Low Complexity Algorithms for Linear Recurrences

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 $O(\sqrt{N}\log^{2}N)$ bit operations; a deterministic one that computes a compact representation of the solution in $O(N\log^{3}N)$ bit operations. Similar speed-ups are obtained in indefinite and definite hypergeometric summation. We describe the results of an implementation.Comment: This is the author's version of the work. It is posted here by permission of ACM for your personal use. Not for redistributio

Serre's reduction of linear systems of partial differential equations with holonomic adjoints

Given a linear functional system (e.g., ordinary/partial di erential system, di erential time-delay system, di erence system), Serre's reduction aims at nding an equivalent linear functional system which contains fewer equations and fewer unknowns. The purpose of this paper is to study Serre's reduction of underdetermined linear systems of partial di erential equations with either polynomial, formal power series or analytic coe cients and with holonomic adjoints in the sense of algebraic analysis. We prove that these linear partial di erential systems can be de ned by means of a single linear partial di erential equation. In the case of polynomial coe cients, we give an algorithm to compute the corresponding equation

Serre's reduction of linear partial differential systems with holonomic adjoints

Given a linear functional system (e.g., ordinary/partial differential systems, differential time-delay systems, difference systems), Serre's reduction aims at finding an equivalent linear functional system which contains fewer equations and fewer unknowns. The purpose of this paper is to study Serre's reduction of underdetermined linear systems of partial differential equations with either polynomial, formal power series or analytic coefficients and with holonomic adjoints in the sense of algebraic analysis. We prove that these linear partial differential systems can be defined by means of a single linear partial differential equation. In the case of polynomial coefficients, we give an algorithm to compute the corresponding equation.Etant donné un système fonctionnel linéaire (e.g., système d'équations différentielles ordinaires, système d'équations aux dérivées partielles, système d'équations différentielles à retard, système d'équations aux différences), la réduction de Serre a pour but de trouver un système fonctionnel linéaire équivalent contenant moins d'équations et d'inconnues. L'objectif de ce papier est l'étude de la réduction de Serre des systèmes linéaires sous-déterminés d'équations aux dérivées partielles à coefficients polynomiaux, séries formelles ou séries localement convergentes, dont les adjoints sont holonomes au sens de l'analyse algébrique. Nous prouvons que de tels systèmes peuvent être définis par une seule équation aux dérivées partielles. Dans le cas des coefficients polynomiaux, nous donnons un algorithme permettant de calculer l'équation correspondante

Symbolic methods for developing new domain decomposition algorithms

The purpose of this work is to show how algebraic and symbolic techniques such as Smith normal forms and Gröbner basis techniques can be used to develop new Schwarz-like algorithms and preconditioners for linear systems of partial differential equationsL'objet de ce travail est de monter comment les techniques algébriques et symboliques telles que les formes normales de Smith et les techniques de bases de Gröbner peuvent être utilisées pour développer de nouveaux algorithmes de type Schwarz et des préconditionneurs pour les systèmes linéaires d'équations aux dérivées partielles

Symbolic preconditioning techniques for linear systems of partial differential equations

Some algorithmic aspects of systems of PDEs based simulations can be better clarified by means of symbolic computation techniques. This is very important since numerical simulations heavily rely on solving systems of PDEs. For the large-scale problems we deal with in today's standard applications, it is necessary to rely on iterative Krylov methods that are scalable (i.e., weakly dependent on the number of degrees on freedom and number of subdomains) and have limited memory requirements. They are preconditioned by domain decomposition methods, incomplete factorizations and multigrid preconditioners. These techniques are well understood and efficient for scalar symmetric equations (e.g., Laplacian, biLaplacian) and to some extent for non-symmetric equations (e.g., convection-diffusion). But they have poor performances and lack robustness when used for symmetric systems of PDEs, and even more so for non-symmetric complex systems (fluid mechanics, porous media ...). As a general rule, the study of iterative solvers for systems of PDEs as opposed to scalar PDEs is an underdeveloped subject. We aim at building new robust and efficient solvers, such as domain decomposition methods and preconditioners for some linear and well-known systems of PDEs

Efficient Algorithms for Computing Rational First Integrals and Darboux Polynomials of Planar Polynomial Vector Fields

International audienceWe present fast algorithms for computing rational first integrals with bounded degree of a planar polynomial vector field. Our approach builds upon a method proposed by Ferragut and Giacomini, whose main ingredients are the calculation of a power series solution of a first order differential equation and the reconstruction of a bivariate polynomial annihilating this power series. We provide explicit bounds on the number of terms needed in the power series. This enables us to transform their method into a certified algorithm computing rational first integrals via systems of linear equations. We then significantly improve upon this first algorithm by building a probabilistic algorithm with arithmetic complexity $\~O(N^{2 \omega})$ and a deterministic algorithm solving the problem in at most $\~O(d^2N^{2 \omega+1})$ arithmetic operations, where~$N$ denotes the given bound for the degree of the rational first integral, and where $d \leq N$ is the degree of the vector field, and $\omega$ the exponent of linear algebra. We also provide a fast heuristic variant which computes a rational first integral, or fails, in $\~O(N^{\omega+2})$ arithmetic operations. By comparison, the best previous algorithm uses at least $d^{\omega+1}\, N^{4\omega +4}$ arithmetic operations. We then show how to apply a similar method to the computation of Darboux polynomials. The algorithms are implemented in a Maple package RationalFirstIntegrals which is available to interested readers with examples showing its efficiency

Using morphism computations for factoring and decomposing linear functional systems

Within a constructive homological algebra approach, we study the factorization and decomposition problems for general linear functional systems (determined, under-determined, over-determined). Using the concept of Ore algebras of functional operators (e.g., ordinary/partial differential operators, shift operators, time-delay operators), we first concentrate on the computation of morphisms from a finitely presented left module $M$ over an Ore algebra to another one $M'$, where $M$ (resp., $M'$) is a module intrinsically associated with the linear functional system $R \, y=0$ (resp., $R' \, z=0$). These morphisms define applications sending solutions of the system $R' \, z=0$ to the ones of $R \, y=0$. We explicitly characterize the kernel, image, cokernel and coimage of a general morphism. We then show that the existence of a non-injective endomorphism of the module $M$ is equivalent to the existence of a non-trivial factorization $R=R_2\,R_1$ of the system matrix $R$. The corresponding system can then be integrated in cascade. Under certain conditions, we also show that the system $R \, y=0$ is equivalent to a system $R'\, z=0$, where $R'$ is a block-triangular matrix of the same size as $R$. We show that the existence of projectors of the ring of endomorphisms of the module $M$ allows us to reduce the integration of the system $R\,y=0$ to the integration of two independent systems $R_1 \, y_1=0$ and $R_2 \, y_2=0$. Furthermore, we prove that, under certain conditions, idempotents provide decompositions of the system $R\,y=0$, i.e., they allow us to compute an equivalent system $R'\, z=0$, where $R'$ is a block-diagonal matrix of the same size as $R$. Many applications of these results in mathematical physics and control theory are given. Finally, the different algorithms of the paper are implemented in a Maple package Morphisms based on the library OreModules

Computation of Koszul homology and application to involutivity of partial differential systems

International audienceThe formal integrability of systems of partial differential equations plays a fundamental role in different analysis and synthesis problems for both linear and nonlinear differential control systems. Following Spencer's theory, to test the formal integrability of a system of partial differential equations, we must study when the symbol of the system, namely, the top-order part of the linearization of the system, is 2-acyclic or involutive, i.e., when certain Spencer cohomology groups vanish. Combining the fact that Spencer cohomology is dual to Koszul homology and symbolic computation methods, we show how to effectively compute the homology modules defined by the Koszul complex of a finitely presented module over a commutative polynomial ring. These results are implemented using the OreMorphisms package. We then use these results to effectively characterize 2-acyclicity and involutivity of the symbol of a linear system of partial differential equations. Finally, we show explicit computations on two standard examples

Etude algorithmique du problème algébrique d'estimation de paramètres pour une classe de perturbations

We consider the algebraic parameter estimation problem for a class of standard perturbations. We assume that the measurement z(t) of a solution x(t) of a linear ordinary differential equation -- whose coefficients depend on a set \theta := {\theta_1, ..., \theta_r} of unknown constant parameters -- is affected by a perturbation \gamma(t) whose structure is supposed to be known (e.g., an unknown bias, an unknown ramp), i.e., z(t)=x(t, \theta)+\gamma(t). We investigate the problem of obtaining closed-form expressions for the parameters \theta_i's in terms of iterative indefinite integrals or convolutions of z. The different results are illustrated by explicit examples computed using the NonA package -- developed in Maple -- in which we have implemented our main contributions.Nous considérons le problème algébrique d'estimation de paramètres pour une classe classique de perturbations. Nous supposons que la mesure z(t) d'une solution x(t) d'une équation différentielle linéaire -- dont les coefficients dépendent d'un ensemble \theta := {\theta_1, ..., \theta_r} de paramètres constants inconnus -- est affectée par une perturbation \gamma(t) dont la structure est supposée connue (par exemple, un biais inconnu, une rampe inconnue), c'est-à-dire, nous supposons que z(t)=x(t, \theta)+\gamma(t). Nous étudions alors le problème d'obtenir des formes closes pour les paramètres \theta_i en fonction d'intégrales indéfinies itérées ou de convolutions de z. Les différents résultats sont illustrés par des exemples explicites calculés grâce au package NonA -- développé en Maple -- dans lequel nous avons implanté nos principales contributions