Thomas Cluzeau
University of Limoges
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Featured researches published by Thomas Cluzeau.
Applicable Algebra in Engineering, Communication and Computing | 2006
Thomas Cluzeau; Mark van Hoeij
We describe a complete algorithm to compute the hypergeometric solutions of linear recurrence relations with rational function coefficients. We use the notion of finite singularities and avoid computations in splitting fields. An implementation is available in Maple 9.
IEEE Transactions on Reliability | 2008
Thomas Cluzeau; Jörg Keller; Winfrid G. Schneeweiss
Many algorithms for computing the reliability of linear or circular consecutive-k-out-of-n:F systems appeared in this Transactions. The best complexity estimate obtained for solving this problem is O(k3 log(n/k)) operations in the case of i.i.d. components. Using fast algorithms for computing a selected term of a linear recurrence with constant coefficients, we provide an algorithm having arithmetic complexity O(k log (k) log(log(k)) log(n)+komega) where 2<omega< 3 is the exponent of linear algebra. This algorithm holds generally for linear, and circular consecutive-k-out-of-n:F systems with independent but not necessarily identical components.
international symposium on symbolic and algebraic computation | 2003
Thomas Cluzeau
We present an algorithm for factoring differential systems with coefficients in <b>F</b><inf><i>p</i></inf>(<i>z</i>). Such an algorithm has already been given by van der Put in [20], [24, 13.1] and [22]. We recast his ideas to handle systems directly and we add some comparisons of strategies, an implementation in Maple<sup>1</sup> and a complexity analysis. The central tool for factoring in characteristic <i>p</i> is the <i>p</i> curvature. We prove the links between the <i>p</i>-curvature and the eigenring and we show how to use these to obtain another algorithm following the exposition of Barkatou in [1].
international symposium on symbolic and algebraic computation | 2005
Alin Bostan; Thomas Cluzeau; Bruno Salvy
We investigate polynomial solutions of homogeneous linear differential equations with coefficients that are polynomials with integer coefficients. The problems we consider are the existence of nonzero polynomial solutions, the determination of the dimension of the vector space of polynomial solutions, the computation of a basis of this space. Previous algorithms have a bit complexity that is at least quadratic in the largest integer valuation N of formal Laurent series solutions at infinity, even for merely detecting the existence of nonzero polynomial solutions. We give a deterministic algorithm that computes a compact representation of a basis of polynomial solutions in O(Nlog3N) bit operations. We also give a probabilistic algorithm that computes the dimension of the space of polynomial solutions in O(√Nlog2N) bit operations. In general, the integer N is not polynomially bounded in the bit size of the input differential equation. We isolate a class of equations for which detecting nonzero polynomial solutions can be performed in polynomial complexity. We discuss implementation issues and possible extensions.
Lecture Notes in Control and Information Sciences | 2009
Thomas Cluzeau; Alban Quadrat
The purpose of this paper is to demonstrate the symbolic package OREMORPHISMS which is dedicated to the implementation of different algorithms and heuristic methods for the study of the factorization, reduction and decomposition problems of general linear functional systems (e.g., systems of partial differential or difference equations, differential time-delay systems). In particular, we explicitly show how to decompose a differential timedelay system (a string with an interior mass [15]) formed by 4 equations in 6 unknowns and prove that it is equivalent to a simple equation in 3 unknowns. We finally give a list of reductions of classical systems of differential time-delay equations and partial differential equations coming from control theory and mathematical physics.
Applicable Algebra in Engineering, Communication and Computing | 2003
Thomas Cluzeau; Evelyne Hubert
Abstract We show that the generic zeros of a differential ideal [A]:H∞A defined by a differential chain A are birationally equivalent to the general zeros of a single regular differential polynomial. This provides a generalization of both the cyclic vector construction for system of linear differential equations and the rational univariate representation of algebraic zero dimensional radical ideals. In order to achieve generality, we prove new results on differential dimension and relative orders which are of independent interest.
international symposium on symbolic and algebraic computation | 2012
Moulay A. Barkatou; Thomas Cluzeau; C. El Bacha; Jacques-Arthur Weil
We present algorithms for computing rational and hyperexponential solutions of linear D-finite partial differential systems written as integrable connections. We show that these types of solutions can be computed recursively by adapting existing algorithms handling ordinary linear differential systems. We provide an arithmetic complexity analysis of the algorithms that we develop. A Maple implementation is available and some examples and applications are given.
Journal of Symbolic Computation | 2011
Moulay A. Barkatou; Thomas Cluzeau; Carole El Bacha
We propose a direct algorithm for computing regular formal solutions of a given higher-order linear differential system near a singular point. With such a system, we associate a matrix polynomial and we say that the system is simple if the determinant of this matrix polynomial does not identically vanish. In this case, we show that the algorithm developed in Barkatou et al. (2009) can be applied to compute a basis of the regular formal solutions space. Otherwise, we develop an algorithm which, given a non-simple system, computes an auxiliary simple one from which the regular formal solutions space of the original system can be recovered. We also give the arithmetic complexity of our algorithms.
international symposium on symbolic and algebraic computation | 2006
Alin Bostan; Frédéric Chyzak; Bruno Salvy; Thomas Cluzeau
We consider two kinds of problems: the computation of polynomial and rational solutions of linear recurrences with coefficients that are polynomials with integer coefficients; indefinite and definite summation of sequences that are hypergeometric over the rational numbers. The algorithms for these tasks all involve as an intermediate quantity an integer N (dispersion or root of an indicial polynomial) that is potentially exponential in the bit size of their input. Previous algorithms have a bit complexity that is at least quadratic in N. We revisit them and propose variants that exploit the structure of solutions and avoid expanding polynomials of degree N. We give two algorithms: a probabilistic one that detects the existence or absence of nonzero polynomial and rational solutions in O(√N log2 N) bit operations; a deterministic one that computes a compact representation of the solution in O(N log3 N) bit operations. Similar speedups are obtained in indefinite and definite hypergeometric summation. We describe the results of an implementation.
Journal of Symbolic Computation | 2004
Thomas Cluzeau; Mark van Hoeij
Abstract We present a new algorithm for computing exponential solutions of differential operators with rational function coefficients. We use a combination of local and modular computations, which allows us to reduce the number of possibilities in the combinatorial part of the algorithm. We also show how unnecessarily large algebraic extensions of the constants can be avoided in the algorithm.