Article Dans Une Revue Journal of Symbolic Computation Année : 2015

Nearly Optimal Refinement of Real Roots of a Univariate Polynomial

Résumé

We assume that a real square-free polynomial A has a degree d, a maximum coefficient bitsize τ and a real root lying in an isolating interval and having no nonreal roots nearby (we quantify this assumption). Then we combine the {\em Double Exponential Sieve} algorithm (also called the {\em Bisection of the Exponents}), the bisection, and Newton iteration to decrease the width of this inclusion interval by a factor of t=2L. The algorithm has Boolean complexity OB (d2τ+dL). This substantially decreases the known bound OB (d3+d2L) and is optimal up to a polylogarithmic factor. Furthermore we readily extend our algorithm to support the same upper bound on the complexity of the refinement of r real roots, for any rd, by incorporating the known efficient algorithms for multipoint polynomial evaluation. The main ingredient for the latter is an efficient algorithm for (approximate) polynomial division; we present a variation based on structured matrix computation with quasi-optimal Boolean complexity.
Fichier principal
Vignette du fichier
pt-refine-jsc.pdf (428) Télécharger le fichier
Origine Fichiers produits par l'(les) auteur(s)
Loading...

Dates et versions

hal-00960896 , version 1 (19-03-2014)
hal-00960896 , version 2 (28-12-2015)

Identifiants

Citer

Victor Y. Pan, Elias Tsigaridas. Nearly Optimal Refinement of Real Roots of a Univariate Polynomial. Journal of Symbolic Computation, 2015, 74, pp.181-204. ⟨10.1016/j.jsc.2015.06.009⟩. ⟨hal-00960896v2⟩
572 Consultations
404 Téléchargements

Altmetric

Partager

More