Effective Formal Reduction of Linear Differential Systems

Abstract. 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].

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“…We need the following well-known technical lemma, which has been stated in [14], see also its use in [15].…”

Section: Remark 2 the Results In Proposition 3 Are Valid For More Gementioning

confidence: 99%

“…We have added the new option 'allSolutions' which ensures that a complete basis of the regular solutions space is computed, by taking into account formula (15). The sequence is then a 1 = a 2 = 3 and a l = l for l ≥ 3, since the indicial polynomial of the system has roots 0 and 3.…”

Section: Methodsmentioning

confidence: 99%

Higher-Order Linear Differential Systems with Truncated Coefficients

Абрамов¹,

Barkatou²,

Pflügel³

2011

Computer Algebra in Scientific Computing

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“…Contrary to system (1), there exist efficient algorithms contributing to the formal reduction of system (2) (see, e.g [Barkatou, 1997;Pflugel , 2000]), that is the algorithmic procedure that computes a change of basis w.r.t. which A(x) has normal form facilitating the construction of formal solutions.…”

Section: Introductionmentioning

confidence: 99%

“…For the univariate case, system (2), the research advanced profoundly in the last two decades making use of methods of modern algebra and topology. The former classical approach is substituted by efficient algorithms (see, e.g., [Barkatou, 1997;Barkatou et al, 2013Barkatou et al, , 2009Pflugel , 2000] and references therein). It was the hope of Wasow [Wasow, 1985], in his 1985 treatise summing up contemporary research directions and results on system (1), that techniques of system (2) be generalized to tackle the problems of system (1).…”

Section: Introductionmentioning

confidence: 99%

On the reduction of singularly-perturbed linear differential systems

Maddah

Barkatou

Abbas

2014

Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation

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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 [Moser, 1960]. Such algorithms have proved their utility in the symbolic resolution of the systems of linear functional equations [Barkatou, 1989;Barkatou et al., 2008Barkatou et al., , 2009, giving rise to the package ISOLDE [Barkatou et al., 2013], as well as in the perturbed algebraic eigenvalue problem [Jeannerod et al., 1999]. In particular, we generalize the Moser-based algorithm described in [Barkatou, 1995]. 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].

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“…Our proof relies on the results in [2] and a generalisation of the Splitting Lemma [7]. This result can be useful when designing algorithms for computing ρ and related formal invariants.…”

mentioning

confidence: 99%