Rectangular eigenvalue problems

TL;DR

This paper presents a method for solving eigenvalue problems discretized with rectangular numerical methods, using QR reduction to convert to square matrix problems, applicable even in the limit of infinite collocation points.

Contribution

It introduces a novel approach to solve eigenvalue problems in rectangular discretizations via QR reduction, extending to quasimatrix cases.

Findings

Effective eigenvalue computation in rectangular discretizations

Applicability to quasimatrix eigenvalue problems

Numerical examples demonstrating the method

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=infinity" of eigenvalue problems for quasimatrices. Numerical examples are presented as well as pointers to some related literature.