Volume 7, Issue 3
Efficient Preconditioner and Iterative Method for Large Complex Symmetric Linear Algebraic Systems

East Asian J. Appl. Math., 7 (2017), pp. 530-547.

Published online: 2018-02

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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

Preconditioner, complex linear algebraic systems, Krylov subspace method, spectral properties, convergence.

15A06, 65F10, 65H10

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@Article{EAJAM-7-530, author = {}, title = {Efficient Preconditioner and Iterative Method for Large Complex Symmetric Linear Algebraic Systems}, journal = {East Asian Journal on Applied Mathematics}, year = {2018}, volume = {7}, number = {3}, pages = {530--547}, 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.

}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.240316.290417a}, url = {http://global-sci.org/intro/article_detail/eajam/10763.html} }
TY - JOUR T1 - Efficient Preconditioner and Iterative Method for Large Complex Symmetric Linear Algebraic Systems JO - East Asian Journal on Applied Mathematics VL - 3 SP - 530 EP - 547 PY - 2018 DA - 2018/02 SN - 7 DO - http://doi.org/10.4208/eajam.240316.290417a UR - https://global-sci.org/intro/article_detail/eajam/10763.html KW - Preconditioner, complex linear algebraic systems, Krylov subspace method, spectral properties, convergence. AB -

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.

Li Dan Liao & Guo Feng Zhang. (2020). Efficient Preconditioner and Iterative Method for Large Complex Symmetric Linear Algebraic Systems. East Asian Journal on Applied Mathematics. 7 (3). 530-547. doi:10.4208/eajam.240316.290417a
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