Optimal preconditioners for systems defined by functions of Toeplitz matrices
For researchers working on iterative solvers for structured matrix functions, this work provides practical preconditioners but is incremental in nature.
The paper proposes circulant preconditioners for solving linear systems involving functions (exponential, sine, cosine) of Toeplitz matrices, demonstrating effectiveness through numerical results.
We propose several circulant preconditioners for systems defined by some functions $g$ of Toeplitz matrices $A_n$. In this paper we are interested in solving $g(A_n)\mathbf{x}=\mathbf{b}$ by the preconditioned conjugate method or the preconditioned minimal residual method, namely in the cases when $g(z)$ are the functions $e^{z}$, $\sin{z}$ and $\cos{z}$. Numerical results are given to show the effectiveness of the proposed preconditioners.