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Improved complexity bounds for real root isolation using Continued Fractions

Abstract : We consider the problem of isolating the real roots of a square-free polynomial with integer coefficients using (variants of) the continued fraction algorithm (CF). We introduce a novel way to compute a lower bound on the positive real roots of univariate polynomials. This allows us to derive a worst case bound of $\sOB( d^6 + d^4\tau^2 + d^3\tau^2)$ for isolating the real roots of a polynomial with integer coefficients using the classic variant \cite{Akritas:implementation} of CF, where $d$ is the degree of the polynomial and $\tau$ the maximum bitsize of its coefficients. This improves the previous bound of Sharma \cite{sharma-tcs-2008} by a factor of $d^3$ and matches the bound derived by Mehlhorn and Ray \cite{mr-jsc-2009} for another variant of CF; it also matches the worst case bound of the subdivision-based solvers.
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https://hal.inria.fr/inria-00524834
Contributor : Elias Tsigaridas <>
Submitted on : Wednesday, June 1, 2011 - 6:48:41 PM
Last modification on : Saturday, September 17, 2016 - 1:20:17 AM
Long-term archiving on: : Sunday, December 4, 2016 - 2:33:58 AM

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  • HAL Id : inria-00524834, version 4
  • ARXIV : 1010.2006

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Elias Tsigaridas. Improved complexity bounds for real root isolation using Continued Fractions. [Research Report] RR2010, Aarhus Universitet. 2010. ⟨inria-00524834v4⟩

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