Root Radii and Subdivision for Polynomial Root-Finding

TL;DR

This paper introduces a novel approach to polynomial root-finding that significantly accelerates subdivision algorithms by optimizing exclusion tests and Taylor's shifts, supported by extensive analysis and numerical validation.

Contribution

It advances the extension of root radius approximation algorithms, providing practical acceleration and new initialization methods for subdivision root-finders.

Findings

Significant reduction in exclusion tests and Taylor's shifts needed

Achieved near optimal Boolean complexity bounds

Validated improvements through numerical experiments

Abstract

We depart from our approximation of 2000 of all root radii of a polynomial, which has readily extended Sch{\"o}nhage's efficient algorithm of 1982 for a single root radius. We revisit this extension, advance it, based on our simple but novel idea, and yield significant practical acceleration of the known near optimal subdivision algorithms for complex and real root-finding of user's choice. We achieve this by means of significant saving of exclusion tests and Taylor's shifts, which are the bottleneck of subdivision root-finders. This saving relies on our novel recipes for the initialization of root-finding iterations of independent interest. We demonstrate our practical progress with numerical tests, provide extensive analysis of the resulting algorithms, and show that, like the preceding subdivision root-finders, they support near optimal Boolean complexity bounds.