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On preconditioned MHSS iteration methods for complex symmetric linear systems

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Abstract

We propose a preconditioned variant of the modified HSS (MHSS) iteration method for solving a class of complex symmetric systems of linear equations. Under suitable conditions, we prove the convergence of the preconditioned MHSS (PMHSS) iteration method and discuss the spectral properties of the PMHSS-preconditioned matrix. Numerical implementations show that the resulting PMHSS preconditioner leads to fast convergence when it is used to precondition Krylov subspace iteration methods such as GMRES and its restarted variants. In particular, both the stationary PMHSS iteration and PMHSS-preconditioned GMRES show meshsize-independent and parameter-insensitive convergence behavior for the tested numerical examples.

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References

  1. Axelsson, O., Kucherov, A.: Real valued iterative methods for solving complex symmetric linear systems. Numer. Linear Algebra Appl. 7, 197–218 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bai, Z.-Z.: Construction and analysis of structured preconditioners for block two-by-two matrices. J. Shanghai Univ. (English Edition) 8, 397–405 (2004)

    Article  Google Scholar 

  3. Bai, Z.-Z.: Structured preconditioners for nonsingular matrices of block two-by-two structures. Math. Comput. 75, 791–815 (2006)

    MATH  Google Scholar 

  4. Bai, Z.-Z., Benzi, M., Chen, F.: Modified HSS iteration methods for a class of complex symmetric linear systems. Computing 87, 93–111 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bai, Z.-Z., Golub, G.H., Li, C.-K.: Convergence properties of preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite matrices. Math. Comput. 76, 287–298 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bai, Z.-Z., Golub, G.H., Ng, M.K.: Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems. SIAM J. Matrix Anal. Appl. 24, 603–626 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bai, Z.-Z., Golub, G.H., Ng, M.K.: On inexact Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems. Linear Algebra Appl. 428, 413–440 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  8. Bai, Z.-Z., Golub, G.H., Pan, J.-Y.: Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems. Numer. Math. 98, 1–32 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  9. Benzi, M., Bertaccini, D.: Block preconditioning of real-valued iterative algorithms for complex linear systems. IMA J. Numer. Anal. 28, 598–618 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  10. Bertaccini, D.: Efficient solvers for sequences of complex symmetric linear systems. Electron. Trans. Numer. Anal. 18, 49–64 (2004)

    MATH  MathSciNet  Google Scholar 

  11. Chen, K.: Matrix Preconditioning Techniques and Applications. Cambridge University Press, Cambridge and New York (2005)

    Book  MATH  Google Scholar 

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Authors and Affiliations

  1. State Key Laboratory of Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, P.O. Box 2719, Beijing, 100190, People’s Republic of China

    Zhong-Zhi Bai

  2. Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, P.O. Box 2719, Beijing, 100190, People’s Republic of China

    Zhong-Zhi Bai

  3. Department of Mathematics and Computer Science, Emory University, Atlanta, GA, 30322, USA

    Michele Benzi

  4. School of Science, Xi’an University of Post and Telecommunications, Xi’an, 710121, People’s Republic of China

    Fang Chen

Authors
  1. Zhong-Zhi Bai
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  2. Michele Benzi
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  3. Fang Chen
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Corresponding author

Correspondence to Zhong-Zhi Bai.

Additional information

Supported by The National Natural Science Foundation for Innovative Research Groups (No. 11021101), The Hundred Talent Project of Chinese Academy of Sciences, The National Basic Research Program (No. 2011CB309703), P.R. China, and by the US National Science Foundation grant DMS-0810862.

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Bai, ZZ., Benzi, M. & Chen, F. On preconditioned MHSS iteration methods for complex symmetric linear systems. Numer Algor 56, 297–317 (2011). https://doi.org/10.1007/s11075-010-9441-6

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  • DOI: https://doi.org/10.1007/s11075-010-9441-6

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