Year: 2015
Numerical Mathematics: Theory, Methods and Applications, Vol. 8 (2015), Iss. 4 : pp. 515–529
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 $f(A)x=b$.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/nmtma.2015.my1340
Numerical Mathematics: Theory, Methods and Applications, Vol. 8 (2015), Iss. 4 : pp. 515–529
Published online: 2015-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 15
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Circulant preconditioners for functions of Hermitian Toeplitz matrices
Hon, Sean
Journal of Computational and Applied Mathematics, Vol. 352 (2019), Iss. P.328
https://doi.org/10.1016/j.cam.2018.11.011 [Citations: 6]