@Article{NMTMA-8-515,
author = {},
title = {Superoptimal Preconditioners for Functions of Matrices},
journal = {Numerical Mathematics: Theory, Methods and Applications},
year = {2015},
volume = {8},
number = {4},
pages = {515--529},
abstract = {<p style="text-align: justify;">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&nbsp;$\mathbb{C}^{n×n}$ to&nbsp;$\mathbb{C}^{n×n}$. 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 $f(A)x=b$.</p>},
issn = {2079-7338},
doi = {https://doi.org/10.4208/nmtma.2015.my1340},
url = {http://global-sci.org/intro/article_detail/nmtma/12421.html}
}