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Superoptimal Preconditioners for Functions of Matrices

Part of: Numerical linear algebra Partial differential equations, boundary value problems Numerical analysis: Ordinary differential equations

Published online by Cambridge University Press:  10 November 2015

Zheng-Jian Bai ,
Xiao-Qing Jin  and
Teng-Teng Yao
Zheng-Jian Bai*
Affiliation:
School of Mathematical Sciences, Xiamen University, Xiamen 361005, P. R. China
Xiao-Qing Jin
Affiliation:
Department of Mathematics, University of Macau, Macao, P. R. China
Teng-Teng Yao
Affiliation:
School of Mathematical Sciences, Xiamen University, Xiamen 361005, P. R. China
*
*Corresponding author. Email address: zjbai@xmu.edu.cn(Z. J. Bai), xqjin@umac.mo(X. Q. Jin), yaotengteng718@163.com (T. T. Yao)
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Abstract

For any given matrix A ∈ℂnxn, a preconditioner tU(A) called the superoptimal preconditioner was proposed in 1992 by Tyrtyshnikov. It has been shown that tU(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 ℂnxn to ℂnxn. For any A ∈ ℂ nxn, one may construct two superoptimal preconditioners for f(A): tU(f(A)) and f(tU(A)). We establish basic properties of tU(f(A)) and f(tU(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.

Keywords

Superoptimal preconditionersfunctions of matricesToeplitz matrixPCG method

MSC classification

Secondary: 65F10: Iterative methods for linear systems 65F15: Eigenvalues, eigenvectors 65L05: Initial value problems 65N22: Solution of discretized equations
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 8 , Issue 4 , November 2015 , pp. 515 - 529
Copyright
Copyright © Global-Science Press 2015 

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