Accelerated Subdivision for Clustering Roots of Polynomials given by Evaluation Oracles

TL;DR

This paper introduces accelerated subdivision algorithms for finding complex roots of polynomials given by evaluation oracles, utilizing Cauchy sums for efficient root clustering and demonstrating competitive performance.

Contribution

It presents novel sub-algorithms for root exclusion, counting, radius approximation, and disc contraction, significantly speeding up root clustering for polynomials with fast evaluation oracles.

Findings

Algorithms efficiently compute root clusters using Cauchy sums.

Prototype implementation outperforms traditional root-finding methods in certain cases.

Methods are effective for polynomials with fast evaluation oracles.

Abstract

In our quest for the design, the analysis and the implementation of a subdivision algorithm for finding the complex roots of univariate polynomials given by oracles for their evaluation, we present sub-algorithms allowing substantial acceleration of subdivision for complex roots clustering for such polynomials. We rely on Cauchy sums which approximate power sums of the roots in a fixed complex disc and can be computed in a small number of evaluations --polylogarithmic in the degree. We describe root exclusion, root counting, root radius approximation and a procedure for contracting a disc towards the cluster of root it contains, called $\varepsilon$-compression. To demonstrate the efficiency of our algorithms, we combine them in a prototype root clustering algorithm. For computing clusters of roots of polynomials that can be evaluated fast, our implementation competes advantageously with user's choice for root finding, MPsolve.