Computing Real Roots of Real Polynomials -- An Efficient Method Based on Descartes' Rule of Signs and Newton Iteration

摘要

Computing the real roots of a polynomial is a fundamental problem of computational algebra. We describe a variant of the Descartes method that isolates the real roots of any real square-free polynomial given through coefficient oracles. A coefficient oracle provides arbitrarily good approximations of the coefficients. The bit complexity of the algorithm matches the complexity of the best algorithm known, and the algorithm is simpler than this algorithm. The algorithm derives its speed from the combination of Descartes method with Newton iteration. Our algorithm can also be used to further refine the isolating intervals to an arbitrary small size. The complexity of root refinement is nearly optimal.