A Geometrical Root Finding Method for Polynomials, with Complexity Analysis

TL;DR

This paper introduces a geometric root-finding method for polynomials based on winding numbers, allowing targeted searches within specific regions, with proven correctness and analyzed computational complexity.

Contribution

It presents a novel geometric approach using winding numbers for polynomial root finding that overcomes limitations of iterative methods and enables region-specific searches.

Findings

Method is formally proven correct.

Algorithm can be parallelized.

Complexity analysis provided.

Abstract

The usual methods for root finding of polynomials are based on the iteration of a numerical formula for improvement of successive estimations. The unpredictable nature of the iterations prevents to search roots inside a pre-specified region of complex plane. This work describes a root finding method that overcomes this disadvantage. It is based on the winding number of plane curves, a geometric construct that is related to the number of zeros of each polynomial. The method can be restrained to search inside a pre-specified region. It can be easily parallelized. Its correctness is formally proven, and its computational complexity is also analyzed.