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Efficient Preconditioner and Iterative Method for Large Complex Symmetric Linear Algebraic Systems

Part of: Numerical linear algebra Nonlinear algebraic or transcendental equations Basic linear algebra

Published online by Cambridge University Press:  07 September 2017

Li Dan Liao  and
Guo Feng Zhang
Li Dan Liao*
Affiliation:
School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China
Guo Feng Zhang*
Affiliation:
School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China
*
*Corresponding author. Email addresses:liaold15@lzu.edu.cn (L.D. Liao), gf_zhang@lzu.edu.cn (G.F. Zhang)
*Corresponding author. Email addresses:liaold15@lzu.edu.cn (L.D. Liao), gf_zhang@lzu.edu.cn (G.F. Zhang)
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Abstract

We discuss an efficient preconditioner and iterative numerical method to solve large complex linear algebraic systems of the form (W + iT)u = c, where W and T are symmetric matrices, and at least one of them is nonsingular. When the real part W is dominantly stronger or weaker than the imaginary part T, we propose a block multiplicative (BM) preconditioner or its variant (VBM), respectively. The BM and VBM preconditioned iteration methods are shown to be parameter-free, in terms of eigenvalue distributions of the preconditioned matrix. Furthermore, when the relationship between W and T is obscure, we propose a new preconditioned BM method (PBM) to overcome this difficulty. Both convergent properties of these new iteration methods and spectral properties of the corresponding preconditioned matrices are discussed. The optimal value of iteration parameter for the PBM method is determined. Numerical experiments involving the Helmholtz equation and some other applications show the effectiveness and robustness of the proposed preconditioners and corresponding iterative methods.

Keywords

Preconditionercomplex linear algebraic systemsKrylov subspace methodspectral propertiesconvergence

MSC classification

Secondary: 15A06: Linear equations 65F10: Iterative methods for linear systems 65H10: Systems of equations
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 7 , Issue 3 , August 2017 , pp. 530 - 547
Copyright
Copyright © Global-Science Press 2017 

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