168 research outputs found
Clustering Complex Zeros of Triangular Systems of Polynomials
This paper gives the first algorithm for finding a set of natural
-clusters of complex zeros of a triangular system of polynomials
within a given polybox in , for any given . Our
algorithm is based on a recent near-optimal algorithm of Becker et al (2016)
for clustering the complex roots of a univariate polynomial where the
coefficients are represented by number oracles.
Our algorithm is numeric, certified and based on subdivision. We implemented
it and compared it with two well-known homotopy solvers on various triangular
systems. Our solver always gives correct answers, is often faster than the
homotopy solver that often gives correct answers, and sometimes faster than the
one that gives sometimes correct results.Comment: Research report V6: description of the main algorithm update
Improved algorithm for computing separating linear forms for bivariate systems
We address the problem of computing a linear separating form of a system of
two bivariate polynomials with integer coefficients, that is a linear
combination of the variables that takes different values when evaluated at the
distinct solutions of the system. The computation of such linear forms is at
the core of most algorithms that solve algebraic systems by computing rational
parameterizations of the solutions and this is the bottleneck of these
algorithms in terms of worst-case bit complexity. We present for this problem a
new algorithm of worst-case bit complexity \sOB(d^7+d^6\tau) where and
denote respectively the maximum degree and bitsize of the input (and
where \sO refers to the complexity where polylogarithmic factors are omitted
and refers to the bit complexity). This algorithm simplifies and
decreases by a factor the worst-case bit complexity presented for this
problem by Bouzidi et al. \cite{bouzidiJSC2014a}. This algorithm also yields,
for this problem, a probabilistic Las-Vegas algorithm of expected bit
complexity \sOB(d^5+d^4\tau).Comment: ISSAC - 39th International Symposium on Symbolic and Algebraic
Computation (2014
Numeric certified algorithm for the topology of resultant and discriminant curves
Let be a real plane algebraic curve defined by the resultant of
two polynomials (resp. by the discriminant of a polynomial). Geometrically such
a curve is the projection of the intersection of the surfaces
(resp. ), and generically its singularities are nodes (resp. nodes and
ordinary cusps). State-of-the-art numerical algorithms compute the topology of
smooth curves but usually fail to certify the topology of singular ones. The
main challenge is to find practical numerical criteria that guarantee the
existence and the uniqueness of a singularity inside a given box , while
ensuring that does not contain any closed loop of . We solve
this problem by first providing a square deflation system, based on
subresultants, that can be used to certify numerically whether contains a
unique singularity or not. Then we introduce a numeric adaptive separation
criterion based on interval arithmetic to ensure that the topology of in is homeomorphic to the local topology at . Our algorithms are
implemented and experiments show their efficiency compared to state-of-the-art
symbolic or homotopic methods
Bivariate systems and topology of plane curves: algebraic and numerical methods
The work presented in this thesis belongs to the domain of non-linear computational geometry in lowdimension. More precisely it focuses on solving bivariate systems and computing the topology of curvesin the plane. When the input is given by polynomials, the natural tools come from computer algebra.Our contributions are algorithms proven efficient in a deterministic or a Las Vegas settings together witha practical efficient software for topology certified drawing of a plane algebraic curve. When the input isnot restricted to be polynomials but given by interval functions, we design algorithms based on certifiednumerical approches using subdivision and interval arithmetic. The input is then required to fulfill somegeneric assumptions and our algorithms are certified in the sense that they terminate if and only if theassumptions are satisfied.Le travail présenté dans cette thèse appartient au domaine de la géométrie computationnelle non linéaireen petite dimension. Plus précisément, il se concentre sur la résolution de systèmes bivariés et le calcul dela topologie des courbes dans le plan. Lorsque l’entrée est donnée par des polynômes, les outils naturelsproviennent du calcul formel. Nos contributions sont des algorithmes dont l’efficacité a été prouvée dansun cadre déterministe ou Las Vegas, ainsi qu’un logiciel efficace pour le dessin certifié de la topologied’une courbe algébrique plane. Lorsque les données d’entrée ne sont pas limitées aux polynômes maissont données par des fonctions d’intervalles, nous concevons des algorithmes basés sur des approchesnumériques certifiées utilisant la subdivision et l’arithmétique d’intervalles. L’entrée doit alors satisfairecertaines hypothèses génériques et nos algorithmes sont certifiés dans le sens où ils se terminent si etseulement si les hypothèses sont satisfaites
HYDRAM : système informatique d'aide à la décision dans l'aménagement des eaux, planification des HYDro-AMénagements
La complexité croissante des systèmes d'eau en Guadeloupe, notamment pour l'irrigation, et la forte variabilité spatio-temporelle des précipitations ont conduit l'ORSTOM à proposer l'élaboration d'un outil d'aide à la décision dans l'aménagement des eaux. Une des originalités du logiciel réside dans la construction interactive des systèmes permettant d'envisager facilement différents scénarios de développement. En se basant sur la confrontation des besoins et des ressources, l'outil permet de simuler le fonctionnement hydrologique des aménagements et de fournir des analyses synthétiques des résultats. Le choix de la conception et programmation par objets autorise toutes les extensions nécessaires pour envisager les multiples facettes d'une gestion rationnelle des ressources en eau. (Résumé d'auteur
Numerical Algorithm for the Topology of Singular Plane Curves
International audienceWe are interested in computing the topology of plane singular curves. For this, the singular points must be isolated. Numerical methods for isolating singular points are efficient but not certified in general. We are interested in developing certified numerical algorithms for isolating the singularities. In order to do so, we restrict our attention to the special case of plane curves that are projections of smooth curves in higher dimensions. In this setting, we show that the singularities can be encoded by a regular square system whose isolation can be certified by numerical methods. This type of curves appears naturally in robotics applications and scientific visualization
La politique du conseil général de l’Indre en faveur du réseau ferré local (1852-1909)
Les premiers projets de l’Empire Après la construction de la ligne Paris–Toulouse, achevée en 1847 à Châteauroux et en 1856 à Limoges, de nombreuses réclamations s’élevèrent pour réclamer de nouvelles voies ferrées dans l’Indre. Un débat s’instaura entre les notables, l’État et les compagnies concessionnaires. Les plus ardents défenseurs du rail berrichon étaient les députés de l’Indre, le comte de Bryas, et le maire de La Châtre et président du conseil général Delavau. Un cercle influent se ..
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