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arXiv · 2508.05948

A Minimal Perturbation Approach For The Rectangular Multiparameter Eigenvalue Problem

Shanheng Han·Lei-Hong Zhang·Ren-Cang Li
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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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Shanheng Han, Lei-Hong Zhang, Ren-Cang Li. 2025-08-08. A Minimal Perturbation Approach For The Rectangular Multiparameter Eigenvalue Problem. https://arxiv.org/abs/2508.05948

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