Equations with infinite delay: numerical stability via truncated Laguerre rules

We propose a pseudospectral approximation of scalar linear equations with infinite delay using truncated Laguerre interpolation and quadrature rules. We study numerically the spurious eigenvalues introduced by the truncated approximation scheme compared to the complete scheme, and the convergence of the approximating eigenvalues to the true ones. The tests show that the convergence order of the truncated scheme is dominated by the quadrature error, and depends on the regularity of the integration kernel. The advantage of truncated rules is that, for kernels with limited regularity, a given tolerance can be achieved with lower-dimensional systems.