Often in applications such as rare events estimation or optimal control it is
required that one calculates the principal eigen-function and eigen-value of a
non-negative integral kernel. Except in the finite-dimensional case, usually
neither the principal eigen-function nor the eigen-value can be computed
exactly. In this paper, we develop numerical approximations for these
quantities. We show how a generic interacting particle algorithm can be used to
deliver numerical approximations of the eigen-quantities and the associated
so-called "twisted" Markov kernel as well as how these approximations are
relevant to the aforementioned applications. In addition, we study a collection
of random integral operators underlying the algorithm, address some of their
mean and path-wise properties, and obtain $L_{r}$ error estimates. Finally,
numerical examples are provided in the context of importance sampling for
computing tail probabilities of Markov chains and computing value functions for
a class of stochastic optimal control problems.