On a generalization of the resolvent condition in the Kreiss matrix theorem

Abstract:

This paper deals with a condition on the resolvent of $s \times s$ matrices A. In one of the equivalent assertions of the Kreiss matrix theorem, the spectral norm of the resolvent of A at $\zeta$ must satisfy an inequality for all $\zeta$ lying outside the unit disk in $\mathbb {C}$. We consider a generalization in which domains different from the unit disk and more general norms are allowed. Under this generalized resolvent condition an upper bound is derived for the norms of the nth powers of $s \times s$ matrices B. Here, B depends on A via a relation $B = \varphi (A)$, where $\varphi$ is an arbitrary rational function. The upper bound grows linearly with $s \geq 1$ and is independent of $n \geq 1$. This generalizes an upper bound occurring in the Kreiss theorem where $B = A$. Like the classical Kreiss theorem, the upper bound derived in this paper can be used in the stability analysis of numerical methods for solving differential equations.