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Superoptimal Preconditioners for Functions of Matrices

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

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. Emailaddress: [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 thesuperoptimal preconditioner was proposed in 1992 by Tyrtyshnikov. It has beenshown that tU(A) is an efficient preconditioner forsolving various structured systems, for instance, Toeplitz-like systems. In thispaper, we construct the superoptimal preconditioners for different functions ofmatrices. 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)) andf(tU(A)) for different functions ofmatrices. Some numerical tests demonstrate that the proposed preconditioners arevery 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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