We present an algorithm for determining the stability of delay differential equations (DDEs) with time-periodic coefficients and time-periodic delays. The DDEs are first posed as an equivalent system of partial differential equations (PDEs) along with a nonlinear boundary condition. A Galerkin approximation is then employed to discretize the PDEs into a set of time-periodic ordinary differential equations (ODEs). Finally, we apply a modified version of the Arnoldi algorithm to extract the dominant eigenvalues of the Floquet transition matrix without computing the entire matrix, thereby reducing the required number of integrations of the ODE system. Five numerical examples demonstrate that our modified Arnoldi algorithm provides reliable approximations of the dominant eigenvalues of the Floquet transition matrix, and does so with substantially less computational effort than the classical Floquet method. The stability charts and bifurcation diagrams generated using our Galerkin–Arnoldi method clearly demonstrate the utility of this approach for establishing the stability of a system of DDEs. © 2015, Springer Science+Business Media Dordrecht.