An effective numerical technique for solving a special class of ordinary difference equations
The authors consider a system of ordinary difference equations of the form \[ Av_{m + 1} + Bv_m = f_{m + 1/2}, \quad m \in \mathbb{Z}. \tag{1} \] Here, \(A\) and \(B\) are square matrices with constant entries, and the right hand side is compactly supported, which means that the equation is homogeneous outside some finite mesh interval. Moreover, it is first assumed that the matrices \(Q = A^{-1} B\) and \(Q^{-1}\) exist, and that each eigenvalue \(\mu\) of \(Q\) satisfies \(|\mu |\neq 1\). Under the additional assumption that \(v_m \to 0\) for \(m \to \pm \infty\), it is shown that there is a unique solution of (1). Then the authors develop a numerical algorithm for the computation of this solution, and give a detailed discussion of the numerical stability and efficiency of their method. A discussion of the case where eigenvalues of \(Q\) of absolute value 1 exist, and some considerations on practical applications of the problem under consideration conclude the paper.
scientific article; zbMATH DE number 826020