Circulant preconditioners for functions of Hermitian Toeplitz matrices

TL;DR

This paper introduces absolute value superoptimal circulant preconditioners for functions of Hermitian Toeplitz matrices, demonstrating their effectiveness through theoretical analysis and numerical experiments, leading to rapid convergence of Krylov methods.

Contribution

It proposes a new class of circulant preconditioners for functions of Toeplitz matrices and extends the results to block Toeplitz matrices, with theoretical and numerical validation.

Findings

Eigenvalues cluster around ±1 after preconditioning

Preconditioners improve convergence of Krylov methods

Theoretical results are supported by numerical examples

Abstract

Circulant preconditioners for functions of matrices have been recently of interest. In particular, several authors proposed the use of the optimal circulant preconditioners as well as the superoptimal circulant preconditioners in this context and numerically illustrated that such preconditioners are effective for certain functions of Toeplitz matrices. Motivated by their results, we propose in this work the absolute value superoptimal circulant preconditioners and provide several theorems that analytically show the effectiveness of such circulant preconditioners for systems defined by functions of Toeplitz matrices. Namely, we show that the eigenvalues of the preconditioned matrices are clustered around $\pm 1$ and rapid convergence of Krylov subspace methods can therefore be expected. Moreover, we show that our results can be extended to functions of block Toeplitz matrices with Toeplitz blocks provided that the optimal block circulant matrices with circulant blocks are used as preconditioners. Numerical examples are given to support our theoretical results.