arXiv subjects

Guillaume Moroz

Publications and source records attributed to Guillaume Moroz.

At least 19 recordsLinked to original sources

arXiv2024

Sparse Tensors and Subdivision Methods for Finding the Zero Set of Polynomial Equations

Finding the solutions to a system of multivariate polynomial equations is a fundamental problem in mathematics and computer science. It involves evaluating the polynomials at many points, often chosen from a grid. In most current methods, such as subdivision, homotopy continuation, or marching cube algorithms, polynomial evaluation is treated as a black box, repeating the process for each point. We propose a new approach that partially evaluates the polynomials, allowing us to efficiently reuse computations across multiple points in a grid. Our method leverages the Compressed Sparse Fiber data structure to efficiently store and process subsets of grid points. We integrated our amortized evaluation scheme into a subdivision algorithm. Experimental results show that our approach is efficient in practice. Notably, our software \texttt{voxelize} can successfully enclose curves defined by two trivariate polynomial equations of degree $100$, a problem that was previously intractable.

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arXiv2023

Fast evaluation and root finding for polynomials with floating-point coefficients

Evaluating or finding the roots of a polynomial $f(z) = f_0 + \cdots + f_d z^d$ with floating-point number coefficients is a ubiquitous problem. By using a piecewise approximation of $f$ obtained with a careful use of the Newton polygon of $f$, we improve state-of-the-art upper bounds on the number of operations to evaluate and find the roots of a polynomial. In particular, if the coefficients of $f$ are given with $m$ significant bits, we provide for the first time an algorithm that finds all the roots of $f$ with a relative condition number lower than $2^m$, using a number of bit operations quasi-linear in the bit-size of the floating-point representation of $f$. Notably, our new approach handles efficiently polynomials with coefficients ranging from $2^{-d}$ to $2^d$, both in theory and in practice.

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arXiv2021

New data structure for univariate polynomial approximation and applications to root isolation, numerical multipoint evaluation, and other problems

We present a new data structure to approximate accurately and efficiently a polynomial $f$ of degree $d$ given as a list of coefficients. Its properties allow us to improve the state-of-the-art bounds on the bit complexity for the problems of root isolation and approximate multipoint evaluation. This data structure also leads to a new geometric criterion to detect ill-conditioned polynomials, implying notably that the standard condition number of the zeros of a polynomial is at least exponential in the number of roots of modulus less than $1/2$ or greater than $2$.Given a polynomial $f$ of degree $d$ with $\|f\|_1 \leq 2^τ$ for $τ\geq 1$, isolating all its complex roots or evaluating it at $d$ points can be done with a quasi-linear number of arithmetic operations. However, considering the bit complexity, the state-of-the-art algorithms require at least $d^{3/2}$ bit operations even for well-conditioned polynomials and when the accuracy required is low. Given a positive integer $m$, we can compute our new data structure and evaluate $f$ at $d$ points in the unit disk with an absolute error less than $2^{-m}$ in $\widetilde O(d(τ+m))$ bit operations, where $\widetilde O(\cdot)$ means that we omit logarithmic factors. We also show that if $κ$ is the absolute condition number of the zeros of $f$, then we can isolate all the roots of $f$ in $\widetilde O(d(τ+ \log κ))$ bit operations. Moreover, our algorithms are simple to implement. For approximating the complex roots of a polynomial, we implemented a small prototype in \verb|Python/NumPy| that is an order of magnitude faster than the state-of-the-art solver \verb/MPSolve/ for high degree polynomials with random coefficients.

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arXiv2021

Fast real and complex root-finding methods for well-conditioned polynomials

Given a polynomial $p$ of degree $d$ and a bound $κ$ on a condition number of $p$, we present the first root-finding algorithms that return all its real and complex roots with a number of bit operations quasi-linear in $d \log^2(κ)$. More precisely, several condition numbers can be defined depending on the norm chosen on the coefficients of the polynomial. Let $p(x) = \sum\_{k=0}^d a\_k x^k = \sum\_{k=0}^d \sqrt{\binom d k} b\_k x^k$. We call the condition number associated with a perturbation of the $a\_k$ the hyperbolic condition number $κ\_h$, and the one associated with a perturbation of the $b\_k$ the elliptic condition number $κ\_e$. For each of these condition numbers, we present algorithms that find the real and the complex roots of $p$ in $O\left(d\log^2(dκ)\ \text{polylog}(\log(dκ))\right)$ bit operations.Our algorithms are well suited for random polynomials since $κ\_h$ (resp. $κ\_e$) is bounded by a polynomial in $d$ with high probability if the $a\_k$ (resp. the $b\_k$) are independent, centered Gaussian variables of variance $1$.

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arXiv2020

Optimal transport between determinantal point processes and application to fast simulation

We analyze several optimal transportation problems between de-terminantal point processes. We show how to estimate some of the distances between distributions of DPP they induce. We then apply these results to evaluate the accuracy of a new and fast DPP simulation algorithm. We can now simulate in a reasonable amount of time more than ten thousands points.

cs.DS↗

arXiv2019

Evaluation of Chebyshev polynomials on intervals and application to root finding

In approximation theory, it is standard to approximate functions by polynomials expressed in the Chebyshev basis. Evaluating a polynomial $f$ of degree n given in the Chebyshev basis can be done in $O(n)$ arithmetic operations using the Clenshaw algorithm. Unfortunately, the evaluation of $f$ on an interval $I$ using the Clenshaw algorithm with interval arithmetic returns an interval of width exponential in $n$. We describe a variant of the Clenshaw algorithm based on ball arithmetic that returns an interval of width quadratic in $n$ for an interval of small enough width. As an application, our variant of the Clenshaw algorithm can be used to design an efficient root finding algorithm.

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arXiv2017

Computing effectively stabilizing controllers for a class of $n$D systems

In this paper, we study the internal stabilizability and internal stabilization problems for multidimensional (nD) systems. Within the fractional representation approach, a multidimen-sional system can be studied by means of matrices with entries in the integral domain of structurally stable rational fractions, namely the ring of rational functions which have no poles in the closed unit polydisc U n = {z = (z 1 ,. .. , z n) $\in$ C n | |z 1 | 1,. .. , |z n | 1}. It is known that the internal stabilizability of a multidimensional system can be investigated by studying a certain polynomial ideal I = p 1 ,. .. , p r that can be explicitly described in terms of the transfer matrix of the plant. More precisely the system is stabilizable if and only if V (I) = {z $\in$ C n | p 1 (z) = $\times$ $\times$ $\times$ = p r (z) = 0} $\cap$ U n = $\emptyset$. In the present article, we consider the specific class of linear nD systems (which includes the class of 2D systems) for which the ideal I is zero-dimensional, i.e., the p i 's have only a finite number of common complex zeros. We propose effective symbolic-numeric algorithms for testing if V (I) $\cap$ U n = $\emptyset$, as well as for computing, if it exists, a stable polynomial p $\in$ I which allows the effective computation of a stabilizing controller. We illustrate our algorithms through an example and finally provide running times of prototype implementations for 2D and 3D systems.

math.NA↗

arXiv2017

The assembly modes of rigid 11-bar linkages

Designing an m-bar linkage with a maximal number of assembly modes is important in robot kinematics, and has further applications in structural biology and computational geometry. A related question concerns the number of assembly modes of rigid mechanisms as a function of their nodes n, which is uniquely defined given m. Rigid 11-bar linkages, where n=7, are the simplest planar linkages for which these questions were still open. It will be proven that the maximal number of assembly modes of such linkages is exactly 56. The rigidity of a linkage is captured by a polynomial system derived from distance, or Cayley-Menger, matrices. The upper bound on the number of assembly modes is obtained as the mixed volume of a 5x5 system. An 11-bar linkage admitting 56 configurations is constructed using stochastic optimisation methods. This yields a general lower bound of $Ω(2.3^n)$ on the number of assembly modes, slightly improving the current record of $Ω(2.289^n)$, while the best known upper bound is roughly $4^n$. Our methods are straightforward and have been implemented in Maple. They are described in general terms illustrating the fact that they can be readily extended to other planar or spatial linkages. The main results have been reported in conference publication [EM11]. This version (2017) typesets correctly the last Figure 5 so as to include all 28 configurations modulo reflection.

cs.RO↗

arXiv2016

A Fast Algorithm for Computing the Truncated Resultant

Let P and Q be two polynomials in K[x, y] with degree at most d, where K is a field. Denoting by R $\in$ K[x] the resultant of P and Q with respect to y, we present an algorithm to compute R mod x^k in O~(kd) arithmetic operations in K, where the O~ notation indicates that we omit polylogarithmic factors. This is an improvement over state-of-the-art algorithms that require to compute R in O~(d^3) operations before computing its first k coefficients.

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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.

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arXiv2015

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.

cs.CG↗

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↗

arXiv2013

Fast polynomial evaluation and composition

The library \emph{fast\_polynomial} for Sage compiles multivariate polynomials for subsequent fast evaluation. Several evaluation schemes are handled, such as Hörner, divide and conquer and new ones can be added easily. Notably, a new scheme is introduced that improves the classical divide and conquer scheme when the number of terms is not a pure power of two. Natively, the library handles polynomials over gmp big integers, boost intervals, python numeric types. And any type that supports addition and multiplication can extend the library thanks to the template design. Finally, the code is parallelized for the divide and conquer schemes, and memory allocation is localized and optimized for the different evaluation schemes. This extended abstract presents the concepts behind the \emph{fast\_polynomial} library. The sage package can be downloaded at \url{http://trac.sagemath.org/sage_trac/ticket/13358}.

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arXiv2012

The monotonicity of f-vectors of random polytopes

Let K be a compact convex body in Rd, let Kn be the convex hull of n points chosen uniformly and independently in K, and let fi(Kn) denote the number of i-dimensional faces of Kn. We show that for planar convex sets, E(f0(Kn)) is increasing in n. In dimension d>=3 we prove that if lim(E((f[d -1](Kn))/(An^c)->1 when n->infinity for some constants A and c > 0 then the function E(f[d-1](Kn)) is increasing for n large enough. In particular, the number of facets of the convex hull of n random points distributed uniformly and independently in a smooth compact convex body is asymptotically increasing. Our proof relies on a random sampling argument.

cs.CG↗

arXiv2012

Cusp Points in the Parameter Space of Degenerate 3-RPR Planar Parallel Manipulators

This paper investigates the conditions in the design parameter space for the existence and distribution of the cusp locus for planar parallel manipulators. Cusp points make possible non-singular assembly-mode changing motion, which increases the maximum singularity-free workspace. An accurate algorithm for the determination is proposed amending some imprecisions done by previous existing algorithms. This is combined with methods of Cylindric Algebraic Decomposition, Gröbner bases and Discriminant Varieties in order to partition the parameter space into cells with constant number of cusp points. These algorithms will allow us to classify a family of degenerate 3-RPR manipulators.

cs.RO↗

arXiv2012

Solution regions in the parameter space of a 3-RRR decoupled robot for a prescribed workspace

This paper proposes a new design method to determine the feasible set of parameters of translational or position/orientation decoupled parallel robots for a prescribed singularity-free workspace of regular shape. The suggested method uses Groebner bases to define the singularities and the cylindrical algebraic decomposition to characterize the set of parameters. It makes it possible to generate all the robot designs. A 3-RRR decoupled robot is used to validate the proposed design method.

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arXiv2011

Computing the Distance between Piecewise-Linear Bivariate Functions

We consider the problem of computing the distance between two piecewise-linear bivariate functions $f$ and $g$ defined over a common domain $M$. We focus on the distance induced by the $L_2$-norm, that is $\|f-g\|_2=\sqrt{\iint_M (f-g)^2}$. If $f$ is defined by linear interpolation over a triangulation of $M$ with $n$ triangles, while $g$ is defined over another such triangulation, the obvious naïve algorithm requires $Θ(n^2)$ arithmetic operations to compute this distance. We show that it is possible to compute it in $Ø(n\log^4 n)$ arithmetic operations, by reducing the problem to multi-point evaluation of a certain type of polynomials. We also present an application to terrain matching.

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