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Randomized methods for computing joint eigenvalues, with applications to multiparameter eigenvalue problems and root
Haoze He1, Daniel Kressner1, Bor Plestenjak2,3
1École Polytechnique Fédérale de Lausanne (EPFL), Institute of Mathematics, 1015 Lausanne, Switzerland.
Numerical Algorithms
|October 14, 2025
Summary
This study introduces a novel randomized method for accurately approximating joint eigenvalues of commuting matrices. This approach enhances the performance of solvers for multiparameter eigenvalue problems and polynomial systems.
Area of Science:
- Numerical Analysis
- Linear Algebra
- Computational Mathematics
Background:
- Commuting matrix families are unitarily triangularizable, with diagonal entries representing joint eigenvalues.
- Computing these joint eigenvalues is crucial for applications like multiparameter eigenvalue problems and solving multivariate polynomial systems.
Purpose of the Study:
- To develop and analyze a numerical method for approximating joint eigenvalues of (nearly) commuting matrix families.
- To demonstrate the effectiveness of the proposed method in improving existing solvers.
Main Methods:
- Proposing a randomized approach that computes eigenvalues using Rayleigh quotients.
- Utilizing eigenvectors from a random linear combination of matrices in the family.
Main Results:
- The randomized method accurately computes semisimple joint eigenvalues.
- Numerical examples demonstrate improved performance in relevant solvers.
Conclusions:
- Randomized Rayleigh quotient methods offer an effective strategy for approximating joint eigenvalues.
- This approach provides a valuable tool for tackling complex eigenvalue problems and polynomial systems.
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