Complexity Analysis of Root Clustering for a Complex Polynomial

Abstract

Let F (z) be an arbitrary complex polynomial. We introduce the local root clustering problem, to compute a set of natural ε-clusters of roots of F (z) in some box region B0 in the complex plane. This may be viewed as an extension of the classical root isolation problem. Our contribution is twofold: 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 [3] and are based on the Pellet test, Graeffe and Newton iterations, and are independent of Schönhage's splitting circle method. Our algorithm is relatively simple and promises to be efficient in practice.

Effective Subdivision Algorithm for Isolating Zeros of Real Systems of Equations, with Complexity Analysis

“…But its complexity would not be a function of the roots in 2B 0 . Such local complexity behavior are provable in the univariate case (e.g., [4]), and we will also show similar local complexity in the algorithm of this paper.…”

“…us, these subdivision algorithms were "effective". For two parallel accounts of this development, see [17,25] for the case of real roots, and to [4,5,14] for complex roots. What is the power conferred by subdivision?…”

Counting Roots of a Polynomial in a Convex Compact Region by Means of Winding Number Calculation via Sampling

“…(1) be a polynomial of degree d with real or complex coefficients. Counting its roots (with their multiplicity) in a fixed domain (such as an interior of a polygon or a disc) is a fundamental problem with an important application to devising efficient root-finders for p(z) on the complex plane, particularly subdivision algorithms, proposed by Hermann Weyl in [10] and then extended and improved in [4], [3], [8], [7], [1], and [2] 1 and recently implemented in [6]. We propose a new algorithm for counting the roots in a fixed convex region on the complex plane by expressing their number as the winding number computed along the boundary of the region, provided that the boundary was sufficiently isolated from the roots of p(z).…”

“…Winding number algorithms have been proposed for counting roots in a disc as parts of root-finding algorithms by Henrici and Gargantini in [4], then by Henrici in [3] and by Renegar in [8]. Pan in [7] used root-radii algorithm by Schönhage [9] for counting roots in a disc, and Becker et al in [1] and [2] performed counting based on Pellet's theorem.…”

Planar Minimization Diagrams via Subdivision with Applications to Anisotropic Voronoi Diagrams

“…Assumption (A) is not essential; see [BCGY12] and [BSS*16] for ways to remove Assumption (A) without giving up our use of soft (numerical) methods The idea is to compute an embedded planar multigraph G * = ( V *, E *) such that there exists a simply‐connected set B * satisfying (1 – ε) B 0 ⊆ B * ⊆ (1 + ε) B 0 , and G * is is ε‐isotopic to Vor( X ) ∩ B *. But in this paper, we focus on generalizing the notion of ε‐approximation in another direction: to take full advantage of the ε parameter, we want the ability to replace a set of ε‐close Voronoi vertices by a single “super vertex” in V .…”