Accelerated Newton Iteration for Roots of Black Box Polynomials

We study the problem of computing the largest root of a real rooted polynomial p(x) to within error 'z' given only black box access to it, i.e., for any x, the algorithm can query an oracle for the value of p(x), but the algorithm is not allowed access to the coefficients of p(x). A folklore result for this problem is that the largest root of a polynomial can be computed in O(n log (1/z)) polynomial queries using the Newton iteration. We give a simple algorithm that queries the oracle at only O(log n log(1/z)) points, where n is the degree of the polynomial. Our algorithm is based on a novel approach for accelerating the Newton method by using higher derivatives.