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Section: New Results

Worst case complexity of the Continued Fraction (CF) algorithm.

In [16] we consider the problem of isolating the real roots of a square-free polynomial with integer coefficients using the classic variant of the continued fraction algorithm (CF), introduced by Akritas. We compute a lower bound on the positive real roots of univariate polynomials using exponential search. This allows us to derive a worst case bound of $\stackrel{˜}{O}\left({d}^{4}{\tau }^{2}\right)$ for isolating the real roots of a polynomial with integer coefficients using the classic variant 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 by a factor of ${d}^{3}$ and matches the bound derived by Mehlhorn and Ray for another variant of CF which is combined with subdivision; it also matches the worst case bound of the classical subdivision-based solvers STURM, DESCARTES, and BERNSTEIN.