Resolvent conditions and bounds on the powers of matrices, with relevance to numerical stability of initial value problems

We deal with the problem of establishing upper bounds for the norm of the nth power of square matrices. This problem is of central importance in the stability analysis of numerical methods for solving (linear) initial value problems for ordinary, partial or delay differential equations. A review is presented of upper bounds which were obtained in the literature under the resolvent condition occurring in the Kreiss matrix theorem, as well as under variants of that condition. Moreover, we prove new bounds, under resolvent conditions which generalize some of the reviewed ones. The paper concludes by applying one of the new upper bounds in a stability analysis of the trapezoidal rule for delay differential equations.