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Papers by Moulay Barkatou

Proceedings of the 1995 international symposium on Symbolic and algebraic computation - ISSAC '95, 1995

ABSTRACT

Proceedings of the 2014 Symposium, Jul 28, 2014

ABSTRACT We consider a matrix polynomial with given complex coefficients. A matrix S is called a ... more ABSTRACT We consider a matrix polynomial with given complex coefficients. A matrix S is called a solvent if P(S) = 0. We explore some approaches to the symbolic and numeric computation of solvents. In particular, we compute formulas for the condition number and backward error of the problem which rely on the contour integral based representation of P(S). Finally, we describe a possible approach for computing exact solvents symbolically.

Let L be a linear difference operator with polynomial coefficients. We consider singularities of ... more Let L be a linear difference operator with polynomial coefficients. We consider singularities of L that correspond to roots of the trailing (resp. leading) coefficient of L. We prove that one can effectively construct a left multiple with polynomial coefficients L' of L such that every singularity of L' is a singularity of L that is not apparent. As a consequence, if all singularities of L are apparent, then L has a left multiple whose trailing and leading coefficients equal 1.

In this paper we give an algorithmic characterization of rank two locally nilpotent derivations i... more In this paper we give an algorithmic characterization of rank two locally nilpotent derivations in dimension three. Together with an algorithm for computing the plinth ideal, this gives a method for computing the rank of a locally nilpotent derivation in dimension three.

We consider linear dierence equations with polynomial coecients over C and their solutions in the... more We consider linear dierence equations with polynomial coecients over C and their solutions in the form of doubly infinite sequences (sequential solutions). We investigate the C-linear space of subanalytic solutions, i.e., those sequential solutions that are the restrictions to Z of some analytic solutions of the original equation. It is shown that this space coincides with the space of the restrictions to Z of entire solutions and the dimension of this space is equal to the order of the original equation. We also consider d-dimensional (d 1) hypergeometric sequences, i.e., sequential resp. sub- analytic solutions of consistent systems of first-order dierence equations for a single unknown function. We show that the dimension of the space of subanalytic solutions is always at most 1, and that this dimension may be equal to 0 for some systems (although the dimension of the space of all sequential solutions is always positive).

We consider the problem of computing regular formal solutions of systems of linear differential e... more We consider the problem of computing regular formal solutions of systems of linear differential equations with analytic coefficients. The classical approach consists in reducing the system to an equivalent scalar linear differential equation and to apply the well-known Frobenius method. This transformation to a scalar equation is not necessarily relevant so we propose a generalization of the Frobenius method to handle directly square linear differential systems. Finally, we investigate the case of rectangular systems and show how their regular formal solutions can be obtained by computing those of an auxiliary square system.

Mathematics in Computer Science, 2010

This paper deals with the local analysis of systems of pseudo-linear equations. We define regular... more This paper deals with the local analysis of systems of pseudo-linear equations. We define regular solutions and use this as a unifying theoretical framework for discussing the structure and existence of regular solutions of various systems of linear functional equations. We then give a generic and flexible algorithm for the computation of a basis of regular solutions. We have implemented this algorithm in the computer algebra system Maple in order to provide novel functionality for solving systems of linear differential, difference and q-difference equations given in various input formats.

Proceedings of the 2010 International Symposium, 2010

ABSTRACT In this paper, we investigate the local analysis of systems of linear differential-algeb... more ABSTRACT In this paper, we investigate the local analysis of systems of linear differential-algebraic equations (DAEs) and second-order linear differential systems. In the first part of the pa-per, we show how one can transform an input linear DAE into a reduced form that allows for ...

Proceedings of the 39th International Symposium, Jul 23, 2014

In this article, we recover singularly-perturbed linear differential systems from their turning p... more In this article, we recover singularly-perturbed linear differential systems from their turning points and reduce their parameter singularity's rank to its minimal integer value. Our treatment is Moser-based; that is to say it is based on the reduction criterion introduced for linear singular differential systems in . Such algorithms have proved their utility in the symbolic resolution of the systems of linear functional equations , giving rise to the package ISOLDE [ Barkatou et al., 2013], as well as in the perturbed algebraic eigenvalue problem . In particular, we generalize the Moser-based algorithm described in . Our algorithm, implemented in the computer algebra system Maple, paves the way for efficient symbolic resolution of singularly-perturbed linear differential systems as well as further applications of Moser-based reduction over bivariate (differential) fields [Barkatou et al., 2014]. * Submitted to ISSAC'14, Kobe, Japan † Enrolled under a joint PhD program with the Lebanese University

Advances in Applied Mathematics, 2015

Proceedings of the 1998 international symposium on Symbolic and algebraic computation - ISSAC '98, 1998

ABSTRACT

Proceedings of the 1999 international symposium on Symbolic and algebraic computation - ISSAC '99, 1999

ABSTRACT

Lecture Notes in Computer Science, 2009

We consider linear homogeneous difference equations with rational-function coefficients. The sear... more We consider linear homogeneous difference equations with rational-function coefficients. The search for solutions in the form of the minterlacing (1 ≤ m ≤ ord L, where L is a given operator) of finite sums of hypergeometric sequences, plays an important role in the Hendriks-Singer algorithm for constructing all Liouvillian solutions of L(y) = 0. We show that Hendriks-Singer's procedure for finding solutions in the form of such m-interlacing can be simplified. We also show that the space of solutions of L(y) = 0 spanned by the solutions of the form of the m-interlacing of hypergeometric sequences possesses a cyclic permutation property. In addition, we describe adjustments of our implementation of the Hendriks-Singer algorithm to utilize the presented results.

Lecture Notes in Computer Science, 2011

We consider the following problem: given a linear differential system with formal Laurent series ... more We consider the following problem: given a linear differential system with formal Laurent series coefficients, we want to decide whether the system has non-zero Laurent series solutions, and find all such solutions if they exist. Let us also assume we need only a given positive integer number l of initial terms of these series solutions. How many initial terms of the coefficients of the original system should we use to construct what we need?

Lecture Notes in Computer Science, 2013

Systems of linear ordinary differential equations of arbitrary orders of full rank are considered... more Systems of linear ordinary differential equations of arbitrary orders of full rank are considered. We study the change in the dimension of the solution space that occurs while differentiating one of the equations. Basing on this, we show a way to compute the dimension of the solution space of a given full rank system. In addition, we show how the change in the dimension can be used to estimate the number of steps of some algorithms to convert a given full rank system into an appropriate form.

Proceedings of the ACM-SIGSAM 1989 international symposium on Symbolic and algebraic computation - ISSAC '89, 1989

This paper deals with linear systems of difference equations whose coefficients admit generalized... more This paper deals with linear systems of difference equations whose coefficients admit generalized factorial series representations at z = ∞. We shall give a criterion by which a given system is determined to have a regular singularity.In the same manner, we give an algorithm, implementable in a computer algebra system, which reduces in a finite number of steps the system

Proceedings of the 1995 international symposium on Symbolic and algebraic computation - ISSAC '95, 1995

ABSTRACT

Proceedings of the 2014 Symposium, Jul 28, 2014

ABSTRACT We consider a matrix polynomial with given complex coefficients. A matrix S is called a ... more ABSTRACT We consider a matrix polynomial with given complex coefficients. A matrix S is called a solvent if P(S) = 0. We explore some approaches to the symbolic and numeric computation of solvents. In particular, we compute formulas for the condition number and backward error of the problem which rely on the contour integral based representation of P(S). Finally, we describe a possible approach for computing exact solvents symbolically.

Let L be a linear difference operator with polynomial coefficients. We consider singularities of ... more Let L be a linear difference operator with polynomial coefficients. We consider singularities of L that correspond to roots of the trailing (resp. leading) coefficient of L. We prove that one can effectively construct a left multiple with polynomial coefficients L' of L such that every singularity of L' is a singularity of L that is not apparent. As a consequence, if all singularities of L are apparent, then L has a left multiple whose trailing and leading coefficients equal 1.

In this paper we give an algorithmic characterization of rank two locally nilpotent derivations i... more In this paper we give an algorithmic characterization of rank two locally nilpotent derivations in dimension three. Together with an algorithm for computing the plinth ideal, this gives a method for computing the rank of a locally nilpotent derivation in dimension three.

We consider linear dierence equations with polynomial coecients over C and their solutions in the... more We consider linear dierence equations with polynomial coecients over C and their solutions in the form of doubly infinite sequences (sequential solutions). We investigate the C-linear space of subanalytic solutions, i.e., those sequential solutions that are the restrictions to Z of some analytic solutions of the original equation. It is shown that this space coincides with the space of the restrictions to Z of entire solutions and the dimension of this space is equal to the order of the original equation. We also consider d-dimensional (d 1) hypergeometric sequences, i.e., sequential resp. sub- analytic solutions of consistent systems of first-order dierence equations for a single unknown function. We show that the dimension of the space of subanalytic solutions is always at most 1, and that this dimension may be equal to 0 for some systems (although the dimension of the space of all sequential solutions is always positive).

We consider the problem of computing regular formal solutions of systems of linear differential e... more We consider the problem of computing regular formal solutions of systems of linear differential equations with analytic coefficients. The classical approach consists in reducing the system to an equivalent scalar linear differential equation and to apply the well-known Frobenius method. This transformation to a scalar equation is not necessarily relevant so we propose a generalization of the Frobenius method to handle directly square linear differential systems. Finally, we investigate the case of rectangular systems and show how their regular formal solutions can be obtained by computing those of an auxiliary square system.

Mathematics in Computer Science, 2010

This paper deals with the local analysis of systems of pseudo-linear equations. We define regular... more This paper deals with the local analysis of systems of pseudo-linear equations. We define regular solutions and use this as a unifying theoretical framework for discussing the structure and existence of regular solutions of various systems of linear functional equations. We then give a generic and flexible algorithm for the computation of a basis of regular solutions. We have implemented this algorithm in the computer algebra system Maple in order to provide novel functionality for solving systems of linear differential, difference and q-difference equations given in various input formats.

Proceedings of the 2010 International Symposium, 2010

ABSTRACT In this paper, we investigate the local analysis of systems of linear differential-algeb... more ABSTRACT In this paper, we investigate the local analysis of systems of linear differential-algebraic equations (DAEs) and second-order linear differential systems. In the first part of the pa-per, we show how one can transform an input linear DAE into a reduced form that allows for ...

Proceedings of the 39th International Symposium, Jul 23, 2014

In this article, we recover singularly-perturbed linear differential systems from their turning p... more In this article, we recover singularly-perturbed linear differential systems from their turning points and reduce their parameter singularity's rank to its minimal integer value. Our treatment is Moser-based; that is to say it is based on the reduction criterion introduced for linear singular differential systems in . Such algorithms have proved their utility in the symbolic resolution of the systems of linear functional equations , giving rise to the package ISOLDE [ Barkatou et al., 2013], as well as in the perturbed algebraic eigenvalue problem . In particular, we generalize the Moser-based algorithm described in . Our algorithm, implemented in the computer algebra system Maple, paves the way for efficient symbolic resolution of singularly-perturbed linear differential systems as well as further applications of Moser-based reduction over bivariate (differential) fields [Barkatou et al., 2014]. * Submitted to ISSAC'14, Kobe, Japan † Enrolled under a joint PhD program with the Lebanese University

Advances in Applied Mathematics, 2015

Proceedings of the 1998 international symposium on Symbolic and algebraic computation - ISSAC '98, 1998

ABSTRACT

Proceedings of the 1999 international symposium on Symbolic and algebraic computation - ISSAC '99, 1999

ABSTRACT

Lecture Notes in Computer Science, 2009

We consider linear homogeneous difference equations with rational-function coefficients. The sear... more We consider linear homogeneous difference equations with rational-function coefficients. The search for solutions in the form of the minterlacing (1 ≤ m ≤ ord L, where L is a given operator) of finite sums of hypergeometric sequences, plays an important role in the Hendriks-Singer algorithm for constructing all Liouvillian solutions of L(y) = 0. We show that Hendriks-Singer's procedure for finding solutions in the form of such m-interlacing can be simplified. We also show that the space of solutions of L(y) = 0 spanned by the solutions of the form of the m-interlacing of hypergeometric sequences possesses a cyclic permutation property. In addition, we describe adjustments of our implementation of the Hendriks-Singer algorithm to utilize the presented results.

Lecture Notes in Computer Science, 2011

We consider the following problem: given a linear differential system with formal Laurent series ... more We consider the following problem: given a linear differential system with formal Laurent series coefficients, we want to decide whether the system has non-zero Laurent series solutions, and find all such solutions if they exist. Let us also assume we need only a given positive integer number l of initial terms of these series solutions. How many initial terms of the coefficients of the original system should we use to construct what we need?

Lecture Notes in Computer Science, 2013

Systems of linear ordinary differential equations of arbitrary orders of full rank are considered... more Systems of linear ordinary differential equations of arbitrary orders of full rank are considered. We study the change in the dimension of the solution space that occurs while differentiating one of the equations. Basing on this, we show a way to compute the dimension of the solution space of a given full rank system. In addition, we show how the change in the dimension can be used to estimate the number of steps of some algorithms to convert a given full rank system into an appropriate form.

Proceedings of the ACM-SIGSAM 1989 international symposium on Symbolic and algebraic computation - ISSAC '89, 1989

This paper deals with linear systems of difference equations whose coefficients admit generalized... more This paper deals with linear systems of difference equations whose coefficients admit generalized factorial series representations at z = ∞. We shall give a criterion by which a given system is determined to have a regular singularity.In the same manner, we give an algorithm, implementable in a computer algebra system, which reduces in a finite number of steps the system