Implementation of a Near-Optimal Complex Root Clustering Algorithm

Let F (z) be an arbitrary complex polynomial. We introduce the local root clustering problem, to compute a set of natural ε-clusters of roots of F (z) in some box region B0 in the complex plane. This may be viewed as an extension of the classical root isolation problem. Our contribution is twofold: we provide an efficient certified subdivision algorithm for this problem, and we provide a bit-complexity analysis based on the local geometry of the root clusters. Our computational model assumes that arbitrarily good approximations of the coefficients of F are provided by means of an oracle at the cost of reading the coefficients. Our algorithmic techniques come from a companion paper [3] and are based on the Pellet test, Graeffe and Newton iterations, and are independent of Schönhage's splitting circle method. Our algorithm is relatively simple and promises to be efficient in practice.

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“…Postscript: Our clustering algorithm has been implemented in [13].…”

Section: Discussionmentioning

confidence: 99%

Complexity Analysis of Root Clustering for a Complex Polynomial

Becker¹,

Sagraloff

Sharma

et al. 2016

Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation

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“…The variant (42) of Ehrlich's iterations has been proposed, tested, and analyzed in [15]. The tests demonstrated the same fast global convergence (right from the start) as for Ehrlich's original iterations (40), (41); the analysis showed significant numerical benefits of root-finding by using expression (42), and the authors traced the origin of this approach to [13], but now we complement it with the reduction of updating parameters v j to FMM.…”

Section: Acceleration With Fmmmentioning

confidence: 89%

Old and New Nearly Optimal Polynomial Root-Finders

Pan

2019

Computer Algebra in Scientific Computing

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Univariate polynomial root-finding has been studied for four millennia and is still the subject of intensive research. Hundreds of efficient algorithms for this task have been proposed. Two of them are nearly optimal. The first one, proposed in 1995, relies on recursive factorization of a polynomial, is quite involved, and has never been implemented. The second one, proposed in 2016, relies on subdivision iterations, was implemented in 2018, and promises to be practically competitive, although user's current choice for univariate polynomial root-finding is the package MPSolve, proposed in 2000, revised in 2014, and based on Ehrlich's functional iterations. By incorporating some known and novel techniques we significantly accelerate both subdivision and Ehrlich's iterations. Moreover our acceleration of the known subdivision root-finders is dramatic in the case of sparse input polynomials. Our techniques can be of some independent interest for the design and analysis of polynomial root-finders.

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“…us, these subdivision algorithms were "effective". For two parallel accounts of this development, see [17,25] for the case of real roots, and to [4,5,14] for complex roots. What is the power conferred by subdivision?…”

Section: How To Derive Effective Algorithmsmentioning

confidence: 99%

Effective Subdivision Algorithm for Isolating Zeros of Real Systems of Equations, with Complexity Analysis

Yap

2019

Proceedings of the 2019 International Symposium on Symbolic and Algebraic Computation

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We describe a new algorithm Miranda for isolating the simple zeros of a function f : R n → R n within a box B 0 ⊆ R n . e function f and its partial derivatives must have interval forms, but need not be polynomial. Our subdivision-based algorithm is "effective" in the sense that our algorithmic description also specifies the numerical precision that is sufficient to certify an implementation with any standard BigFloat number type. e main predicate is the Moore-Kioustelides (MK) test, based on Miranda's eorem (1940). Although the MK test is well-known, this paper appears to be the first synthesis of this test into a complete root isolation algorithm.We provide a complexity analysis of our algorithm based on intrinsic geometric parameters of the system. Our algorithm and complexity analysis are developed using 3 levels of description (Abstract, Interval, Effective). is methodology provides a systematic pathway for achieving effective subdivision algorithms in general.

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Implementation of a Near-Optimal Complex Root Clustering Algorithm

Cited by 27 publications

References 15 publications

Complexity Analysis of Root Clustering for a Complex Polynomial

Complexity Analysis of Root Clustering for a Complex Polynomial

Old and New Nearly Optimal Polynomial Root-Finders

Effective Subdivision Algorithm for Isolating Zeros of Real Systems of Equations, with Complexity Analysis

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