Guessing Linear Recurrence Relations of Sequence Tuples and P-recursive Sequences with Linear Algebra
Résumé
Given several n-dimensional sequences, we first present an algorithm
for computing the Gröbner basis of their module of linear recurrence
relations.
A P-recursive sequence (ui)i∈Nn
satisfies linear recurrence relations with polynomial coefficients in
i, as defined by Stanley in 1980. Calling directly
the aforementioned algorithm on the tuple of
sequences ((ijui)i∈Nn)j
for retrieving the relations yields redundant relations.
Since the module of relations of a
P-recursive sequence also has an extra structure of a 0-dimensional right
ideal of an Ore algebra, we design a more efficient algorithm that takes
advantage of this extra structure for
computing the relations.
Finally, we show how to incorporate Gröbner bases computations in an
Ore algebra K⟨t1,…,tn,x1,…,xn⟩, with
commutators xkxℓ−xℓxk=tktℓ−tℓtk=tkxℓ−xℓtk=0 for k≠ℓ and
tkxk−xktk=xk, into the algorithm designed for P-recursive
sequences. This allows us to compute faster the Gr\"obner basis of the ideal spanned by the first relations,
such as in \textsc{2D}/\textsc{3D}-space walks examples.
Domaines
Calcul formel [cs.SC]Origine | Fichiers produits par l'(les) auteur(s) |
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