A descartes algorithm for polynomials with bit-stream coefficients

  • Authors:

  • Arno Eigenwillig;Lutz Kettner;Werner Krandick;Kurt Mehlhorn;Susanne Schmitt;Nicola Wolpert

  • Affiliations:

  • Max-Planck-Institut für Informatik, Saarbrücken, Germany;Max-Planck-Institut für Informatik, Saarbrücken, Germany;Dept. of Computer Science, Drexel University, Philadelphia, PA;Max-Planck-Institut für Informatik, Saarbrücken, Germany;Max-Planck-Institut für Informatik, Saarbrücken, Germany;Max-Planck-Institut für Informatik, Saarbrücken, Germany

  • Venue:
  • CASC'05 Proceedings of the 8th international conference on Computer Algebra in Scientific Computing
  • Year:
  • 2005

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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 |qn|≥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(n4 (log(1/ρ) + τ)2) bit operations.