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A new block preconditioner for complex symmetric indefinite linear systems

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Abstract

Using the equivalent block two-by-two real linear systems and relaxing technique, we establish a new block preconditioner for a class of complex symmetric indefinite linear systems. The new preconditioner is much closer to the original block two-by-two coefficient matrix than the Hermitian and skew-Hermitian splitting (HSS) preconditioner. We analyze the spectral properties of the new preconditioned matrix, discuss the eigenvalue distribution and derive an upper bound for the degree of its minimal polynomial. Finally, some numerical examples are provided to show the effectiveness and robustness of our proposed preconditioner.

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References

  1. Arridge, S.R.: Optical tomography in medical imaging. Inverse Probl. 15, R41–R93 (1999)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Bai, Z.-Z.: Rotated block triangular preconditioning based on PMHSS. Sci. China Math. 56, 2523–2538 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bai, Z.-Z.: On preconditioned iteration methods for complex linear systems. J. Eng. Math. 93, 41–60 (2015)

    Article  MathSciNet  Google Scholar 

  5. 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  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Bai, Z.-Z., Benzi, M., Chen, F., Wang, Z.-Q.: 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 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bai, Z.-Z., Golub, G.H.: Accelerated Hermitian and skew-Hermitian splitting iteration methods for saddle-point problems. IMA J. Numer. Anal. 27, 1–23 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bai, Z.-Z., Golub, G.H., Lu, L.-Z., Yin, J.-F.: Block triangular and skew-Hermitian splitting method for positive definite linear systems. SIAM J. Sci. Comput. 26, 844–863 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. 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  MATH  Google Scholar 

  11. Bai, Z.-Z., Golub, G.H., Ng, M.K.: On successive-overrelaxation acceleration of the Hermitian and skew-Hermitian splitting iterations. Numer. Linear Algebra Appl. 14, 319–335 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. 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  MathSciNet  MATH  Google Scholar 

  13. 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  MathSciNet  MATH  Google Scholar 

  14. Bai, Z.-Z., Ng, M.K.: On inexact preconditioners for nonsymmetric matrices. SIAM J. Sci. Comput. 26, 1710–1724 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  17. Bao, G., Sun, W.-W.: A fast algorithm for the electromagnetic scattering from a large cavity. SIAM J. Sci. Comput. 27, 553–574 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  18. Benzi, M.: A generalization of the Hermitian and skew-Hermitian splitting iteration. SIAM J. Matrix Anal. Appl. 31, 360–374 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. 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  MATH  Google Scholar 

  20. Cao, Y., Dong, J.-L., Wang, Y.-M.: A relaxed deteriorated PSS preconditioner for nonsymmetric saddle point problems from the steady Navier–Stokes equation. J. Comput. Appl. Math. 273, 41–60 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. 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  Google Scholar 

  22. 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  MATH  Google Scholar 

  23. Cao, Y., Yao, L.-Q., Jiang, M.-Q., Niu, Q.: A relaxed HSS preconditioner for saddle point problems from meshfree discretization. J. Comput. Math. 31, 398–421 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  25. 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, 761–776 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  26. Lang, C., Ren, Z.-R.: Inexact rotated block triangular preconditioners for a class of block two-by-two matrices. J. Eng. Math. 93, 87–98 (2015)

    Article  MathSciNet  Google Scholar 

  27. Li, X., Yang, A.-L., Wu, Y.-J.: Lopsided PMHSS iteration method for a class of complex symmetric linear systems. Numer. Algor. 66, 555–568 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  28. 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  MATH  Google Scholar 

  29. Pan, J.-Y., Ng, M.K., Bai, Z.-Z.: New preconditioners for saddle point problems. Appl. Math. Comput. 172, 762–771 (2006)

    MathSciNet  MATH  Google Scholar 

  30. Parlett, B.N.: The Symmetric Eigenvalue Problem. SIAM, Philadelphia (1998)

    Book  MATH  Google Scholar 

  31. Saad, Y.: Iterative Methods for Sparse Linear Systems, 2nd edn. SIAM, Philadelphia (2003)

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  34. van Dijk, W., Toyama, F.M.: Accurate numerical solutions of the time-dependent Schrödinger equation. Phys. Rev. E. 75, 036707–1–036707-10 (2007)

    Article  Google Scholar 

  35. Wu, S.-L.: Several variants of the Hermitian and skew-Hermitian splitting method for a class of complex symmetric linear systems. Numer. Linear Algebra Appl. 22, 338–356 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  36. Xu, W.-W.: A generalization of preconditioned MHSS iteration method for complex symmetric indefinite linear systems. Appl. Math. Comput. 219, 10510–10517 (2013)

    MathSciNet  MATH  Google Scholar 

  37. Yan, H.-Y., Huang, Y.-M.: Splitting-based block preconditioning methods for block two-by-two matrices of real square blocks. Appl. Math. Comput. 243, 825–837 (2014)

    MathSciNet  MATH  Google Scholar 

  38. Zhang, J.-H., Dai, H.: A new splitting preconditioner for the iterative solution of complex symmetric indefinite linear systems. Appl. Math. Lett. 49, 100–106 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  39. Zhang, G.-F., Zheng, Z.: A parameterized splitting iteration method for complex symmetric linear systems. J.pan J. Indust. Appl. Math. 31, 265–278 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  40. Zheng, Q.-Q., Ma, C.-F.: Accelerated PMHSS iteration methods for complex symmetric linear systems. Numer. Algor. in press (2016)

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Authors and Affiliations

  1. School of Science, East China University of Technology, Nanchang, Jiangxi, 330013, China

    Jian-Hua Zhang

  2. Department of Mathematics, Anhui Science and Technology University, Fengyang, 233100, China

    Jian-Hua Zhang

  3. Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China

    Hua Dai

Authors
  1. Jian-Hua Zhang
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  2. Hua Dai
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Correspondence to Jian-Hua Zhang.

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Zhang, JH., Dai, H. A new block preconditioner for complex symmetric indefinite linear systems. Numer Algor 74, 889–903 (2017). https://doi.org/10.1007/s11075-016-0175-y

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  • DOI: https://doi.org/10.1007/s11075-016-0175-y

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