A Contour Method for Multiparameter Eigenvalue Problems
Abstract
Multiparameter eigenvalue problems arise in boundary value problems, stability analysis, and delay-differential equations. Despite their importance, existing methods either require solving extremely large global problems or rely on local iterative techniques that only recover a few eigenvalues at a time. In this work we develop the first contour method for analytic multiparameter eigenvalue problems. The key theoretical ingredient is a new residue formula for multivariate matrix-valued analytic functions, extending Beyn's Keldysh-based residue theorem for meromorphic operator functions to several complex variables. Using this, we derive a multidimensional analogue of Beyn's contour method that computes all the eigenvalues of a multiparameter eigenvalue problem contained in a prescribed region of ℂd. The resulting algorithm targets eigenvalues in a region without constructing preposterously large matrices and is embarrassingly parallelizable. Numerical experiments demonstrate that we can now successfully solve large multiparameter problems arising in applications where existing approaches struggle.
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BibTeX
@misc{graf2026contour,
title = {{A Contour Method for Multiparameter Eigenvalue Problems}},
author = {Emil Graf and Alex Townsend},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24943},
eprint = {2609.24943},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24943},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Contour Method for Multiparameter Eigenvalue Problems
AU - Emil Graf
AU - Alex Townsend
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.24943
UR - https://arxiv.org/abs/2609.24943
AB - Multiparameter eigenvalue problems arise in boundary value problems, stability analysis, and delay-differential equations. Despite their importance, existing methods either require solving extremely large global problems or rely on local iterative techniques that only recover a few eigenvalues at a time. In this work we develop the first contour method for analytic multiparameter eigenvalue problems. The key theoretical ingredient is a new residue formula for multivariate matrix-valued analytic functions, extending Beyn's Keldysh-based residue theorem for meromorphic operator functions to several complex variables. Using this, we derive a multidimensional analogue of Beyn's contour method that computes all the eigenvalues of a multiparameter eigenvalue problem contained in a prescribed region of ℂ^d. The resulting algorithm targets eigenvalues in a region without constructing preposterously large matrices and is embarrassingly parallelizable. Numerical experiments demonstrate that we can now successfully solve large multiparameter problems arising in applications where existing approaches struggle.
ER -