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Acceleration of Subdivision Root-Finding for Sparse Polynomials

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Computer Algebra in Scientific Computing (CASC 2020)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12291))

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Abstract

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. A recent front-running algorithm relies on subdivision iterations. Already its initial implementation of 2018 has competed for user’s choice for root-finding in a region that contains a small number of roots. Recently, we significantly accelerated the basic blocks of these iterations, namely root-counting and exclusion tests. In  [18], we solidified this approach and made our acceleration dramatic in the case of sparse polynomials and other ones defined by a black box for their fast evaluation. Our techniques are novel and should be of independent interest. In the present paper and its companion  [19], we expose a substantial part of that work.

This research has been supported by NSF Grants CCF–1563942 and CCF–1733834 and PSC CUNY Award 69813 00 48.

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Notes

  1. 1.

    It becomes the second such root-finder. The first one, of  [13, 16], has provided nearly optimal solution also for numerical factorization of a polynomial into the product of its linear factors, which is a problem of high independent interest  [18].

  2. 2.

    Unlike paper  [21], this result is deduced in [18] from Theorem 2, which is also the basis for correctness proof of Sect. 5 for our probabilistic root-counter.

  3. 3.

    One can extend the algorithm by applying Algorithm 1a to a disc \(D(0, \theta )\) for smaller \(\theta >1\) and modifying bound (9) accordingly. See a refined version of this algorithm in [18].

  4. 4.

    One can slightly strengthen our estimates based on the observation that \(|\widehat{p}_0|^2\) and \(||\widehat{\mathbf{p}}_1||^2\) are \(\chi ^2\)-functions of dimension 1 and \(q-1\), respectively.

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Acknowledgments

This research has been supported by NSF Grants CCF–1563942 and CCF–1733834 and by PSC CUNY Award 69813 00 48.

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Authors and Affiliations

  1. Department of Computer Science, Lehman College of the City University of New York, Bronx, Ny, 10468, USA

    Victor Y. Pan

  2. Ph.D. Programs in Mathematics and Computer Science, The Graduate Center of the City University of New York, New York, NY, 10036, USA

    Victor Y. Pan

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  1. Victor Y. Pan
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Correspondence to Victor Y. Pan .

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Editors and Affiliations

  1. University of Lille, Villeneuve d’Ascq, France

    François Boulier

  2. Coventry University, Coventry, UK

    Matthew England

  3. Plekhanov Russian University of Economics, Moscow, Russia

    Timur M. Sadykov

  4. Institute of Theoretical and Applied Mechanics, Novosibirsk, Russia

    Evgenii V. Vorozhtsov

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Pan, V.Y. (2020). Acceleration of Subdivision Root-Finding for Sparse Polynomials. In: Boulier, F., England, M., Sadykov, T.M., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2020. Lecture Notes in Computer Science(), vol 12291. Springer, Cham. https://doi.org/10.1007/978-3-030-60026-6_27

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  • DOI: https://doi.org/10.1007/978-3-030-60026-6_27

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