Optimal preconditioners for systems defined by functions of Toeplitz matrices

TL;DR

This paper introduces circulant preconditioners tailored for systems involving functions of Toeplitz matrices, enhancing the efficiency of iterative solvers like conjugate gradient and minimal residual methods.

Contribution

It presents novel circulant preconditioners specifically designed for functions of Toeplitz matrices such as exponential, sine, and cosine functions.

Findings

Numerical results demonstrate improved convergence with the proposed preconditioners.

Preconditioners are effective for functions like e^z, sin z, and cos z.

The methods outperform standard approaches in tested cases.

Abstract

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.