An efficient relaxed shift-splitting preconditioner for a class of complex symmetric indefinite linear systems

Qian Li; Qianqian Yuan; Jianhua Chen; Qian Li; Qianqian Yuan; Jianhua Chen

doi:10.3934/math.2022942

AIMS Mathematics

2022, Volume 7, Issue 9: 17123-17132. doi: 10.3934/math.2022942

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Research article

An efficient relaxed shift-splitting preconditioner for a class of complex symmetric indefinite linear systems

School of Mathematics and Statistics, Xinyang College, Xinyang 464000, China

Received: 26 May 2022 Revised: 08 July 2022 Accepted: 19 July 2022 Published: 21 July 2022
MSC : 65F10, 65F50, 65W05

In this work, by introducing a scalar matrix $ \alpha I $, we transform the complex symmetric indefinite linear systems $ (W+i T)x = b $ into a block two-by-two complex equations equivalently, and propose an efficient relaxed shift-splitting (ERSS) preconditioner. By adopting the relaxation technique, the ERSS preconditioner is not only a computational advantage but also closer to the original two-by-two of complex coefficient matrix. The eigenvalue distributions of the preconditioned matrix are analysed. An efficient and practical formula for computing the parameter value $ \alpha $ is also derived by computing the Frobenius norm of symmetric indefinite matrix $ T $. Numerical examples on a few model problems are illustrated to verify the performances of the ERSS preconditioner.
- symmetric indefinite systems,
- scalar matrix,
- ERSS preconditioner,
- parameter value,
- Frobenius norm
Citation: Qian Li, Qianqian Yuan, Jianhua Chen. An efficient relaxed shift-splitting preconditioner for a class of complex symmetric indefinite linear systems[J]. AIMS Mathematics, 2022, 7(9): 17123-17132. doi: 10.3934/math.2022942

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Abstract

In this work, by introducing a scalar matrix $ \alpha I $, we transform the complex symmetric indefinite linear systems $ (W+i T)x = b $ into a block two-by-two complex equations equivalently, and propose an efficient relaxed shift-splitting (ERSS) preconditioner. By adopting the relaxation technique, the ERSS preconditioner is not only a computational advantage but also closer to the original two-by-two of complex coefficient matrix. The eigenvalue distributions of the preconditioned matrix are analysed. An efficient and practical formula for computing the parameter value $ \alpha $ is also derived by computing the Frobenius norm of symmetric indefinite matrix $ T $. Numerical examples on a few model problems are illustrated to verify the performances of the ERSS preconditioner.

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