The stability of an equilibrium point of a dynamical system is determined by the position in the complex plane of the so-called characteristic values of the line- arization around the equilibrium. This paper presents an approach for the computation of characteristic values of partial differential equations of evolution involving time delay, which is based on a pseudospectral method coupled with a spectral method. The convergence of the computed characteristic values is of infinite order with respect to the pseudospectral discretization and of finite order with respect to the spectral one. However, for one dimensional reaction diffusion equations, the finite order of the spectral discretization is proved to be so high that the convergence turns out to be as fast as one of infinite order.
