Accelerated Newton Iteration: Roots of Black Box Polynomials and Matrix Eigenvalues

We study the problem of computing the largest root of a real rooted polynomial p(x) to within error ε 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/ε)) polynomial queries using the Newton iteration. We give a simple algorithm that queries the oracle at only O (log n log(1/ε)) 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. As a special case, we consider the problem of computing the top eigenvalue of a symmetric matrix in ‘n×n to within error ε in time polynomial in the input description, i.e., the number of bits to describe the matrix and log(1/ε). Well-known methods such as the power iteration and Lanczos iteration incur running time polynomial in 1/ε, while Gaussian elimination takes Ω(n4) bit operations. As a corollary of our main result, we obtain a Õ ( n log(‖A‖F /ε) ) bit complexity algorithm to compute the top eigenvalue of the matrix A or to check if it is approximately PSD (A − I). ∗Supported by the Simons Collaboration on Algorithms and Geometry. †Supported in part by NSF awards CCF-1217793 and EAGER-1555447. 1 ar X iv :1 51 1. 03 18 6v 2 [ cs .D S] 3 J an 2 01 6