OreModules (click here) is a Maple package dedicated to module theory and homological algebra for finitely presented modules defined over an Ore algebra of functional operators (e.g., ordinary or partial differential operators, shift operators, time-delay operators, difference operators) available in the Maple package Ore_algebra, and to their applications in mathematical systems theory and mathematical physics.

Within this algebraic unified framework, OreModules handles the following classes of linear functional systems:

  1. ordinary differential systems,
  2. partial differential systems,
  3. multidimensional discrete systems,
  4. differential time-delay systems,
  5. multidimensional convolutional codes...

These linear functional systems may be with constant, polynomial or rational coefficients.

The OreModules package is an implementation of algorithms which compute different invariants or algebraic objects associated with a finitely presented left module over an Ore algebra such that:

  1. (shortest) free resolution,
  2. projective dimension,
  3. Hilbert series,
  4. extension modules with values in the ring,
  5. (minimal, chain, injective) parametrizations...

It can also be used to check whether or not a finitely presented module admits torsion elements and if so, to compute a generating set, or if it is torsion-free, reflexive, projective, stably free or free. If the module is torsion-free, then a parametrization which parametrizes its solutions space can be computed.

In the context of linear control systems, the main features of OreModules are the following:

  1. compute a family of generators for the autonomous elements,
  2. decide controllability and parametrizability,
  3. compute (minimal, chain, injective) parametrizations,
  4. compute Monge parametrizations,
  5. compute Bezout identities (left/right/generalized inverses),
  6. decide flatness (also pi-freeness),
  7. study linear quadratic problems...

For more details, see:

  1. F. Chyzak, A. Quadrat, D. Robertz, Effective algorithms for parametrizing linear control systems over Ore algebras, Applicable Algebra in Engineering, Communications and Computing, 16 (2005), 319-376.
  2. F. Chyzak, A. Quadrat, D. Robertz, OreModules: A symbolic package for the study of multidimensional linear systems, in Applications of Time-Delay Systems, J. Chiasson, J.-J. Loiseau (Eds.), Lecture Notes in Control and Information Sciences 352, Springer, 2007, 233-264.

The package OreModules was initiated by F. Chyzak and A. Quadrat, and further developed by D. Robertz and A. Quadrat.

A Mathematica version of the OreModules package, called OreAlgebraicAnalysis, is also available. It was developed by Maris Tõnso, T. Cluzeau and A. Quadrat within the PHC Parrot project CASCAC. The Mathematica version of the OreModules package is based on the implementation of Gröbner bases over Ore algebras available in the Mathematica HolonomicFunctions package developed by Christoph Koutschan.

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