On the worst-case arithmetic complexity of approximating zeros of polynomials
Journal of Complexity
On condition numbers and the distance to the nearest III-posted problem
Numerische Mathematik
Polynomial and matrix computations (vol. 1): fundamental algorithms
Polynomial and matrix computations (vol. 1): fundamental algorithms
Computational Complexity of Two-Dimensional Regions
SIAM Journal on Computing
Complexity and real computation
Complexity and real computation
A Global Bisection Algorithm for Computing the Zeros of Polynomials in the Complex Plane
Journal of the ACM (JACM)
Euro-Par'13 Proceedings of the 19th international conference on Parallel Processing
Finding the number of roots of a polynomial in a plane region using the winding number
Computers & Mathematics with Applications
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Many methods to compute the winding number of plane curves have been proposed, often with the aim of counting the number of roots of polynomials (or, more generally, zeros of analytic functions) inside some region by using the principle of argument. In this paper we propose another method, which are not based on numerical integration, but on discrete geometry. We give conditions that ensure its correct behavior, and a complexity bound based on the distance of the curve to singular cases. Besides, we provide a modification of the algorithm that detects the proximity to a singular case of the curve. If this proximity is such that the number of operations required grows over certain threshold, set by the user, the modified algorithm returns without winding number computation, but with information about the distance to singular case. When the method is applied to polynomials, this information refers to the localization of the roots placed near the curve.