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In our quest for the design, the analysis and the implementation of asubdivision algorithm for finding the complex roots of univariate polynomialsgiven by oracles for their evaluation, we present sub-algorithms allowingsubstantial acceleration of subdivision for complex roots clustering for suchpolynomials. We rely on Cauchy sums which approximate power sums of the rootsin a fixed complex disc and can be computed in a small number of evaluations--polylogarithmic in the degree. We describe root exclusion, root counting,root radius approximation and a procedure for contracting a disc towards thecluster of root it contains, called $\varepsilon$-compression. To demonstratethe efficiency of our algorithms, we combine them in a prototype rootclustering algorithm. For computing clusters of roots of polynomials that canbe evaluated fast, our implementation competes advantageously with user'schoice for root finding, MPsolve.

Accelerated Subdivision for Clustering Roots of Polynomials given by Evaluation Oracles