arXiv subjects

Fabrice Rouillier

Publications and source records attributed to Fabrice Rouillier.

At least 19 recordsLinked to original sources

arXiv2026

Fast Rational Univariate Representation via Gaussian Elimination

In this note, we present RationalUnivariateRepresentation$.$jl (https://newrur.gitlabpages.inria.fr/RationalUnivariateRepresentation.jl/), a Julia package for computing rational univariate representations of zero-dimensional polynomial systems. The package uses dense linear algebra and Gaussian elimination for the FGLM-like stage. The purpose of this contribution is to advocate for this choice and explain the implementation details that turn the algorithm into practical software. In particular, we show that our implementation can compute guaranteedly correct parametrizations of ideals with thousands of solutions within seconds.

cs.SC↗

arXiv2025

Reading Rational Univariate Representations on lexicographic Groebner bases

In this contribution, we consider a zero-dimensional polynomial system in $n$ variables defined over a field $\mathbb{K}$. In the context of computing a Rational Univariate Representation (RUR) of its solutions, we address the problem of certifying a separating linear form and, once certified, calculating the RUR that comes from it, without any condition on the ideal else than being zero-dimensional. Our key result is that the RUR can be read (closed formula) from lexicographic Groebner bases of bivariate elimination ideals, even in the case where the original ideal that is not in shape position, so that one can use the same core as the well known FGLM method to propose a simple algorithm. Our first experiments, either with a very short code (300 lines) written in Maple or with a Julia code using straightforward implementations performing only classical Gaussian reductions in addition to Groebner bases for the degree reverse lexicographic ordering, show that this new method is already competitive with sophisticated state of the art implementations which do not certify the parameterizations.

cs.SC↗

arXiv2025

Symbolic and Numerical Tools for $L_{\infty}$-Norm Calculation

The computation of the $L_\infty $-norm is an important issue in $H_{\infty}$ control, particularly for analyzing system stability and robustness. This paper focuses on symbolic computation methods for determining the $L_{\infty} $-norm of finite-dimensional linear systems, highlighting their advantages in achieving exact solutions where numerical methods often encounter limitations. Key techniques such as Sturm-Habicht sequences, Rational Univariate Representations (RUR), and Cylindrical Algebraic Decomposition (CAD) are surveyed, with an emphasis on their theoretical foundations, practical implementations, and specific applicability to $ L_{\infty} $-norm computation. A comparative analysis is conducted between symbolic and conventional numerical approaches, underscoring scenarios in which symbolic computation provides superior accuracy, particularly in parametric cases. Benchmark evaluations reveal the strengths and limitations of both approaches, offering insights into the trade-offs involved. Finally, the discussion addresses the challenges of symbolic computation and explores future opportunities for its integration into control theory, particularly for robust and stable system analysis.

math.OC↗

arXiv2025

Some Computational Tools for Solving a Selection of Problems in Control Theory

This paper demonstrates how certified computational tools can be used to address various problems in control theory. In particular, we introduce PACE.jl, a Julia package that implements symbolic elimination techniques, including (among others) discriminant varieties and Rational Univariate Representation, while also supporting multi-precision interval computations. We showcase its applications to key control theory problems, including identification, stability analysis, and optimization, for both parameter-dependent and parameter-free systems.

cs.SC↗

arXiv2025

Parameter Estimation in ODE Models with Certified Polynomial System Solving

We consider dynamical models given by rational ODE systems. Parameter estimation is an important and challenging task of recovering parameter values from observed data. Recently, a method based on differential algebra and rational interpolation was proposed to express parameter estimation in terms of polynomial system solving. Typically, polynomial system solving is a bottleneck, hence the choice of the polynomial solver is crucial. In this contribution, we compare two polynomial system solvers applied to parameter estimation: homotopy continuation solver from HomotopyContinuation.jl and our new implementation of a certified solver based on rational univariate representation (RUR) and real root isolation. We show how the new RUR solver can tackle examples that are out of reach for the homotopy methods and vice versa.

cs.SC↗

arXiv2024

On the Certification of the Kinematics of 3-DOF Spherical Parallel Manipulators

This paper aims to study a specific kind of parallel robot: Spherical Parallel Manipulators (SPM) that are capable of unlimited rolling. A focus is made on the kinematics of such mechanisms, especially taking into account uncertainties (e.g. on conception & fabrication parameters, measures) and their propagations. Such considerations are crucial if we want to control our robot correctly without any undesirable behavior in its workspace (e.g. effects of singularities). In this paper, we will consider two different approaches to study the kinematics and the singularities of the robot of interest: symbolic and semi-numerical. By doing so, we can compute a singularity-free zone in the work- and joint spaces, considering given uncertainties on the parameters. In this zone, we can use any control law to inertially stabilize the upper platform of the robot.

cs.RO↗

arXiv2024

Inertial Line-Of-Sight Stabilization Using a 3-DOF Spherical Parallel Manipulator with Coaxial Input Shafts

This article dives into the use of a 3-RRR Spherical Parallel Manipulator (SPM) for the purpose of inertial Line Of Sight (LOS) stabilization. Such a parallel robot provides three Degrees of Freedom (DOF) in orientation and is studied from the kinematic point of view. In particular, one guarantees that the singular loci (with the resulting numerical instabilities and inappropriate behavior of the mechanism) are far away from the prescribed workspace. Once the kinematics of the device is certified, a control strategy needs to be implemented in order to stabilize the LOS through the upper platform of the mechanism. Such a work is done with MATLAB Simulink using a SimMechanics model of our robot.

cs.RO↗

arXiv2023

${L}^{\infty}$-norm computation for linear time-invariant systems depending on parameters

This paper focuses on representing the $L^{\infty}$-norm of finite-dimensional linear time-invariant systems with parameter-dependent coefficients. Previous studies tackled the problem in a non-parametric scenario by simplifying it to finding the maximum $y$-projection of real solutions $(x, y)$ of a system of the form $Σ=\{P=0, \, \partial P/\partial x=0\}$, where $P \in \Z[x, y]$. To solve this problem, standard computer algebra methods were employed and analyzed \cite{bouzidi2021computation}. In this paper, we extend our approach to address the parametric case. We aim to represent the "maximal" $y$-projection of real solutions of $Σ$ as a function of the given parameters. %a set of parameters $α$. To accomplish this, we utilize cylindrical algebraic decomposition. This method allows us to determine the desired value as a function of the parameters within specific regions of parameter space.

cs.SC↗

arXiv2023

On Isolating Roots in a Multiple Field Extension

We address univariate root isolation when the polynomial's coefficients are in a multiple field extension. We consider a polynomial $F \in L[Y]$, where $L$ is a multiple algebraic extension of $\mathbb{Q}$. We provide aggregate bounds for $F$ and algorithmic and bit-complexity results for the problem of isolating its roots. For the latter problem we follow a common approach based on univariate root isolation algorithms. For the particular case where $F$ does not have multiple roots, we achieve a bit-complexity in $\tilde{\mathcal{O}}_B(n d^{2n+2}(d+nτ))$, where $d$ is the total degree and $τ$ is the bitsize of the involved polynomials.In the general case we need to enhance our algorithm with a preprocessing step that determines the number of distinct roots of $F$. We follow a numerical, yet certified, approach that has bit-complexity $\tilde{\mathcal{O}}_B(n^2d^{3n+3}τ+ n^3 d^{2n+4}τ)$.

cs.SC↗

arXiv2022

PTOPO: Computing the Geometry and the Topology of Parametric Curves

We consider the problem of computing the topology and describing the geometry of a parametric curve in $\mathbb{R}^n$. We present an algorithm, PTOPO, that constructs an abstract graph that is isotopic to the curve in the embedding space. Our method exploits the benefits of the parametric representation and does not resort to implicitization. Most importantly, we perform all computations in the parameter space and not in the implicit space. When the parametrization involves polynomials of degree at most $d$ and maximum bitsize of coefficients $τ$, then the worst case bit complexity of PTOPO is $ \tilde{\mathcal{O}}_B(nd^6+nd^5τ+d^4(n^2+nτ)+d^3(n^2τ+ n^3)+n^3d^2τ)$. This bound matches the current record bound $\tilde{\mathcal{O}}_B(d^6+d^5τ)$ for the problem of computing the topology of a plane algebraic curve given in implicit form. For plane and space curves, if $N = \max\{d, τ\}$, the complexity of PTOPO becomes $\tilde{\mathcal{O}}_B(N^6)$, which improves the state-of-the-art result, due to Alcázar and Díaz-Toca [CAGD'10], by a factor of $N^{10}$. In the same time complexity, we obtain a graph whose straight-line embedding is isotopic to the curve. However, visualizing the curve on top of the abstract graph construction, increases the bound to $\tilde{\mathcal{O}}_B(N^7)$. For curves of general dimension, we can also distinguish between ordinary and non-ordinary real singularities and determine their multiplicities in the same expected complexity of PTOPO by employing the algorithm of Blasco and Pérez-Díaz [CAGD'19]. We have implemented PTOPO in Maple for the case of plane and space curves. Our experiments illustrate its practical nature.

cs.SC↗

arXiv2021

Bounds for polynomials on algebraic numbers and application to curve topology

Let $P \in \mathbb{Z} [X, Y]$ be a given square-free polynomial of total degree $d$ with integer coefficients of bitsize less than $τ$, and let $V_{\mathbb{R}} (P) := \{ (x,y) \in \mathbb{R}^2, P (x,y) = 0 \}$ be the real planar algebraic curve implicitly defined as the vanishing set of $P$. We give a deterministic and certified algorithm to compute the topology of $V_{\mathbb{R}} (P)$ in terms of a straight-line planar graph $\mathcal{G}$ that is isotopic to $V_{\mathbb{R}} (P)$. Our analysis yields the upper bound $\tilde O (d^5 τ+ d^6)$ on the bit complexity of our algorithm, which matches the current record bound for the problem of computing the topology of a planar algebraic curve However, compared to existing algorithms with comparable complexity, our method does not consider any change of coordinates, and the returned graph $\mathcal{G}$ yields the cylindrical algebraic decomposition information of the curve. Our result is based on two main ingredients: First, we derive amortized quantitative bounds on the the roots of polynomials with algebraic coefficients as well as adaptive methods for computing the roots of such polynomials that actually exploit this amortization. Our second ingredient is a novel approach for the computation of the local topology of the curve in a neighborhood of all critical points.

math.AG↗

arXiv2017

Computing Chebyshev knot diagrams

A Chebyshev curve $\mathcal{C}(a,b,c,ϕ)$ has a parametrization of the form$ x(t)=T\_a(t)$; \ $y(t)=T\_b(t)$; $z(t)= T\_c(t + ϕ)$, where $a,b,c$are integers, $T\_n(t)$ is the Chebyshev polynomialof degree $n$ and $ϕ\in \mathbb{R}$. When $\mathcal{C}(a,b,c,ϕ)$ is nonsingular,it defines a polynomial knot. We determine all possible knot diagrams when $ϕ$ varies. Let $a,b,c$ be integers, $a$ is odd, $(a,b)=1$, we show that one can list all possible knots $\mathcal{C}(a,b,c,ϕ)$ in$\tilde{\mathcal{O}}(n^2)$ bit operations, with $n=abc$.

cs.SC↗

arXiv2016

Computing Real Roots of Real Polynomials ... 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.

cs.MS↗

arXiv2015

An algebraic method to check the singularity-free paths for parallel robots

Trajectory planning is a critical step while programming the parallel manipulators in a robotic cell. The main problem arises when there exists a singular configuration between the two poses of the end-effectors while discretizing the path with a classical approach. This paper presents an algebraic method to check the feasibility of any given trajectories in the workspace. The solutions of the polynomial equations associated with the tra-jectories are projected in the joint space using Gr{ö}bner based elimination methods and the remaining equations are expressed in a parametric form where the articular variables are functions of time t unlike any numerical or discretization method. These formal computations allow to write the Jacobian of the manip-ulator as a function of time and to check if its determinant can vanish between two poses. Another benefit of this approach is to use a largest workspace with a more complex shape than a cube, cylinder or sphere. For the Orthoglide, a three degrees of freedom parallel robot, three different trajectories are used to illustrate this method.

cs.RO↗

arXiv2015

Workspace and Singularity analysis of a Delta like family robot

Workspace and joint space analysis are essential steps in describing the task and designing the control loop of the robot, respectively. This paper presents the descriptive analysis of a family of delta-like parallel robots by using algebraic tools to induce an estimation about the complexity in representing the singularities in the workspace and the joint space. A Gr{ö}bner based elimination is used to compute the singularities of the manipulator and a Cylindrical Algebraic Decomposition algorithm is used to study the workspace and the joint space. From these algebraic objects, we propose some certified three dimensional plotting describing the the shape of workspace and of the joint space which will help the engineers or researchers to decide the most suited configuration of the manipulator they should use for a given task. Also, the different parameters associated with the complexity of the serial and parallel singularities are tabulated, which further enhance the selection of the different configuration of the manipulator by comparing the complexity of the singularity equations.

cs.RO↗

arXiv2014

Character varieties for SL(3,C): the figure eight knot

We give a description of several representation varieties of the fundamental group of the complement of the figure eight knot in PGL(3,C) or SL(3,C). We moreover obtain an explicit parametrization of matrices generating the representation and a description of the projection of the representation variety into the character variety of the boundary torus into SL(3,C).

math.GT↗

arXiv2014

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 $\sOB(d^7+d^6τ)$ where $d$ and $τ$ denote respectively the maximum degree and bitsize of the input (and where $\sO$ refers to the complexity where polylogarithmic factors are omitted and $O_B$ refers to the bit complexity). This algorithm simplifies and decreases by a factor $d$ 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 $\sOB(d^5+d^4τ)$.

cs.CG↗

arXiv2014

Non-singular assembly mode changing trajectories in the workspace for the 3-RPS parallel robot

Having non-singular assembly modes changing trajectories for the 3-RPS parallel robot is a well-known feature. The only known solution for defining such trajectory is to encircle a cusp point in the joint space. In this paper, the aspects and the characteristic surfaces are computed for each operation mode to define the uniqueness of the domains. Thus, we can easily see in the workspace that at least three assembly modes can be reached for each operation mode. To validate this property, the mathematical analysis of the determinant of the Jacobian is done. The image of these trajectories in the joint space is depicted with the curves associated with the cusp points.

cs.RO↗