Counting Roots of a Polynomial in a Convex Compact Region by Means of Winding Number Calculation via Sampling
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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-fnding, and would be especially useful in application scenarios where high-precision polynomial cofficients 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 as well as a proof of its correctness.
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