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An improved preconditioner for \(2\times 2\) block linear system arising from complex linear system

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Abstract

In this paper, an improved preconditioner for 2 \(\times \) 2 block linear system, which arises from complex linear system, is proposed. Some properties of the preconditioned matrix, including the distribution of eigenvalues and eigenvectors and an upper bound of the minimum polynomial degree, are studied. Finally, numerical examples are given to show the efficiency of the proposed preconditioner.

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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  MathSciNet  Google Scholar 

  2. Axelsson, O., Neytcheva, M., Ahmad, B.: A comparison of iterative methods to solve complex valued linear algebraic systems. Numer. Algorithms. 66, 811–841 (2014)

    Article  MathSciNet  Google Scholar 

  3. Bai, Z.-Z.: Sharp error bounds of some Krylov subspace methods for non-Hermitian linear systems. Appl. Math. Comput. 109, 273–285 (2000)

    MathSciNet  MATH  Google Scholar 

  4. Bai, Z.-Z.: Motivations and realizations of Krylov subspace methods for large sparse linear systems. J. Comput. Appl. Math. 283, 71–78 (2015)

    Article  MathSciNet  Google Scholar 

  5. 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  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  8. Cao, Y., Ren, Z.-R.: Two variants of the PMHSS iteration method for a class of complex symmetric indefinite linear systems. Appl. Math. Comput. 264, 61–71 (2015)

    MathSciNet  MATH  Google Scholar 

  9. Cao, Y., Ren, Z.-R., Shi, Q.: A simplified HSS preconditioner for generalized saddle point problems. BIT Numer. Math. 56, 423–439 (2016)

    Article  MathSciNet  Google Scholar 

  10. Cao, Y., Wang, A., Chen, Y.-J.: A modified relaxed positive-semidefinite and Skew-Hermitian splitting preconditioner for generalized saddle point problems. East Asian J. Appl. Math. 7, 192–210 (2017)

    Article  MathSciNet  Google Scholar 

  11. Day, D., Heroux, M.A.: Solving complex-valued linear systems via equivalent real formulations. SIAM J. Sci. Comput. 23(2), 480–498 (2001)

    Article  MathSciNet  Google Scholar 

  12. Feriani, A., Perotti, F., Simoncini, V.: Iterative system solvers for the frequency analysis of linear mechanical systems. Comput. Methods Appl. Mech. Engrg. 190, 1719–1739 (2000)

    Article  Google Scholar 

  13. Hezari, D., Edalatpour, V., Salkuyeh, D.K.: Preconditioned GSOR iterative method for a class of complex symmetric system of linear equations. Numer. Linear Algebra Appl. 22, 338–356 (2015)

    Article  MathSciNet  Google Scholar 

  14. Howle, V.E., Vavasis, S.A.: An iterative method for solving complex-symmetric systems arising in electrical power modeling. SIAM J. Matrix Anal. Appl. 26, 1150–1178 (2005)

    Article  MathSciNet  Google Scholar 

  15. Huang, Y.-M.: A practical formula for computing optimal parameters in the HSS iteration methods. J. Comput. Appl. Math. 225, 142–149 (2014)

    Article  MathSciNet  Google Scholar 

  16. Huang, Z.-G., Wang, L.-G., Xu, Z., Cui, J.-J.: An efficient preconditioned variant of the PSS preconditioner for generalized saddle point problems. Appl. Math. Comput. 376, art. 125110 (2020)

    MathSciNet  MATH  Google Scholar 

  17. Liang, Z.-Z., Zhang, G.-F.: On SSOR iteration method for a class of block two-by-two linear systems. Numer. Algor. 71, 655–671 (2016)

    Article  MathSciNet  Google Scholar 

  18. Miao, S.-X.: A new preconditioner for a class of \(2\times 2\) block linear systems. Japan J. Indust. Appl. Math. 37, 913–928 (2020)

    Article  Google Scholar 

  19. Saad, Y.: Iterative Methods for Sparse Linear Systems. SIAM, Philadephia (2003)

    Book  Google Scholar 

  20. Saad, Y., Schultz, M.H.: GMRES: a generalised minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7(3), 856–869 (1986)

    Article  Google Scholar 

  21. Shen, Q.-Q., Shi, Q.: A variant of the HSS preconditioner for complex symmetric indefinite linear systems. Comput. Math. Appl. 75, 850–863 (2018)

    Article  MathSciNet  Google Scholar 

  22. Sogabe, T., Zhang, S.-L.: A COCR method for solving complex symmetric linear systems. J. Comput. Appl. Math. 199, 297–303 (2007)

    Article  MathSciNet  Google Scholar 

  23. van der Vorst, H.A., Melissen, J.B.M.: A Petrov-Galerkin type method for solving \(A x = b\), where A is symmetric complex. IEEE Trans. Mag. 26, 706–708 (1990)

    Article  Google Scholar 

  24. Wang, T., Lu, L.-Z.: Alternating-directional PMHSS iteration method for a class of two-by-two block linear systems. Appl. Math. Lett. 58, 159–164 (2016)

    Article  MathSciNet  Google Scholar 

  25. Zhang, J.-H., Dai, H.: A new block preconditioner for complex symmetric indefinite linear systems. Numer. Algorithms. 74, 889–903 (2017)

    Article  MathSciNet  Google Scholar 

  26. Zheng, Q.-Q., Lu, L.-Z.: A shift-splitting preconditioner for a class of block two-by-two linear systems. Appl. Math. Lett. 66, 54–60 (2017)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. College of Mathematics and Statistics, Northwest Normal University, Lanzhou, 730070, People’s Republic of China

    Wen-Bin Bao

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  1. Wen-Bin Bao
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Bao, WB. An improved preconditioner for \(2\times 2\) block linear system arising from complex linear system. Japan J. Indust. Appl. Math. 38, 859–875 (2021). https://doi.org/10.1007/s13160-021-00463-1

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  • DOI: https://doi.org/10.1007/s13160-021-00463-1

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