Randomized methods for computing joint eigenvalues, with applications to multiparameter eigenvalue problems and root finding

TL;DR

A simple approach that computes eigenvalues as one-sided or two-sided Rayleigh quotients from eigenvectors of a random linear combination of the matrices in the family is proposed and analyzed.

Abstract

It is well known that a family of $n\times n$ commuting matrices can be simultaneously triangularized by a unitary similarity transformation. The diagonal entries of the triangular matrices define the $n$ joint eigenvalues of the family. In this work, we consider the task of numerically computing approximations to such joint eigenvalues for a family of (nearly) commuting matrices. This task arises, for example, in solvers for multiparameter eigenvalue problems and systems of multivariate polynomials, which are our main motivations. We propose and analyze a simple approach that computes eigenvalues as one-sided or two-sided Rayleigh quotients from eigenvectors of a random linear combination of the matrices in the family. We provide some analysis and numerous numerical examples, showing that such randomized approaches can compute semisimple joint eigenvalues accurately and lead to improved performance of existing solvers.

Figures (9)

• Figure 1: Distribution of absolute errors for the eigenvalue ${\boldsymbol \lambda}^{(1)}$ from Example \ref{['ex:ex1']} using double precision (left) and extended precision (right) to compute the eigenvalues of $\widetilde{A}(\mu)$ using one-sided (dashed lines) and two-sided (solid lines) Rayleigh quotients.
• Figure 2: Absolute errors for the eigenvalue ${\boldsymbol \lambda}^{(1)}$ from Example \ref{['ex:ex1']} using one-sided Rayleigh quotient vs. bound \ref{['eqdet:Asimple1s']} (left) and two-sided Rayleigh quotient vs. bound \ref{['eqdet:Asemisimple']} (right).
• Figure 3: Empirical probabilities compared to the bounds for \ref{['lemeq:d_mu_prob_bound_ex']} (left) and \ref{['eq:boundR_semisimple_ex']} (right) for the data from Example \ref{['ex:empProb']}.
• Figure 4: Distribution of absolute errors for the eigenvalues ${\boldsymbol \lambda}^{(1)}=(1,1)$ (left) and ${\boldsymbol \lambda}^{(4)}=(2,1)$ (right) from Example \ref{['ex:ex2']} using one-sided (dashed lines) and two-sided (solid lines) Rayleigh quotients.
• Figure 5: Distribution of absolute errors for eigenvalue ${\boldsymbol \lambda}^{(1)}$ from Example \ref{['ex:ex3']} for $\delta=10^{-6}$ (left) and $\delta=10^{-12}$ (right), using one-sided (dashed lines) and two-sided (solid lines) Rayleigh quotients.

Theorems & Definitions (40)

• Theorem 2.1: horn13, theoryofmatrices
• Definition 2.2: AtkinsonBook, kosirPlenstenjak02
• Lemma 2.3
• Lemma 2.4
• proof : Proof of Lemma \ref{['lem:semisimple_partition']}
• Theorem 2.5
• proof
• Theorem 2.6
• proof
• Definition 2.7