Counting Roots of a Polynomial in a Convex Compact Region by Means of Winding Number Calculation via Sampling

Abstract

In this paper, we propose a novel efficient algorithm for calculating winding numbers, aiming at counting the number of roots of a given polynomial in a convex region on the complex plane. This algorithm can be used for counting and exclusion tests in a subdivision algorithms for polynomial root-finding, and would be especially useful in application scenarios where high-precision polynomial coefficients are hard to obtain but we succeed with counting already by using polynomial evaluation with lower precision. We provide the pseudo code of the algorithm, proof of its correctness as well as estimation of its complexity.