ANewDsc swMATH ID: 19984 Software Authors: Kobel, Alexander; Rouillier, Fabrice; Sagraloff, Michael Description: Computing real roots of real polynomials dots and now for real! 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. Homepage: https://anewdsc.mpi-inf.mpg.de/ Keywords: Descartes method; Newton’s method; approximate arithmetic; certified computation; real roots; root finding; root isolation; univariate polynomials Related Software: ISOLATE; Ccluster; Arb; SqFreeEVAL; SLV; Maple; Divisor; Macaulay2; Lgp; HomotopyContinuation; Julia; NAG4M2; Bertini; PHCpack; HOM4PS; SINGULAR; RegularChains; na10; Kronecker; NumGfun Cited in: 12 Publications all top 5 Cited by 23 Authors 2 Imbach, Rémi 2 Sagraloff, Michael 2 Yap, Chee-Keng 1 Becker, Ruben 1 Bouzidi, Yacine 1 Dai, Liyun 1 Fan, Zhe 1 Kobel, Alexander 1 Manevich, Dimitri 1 Melczer, Stephen 1 Pan, Victor Yakovlevich 1 Plaumann, Daniel 1 Poteaux, Adrien 1 Pouget, Marc 1 Quadrat, Alban 1 Rangarajan, Ramsharan 1 Rouillier, Fabrice 1 Salvy, Bruno 1 Sharma, Vikram 1 Tsigaridas, Elias P. 1 Wang, Dongming 1 Xia, Bican 1 Zhang, Hanwen all top 5 Cited in 7 Serials 2 Journal of Computational and Applied Mathematics 2 Journal of Symbolic Computation 1 Le Matematiche 1 Computational Optimization and Applications 1 Journal of Systems Science and Complexity 1 Mathematics in Computer Science 1 ACM Communications in Computer Algebra all top 5 Cited in 11 Fields 6 Numerical analysis (65-XX) 6 Computer science (68-XX) 3 Functions of a complex variable (30-XX) 2 Algebraic geometry (14-XX) 1 Combinatorics (05-XX) 1 Field theory and polynomials (12-XX) 1 Commutative algebra (13-XX) 1 Real functions (26-XX) 1 Calculus of variations and optimal control; optimization (49-XX) 1 Operations research, mathematical programming (90-XX) 1 Systems theory; control (93-XX) Citations by Year