A Descartes Algorithm for Polynomials with Bit-Stream Coefficients

  • Arno Eigenwillig
  • Lutz Kettner
  • Werner Krandick
  • Kurt Mehlhorn
  • Susanne Schmitt
  • Nicola Wolpert
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3718)

Abstract

The Descartes method is an algorithm for isolating the real roots of square-free polynomials with real coefficients. We assume that coefficients are given as (potentially infinite) bit-streams. In other words, coefficients can be approximated to any desired accuracy, but are not known exactly. We show that a variant of the Descartes algorithm can cope with bit-stream coefficients. To isolate the real roots of a square-free real polynomial \(q(x)=q_{n^{x^{n}}}+...+q_{0}\) with root separation ρ, coefficients |q n | ≥ 1 and \(|q_{i}|\leq 2^{\tau}\), it needs coefficient approximations to O(n(log(1/ρ) + τ)) bits after the binary point and has an expected cost of O(n 4 (log(1/ρ) + τ)2) bit operations.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Arno Eigenwillig
    • 1
  • Lutz Kettner
    • 1
  • Werner Krandick
    • 2
  • Kurt Mehlhorn
    • 1
  • Susanne Schmitt
    • 1
  • Nicola Wolpert
    • 1
  1. 1.Max-Planck-Institut für InformatikSaarbrückenGermany
  2. 2.Dept. of Computer ScienceDrexel UniversityPhiladelphiaUSA

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