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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



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Author Details

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

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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

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.
Keywords
  • Root Isolation
  • Interval Arithmetic
  • Descartes Algorithm

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