Computing Real Roots of Real Polynomials ... and now For Real!

TL;DR

This paper presents an efficient implementation of the ANewDsc algorithm for real root isolation of polynomials, demonstrating significant practical performance improvements over existing methods, especially for challenging clustered roots.

Contribution

The paper introduces a high-performance implementation of the ANewDsc algorithm integrated with RS root isolator, maintaining theoretical complexity while outperforming competitors in practice.

Findings

ANewDsc outperforms RS and other solvers on hard instances with clustered roots.

Implementation achieves significant speedups in practical benchmarks.

Theoretical complexity gains translate into real-world performance improvements.

Abstract

Very recent work 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. It 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 the article at hand, we report on an implementation of ANewDsc on top of the RS root isolator. RS is a highly efficient realization of the classical Descartes method and currently serves as the default real root solver in Maple. We describe crucial design changes within ANewDsc and RS that led to a high-performance implementation without harming the theoretical complexity of the underlying algorithm. With an excerpt of our extensive collection of benchmarks, available online at http://anewdsc.mpi-inf.mpg.de/, we illustrate that the theoretical gain in performance of ANewDsc over other subdivision methods also transfers into practice. These experiments also show that our new implementation outperforms both RS and mature competitors by magnitudes for notoriously hard instances with clustered roots. For all other instances, we avoid almost any overhead by integrating additional optimizations and heuristics.