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Root Radii and Subdivision for Polynomial Root-Finding

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Computer Algebra in Scientific Computing (CASC 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12865))

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Abstract

We depart from our approximation of 2000 of all root radii of a polynomial, which has readily extended Schönhage’s efficient algorithm of 1982 for a single root radius. We revisit this extension, advance it, based on our simple but novel idea, and yield significant practical acceleration of the known near optimal subdivision algorithms for complex and real root-finding of user’s choice. We achieve this by means of significant saving of exclusion tests and Taylor’s shifts, which are the bottleneck of subdivision root-finders. This saving relies on our novel recipes for the initialization of root-finding iterations of independent interest. We demonstrate our practical progress with numerical tests, provide extensive analysis of the resulting algorithms, and show that, like the preceding subdivision root-finders, they support near optimal Boolean complexity bounds.

Victor’s work is supported by NSF Grants CCF 1563942 and CCF 1733834 and PSC CUNY Award 63677 00 51.

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Notes

  1. 1.

    https://github.com/rimbach/Ccluster.

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Author information

Authors and Affiliations

  1. Courant Institute of Mathematical Sciences, New York University, New York, USA

    Rémi Imbach

  2. Lehman College and Graduate Center of City University of New York, New York, USA

    Victor Y. Pan

Authors
  1. Rémi Imbach
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  2. Victor Y. Pan
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Corresponding author

Correspondence to Rémi Imbach .

Editor information

Editors and Affiliations

  1. Université de 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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Imbach, R., Pan, V.Y. (2021). Root Radii and Subdivision for Polynomial Root-Finding. In: Boulier, F., England, M., Sadykov, T.M., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2021. Lecture Notes in Computer Science(), vol 12865. Springer, Cham. https://doi.org/10.1007/978-3-030-85165-1_9

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  • DOI: https://doi.org/10.1007/978-3-030-85165-1_9

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-85164-4

  • Online ISBN: 978-3-030-85165-1

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