Computing super-irreducible forms of systems of linear differential equations via moser-reduction

Barkatou, Moulay A.; Pflügel, Eckhard

doi:10.1145/1277548.1277550

Cited by 18 publications

(30 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Having clarified the situation if a system is simple, we now solve Problem 1 and Problem 2 for the case of a general first order system of the form (9). Define the span of an invertible matrix T (x) ∈ Mat m (k((x))) by…”

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

confidence: 99%

See 1 more Smart Citation

Higher-Order Linear Differential Systems with Truncated Coefficients

Абрамов¹,

Barkatou²,

Pflügel³

2011

Computer Algebra in Scientific Computing

View full text Add to dashboard Cite

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].

show abstract

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

confidence: 99%

“…(i) The algorithm from [13] computes the so-called super-irreducible form of a given system (9). It was shown in [3] that if a system has the super-irreducible form then it can be written as a simple system.…”

Section: Which Is Simple Letĩ(λ) Denote the Indicial Polynomial Of Tmentioning

confidence: 99%

Higher-Order Linear Differential Systems with Truncated Coefficients

Абрамов¹,

Barkatou²,

Pflügel³

2011

Computer Algebra in Scientific Computing

View full text Add to dashboard Cite

show abstract

“…[12]. The specific algorithm is given by Barkatou&Pflügel [13,14]. In fact, this algorithm allows one the reduce the system to Fuchsian form in all finite regular singular points.…”

Section: Reducing To Fuchsian Form and Normalizating Eigenvaluesmentioning

confidence: 99%

Multiloop calculations with differential equations in ε-form

Lee

2016

EPJ Web Conf.

View full text Add to dashboard Cite

show abstract

“…rational solutions of a linear firstorder system of differential equations directly. So far only simple systems (for the definition see [3,11]) can be treated; there are several algorithms to transform a system that not simple into a simple one [12,13] but this seems to be a very cumbersome procedure.…”

Section: Rational Solutions Of Systems Of Differential and Differencementioning

confidence: 99%

Advanced applications of the holonomic systems approach

Koutschan

2010

ACM Commun. Comput. Algebra

107

174

View full text Add to dashboard Cite

Ich erkläre an Eides statt, dass ich die vorliegende Dissertation selbstständig und ohne fremde Hilfe verfasst, andere als die angegebenen Quellen und Hilfsmittel nicht benutzt bzw. die wörtlich oder sinngemäß entnommenen Stellen als solche kenntlich gemacht habe. Linz, September 2009Christoph Koutschan AbstractThe holonomic systems approach was proposed in the early 1990s by Doron Zeilberger. It laid a foundation for the algorithmic treatment of holonomic function identities. Frédéric Chyzak later extended this framework by introducing the closely related notion of ∂-finite functions and by placing their manipulation on solid algorithmic grounds. For practical purposes it is convenient to take advantage of both concepts which is not too much of a restriction: The class of functions that are holonomic and ∂-finite contains many elementary functions (such as rational functions, algebraic functions, logarithms, exponentials, sine function, etc.) as well as a multitude of special functions (like classical orthogonal polynomials, elliptic integrals, Airy, Bessel, and Kelvin functions, etc.). In short, it is composed of functions that can be characterized by sufficiently many partial differential and difference equations, both linear and with polynomial coefficients. An important ingredient is the ability to execute closure properties algorithmically, for example addition, multiplication, and certain substitutions. But the central technique is called creative telescoping which allows to deal with summation and integration problems in a completely automatized fashion. Part of this thesis is our Mathematica package HolonomicFunctions in which the above mentioned algorithms are implemented, including more basic functionality such as noncommutative operator algebras, the computation of Gröbner bases in them, and finding rational solutions of parameterized systems of linear differential or difference equations.Besides standard applications like proving special function identities, the focus of this thesis is on three advanced applications that are interesting in their own right as well as for their computational challenge. First, we contributed to translating Takayama's algorithm into a new context, in order to apply it to an until then open problem, the proof of Ira Gessel's lattice path conjecture. The computations that completed the proof were of a nontrivial size and have been performed with our software. Second, investigating basis functions in finite element methods, we were able to extend the existing algorithms in a way that allowed us to derive various relations which generated a considerable speed-up in the subsequent numerical simulations, in this case of the propagation of electromagnetic waves. The third application concerns a computer proof of the enumeration formula for totally symmetric plane partitions, also known as Stembridge's theorem. To make the underlying computations feasible we employed a new approach for finding creative telescoping operators. KurzzusammenfassungIn den frühen 1990er Jahren beschr...

show abstract

Computing super-irreducible forms of systems of linear differential equations via moser-reduction

Cited by 18 publications

References 14 publications

Higher-Order Linear Differential Systems with Truncated Coefficients

Higher-Order Linear Differential Systems with Truncated Coefficients

Multiloop calculations with differential equations in ε-form

Advanced applications of the holonomic systems approach

Contact Info

Product

Resources

About