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October 16, 2025Open Access

A Minimal Perturbation Approach For The Rectangular Multiparameter Eigenvalue Problem

Key Points

The minimal perturbation framework successfully defines approximate solutions for RMEP, addressing potential non-solutions.
An alternating iterative scheme shows convergence for computing a single approximate eigen-tuple, enhancing solution methods.
The method leads to numerical solutions for RMEP by converting it into a standard eigenvalue problem with square matrices.
Validation of the approach is demonstrated on RMEPs derived from the multiparameter 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.

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Cite This Study

Han et al. (Fri,) studied this question.

synapsesocial.com/papers/68f0f51d8dd8ea469b1d71a6 https://doi.org/https://doi.org/10.48550/arxiv.2508.05948

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