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Univariate Polynomial Real Root Isolation: Continued Fractions Revisited

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Algorithms – ESA 2006 (ESA 2006)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4168))

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Abstract

We present algorithmic, complexity and implementation results concerning real root isolation of integer univariate polynomials using the continued fraction expansion of real numbers. We improve the previously known bound by a factor of dĪ„, where d is the polynomial degree and Ī„ bounds the coefficient bitsize, thus matching the current record complexity for real root isolation by exact methods. Namely, the complexity bound is \({{\widetilde{\mathcal{O}}_B}(d^4 \tau^2)}\) using a standard bound on the expected bitsize of the integers in the continued fraction expansion. We show how to compute the multiplicities within the same complexity and extend the algorithm to non square-free polynomials. Finally, we present an efficient open-source C++ implementation in the algebraic library synaps, and illustrate its efficiency as compared to other available software. We use polynomials with coefficient bitsize up to 8000 and degree up to 1000.

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Authors and Affiliations

  1. Department of Informatics and Telecommunications, National Kapodistrian University of Athens, HELLAS

    Elias P. Tsigaridas & Ioannis Z. Emiris

Authors
  1. Elias P. Tsigaridas
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  2. Ioannis Z. Emiris
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Editors and Affiliations

  1. Microsoft Research, One Microsoft Way, Redmond, WA 98052, USA and School of Computer Science, Tel-Aviv University, 69978, Tel-Aviv, Israel

    Yossi Azar

  2. Department of Computer Science, University of Leicester, University Road, LE1 7RH, Leicester, UK

    Thomas Erlebach

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Tsigaridas, E.P., Emiris, I.Z. (2006). Univariate Polynomial Real Root Isolation: Continued Fractions Revisited. In: Azar, Y., Erlebach, T. (eds) Algorithms – ESA 2006. ESA 2006. Lecture Notes in Computer Science, vol 4168. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11841036_72

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  • DOI: https://doi.org/10.1007/11841036_72

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-38875-3

  • Online ISBN: 978-3-540-38876-0

  • eBook Packages: Computer ScienceComputer Science (R0)

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