Polynomial Root Isolation by Means of Root Radii Approximation

Abstract

Univariate polynomial root-finding is a classical subject, still important for modern computing. Frequently one seeks just the real roots of a real coefficient polynomial. They can be approximated at a low computational cost if the polynomial has no nonreal roots, but for high degree polynomials, nonreal roots are typically much more numerous than the real ones. The challenge is known for long time, and the subject has been intensively studied. The Boolean cost bounds for the refinement of the simple and isolated real roots have been decreased to nearly optimal, but the success has been more limited at the stage of the isolation of real roots. We obtain substantial progress by applying the algorithm of of 1982 by Schoenhage for the approximation of the root radii, that is, the distances of the roots to the origin. Namely we isolate the simple and well-conditioned real roots of a polynomial at the Boolean cost dominated by the nearly optimal bounds for the refinement of such roots. We also extend our algorithm to the isolation of complex, possibly multiple, roots and root clusters staying within the same (nearly optimal) asymptotic Boolean cost bound. Our numerical tests with benchmark polynomials performed with the IEEE standard double precision show that our nearly optimal real root-finder is practically promising. Our techniques are simple, and their power and application range may increase in combination with the known efficient methods.