Numerical Stability of Difference Equations with Matrix Coefficients

In this paper, we consider the homogeneous difference equation $\sum^k_{j = 0} \alpha_j y_{n - j} = 0, \quad n = k, k + 1, k + 2, \ldots,$ with initial values $y_j = q_j, \quad j = 0(1)k - 1.$ The yj are d-component column vectors, the αj are d × d matrices independent of n. We derive algebraic criteria for numerical stability of the difference equation, which is understood in the sense that the solution {yj} and its difference quotients up to order s ∈ {0, 1, 2, 3, ...} depend continuously on the initial values {qj}. This generalizes the well-known case where s = 0 and the αj are diagonal matrices.