Polynomial Root-Finding and Algebraic Eigenvalue Problem

Abstract: Univariate polynomial root-finding has been studied for four millennia and very intensively in the last decades. Our new near-optimal root-finders approximate all zeros of a polynomial p almost as fast as one accesses its coefficients with the precision required for the solution within a prescribed error bound. Furthermore, our root-finders can be applied to a black box polynomial, defined by an oracle (black box subroutine) for its evaluation rather than by its coefficients. Due to this feature our root-finders support approximation of the eigenvalues of a matrix in a record Las Vegas expected bit operation time and are particularly fast for a polynomial that can be evaluated fast such as the sum of a few shifted monomials or a Mandelbrot-like polynomial defined by a recurrence. Our divide and conquer algorithm of ACM STOC 1995 is the only other known near-optimal polynomial root-finder, but it extensively uses the coefficients, is quite involved, and has never been implemented, while according to extensive numerical experiments with standard test polynomials, already a slower initial implementation of our new root-finders competes with user's choice package of root-finding subroutines MPSolve and supersedes it more and more significantly as the degree of a polynomial grows large. We elaborate upon the design and analysis of our algorithms, comment on their potential heuristic acceleration, and briefly cover polynomial root-finding by means of functional iterations. Our techniques can be of independent interest.