New practical advances in polynomial root clustering

TL;DR

This paper introduces improved polynomial root clustering algorithms that are robust, efficient, and require fewer evaluations, especially benefiting sparse polynomials and simplifying contour conditions.

Contribution

The paper presents a new counting test based on polynomial and derivative evaluations, reducing complexity and evaluation points, and relaxes contour conditions for root counting.

Findings

Efficient root counting with evaluations at about log(d) points.

Robust algorithms for polynomials with multiple roots and black-box coefficients.

Enhanced subdivision algorithms for real coefficient polynomials.

Abstract

We report an ongoing work on clustering algorithms for complex roots of a univariate polynomial $p$ of degree $d$ with real or complex coefficients. As in their previous best subdivision algorithms our root-finders are robust even for multiple roots of a polynomial given by a black box for the approximation of its coefficients, and their complexity decreases at least proportionally to the number of roots in a region of interest (ROI) on the complex plane, such as a disc or a square, but we greatly strengthen the main ingredient of the previous algorithms. Namely our new counting test essentially amounts to the evaluation of a polynomial $p$ and its derivative $p'$, which is a major benefit, e.g., for sparse polynomials $p$. Moreover with evaluation at about $\log(d)$ points (versus the previous record of order $d$) we output correct number of roots in a disc whose contour has no roots of $p$ nearby. Moreover we greatly soften the latter requirement versus the known subdivision algorithms. Our second and less significant contribution concerns subdivision algorithms for polynomials with real coefficients. Our tests demonstrate the power of the proposed algorithms.