Main functions of OreAlgebraicAnalysis

 

Following is a table of the main functions of the Mathematica package OreAlgebraicAnalysis. For more details on the inputs and outputs of each function, see the different examples in Library of Examples.

In this table, we denote by:

  1. A an Ore algebra handled by the Mathematica package HolonomicFunctions,

  2. R (resp., S) a matrix with entries in A defining the linear functional system R η =0,

  3. ker(.R) the left A-module generated by the vectors λ such that of λ R=0,

  4. im(.R) (resp., im(.S)) the left A-module generated by the rows of R (resp., S),

  5. M (resp., N) is a left A-module finitely presented by R (resp., S), i.e., M=coker(.R) (resp., N=coker(.S)),

  6. v a list of functions,

  7. η the list of system variables,

  8. τ and ξ two symbols,

  9. j, d1 and d2 three non-negative integers,

  10. f a list of functions,

  11. P, Q and Z are matrices with entries in A.

 
 
Some useful low-level functions
ModelToReplacementRules
 Rewriting rules for nonlinear systems
OreAlgebraWithRelations
 Define an Ore algebra which coefficients satisfy relations coming from a nonlinear system
OreSimplify
 Simplify the coefficients of a polynomial in an Ore algebra which coefficients satisfy a nonlinear system
OreDot[R, S, A]
 Multiply two matrices R and S
Functions for the study of matrices with entries in A
Involution[R, A]
 Apply an involution to a matrix R
LeftKernel[R, A]
 Compute the left kernel of a matrix R, i.e., find L such that ker(.R)=im(.L), or return INJ if R has full row rank
LeftFactorize[R, S, A]
 Factorize (if possible) a matrix R by a matrix S, i.e., find L such that R=L S
LeftInverse[R, A] (RightInverse[R , A])
 Compute (if it exists) a left (right) inverse of a matrix R, i.e., find L such that L R=I (R L=I)
GeneralizedInverse[R, A]
 Compute (if it exists) a generalized inverse of a matrix R, i.e. find L such that R L R =R
ApplyMatrix[R, v, A]
 Apply R to v
Functions for the study of modules over an Ore algebra A
OreRank[R, A]
 Compute the rank of M
LeftProjectiveDimension[R, A]
 Compute the left projective dimension of M
AutonomousElements[R, η, τ, A]
 Find a generating set of the autonomous elements τ[i] of the linear functional system R η =0
Parametrization[R, ξ, A]
Find a parametrization of the linear functional system R η =0 in ξ
MinimalParametrization(s)[R, A]
Find a (some) minimal parametrization(s) of the linear functional system R η =0
ObstructionToProjectiveness[R, A]
 Find the obstructions for M to be projective
IsTorsion[R, A]
 Check whether or not M is torsion
IsTorsionFree[R, A]
 Check whether or not M is torsion-free
AnnihilatorTorsionElements[R, S, A]
 Compute the annihilator of torsion elements defined by the residue classes of the rows of R in N
AnnihilatorTorsionModule[R, A]
 Compute the left annihilator of the left torsion module M
IntersectionModule[R, S, A]
 Compute the intersection of the left A-modules im(.R) and im(.S)
Functions for homological algebra A
Resolution[R, A, j]
 Compute the first j terms of a free resolution of M
FreeResolution[R, A]
 Compute a free resolution of M
ShorterFreeResolution[R, A]
 Compute a shorter free resolution of M
ShortestFreeResolution[R, A]
 Compute a free resolution of M of minimal size
Ext1[R, A]
 Compute ext1(M, A)
Exti[R, A, j]
 Compute extj(M, A) for j > 0
Ext[R, A]
 Compute the exti(M, A)'s for i > 0
Functions for computing A-homomorphisms between M and N and idempotent A-endomorphisms of M
Morphisms[R, S, {d1, d2}, τ, A]
 Compute a matrix which defines an A-homomorphism from M to N
GeneralMorphisms[R, S, d1, τ, f, A]
 Compute the obstructions for a given ansatz to define an A-homomorphism from M to N
Riccati[R, P, {d1, d2}, ξ, A]
 Compute solutions Λ of an algebraic Riccati equation Λ R Λ +(P-I) Λ+ Λ Q+Z=0, where Q and Z are defined by the relations R P=Q R and P2=P + Z R
IdempotentMorphisms[P, R, τ, A]
 Given P defining an A-endomorphism of M, compute a list of idempotent matrices which define idempotent A-endomorphisms of M
Function for decomposing a linear functional system
Decomposition[R, P, A]
 Compute (if possible) a decomposition of R. The heuristic part corresponds to the computation of bases of the different free left A-modules

 
Please, see also the Library of Examples.