A Descartes Algorithms for Polynomials with Bit-Stream Coefficients

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

doi:10.4230/DagSemProc.06021.3

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A Descartes Algorithms for Polynomials with Bit-Stream Coefficients

Authors Kurt Mehlhorn, Arno Eigenwillig, Lutz Kettner, Werner Krandick, Susanne Schmitt, Nicola Wolpert

Part of: Volume: Dagstuhl Seminar Proceedings, Volume 6021
Part of: Series: Dagstuhl Seminar Proceedings (DagSemProc)
License: Creative Commons Attribution 4.0 International license
Publication Date: 2006-09-13

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Document Identifiers

DOI: 10.4230/DagSemProc.06021.3
URN: urn:nbn:de:0030-drops-7157

Subject Classification

Keywords

Root Isolation
Interval Arithmetic
Descartes Algorithm

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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 $q(x) = q_nx^n+ldots+q_0$ with root
  separation $
ho$, coefficients $abs{q_n}ge1$ and $abs{q_i} le 2^	au$,
  it needs coefficient approximations to $O(n(log(1/
ho) + 	au))$
  bits after the binary point and has an expected cost of
  $O(n^4 (log(1/
ho) + 	au)^2)$ bit operations.

Cite As Get BibTex

Kurt Mehlhorn, Arno Eigenwillig, Lutz Kettner, Werner Krandick, Susanne Schmitt, and Nicola Wolpert. A Descartes Algorithms for Polynomials with Bit-Stream Coefficients. In Reliable Implementation of Real Number Algorithms: Theory and Practice. Dagstuhl Seminar Proceedings, Volume 6021, pp. 1-12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006) https://doi.org/10.4230/DagSemProc.06021.3

Author Details

Kurt Mehlhorn

Arno Eigenwillig

Lutz Kettner

Werner Krandick

Susanne Schmitt

Nicola Wolpert

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