A near-optimal subdivision algorithm for complex root isolation based on the Pellet test and Newton iteration

Abstract

We describe a subdivision algorithm for isolating the complex roots of a polynomial F ∈ C [x]. Given an oracle that provides approximations of each of the coefficients of F to any absolute error bound and given an arbitrary square B in the complex plane containing only simple roots of F , our algorithm returns disjoint isolating disks for the roots of F in B.Our complexity analysis bounds the absolute error to which the coefficients of F have to be provided, the total number of iterations, and the overall bit complexity. It further shows that the complexity of our algorithm is controlled by the geometry of the roots in a near neighborhood of the input square B, namely, the number of roots, their absolute values and pairwise distances. The number of subdivision steps is near-optimal. For the benchmark problem, namely, to isolate all the roots of a polynomial of degree n with integer coefficients of bit size less than τ , our algorithm needsÕ(n 3 + n 2 τ ) bit operations, which is comparable to the record bound of Pan (2002). It is the first time that such a bound has been achieved using subdivision methods, and independent of divide-and-conquer techniques such as Schönhage's splitting circle technique.Our algorithm uses the quadtree construction of Weyl (1924) with two key ingredients: using Pellet's Theorem (1881) combined with Graeffe iteration, we derive a "soft-test" to count the number of roots in a disk. Using Schröder's modified Newton operator combined with bisection, in a form inspired by the quadratic interval method from Abbot (2006), we achieve quadratic convergence towards root clusters. Relative to the divide-conquer algorithms, our algorithm is quite simple with the potential of being practical. This paper is self-contained: we provide pseudo-code for all subroutines used by our algorithm.

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

“…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?…”

Implementation of a Near-Optimal Complex Root Clustering Algorithm

“…(1) requires to know 2 N−1 + 1 coefficients of f [1] ∆ , 2 N−2 + 1 coefficients of f ∆ such that 2 N−i ≤ d, and it is possible to do it more efficiently that with eq. (1) (for instance with the formula given in definition 2 of [2]).…”

“…The local root clustering algorithm for analytic functions of [16] has termination proof, but no complexity analysis. By restricting f (z) to a polymomial, Becker et al [2] succeeded in giving an algorithm and also its complexity analysis based on the geometry of the roots. When applied to the benchmark problem, where f (z) is an integer polynomial of degree d with L-bit coefficients, the algorithm can isolate all the roots of f (z) with bit complexity O(d 2 (L + d)).…”

“…The clustering algorithm studied in this paper comes from [1], which in turn is based on [2]. Previously, the Pan-Schönhage algorithm has achieved near-optimal bounds with divide-and-conquer methods [13], but [2,1] was the first subdivision algorithm to achieve the near-optimal bound for complex roots. For real roots, Sagraloff-Mehlhorn [15] had earlier achieved near-optimal bound via 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).…”