry.

Contribution

It presents a novel algorithm for local root clustering of complex polynomials along with a bit-complexity analysis considering root cluster geometry.

Findings

Algorithm is efficient and practically promising.

Provides a bit-complexity analysis based on local root geometry.

Extends classical root isolation to local clustering.

Abstract

Let $F(z)$ be an arbitrary complex polynomial. We introduce the local root clustering problem, to compute a set of natural $\varepsilon$-clusters of roots of $F(z)$ in some box region $B_0$ in the complex plane. This may be viewed as an extension of the classical root isolation problem. Our contribution is two-fold: we provide an efficient certified subdivision algorithm for this problem, and we provide a bit-complexity analysis based on the local geometry of the root clusters. Our computational model assumes that arbitrarily good approximations of the coefficients of $F$ are provided by means of an oracle at the cost of reading the coefficients. Our algorithmic techniques come from a companion paper (Becker et al., 2018) and are based on the Pellet test, Graeffe and Newton iterations, and are independent of Sch\"onhage's splitting circle method. Our algorithm is relatively simple and promises to be efficient in practice.