Fast parallel algorithm for determining all roots of a polynomial with real roots

Given a polynomial p(z) of degree n with m bit integer coefficients and an integer μ, the problem of determining all its roots with error less than 2^-μ is considered. It is shown that this problem is in the class NC if p(z) has all real roots. Some very interesting properties of a Sturm sequence of a polynomial with distinct real roots are proved and used in the design of a fast parallel algorithm for this problem. Using Newton identities and a novel numerical integration scheme for evaluating a contour integral to high precision, this algorithm determines good approximations to the linear factors of p(z).