Improved complexity bounds for real root isolation using Continued Fractions
Résumé
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(d6+d4τ2+d3τ2) 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 τ the maximum bitsize of its coefficients. This improves the previous bound of Sharma \cite{sharma-tcs-2008} by a factor of d3 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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