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International Workshop on Computer Algebra in Scientific Computing

CASC 2014: Computer Algebra in Scientific Computing pp 202–217Cite as

A Note on Global Newton Iteration Over Archimedean and Non-Archimedean Fields

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8660))

Abstract

In this paper, we study iterative methods on the coefficients of the rational univariate representation (RUR) of a given algebraic set, called a global Newton iterations. We compare two natural approaches to define locally quadratically convergent iterations: the first one involves Newton iteration applied to the approximate roots individually and then interpolation to find the RUR of these approximate roots; the second one considers the coefficients in the exact RUR as zeroes of a high dimensional map defined by polynomial reduction and applies Newton iteration on this map. We prove that over fields with a p-adic valuation these two approaches give the same iteration function. However, over fields equipped with the usual Archimedean absolute value they are not equivalent. In the latter case, we give explicitly the iteration function for both approaches. Finally, we analyze the parallel complexity of the different versions of the global Newton iteration, compare them, and demonstrate that they can be efficiently computed. The motivation for this study comes from the certification of approximate roots of overdetermined and singular polynomial systems via the recovery of an exact RUR from approximate numerical data.

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Authors and Affiliations

  1. North Carolina State University, USA

    Jonathan D. Hauenstein & Agnes Szanto

  2. Lehman College - City University of New York, USA

    Victor Y. Pan

Authors
  1. Jonathan D. Hauenstein
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  2. Victor Y. Pan
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  3. Agnes Szanto
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Editor information

Editors and Affiliations

  1. Laboratory of Information Technologies (LIT), Joint Institute for Nuclear Research, 141980, Dubna, Russia

    Vladimir P. Gerdt

  2. Institut für Mathematik, Universität Kassel, Heinrich-Plett-Straße 40, 34132, Kassel, Germany

    Wolfram Koepf

  3. Institut für Mathematik, Universität Kassel, 34132, Kassel, Germany

    Werner M. Seiler

  4. Institute of Theoretical and Applied Mechanics, Russian Academy of Sciences, 630090, Novosibirsk, Russia

    Evgenii V. Vorozhtsov

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Hauenstein, J.D., Pan, V.Y., Szanto, A. (2014). A Note on Global Newton Iteration Over Archimedean and Non-Archimedean Fields. In: Gerdt, V.P., Koepf, W., Seiler, W.M., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2014. Lecture Notes in Computer Science, vol 8660. Springer, Cham. https://doi.org/10.1007/978-3-319-10515-4_15

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  • DOI: https://doi.org/10.1007/978-3-319-10515-4_15

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-10514-7

  • Online ISBN: 978-3-319-10515-4

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