Univariate polynomial real root isolation: Continued Fractions revisited

Abstract

We present algorithmic, complexity and implementation results concerning real root isolation of integer univariate polynomials using the continued fraction expansion of real algebraic numbers. One motivation is to explain the method's good performance in practice. We improve the previously known bound by a factor of $d τ$, where $d$ is the polynomial degree and $τ$ bounds the coefficient bitsize, thus matching the current record complexity for real root isolation by exact methods. Namely, the complexity bound is $\sOB(d^4 τ^2)$ using the standard bound on the expected bitsize of the integers in the continued fraction expansion. We show how to compute the multiplicities within the same complexity and extend the algorithm to non square-free polynomials. Finally, we present an efficient open-source \texttt{C++} implementation in the algebraic library \synaps, and illustrate its efficiency as compared to other available software. We use polynomials with coefficient bitsize up to 8000 and degree up to 1000.