Efficient computation of characteristic roots of delay differential equations using LMS methods

We aim at the efficient computation of the rightmost characteristic roots of a system of delay differential equations. The approach we use is based on the discretization of the solution operator by linear multistep (LMS) methods. This results in an eigenvalue problem whose size is inversely proportional to the steplength used in the discretization. We use theoretical results on the location and numerical preservation of roots obtained in earlier work. Furthermore, we construct special-purpose LMS methods emphasising the accuracy of approximation of characteristic roots. We present a novel procedure that computes efficiently and accurately all roots in any right half-plane. In particular, no roots with large imaginary parts can be overlooked. The performance of the new procedure is demonstrated for small and large-scale systems of delay differential equations.