Rectangular eigenvalue problems

Often the easiest way to discretize an ordinary or partial differential equation is by a <i>rectangular numerical method</i>, in which <i>n</i> basis functions are sampled at <i>m</i> ≫ <i>n</i> collocation points. We show how eigenvalue problems can be solved in this setting by QR reduction to square matrix generalized eigenvalue problems. The method applies equally in the limit “<i>m</i> = ∞” of eigenvalue problems for quasimatrices. Numerical examples are presented as well as pointers to related literature.