A Minimal Perturbation Approach For The Rectangular Multiparameter Eigenvalue Problem

TL;DR

This paper introduces a minimal perturbation method for solving the rectangular multiparameter eigenvalue problem, providing iterative algorithms for approximate solutions and validating on discretized differential equations.

Contribution

It proposes a novel minimal perturbation framework for RMEPs, including algorithms for single and complete eigen-tuple approximations, validated on spectral discretizations.

Findings

Convergent iterative scheme for single eigen-tuple approximation.

Framework effectively computes approximate eigenvalues for discretized PDEs.

Validated on Sturm-Liouville and Helmholtz equations.

Abstract

The rectangular multiparameter eigenvalue problem (RMEP) involves rectangular coefficient matrices (usually with more rows than columns) and may potentially have no solution in its original form. A minimal perturbation framework is proposed to defines approximate solutions. Computationally, two particular scenarios are considered: computing one approximate eigen-tuple or a complete set of approximate eigen-tuples. For computing one approximate eigen-tuple, an alternating iterative scheme with proven convergence is devised, while for a complete set of approximate eigen-tuples, the framework leads to a standard MEP (RMEP with square coefficient matrices) for numerical solutions. The proposed approach is validated on RMEPs from discretizing the multiparameter Sturm-Liouville equation and the Helmholtz equations by the least-squares spectral method.