8 research outputs found
An Elimination Method for Solving Bivariate Polynomial Systems: Eliminating the Usual Drawbacks
We present an exact and complete algorithm to isolate the real solutions of a
zero-dimensional bivariate polynomial system. The proposed algorithm
constitutes an elimination method which improves upon existing approaches in a
number of points. First, the amount of purely symbolic operations is
significantly reduced, that is, only resultant computation and square-free
factorization is still needed. Second, our algorithm neither assumes generic
position of the input system nor demands for any change of the coordinate
system. The latter is due to a novel inclusion predicate to certify that a
certain region is isolating for a solution. Our implementation exploits
graphics hardware to expedite the resultant computation. Furthermore, we
integrate a number of filtering techniques to improve the overall performance.
Efficiency of the proposed method is proven by a comparison of our
implementation with two state-of-the-art implementations, that is, LPG and
Maple's isolate. For a series of challenging benchmark instances, experiments
show that our implementation outperforms both contestants.Comment: 16 pages with appendix, 1 figure, submitted to ALENEX 201
On the Boolean complexity of real root refinement
International audienceWe assume that a real square-free polynomial has a degree , a maximum coefficient bitsize 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 . The algorithm has Boolean complexity . Our algorithms support the same complexity bound for the refinement of roots, for any
Nearly Optimal Refinement of Real Roots of a Univariate Polynomial
International audienceWe assume that a real square-free polynomial has a degree , a maximum coefficient bitsize 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 . The algorithm has Boolean complexity . This substantially decreases the known bound 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 real roots, for any , 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
Geometric algorithms for algebraic curves and surfaces
This work presents novel geometric algorithms dealing with algebraic curves and surfaces of arbitrary degree. These algorithms are exact and complete â they return the mathematically true result for all input instances. Efficiency is achieved by cutting back expensive symbolic computation and favoring combinatorial and adaptive numerical methods instead, without spoiling exactness in the overall result. We present an algorithm for computing planar arrangements induced by real algebraic curves. We show its efficiency both in theory by a complexity analysis, as well as in practice by experimental comparison with related methods. For the latter, our solution has been implemented in the context of the Cgal library. The results show that it constitutes the best current exact implementation available for arrangements as well as for the related problem of computing the topology of one algebraic curve. The algorithm is also applied to related problems, such as arrangements of rotated curves, and arrangments embedded on a parameterized surface. In R3, we propose a new method to compute an isotopic triangulation of an algebraic surface. This triangulation is based on a stratification of the surface, which reveals topological and geometric information. Our implementation is the first for this problem that makes consequent use of numerical methods, and still yields the exact topology of the surface.Diese Arbeit stellt neue Algorithmen fĂŒr algebraische Kurven und FlĂ€chen von beliebigem Grad vor. Diese Algorithmen liefern fĂŒr alle Eingaben das mathematisch korrekte Ergebnis. Wir erreichen Effizienz, indem wir aufwendige symbolische Berechnungen weitesgehend vermeiden, und stattdessen kombinatorische und adaptive numerische Methoden einsetzen, ohne die Exaktheit des Resultats zu zerstören. Der Hauptbeitrag ist ein Algorithmus zur Berechnung von planaren Arrangements, die durch reelle algebraische Kurven induziert sind. Wir weisen die Effizienz des Verfahrens sowohl theoretisch durch eine KomplexitĂ€tsanalyse, als auch praktisch durch experimentelle Vergleiche nach. Dazu haben wir unser Verfahren im Rahmen der Softwarebibliothek Cgal implementiert. Die Resultate belegen, dass wir die zur Zeit beste verfĂŒgbare exakte Software bereitstellen. Der Algorithmus wird zur Arrangementberechnung rotierter Kurven, oder fĂŒr Arrangements auf parametrisierten OberflĂ€chen eingesetzt. Im R3 geben wir ein neues Verfahren zur Berechnung einer isotopen Triangulierung einer algebraischen OberflĂ€che an. Diese Triangulierung basiert auf einer Stratifizierung der OberflĂ€che, die topologische und geometrische Informationen berechnet. Unsere Implementierung ist die erste fĂŒr dieses Problem, welche numerische Methoden konsequent einsetzt, und dennoch die exakte Topologie der OberflĂ€che liefert
On the Complexity of Reliable Root Approximation
This is an author-prepared version of the article. The original publication is available at www.springerlink.com This work addresses the problem of computing a certified Ç«-approximation of all real roots of a square-free integer polynomial. We proof an upper bound for its bit complexity, by analyzing an algorithm that first computes isolating intervals for the roots, and subsequently refines them using Abbottâs Quadratic Interval Refinement method. We exploit the eventual quadratic convergence of the method. The threshold for an interval width with guaranteed quadratic convergence speed is bounded by relating it to well-known algebraic quantities.