Polynomial Real 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 and is challenged to approximate them faster. The challenge is known for long time, and the subject has been intensively studied. The real roots can be approximated at a low computational cost if the polynomial has no non-real roots, but for high degree polynomials, non-real roots are typically much more numerous than the real ones. The bounds on the Boolean cost of 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. By revisiting the algorithm of 1982 by Schönhage for the approximation of the root radii, that is, the distances between the roots and the origin, we obtain substantial progress: 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. Our numerical tests with benchmark polynomials performed with the IEEE standard double precision show that the resulting 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. At the end we point out some promising directions to the isolation of complex roots and root clusters.