Eckhard Pflügel

An algorithm computing the regular formal solutions of a system of linear differential equations

We propose a method for computing the regular singular formal solutions of a linear differential system in the neighbourhood of a singular point. This algorithm avoids the use of cyclic vectors and has been implemented11The program is contained in the package ISOLDE at http://www-lmc.imag.fr/CF/logiciel.htmlin the computer algebra system Maple.

On the equivalence problem of linear differential systems and its application for factoring completely reducible systems

Given two linear di erential systems with rational function coe cients, we give an algorithm to decide whether these two systems are equivalent and to compute the corresponding transformation matrices. In the second part of the paper, we use this for computing factorizations of completely reducible systems. In [20] algorithms for solving these problems in the case of scalar di erential equations have been given. They are based upon the local analysis of the singularities of the equation. Our method uses local methods as well, but it avoids converting to the scalar case. The algorithms are implemented and available.

An algorithm for computing exponential solutions of first order linear differential systems

This article deals with the computation of exponential solutions of first order linear differential systems with rational function coefficients. We develop an algorithm working dlrectly at the system in contrast to the standard approach of cyclic vectors which transforms the system in an equivalent nt h order scalar linear differential equation. We have implemented our method in the computer algebra system MAPLE V, and it turns out that it is in general more efficient than the cyclic vector approach.

Computing super-irreducible forms of systems of linear differential equations via moser-reduction: a new approach

The notion of irreducible forms of systems of linear differential equations as defined by Moser [14 ] and its generalisation, the super-irreducible forms introduced by Hilali/Wazner in [9 ] are important concepts in the context of the symbolic resolution of systems of linear differential equations [3,15,16 ]. In this paper, we give a new algorithm for computing, given an arbitrary linear differential system with formal power series coefficients as input, an equivalent system which is super-irreducible. Our algorithm is optimal in the sense that it computes transformation matrices which obtain a maximal reduction of rank in each step of the algorithm. This distinguishes it from the algorithms in [9,14,2] and generalises [7].

On simultaneous row and column reduction of higher-order linear differential systems

In this paper, we define simultaneously row and column reduced forms of higher-order linear differential systems with power series coefficients and give two algorithms, along with their complexities, for their computation. We show how the simultaneously row and column reduced form can be used to transform a given higher-order input system into a first-order system. Finally, we show that the algorithm can be used to compute Two-Sided Block Popov forms as given in Barkatou et al. (2010). These results extend the previous work in Barkatou et al. (2010), on second-order systems, and Harris et al. (1968), on first-order systems, to systems of arbitrary order.

A reduction algorithm for matrices depending on a parameter

In this articlc, \ve study square matrices pcrturbcd by a pararncter E. An efficient algorithm conlputing the z-espansiou of the eigcnvalucs in forinal Laurent-Puiseux series is provided, for \vhicli the computation of the characteristic polynomial is not rccluired. 15;e show 110~ to reduce the init,ial mat.ris so t.hat. the Lidskii-Edelman-Yla perturbat,iou theory [16] can be applied. We also explain why this approach lnily simplify t,he pcrturbcd cigenvcctor problem. The implement,ation of the algorithm in the comput.er algebra syst.em X~APIX has been used in a quantun~ mechani& cont.est to diagonalize sonle perturbed ruat.riccs and is available.

Simultaneously row- and column-reduced higher-order linear differential systems

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 paper, we show how one can transform an input linear DAE into a reduced form that allows for the decoupling of the differential and algebraic components of the system. Classification of singularities of linear DAEs are defined and discussed. In the second part of the paper, we extend this approach to second-order linear differential systems and discuss two applications: the classification of singularities and the computation of regular solutions. The present paper is the first step towards a generalisation of the formal reduction of first-order ODEs to higher-order systems. Our algorithm has been implemented in the computer algebra system Maple as part of the ISOLDE package.

Regular systems of linear functional equations and applications

The algorithmic classification of singularities of linear differential systems via the computation of Moser- and super-irreducible forms as introduced in [21] and [16] respectively has been widely studied in Computer Algebra ([8, 12, 22, 6, 10]). Algorithms have subsequently been given for other forms of systems such as linear difference systems [4, 3] and the perturbed algebraic eigenvalue problem [18]. In this paper, we extend these concepts to the general class of systems of linear functional equations. We derive a definition of regularity for these type of equations, and an algorithm for recognizing regular systems. When specialised to q-difference systems, our results lead to new algorithms for computing polynomial solutions and regular formal solutions.

Higher-order linear differential systems with truncated coefficients

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? Supposing that the series coefficients of the original systems are represented algorithmically, we show that these questions are undecidable in general. However, they are decidable in the scalar case and in the case when we know in advance that a given system has an invertible leading matrix. We use our results in order to improve some functionality of the Maple [17] package ISOLDE [11].

A Monomial-by-Monomial Method for Computing Regular Solutions of Systems of Pseudo-Linear Equations

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.