Abstract
In this paper, based on the complex-symmetric and skew-Hermitian splitting (CSS) of the coefficient matrix, a modified complex-symmetric and skew-Hermitian-splitting (MCSS) iteration method is presented to solve a class of complex-symmetric indefinite linear systems from the classical state-space formulation of frequency analysis of the degree-of-freedom discrete system. The convergence properties of the MCSS method are obtained. The corresponding MCSS preconditioner is proposed and some useful properties of the preconditioned matrix are established. Numerical experiments are reported to verify the efficiency of both the MCSS method and the MCSS preconditioner.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bayliss, A., Goldstein, C.I., Turkel, E.: An iterative method for Helmholtz equation. J. Comput. Phys. 49, 443–457 (1983)
Guo, C.-H.: Incomplete block factorization preconditioner for linear systems arising in the numerical solution of the Helmholtz equation. Appl. Numer. Math. 19, 495–508 (1996)
Wu, S.-L., Huang, T.-Z., Li, L., Xiong, L.-L.: Positive stable preconditioners for symmetric indefinite linear systems arising from Helmholtz equations. Phys. Lett. A. 373, 2401–2407 (2009)
Wu, S.-L., Li, C.-X.: A modified SSOR preconditioning strategy for Helmholtz equations. J. Appl. Math. 2012(4), 254–263 (2012)
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)
Bai, Z.-Z., Benzi, M., Chen, F.: On preconditioned MHSS iteration methods for complex symmetric linear systems. Numer. Algor. 56, 297–317 (2011)
Bai, Z.-Z.: Block alternating splitting implicit iteration methods for saddle-point problems from time-harmonic eddy current models. Numer. Linear Algebra Appl. 19, 914–936 (2012)
Day, D., Heroux, M.A.: Solving complex-valued linear systems via equivalent real formulations. SIAM J. Sci. Comput. 23, 480–498 (2001)
Benzi, M., Bertaccini, D.: Block preconditioning of real-valued iterative algorithms for complex linear systems. IMA J. Numer. Anal. 28, 598–618 (2008)
Feriani, A., Perotti, F., Simoncini, V.: Iterative system solvers for the frequency analysis of linear mechanical systems. Comput. Methods Appl. Mech. Eng. 190, 1719–1739 (2000)
Li, L., Huang, T.-Z., Jing, Y.-F., Zhang, Y.: Application of the incomplete Cholesky factorization preconditioned Krylov subspace method to the vector finite element method for 3-D electromagnetic scattering problems. Comput. Phys. Commun. 181, 271–276 (2010)
Sogabe, T., Zhang, S.-L.: A COCR method for solving complex symmetric linear systems. J. Comput. Appl. Math. 199, 297–303 (2007)
Axelsson, O., Kucherov, A.: Real valued iterative methods for solving complex symmetric linear systems. Numer. Linear Algebra Appl. 7, 197–218 (2000)
Axelsson, O., Neytcheva, M., Ahmad, B.: A comparison of iterative methods to solve complex valued linear algebraic systems. Numer. Algor. 66, 811–841 (2014)
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)
Bai, Z.-Z., Benzi, M., Chen, F., Modified, H S S: iteration methods for a class of complex symmetric linear systems. Computer 87, 93–111 (2010)
Bai, Z.-Z.: On preconditioned iteration methods for complex linear systems. J. Eng. Math. 93, 41–60 (2015)
Wu, S.-L., Li, C.-X.: A splitting iterative method for the discrete dynamic linear systems. J. Comput. Appl. Math. 267, 49–60 (2014)
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)
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)
Benzi, M.: A generalization of the Hermitian and skew-Hermitian splitting iteration. SIAM J. Matrix Anal. Appl. 31, 360–374 (2009)
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)
Bai, Z.-Z., Golub, G.H., Lu, L.-Z., Yin, J.-F.: Block triangular and skew-Hermitian splitting methods for positive-definite linear systems. SIAM J. Sci. Comput. 26, 844–863 (2005)
Xu, W.-W.: A generalization of preconditioned MHSS iteration method for complex symmetric indefinite linear systems. Appl. Math. Comput. 219, 10510–10517 (2013)
Bai, Z.-Z., Golub, G.H., Li, C.-K.: Optimal parameter in Hermitian and skew-Hermitian splitting method for certain two-by-two block matrices. SIAM J. Sci. Comput. 28, 583–603 (2006)
Bai, Z.-Z.: Optimal parameters in the HSS-like methods for saddle-point problems. Numer. Linear Algebra Appl 16, 447–479 (2009)
Huang, Y.-M.: On m-step Hermitian and skew-Hermitian splitting preconditioning methods. J. Eng. Math. 93, 77–86 (2015)
Huang, Y.-M.: A practical formula for computing optimal parameters in the HSS iteration methods. J. Comput. Appl. Math. 255, 142–149 (2014)
Chen, F.: On choices of iteration parameter in HSS method. Appl. Math. Comput 271, 832–837 (2015)
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was supported by the NSFC (11301109), 17HASTIT012 Natural Science Foundations of Henan Province (No.15A110007), Project of Young Core Instructor of Universities in Henan Province (No. 2015GGJS-003).
Rights and permissions
About this article
Cite this article
Wu, SL., Li, CX. Modified complex-symmetric and skew-Hermitian splitting iteration method for a class of complex-symmetric indefinite linear systems. Numer Algor 76, 93–107 (2017). https://doi.org/10.1007/s11075-016-0245-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-016-0245-1
Keywords
- Complex-symmetric linear system
- Complex-symmetric matrix
- Skew-Hermitian matrix
- Matrix splitting
- CSS method
- Convergence