Rectangular eigenvalue problems

Abstract

Often the easiest way to discretize an ordinary or partial differential equation is by a rectangular numerical method, in which n basis functions are sampled at m ≫ n 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 “m= ∞” of eigenvalue problems for quasimatrices. Numerical examples are presented as well as pointers to related literature.