ICMS 2016 Session: Software for the Symbolic Study of Functional Equations
Aim and Scope
The aim of this session is to present softwares or packages dedicated
to the symbolic or symbolic-numeric treatment of (systems of)
functional equations such as ordinary or partial differential
equations, (q-)difference equations, differential time-delay
equations, discrete-time equations, ... The presentations are expected
to enhance interactions between developers and potential users on the
one hand, and amongst developers/researchers on the other hand. Survey
talks of research groups, reviews of the state of the art, works which
make substantial use of such existing softwares for miscellaneous
applications, and demonstrations of softwares under development are
Topics (including, but not limited to)
- Symbolic resolution of linear or nonlinear functional equations or systems,
- Symbolic manipulation of functional operators,
- Qualitative study of functional systems,
- The submission of short abstracts is closed. We kindly refer to the guideline
for details on submitting an extended abstract.
The duration of each talk is 30 minutes. For more details, we kindly refer to the daily program.
First slot :
- Suzy S. Maddah
In this talk, we give an overview of the packages which have been
developed within the Computer Algebra (Calcul Formel) group at the
University of Limoges, for the symbolic treatment of (systems of)
functional equations. We illustrate with examples on the analysis of
singularities and the computation of local formal, rational, and
hyper-exponential solutions, of certain classes of ordinary (perturbed)
and partial linear differential systems.
- Thomas Cluzeau
Algorithms and related Maple packages for integrable connections and planar polynomial vector fields
In the previous talk, Suzy S. Maddah presents algorithms (and related
packages) handling first order systems of linear differential
equations. In this talk, I will consider two kind of generalizations of
linear differential systems.
(a) Integrable connections, i.e., a class of systems of first order
linear partial differential systems. I will present algorithms for
computing closed form (e.g., rational and hyperexponential) solutions
of such systems and illustrate a dedicated Maple package. This
work has been done within the Computer Algebra group at the University
of Limoges (France) in collaboration with M. A. Barkatou, C. El Bacha
and J.-A. Weil. (Ref: Computing Closed-Form Solutions of Integrable
Connections, Proc. of ISSAC 2012: 43-50)
(b) Planar polynomial vector fields, i.e., a class of systems of
two non linear (polynomial) differential equations in two dependent
variables of one independent variable (e.g., time). I will give the
idea of an efficient algorithm for computing rational first integrals
of such vector fields. A Maple implementation of this algorithm within
a dedicated package will be demonstrated. This is a joint work with A.
Bostan, G. Chèze and J.-A. Weil. (Ref: Efficient algorithms for
computing rational first integrals and Darboux polynomials of planar
polynomial vector fields, Math. of Comp. 85: 1393-1425, 2016)
- Jamal Hossein Poor
Normal forms for operators via Gröbner bases in tensor algebras
We propose a general algorithmic approach to noncommutative operator
algebras generated by linear operators. Many cases arising in
applications, like differential and difference operators, are covered
by skew polynomials. Integral operators, however, can not be treated in
this framework since their normal forms do not match the normal forms
of skew polynomials. In our approach, we model operator algebras as
suitable quotients of tensor algebras instead. They naturally capture
the multilinearity of composition of linear operators and make a
basis-free treatment of multiplication operators possible. In order to
work with reduction systems in tensor algebras, Bergman’s setting
provides a tensor analog of Gröbner bases. Given a Gröbner basis, the
structure of normal forms can be determined via a combinatorial word
We introduce a refinement of Bergman’s setting, which allows for
smaller reduction systems and tends to make computations more
efficient. Verification of the confluence criterion based on
S-polynomials defined in terms of pure tensors has been implemented as
a Mathematica package. Our implementation can also be used for
computer-assisted construction of Gröbner bases starting from basic
identities of operators. We illustrate our approach and the software
using integro-differential operators as an example.
Joint work with Clemens G. Raab, and Georg Regensburger.
- Albert Heinle
Factoring Elements in G-Algebras with 'ncfactor.lib'
The library `ncfactor.lib' initiated the support for non-commutative
factorization in the computer algebra system Singular. The last major
update added functions to factor elements in linear partial
differential, difference and certain q-differential operators in n
variables (Giesbrecht, Heinle, Levandovskyy, ISSAC'14 and the follow-up
paper in JSC 2015). Now we announce an algorithm to factor
elements in any given G-algebra, with minor assumptions on the
underlying field. We are going to present our implementation of
this algorithm, and discuss how our findings enable us to generalize
methods formerly only defined for commutative rings, like e.g. the
factorized Groebner basis algorithm.
Unsurprisingly, for certain choices of G-algebras, there exist better
factorization techniques that leverage specific structural properties.
We will show certain characteristics of a given algebra that can be
used to cut down computational costs while computing factorizations.
The key mathematical result, providing the termination on our
factorization algorithm is a recent discovery (Bell, Heinle,
Levandovskyy, Trans. AMS 2015) that a big class of algebras, including
G-algebras, are finite factorization domains. We will outline
further consequences and future research possibilities.
Joint work with Viktor Levandovskyy.
- Viktor Levandovskyy
Algorithms for systems of linear functional equations and their implementation in Singular
A system Singular:Plural provides a user with the wide range of
implemented algorithms for noncommutative G-algebras. In particular,
classical algebras of linear functional operators as well as their
tensor products are supported.
We give an overview of recent developments both in algorithms and in
the implementation in Singular:Plural, focusing on Ore localization,
dimensions, Weyl closure and various constructions with (not only)
differential modules with applications to the study of their solution
- Thomas Cluzeau and Christoph Koutschan
Effective algebraic analysis approach to linear systems over Ore algebras
Physical systems are typically modeled by systems of functional
equations; these can be differential equations, differential time-delay
systems, discrete-time systems, etc. The algebraic analysis approach
studies such systems from a purely algebraic viewpoint, using D-module
theory and homological algebra techniques. The different types of
systems can be uniformly described by using the general concept of Ore
algebras. Recently, we have developed the Mathematica package
OreAlgebraicAnalysis that implements (1) Groebner basis techniques for
finitely presented left modules over Ore algebras, (2) algorithms for
deciding module-theoretic properties, such as torsion-freeness,
reflexivity, projectiveness, freeness, and (3) algorithms from
homological algebra, such as computation of free resolutions and
projective dimension. The OreAlgebraicAnalysis package makes use of the
very generic implementation of Ore algebras in the package
HolonomicFunctions, that also provides the computation of Groebner
bases in such rings. We present and demonstrate these packages by means
of examples from control theory.
Joint work with Alban Quadrat, Maris Tonso.