  In our quest for the design, the analysis and the implementation of a
subdivision algorithm for finding the complex roots of univariate polynomials
given by oracles for their evaluation, we present sub-algorithms allowing
substantial acceleration of subdivision for complex roots clustering for such
polynomials. We rely on Cauchy sums which approximate power sums of the roots
in 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 the
cluster of root it contains, called $\varepsilon$-compression. To demonstrate
the efficiency of our algorithms, we combine them in a prototype root
clustering algorithm. For computing clusters of roots of polynomials that can
be evaluated fast, our implementation competes advantageously with user's
choice for root finding, MPsolve.
