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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
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Summary
This summary is machine-generated.

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.

Keywords:
Commuting matricesJoint eigenvalueMultiparameter eigenvalue problemPolynomial systemRandomized numerical linear algebraRayleigh quotient

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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.