GMRES for the Differentiation Operator

We investigate using the gmres method with the differentiation operator. This operator is unbounded, and thus does not fall into the framework of existing Krylov subspace theory. We establish conditions under which a function can be approximated by its own derivatives in a domain of the complex plane. These conditions are used to determine when gmres converges. This algorithm outperforms traditional quadrature schemes for a large class of highly oscillatory integrals, even when the kernel of oscillations is unknown.