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Main functions for the treatment of linear systems over Ore algebras D
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| 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
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| 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 |