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Abstract

We describe a subdivision algorithm for isolating the complex roots of a polynomial $F\in\mathbb{C}[x]$. Our model assumes that each coefficient of $F$ has an oracle to return an approximation to any absolute error bound. Given any box $\mathcal{B}$ in the complex plane containing only simple roots of $F$, our algorithm returns disjoint isolating disks for the roots in $\mathcal{B}$. Our complexity analysis bounds the absolute error to which the coefficients of $F$ have to be provided, the total number of iterations, and the overall bit complexity. This analysis shows that the complexity of our algorithm is controlled by the geometry of the roots in a near neighborhood of the input box $\mathcal{B}$, namely, the number of roots and their pairwise distances. The number of subdivision steps is near-optimal. For the \emph{benchmark problem}, namely, to isolate all the roots of an integer polynomial of degree $n$ with coefficients of bitsize less than $\tau$, our algorithm needs $\tilde{O}(n^3+n^2\tau)$ bit operations, which is comparable to the record bound of Pan (2002). It is the first time that such a bound has been achieved using subdivision methods, and independent of divide-and-conquer techniques such as Sch\"onhage's splitting circle technique. Our algorithm uses the quadtree construction of Weyl (1924) with two key ingredients: using Pellet's Theorem (1881) combined with Graeffe iteration, we derive a soft test to count the number of roots in a disk. Using Newton iteration combined with bisection, in a form inspired by the quadratic interval method from Abbot (2006), we achieve quadratic convergence towards root clusters. Relative to the divide-conquer algorithms, our algorithm is simple with the potential of being practical. This paper is self-contained: we provide pseudo-code for all subroutines used by our algorithm.