Volume 8, Issue 4
Superoptimal Preconditioners for Functions of Matrices

Zhengjian Bai, Xiaoqing Jin & Tengteng Yao

Numer. Math. Theor. Meth. Appl., 8 (2015), pp. 515-529.

Published online: 2015-08

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  • 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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@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 = {

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 $f(A)x=b$.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2015.my1340}, url = {http://global-sci.org/intro/article_detail/nmtma/12421.html} }
TY - JOUR T1 - Superoptimal Preconditioners for Functions of Matrices JO - Numerical Mathematics: Theory, Methods and Applications VL - 4 SP - 515 EP - 529 PY - 2015 DA - 2015/08 SN - 8 DO - http://doi.org/10.4208/nmtma.2015.my1340 UR - https://global-sci.org/intro/article_detail/nmtma/12421.html KW - AB -

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 $f(A)x=b$.

Zhengjian Bai, Xiaoqing Jin & Tengteng Yao. (2020). Superoptimal Preconditioners for Functions of Matrices. Numerical Mathematics: Theory, Methods and Applications. 8 (4). 515-529. doi:10.4208/nmtma.2015.my1340
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