Homotopy for Rectangular Multiparameter Eigenvalue Problems
Abstract
The rectangular multiparameter eigenvalue problem (rMEP) is a recently introduced form of an eigenvalue problem, in which we are looking for the combinations of parameters where the multivariate rectangular matrix loses full column rank. Existing numerical methods for rMEPs typically require substantial memory and are not suitable for problems with a large number of eigenvalues. We propose a novel numerical approach based on the homotopy method that has low memory requirements and can compute all eigenvalues of an rMEP. Moreover, the homotopy method is naturally parallelizable, providing a simple and efficient way to accelerate computations. An implementation, together with a collection of test examples, is available in a public repository.
Related
- A Contour Method for Multiparameter Eigenvalue Problems
- Porteous Meets Walsh: Counting Critical Points for ℋ2-Norm Model Order Reduction Using Algebraic Geometry Behind Multiparameter Eigenvalue Problems
- Advancements in Spectral Collocation Methods for High-Order Eigenvalue Problems
- Conditioning and backward errors for nonlinear eigenvalue problems with eigenvector nonlinearities
- Characterization of the Tau Spectrum for Eigenvalue Problems
BibTeX
@misc{plestenjak2026homotopy,
title = {{Homotopy for Rectangular Multiparameter Eigenvalue Problems}},
author = {Bor Plestenjak and Christof Vermeersch},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15040},
eprint = {2610.15040},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15040},
}
RIS
TY - UNPB
TI - Homotopy for Rectangular Multiparameter Eigenvalue Problems
AU - Bor Plestenjak
AU - Christof Vermeersch
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2610.15040
UR - https://arxiv.org/abs/2610.15040
AB - The rectangular multiparameter eigenvalue problem (rMEP) is a recently introduced form of an eigenvalue problem, in which we are looking for the combinations of parameters where the multivariate rectangular matrix loses full column rank. Existing numerical methods for rMEPs typically require substantial memory and are not suitable for problems with a large number of eigenvalues. We propose a novel numerical approach based on the homotopy method that has low memory requirements and can compute all eigenvalues of an rMEP. Moreover, the homotopy method is naturally parallelizable, providing a simple and efficient way to accelerate computations. An implementation, together with a collection of test examples, is available in a public repository.
ER -