A Subdivision Solver for Systems of Large Dense Polynomials

We describe here the package {\tt subdivision\\_solver} for the mathematical software {\tt SageMath}. It provides a solver on real numbers for square systems of large dense polynomials. By large polynomials we mean multivariate polynomials with large degrees, which coefficients have large bit-size. While staying robust, symbolic approaches to solve systems of polynomials see their performances dramatically affected by high degree and bit-size of input polynomials.Available numeric approaches suffer from the cost of the evaluation of large polynomials and their derivatives.Our solver is based on interval analysis and bisections of an initial compact domain of $\R^n$ where solutions are sought. Evaluations on intervals with Horner scheme is performed by the package {\tt fast\\_polynomial} for {\tt SageMath}.The non-existence of a solution within a box is certified by an evaluation scheme that uses a Taylor expansion at order 2, and existence and uniqueness of a solution within a box is certified with krawczyk operator.The precision of the working arithmetic is adapted on the fly during the subdivision process and we present a new heuristic criterion to decide if the arithmetic precision has to be increased

Clustering Complex Zeros of Triangular Systems of Polynomials

This paper gives the first algorithm for finding a set of natural $\epsilon$-clusters of complex zeros of a triangular system of polynomials within a given polybox in $\mathbb{C}^n$, for any given $\epsilon>0$. 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

A Robust and Efficient Method for Solving Point Distance Problems by Homotopy

The goal of Point Distance Solving Problems is to find 2D or 3D placements of points knowing distances between some pairs of points. The common guideline is to solve them by a numerical iterative method (\emph{e.g.} Newton-Raphson method). A sole solution is obtained whereas many exist. However the number of solutions can be exponential and methods should provide solutions close to a sketch drawn by the user.Geometric reasoning can help to simplify the underlying system of equations by changing a few equations and triangularizing it.This triangularization is a geometric construction of solutions, called construction plan. We aim at finding several solutions close to the sketch on a one-dimensional path defined by a global parameter-homotopy using a construction plan. Some numerical instabilities may be encountered due to specific geometric configurations. We address this problem by changing on-the-fly the construction plan.Numerical results show that this hybrid method is efficient and robust

Numeric certified algorithm for the topology of resultant and discriminant curves

Let $\mathcal C$ 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 $P(x,y,z)=Q(x,y,z)=0$ (resp. $P(x,y,z)=\frac{\partial P}{\partial z}(x,y,z)=0$), 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 $B$, while ensuring that $B$ does not contain any closed loop of $\mathcal{C}$. We solve this problem by first providing a square deflation system, based on subresultants, that can be used to certify numerically whether $B$ contains a unique singularity $p$ or not. Then we introduce a numeric adaptive separation criterion based on interval arithmetic to ensure that the topology of $\mathcal C$ in $B$ is homeomorphic to the local topology at $p$. Our algorithms are implemented and experiments show their efficiency compared to state-of-the-art symbolic or homotopic methods

Reliable location with respect to the projection of a smooth space curve

Consider a plane curve B defined as the projection of the intersection of two analytic surfaces in R 3 or as the apparent contour of a surface. In general, B has node or cusp singular points and thus is a singular curve. Our main contribution is the computation of a data structure answering point location queries with respect to the subdivision of the plane induced by B. This data structure is composed of an approximation of the space curve together with a topological representation of its projection B. Since B is a singular curve, it is challenging to design a method only based on reliable numerical algorithms. In a recent work, the authors show how to describe the set of singularities of B as regular solutions of a so-called ball system suitable for a numerical subdivision solver. Here, the space curve is first enclosed in a set of boxes with a certified path-tracker to restrict the domain where the ball system is solved. Boxes around singular points are then computed such that the correct topology of the curve inside these boxes can be deduced from the intersections of the curve with their boundaries. The tracking of the space curve is then used to connect the smooth branches to the singular points. The subdivision of the plane induced by B is encoded as an extended planar combinatorial map allowing point location. We experimented our method and show that our reliable numerical approach can handle classes of examples that are not reachable by symbolic methods

Clustering Complex Zeros of Triangular System of Polynomials

International audienceThis paper gives the first algorithm for finding a set of natural ε-clusters of complex zeros of a triangular system of polynomials within a given polybox in C^n, for any given ε > 0. 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, produces guaranteed results and is based on subdivision. We implemented it and compared it with state of the art solvers on various triangular systems, including systems with clusters of solutions or multiple solutions

Algorithmes numériques certifiés pour la topologie d'une courbe résultante ou discriminante

Let $\mathcal C$ 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 $P(x,y,z)=Q(x,y,z)=0$ (resp. $P(x,y,z)=\frac{\partial P}{\partial z}(x,y,z)=0$), 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 $B$, while ensuring that $B$ does not contain any closed loop of $\mathcal{C}$. We solve this problem by first providing a square deflation system, based on subresultants, that can be used to certify numerically whether $B$ contains a unique singularity $p$ or not. Then we introduce a numeric adaptive separation criterion based on interval arithmetic to ensure that the topology of $\mathcal C$ in $B$ is homeomorphic to the local topology at $p$. Our algorithms are implemented and experiments show their efficiency compared to state-of-the-art symbolic or homotopic methods.Bien que francophones et très attachés à notre langue maternelle, nous avons pensé et rédigé ce travail en anglais comme la grande majorité de la production scientifique mondiale. Dans ce contexte, il est clair que cette version française de l' ''abstract'' n'a aucun interêt pour notre communauté, et nous avons peu d'espoir qu'il puisse en être autrement même en dehors de notre communauté. Nous proposons néanmoins quelques pistes en français pour cet improbable lecteur et serions comblés si celui-ci en venait à apprendre l'anglais pour pouvoir lire notre prose. Nous \'etudions la topologie d'une courbe plane issue de la projection d'une courbe lisse dans l'espace. Génériquement, la projection présente des singularités de type noeud et cusp (dans le cas d'un discriminant seulement). Les algorithmes numériques de l'état de l'art ne calculent la topologie que dans le cas de courbes lisses. L'enjeu est donc de concevoir des critères numériques garantissant l'existence et l'unicité d'une singularité dans une boite donnée, tout en assurant que cette boite ne contienne pas d'autre partie de la courbe non connectée à ce point dans la boite. Nous proposons une déflation basée sur les sous-résultants pour le premier problème ainsi qu'un critère de séparation basée sur de l'arithmétique d'intervalles pour le second problème