Computational and Applied Math Proseminar

Department of Mathematics and Statistics
Arizona State University

Thursday, October 10, 2002, 12:15 p.m. in GWC 604

Marino Zennaro

Division of Mathematical Sciences
University of Trieste

The Joint Spectral Radius Approach for Numerical Stability Investigations

Abstract The stability analysis of numerical methods for ODEs often leads to study the asymptotic behaviour of the solutions of linear difference equations with variable coefficients. Therefore, it is important to have efficient methods available for such investigations.

It is known that the asymptotic behaviour of the solutions of a linear difference equation may be described by studying the asymptotic behaviour of the products of the companion matrices associated to the difference equation (as the number of factors goes to infinity). When the difference equation has constant coefficients, there is just one constant companion matrix for which it is sufficient to evaluate its spectral radius. On the contrary, when the coefficients are variable, the companion matrices may be even infinitely many and, in any case, they do not reduce to one constant matrix. Therefore, a satisfactory stability analysis may be done only by evaluating (or, at least, by approximating sufficiently well) the so called joint spectral radius (j.s.r.) of the family of all the companion matrices. In particular, the asymptotic stability of all the solutions is guaranteed by the condition j.s.r. < 1.

In this talk, after outlining some basic results concerning the j.s.r. available in the literature and some new results and conjectures recently obtained jointly with N. Guglielmi, we illustrate how the j.s.r. approach may be successfully applied to the zero-stability analysis of linear multistep methods for ODEs. As an example, we treat the case of the 3-step backward differentiation formula of order 3 with variable stepsize and obtain almost-optimal results.

For further information please contact: mittelmann@asu.edu