Polynomial root clustering and explicit deflation

TL;DR

This paper improves polynomial root-finding algorithms by focusing on roots within a specific region and introducing deflation techniques to reduce computational complexity, especially for polynomials with real coefficients.

Contribution

It presents two novel enhancements to existing subdivision-based root finders, including a region-focused approach for real-coefficient polynomials and a deflation method to lower polynomial degree and computational effort.

Findings

Enhanced efficiency for roots in a specified ROI.

Effective handling of polynomials with real coefficients.

Reduced computational cost through polynomial deflation.

Abstract

We seek complex roots of a univariate polynomial $P$ with real or complex coefficients. We address this problem based on recent algorithms that use subdivision and have a nearly optimal complexity. They are particularly efficient when only roots in a given Region Of Interest (ROI) are sought. We propose two improvements for root finders. The first one is applied to polynomials having only real coefficients; their roots are either real or appear in complex conjugate pairs. We show how to adapt the subdivision scheme to focus the computational effort on the imaginary positive part of the ROI. In our second improvement we deflate $P$ to decrease its degree and the arithmetic cost of the subdivision.