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Superoptimal Preconditioners for Functions of Matrices

Published online by Cambridge University Press:  10 November 2015

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: [email protected](Z. J. Bai), [email protected](X. Q. Jin), [email protected] (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.

Type
Research Article
Copyright
Copyright © Global-Science Press 2015 

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