Stability of numerical methods for the solution of initial value problems
by
Marc N. Spijker
Leiden University, Netherlands

This talk addresses the problem of establishing suitable upper bounds for the norm of the powers of a given square matrix. It is shown that such bounds are highly relevant to the stability analysis of numerical methods for solving initial value problems, in linear partial differential equations.

One of the equivalent statements, in the famous Kreiss matrix theorem, consists in a condition on the resolvent of a matrix. This condition implies an upper bound for the norm of the powers of the matrix. In this talk a review is presented of old and recent results, as well as of open problems, concerning the structure of this upper bound. Special attention will be paid to the question in which manner the bound depends on the order of the matrix, as well as on the so-called Kreiss constant and on the norm under consideration.

Date received: February 15, 1998