Accelerated Subdivision for Clustering Roots of Polynomials Given by Evaluation Oracles

Abstract

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 approximation of the power sums of the roots in a fixed complex disc by Cauchy sums, each computed in a small number of evaluations of an input polynomial and its derivative, that is, in a polylogarithmic number 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.