Package of Procedures for Inverting Matrices Whose Entries are Linear Difference Operators

Abstract

The matrices considered in this paper belong to ${\text{Ma}}{{{\text{t}}}_{n}}(\mathbb{K}[\sigma ,{{\sigma }^{{ - 1}}}])$, i.e., to the ring of $n \times n$-matrices whose entries are scalar difference operators with the coefficients from the difference field $\mathbb{K}$ of characteristic 0 with automorphism (“shift”) $\sigma$. A family of algorithms is discussed that allow one to check whether there exists an inverse matrix for a given matrix from ${\text{Ma}}{{{\text{t}}}_{n}}(\mathbb{K}[\sigma ,{{\sigma }^{{ - 1}}}])$ in this ring and, if exists, to construct it. These algorithms are made to correspond to complexities in terms of the number of arithmetic operations and the number of shifts (i.e., applications of σ and ${{\sigma }^{{ - 1}}}$) in the field $\mathbb{K}$. The algorithms are implemented in the form of Maple-procedures. This makes it possible to experimentally compare them in terms of time spent. The selection of the best algorithm based on these experiments does not always coincide with the complexity-based selection. An attempt is made to find out why this happens. A package of procedures for solving the considered problems is suggested, where the main procedure includes a parameter that specifies which algorithm is to be applied. If this parameter is lacking, than an a priori specified algorithm is selected that is relatively good both from the complexity and experimental standpoint compared to the others.