Delays often appear in mathematical models. Examples from biology include maturation in population models, incubation and latent periods in epidemics models, impulse transmission time in neurological models, and transcription delays in gene regulatory networks. Some of these delays may depend on the state of the system: as an example, the maturation age of an individual may depend on the availability of resources during the individual's life. Models with state-dependent delays appear also in other context, such as control theory, economics, lasers and climate (see [3] and the references therein). A common problem in the study of dynamical systems is the determination of the asymptotic stability of particular invariant sets of states, such as equilibria and periodic orbits. In both cases the stability properties can be investigated by analyzing the spectra of the evolution operators associated to linearizations of the system; in particular, for periodic orbits, a stability indicator is given by the largest modulus of the Floquet multipliers, which constitute the spectrum of the monodromy operators. The method of [1] applies a pseudospectral collocation technique to the evolution operators of linear(ized) delay differential equations (DDEs) with constant delays, obtaining finite-rank approximating operators (converging in norm). These can be represented as matrices, and their eigenvalues, computed with standard methods, are taken as approximations of the desired spectrum. Our goal is to extend this approach to DDEs with state-dependent delays. We consider the one proposed in [3], i.e. $u'(t)=\alpha u(t)+\beta u(t-1-u(t-b))$. After a brief summary of the relevant theoretical background [2, 4], we will describe the numerical method and its implementation. We will then present some experiments comparing the results with those obtained in [3] using DDE-BIFTOOL, and we will exemplify the convergence properties of the method.