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Subdivision methods for solving polynomial equations

Bernard Mourrain 1 Jean-Pascal Pavone 1
1 GALAAD - Geometry, algebra, algorithms
CRISAM - Inria Sophia Antipolis - Méditerranée , UNS - Université Nice Sophia Antipolis (... - 2019), CNRS - Centre National de la Recherche Scientifique : UMR6621
Abstract : This paper presents a new algorithm for solving a system of polynomials, in a domain of $ ^n$. It can be seen as an improvement of the Interval Projected Polyhedron algorithm proposed by Sherbrooke and Patrikalakis . It uses a powerful reduction strategy based on univariate root finder using Bernstein basis representation and Descarte's rule. We analyse the behavior of the method, from a theoretical point of view, shows that for simple roots, it has a local quadratic convergence speed and gives new bounds for the complexity of approximating real roots in a box of $ ^n$. The improvement of our approach, compared with classical subdivision methods, is illustrated on geometric modeling applications such as computing intersection points of implicit curves, self-intersection points of rational curves, and on the classical parallel robot benchmark problem.
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Bernard Mourrain, Jean-Pascal Pavone. Subdivision methods for solving polynomial equations. Journal of Symbolic Computation, Elsevier, 2009, 44 (3), pp.292-306. ⟨inria-00070350⟩

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