Randomized FEAST Algorithm for Generalized Hermitian Eigenvalue Problems with Probabilistic Error Analysis

Randomized NLA methods have recently gained popularity because of their easy implementation, computational efficiency, and numerical robustness. We propose a randomized version of a well-established FEAST eigenvalue algorithm that enables computing the eigenvalues of the Hermitian matrix pencil $(\textbf{A},\textbf{B})$ located in the given real interval $\mathcal{I} \subset [\lambda_{min}, \lambda_{max}]$. In this talk, we will present deterministic as well as probabilistic error analysis of the accuracy of approximate eigenpair and subspaces obtained using the randomized FEAST algorithm. First, we derive bounds for the canonical angles between the exact and the approximate eigenspaces corresponding to the eigenvalues contained in the interval $\mathcal{I}$. Then, we present bounds for the accuracy of the eigenvalues and the corresponding eigenvectors. This part of the analysis is independent of the particular distribution of an initial subspace, therefore we denote it as deterministic. In the case of the starting guess being a Gaussian random matrix, we provide more informative, probabilistic error bounds. Finally, we will illustrate numerically the effectiveness of all the proposed error bounds.

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