Superoptimal Preconditioners for Functions of Matrices

Abstract

For any given matrix $A∈\mathbb{C}^{n×n}$, a preconditioner $t_U(A)$ called the superoptimal preconditioner was proposed in 1992 by Tyrtyshnikov. It has been shown that $t_U(A)$ is an efficient preconditioner for solving various structured systems, for instance, Toeplitz-like systems. In this paper, we construct the superoptimal preconditioners for different functions of matrices. Let $f$ be a function of matrices from $\mathbb{C}^{n×n}$ to $\mathbb{C}^{n×n}$. For any $A∈\mathbb{C}^{n×n}$, one may construct two superoptimal preconditioners for $f(A)$: $t_U(f(A))$ and $f(t_U(A))$. We establish basic properties of $t_U(f(A))$) and $f(t_U(A))$ for different functions of matrices. Some numerical tests demonstrate that the proposed preconditioners are very efficient for solving the system $f(A)x=b$.