Main commands of OreModules

 
Following is a table of the main commands of the Maple package OreModules. Many of them come in pairs: one deals with systems that have polynomial coefficients, the other one ("Rat") copes with rational coefficients.  
 
 
Main functions for the treatment of linear systems over Ore algebras D
Parametrization(Rat) Find a parametrization of the system in terms of functions
MinimalParametrization(s)(Rat) Find a (some) minimal parametrization(s) of the system
AutonomousElements(Rat) Find generating set of autonomous elements of the system (i.e. solve the system of equations for the torsion elements) in case of Weyl algebras D = An (i.e. PDEs)
Brunovsky(Rat) Brunovský canonical form for 1-D systems
LeftInverse(Rat) Left inverse for matrices over D
LocalLeftInverse Given a non-zero polynomial π in k[x1, ..., xn], compute a left inverse for a matrix over k[x1, ..., xn, π^(-1)]
RightInverse(Rat) Right inverse for matrices over D
GeneralizedInverse(Rat) Compute a generalized inverse matrix over D
PiPolynomial Given a system matrix R over a commutative polynomial ring D and a variable xi in D, compute the ideal of all the π-polynomials in xi for the given system
FirstIntegral In the case of ordinary differential equations, find first integrals of motion
LQEquations Euler-Lagrange equations for linear quadratic problems of optimal control (ordinary differential equations)
Module theory over Ore algebras D
TorsionElements(Rat) Compute the torsion submodule of a left f.p. D-module
Exti(Rat) Given a f.p. left D-module M and j, compute ext^j_D(M, D)
Extn(Rat) Given a f.p. left D-module M and m, compute ext^i_D(M, D) for 0 <= i <= m
Quotient(Rat) Compute the quotient module of two left D-modules defined as images of two matrices
SyzygyModule(Rat) Compute the first syzygy module of a f.p. left D-module
Elimination(Rat) Elimination of variables (useful for observability test and elimination of latent variables)
Resolution(Rat) Given i, compute the first ith terms of a free resolution of a f.p. left D-module
FreeResolution(Rat) Compute a free resolution of a f.p. left D-module
OreRank(Rat) Compute the rank of a f.p. left module over D
Some low-level functions of OreModules
DefineOreAlgebra Set up an Ore algebra D in OreModules
Involution Apply an involution to a matrix over D (e.g. compute the adjoint of an operator in the case of Weyl algebras)
Factorize(Rat) Factorize, if possible, one matrix over D by a second one having the same number of columns
Mult Multiply two or more matrices over D
ApplyMatrix Apply (matrices of) operators in D to (vectors of) functions

 
Please, see also the Library of Examples.
 

Download Maple package OreModules

 
OreModules is available for Maple 8, Maple 9, and Maple 10. This is the second release of OreModules: If you need OreModules for Maple V Release 5 or Maple 6, then please contact one of the authors.
 
After downloading the Maple package, you can follow the installation guide below.
 
OreModules requires the Maple library Mgfun. This library is part of the standard library of Maple since version 9. If you want to use OreModules with earlier versions of Maple, then please download and install Mgfun as part of the library algolib.
 
After installing OreModules, it would be helpful if you could send us a short e-mail which explains for what purpose OreModules is beneficial for you and which Maple version you use.
 
If you encounter any problem with OreModules, don't hesitate to contact us.
 

Installing OreModules

 
  1. Copy the OreModules library files "OreModules.ind", "OreModules.lib" and the OreModules help data base "OreModules.hdb" for your Maple version (see download) into a directory called "OreModules".

  2. If you use Maple 8, please copy the library "algolib" (available at the algolib web page) into a directory called "Algolib".
    If you use Maple 9 or a newer version, then don't mind.

  3. Type
     
    libname;
     
    in Maple.

  4. Write
     
    libname := "the global path of the directory Algolib", "the global path of the directory OreModules", the result of step 3:
     
    For more details, see ?libname in Maple.
     
  5. Try
     
    with(Ore_algebra);
    with(OreModules);
    Alg := DefineOreAlgebra(diff=[D,t], polynom=[t]):
     
    If you encounter a problem, then most probably the definition of libname in step 4 is wrong in the sense that its value does not point to the correct directory where your library files reside.
    A reasonable way to check your installation is to run one of the example worksheets of the Library of Examples.
    If you still have problems concerning the installation of OreModules, please contact us.