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An augmentation preconditioner for a class of complex symmetric linear systems

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Abstract

In this paper, we propose an efficient augmentation-based preconditioner for solving complex symmetric linear systems, which is obtained by augmenting the (2,2) block of the coefficient matrix. Then, the convergence of the corresponding iteration method is analyzed, and several spectral properties of the preconditioned matrices, such as eigenvalue distributions and eigenvectors, are also discussed. Numerical experiments demonstrate that our proposed preconditioner is more effective than some existing block preconditioners.

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References

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  • Bai Z-Z, Golub GH, Ng MK (2003) Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems. SIAM J Matrix Anal Appl 24:603–626

    Article  MathSciNet  Google Scholar 

  • Bai Z-Z, Parlett BN, Wang Z-Q (2005) On generalized successive overrelaxation methods for augmented linear systems. Numer Math 102:1–38

    Article  MathSciNet  Google Scholar 

  • Bai Z-Z, Wang Z-Q (2008) On parameterized inexact Uzawa methods for generalized saddle point problems. Linear Algebra Appl 428:2900–2932

    Article  MathSciNet  Google Scholar 

  • Bai Z-Z (2009) Optimal parameters in the HSS-like methods for saddle-point problmes. Numer Linear Algebra Appl 16:447–479

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Bai Z-Z, Benzi M, Chen F (2011) On preconditioned MHSS iteration methods for complex symmetric linear systems. Numer Algor 56:297–317

    Article  MathSciNet  Google Scholar 

  • Bai Z-Z, Benzi M, Chen F, Wang Z-Q (2013) Preconditioned MHSS iteration methods for a class of block two-by-two linear systems with applications to distributed control problems. IMA J Numer Anal 33:343–369

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Bai Z-Z, Benzi M (2017) Regularized HSS iteration methods for saddle-point linear systems. BIT Numer Math 57:287–311

    Article  MathSciNet  Google Scholar 

  • Bai Z-Z (2019) Regularized HSS iteration methods for stabilized saddle-point problems. IMA J Numer Anal 39:1888–1923

    Article  MathSciNet  Google Scholar 

  • Bai Z-Z, Pan J-Y (2021) Matrix analysis and computations. SIAM, Philadelphia, PA

    Book  Google Scholar 

  • Bai Z-Z (2023) A two-step matrix splitting iteration paradigm based on one single splitting for solving systems of linear equations. Numer Linear Algebra Appl. https://doi.org/10.1002/nla.2510

    Article  Google Scholar 

  • Balani FB, Hajarian M (2023) Modified block product preconditioner for a class of complex symmetric linear systems. Linear Multilinear Algebra 71:1521–1535

    Article  MathSciNet  Google Scholar 

  • Benzi M, Golub GH, Liesen J (2005) Numerical solution of saddle point problems. Acta Numer 14:1–137

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Cao Z-H (2008) Augmentation block preconditioners for saddle point-type matrices with singular (1,1) blocks. Numer Linear Algebra Appl 15:515–533

    Article  MathSciNet  Google Scholar 

  • Chen F (2015) On choices of iteration parameter in HSS method. Appl Math Comput 271:832–837

    MathSciNet  Google Scholar 

  • Golub GH, Greif C (2003) On solving block-structured indefinite linear systems. SIAM J Sci Comput 24:2076–2092

    Article  MathSciNet  Google Scholar 

  • Golub GH, Greif C, Varah JM (2006) An algebraic analysis of a block diagonal preconditioner for saddle point systems. SIAM J Matrix Anal Appl 27:779–792

    Article  MathSciNet  Google Scholar 

  • Greif C, Schötzau D (2007) Preconditioners for the discretized time-harmonic Maxwell equations in mixed form. Numer Linear Algebra Appl 14:281–297

    Article  MathSciNet  Google Scholar 

  • He J, Huang T-Z (2011) Two augmentation preconditioners for nonsymmetric and indefinite saddle point linear systems with singular (1,1) blocks. Comput Math Appl 62:87–92

    Article  MathSciNet  Google Scholar 

  • Hezari D, Edalatpour V, Salkuyeh DK (2015) Preconditioned GSOR iterative method for a class of complex symmetric system of linear equation. Numer Linear Algebra Appl 22:761–776

    Article  MathSciNet  Google Scholar 

  • Hezari D, Salkuyeh DK, Edalatpour V (2016) A new iterative method for solving a class of complex symmetric system of linear equations. Numer Algor 73:927–955

    Article  MathSciNet  Google Scholar 

  • Huang Y-M (2014) A practical formula for computing optimal parameters in the HSS iteration methods. J Comput Appl Math 255:142–149

    Article  MathSciNet  Google Scholar 

  • Huang Z-G, Xu Z, Cui J-J (2016) Preconditioned triangular splitting iteration method for a class of complex symmetric linear systems. Calcolo 56:22

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Lund J, Bowers K (1992) Sinc methods for quadrature and differential equations. SIAM, Philadephia

    Book  Google Scholar 

  • Saad Y (2003) Iterative methods for sparse linear systems, 2nd edn. SIAM, PA

    Book  Google Scholar 

  • Salkuyeh DK, Hezari D, Edalatpour V (2015) Generalized SOR iterative method for a class of complex symmetric linear system of equations. Int J Comput Math 92:802–815

    Article  MathSciNet  Google Scholar 

  • Salkuyeh DK, Siahkolaei TS (2018) Two-parameter TSCSP method for solving complex symmetric system of linear equations. Calcolo 55:8

    Article  MathSciNet  Google Scholar 

  • Shen S-Q, Huang T-Z, Zhang J-S (2012) Augmentation block triangular precondition- ers for regularized saddle point problems. SIAM J Matrix Anal Appl 33:721–741

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Wang T, Zheng Q-Q, Lu L-Z (2017) A new iteration method for a class of complex symmetric linear systems. J Comput Appl Math 325:188–197

    Article  MathSciNet  Google Scholar 

  • Xiao X-Y, Qi X, Zhao Y-C (2022) Improved CRI iteration methods for a class of complex symmetric linear systems. Calcolo 59:20

    Article  MathSciNet  Google Scholar 

  • Zhang J-H, Dai H (2017) A new block preconditioner for complex symmetric indefinite linear systems. Numer Algor 74:1–15

    Article  MathSciNet  Google Scholar 

  • Zhang J-H, Zhao J (2018) A new matrix splitting preconditioner for generalized saddle point problems. Comput Math Appl 75:153–160

    Article  MathSciNet  Google Scholar 

  • Zhang J-L, Fan H-T, Gu C-Q (2018) An improved block splitting preconditioner for complex symmetric indefinite linear systems. Numer Algor 77:451–478

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors are very thankful to referees for their constructive comments and meritorious suggestions, which greatly improved the original manuscript of this paper. This work was supported by the Postgraduate Scientific Research Innovation Project of Hunan Province (Grant No. CX20220104).

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  1. School of Mathematics and Statistics, Central South University, Changsha, 410083, Hunan, People’s Republic of China

    Hongyu Wu

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  1. Hongyu Wu
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Correspondence to Hongyu Wu.

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Communicated by Zhong-Zhi Bai.

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Wu, H. An augmentation preconditioner for a class of complex symmetric linear systems. Comp. Appl. Math. 43, 157 (2024). https://doi.org/10.1007/s40314-024-02622-9

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  • DOI: https://doi.org/10.1007/s40314-024-02622-9

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