By taking as a “prototype problem” a one-delay linear autonomous system of delay differential equations we
present the problem of computing the characteristic roots of a retarded functional differential equation as an
eigenvalue problem for a derivative operator with non-local boundary conditions given by the particular system
considered. This theory can be enlarged to more general classes of functional equations such as neutral delay
equations, age-structured population models and mixed-type functional differential equations.
It is thus relevant to have a numerical technique to approximate the eigenvalues of derivative operators under nonlocal
boundary conditions. In this paper we propose to discretize such operators by pseudospectral techniques and
turn the original eigenvalue problem into a matrix eigenvalue problem. This approach is shown to be particularly
efficient due to the well-known “spectral accuracy” convergence of pseudospectral methods. Numerical examples
are given.
