Least-squares spectral methods for ODE eigenvalue problems

TL;DR

This paper introduces a flexible least-squares spectral method for solving ODE eigenvalue problems, capable of handling complex boundary conditions and improving accuracy through reformulations.

Contribution

It extends least-squares spectral methods to generalized eigenvalue problems involving quasimatrices, allowing arbitrary basis functions and enhanced accuracy.

Findings

Effective handling of eigenvalue-dependent boundary conditions

High accuracy achieved through reformulation as integral equations

Extension of methods to quasimatrices and mixed objects

Abstract

We develop spectral methods for ODEs and operator eigenvalue problems that are based on a least-squares formulation of the problem. The key tool is a method for rectangular generalized eigenvalue problems, which we extend to quasimatrices and objects combining quasimatrices and matrices. The strength of the approach is its flexibility that lies in the quasimatrix formulation allowing the basis functions to be chosen arbitrarily (e.g. those obtained by solving nearby problems), and often giving high accuracy. We also show how our algorithm can easily be modified to solve problems with eigenvalue-dependent boundary conditions, and discuss reformulations as an integral equation, which often improves the accuracy.