Fast real and complex root-finding methods for well-conditioned polynomials

Abstract

Given a polynomial of degree and a bound on a condition number of , we present the first root-finding algorithms that return all its real and complex roots with a number of bit operations quasi-linear in log 2 ( ). More precisely, several condition numbers can be defined depending on the norm chosen on the coefficients of the polynomial. Let ( )We call the condition number associated with a perturbation of the the hyperbolic condition number ℎ , and the one associated with a perturbation of the the elliptic condition number . For each of these condition numbers, we present algorithms that find the real and the complex roots of in log 2 ( ) polylog(log( )) bit operations.Our algorithms are well suited for random polynomials since ℎ (resp. ) is bounded by a polynomial in with high probability if the (resp. the ) are independent, centered Gaussian variables of variance 1.