| Recent work by Michael S. and Kurt introduces an asymptotically fast subdivision algorithm, denoted ANewDsc, for isolating the real roots of a univariate real polynomial. The method combines Descartes' Rule of Signs to test intervals for the existence of roots, Newton iteration to speed up convergence against clusters of roots, and approximate computation to decrease the required precision. ANewDsc achieves record bounds on the worst-case complexity for the considered problem, matching the complexity of Pan's method for computing all complex roots and improving upon the complexity of other subdivision methods by several magnitudes.
In this talk, we report on an implementation of ANewDsc on top of the RS root isolator. At the current stage, RS is the most efficient realization of the classical Descartes method, and also constitutes the default real root solver for polynomials in Maple. We describe several crucial design changes within ANewDsc as well as within RS that have led to a high-performance implementation without harming the theoretical complexity of the underlying algorithm. Testing our implementation on numerous benchmark instances shows that the theoretical gain in performance of ANewDsc over other subdivision methods also transfers into practice. Experiments show that our new implementation outperforms RS by magnitudes for notoriously hard instances with clustered roots. For all other instances, there is almost no overhead due to the integration of additional techniques. |
