Abstract
By utilizing the equivalent real block two-by-two linear systems and the shift-splitting techniques, we establish an efficient parameterized rotated shift-splitting (PRSS) preconditioner for solving a class of complex symmetric linear systems. The proposed preconditioner is extracted from a stationary iteration method which is unconditionally convergent. Moreover, some spectral properties of the corresponding preconditioned matrix are studied in detail. Finally, numerical results are presented to show the feasibility and effectiveness of the proposed preconditioner.
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Axelsson, O., Kucherov, A.: Real valued iterative methods for solving complex symmetric linear systems. Numer. Linear Algebra Appl. 7, 197–218 (2000)
Arridge, S.R.: Optical tomography in medical imaging. Inverse Prob. 15, 41–93 (1999)
Bertaccini, D.: Efficient solvers for sequences of complex symmetric linear systems. Electron. Trans. Numer. Anal. 18, 49–64 (2004)
Bertaccini, D.: Efficient preconditioning for sequences of parametric complex symmetric linear systems. Electron. Tran. Numer. Anal. 18, 49–64 (2004)
Benzi, M., Bertaccini, D.: Block preconditioning of real-valued iterative algorithms for complex linear systems. IMA J. Numer. Anal. 28, 598–618 (2008)
Benzi, M., Simoncini, V.: On the eigenvalues of a class of saddle point matrices. Numer. Math. 103, 173–196 (2006)
Bai, Z.-Z.: Structured preconditioners for nonsingular matrices of block two-by-two structures. Math. Comput. 75, 791–815 (2006)
Bai, Z.-Z., Benzi, M., Chen, F.: Modified HSS iteration methods for a class of complex symmetric linear systems. Computing 87, 93–111 (2010)
Bai, Z.-Z., Benzi, M., Chen, F.: On preconditioned MHSS iteration methods for complex symmetric linear systems. Numer. Algorithms 56, 297–317 (2011)
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)
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.: Rotated block triangular preconditioning based on PMHSS. Sci. China Math. 56, 2523–2538 (2013)
Bai, Z.-Z.: On preconditioned iteration methods for complex linear systems. J. Eng. Math. 93, 41–60 (2015)
Bai, Z.-Z., Chen, F., Wang, Z.-Q.: Additive block diagonal preconditioning for block two-by-two linear systems of skew-Hamiltonian coefficient matrices. Numer. Algorithms 64, 655–675 (2013)
Bai, Z.-Z.: Block preconditioners for elliptic PDE-constrained optimization problems. Computing 91, 379–395 (2011)
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., Yin, J.-F., Su, Y.-F.: A shift-splitting preconditioner for non-Hermitian positive definite matrices. J. Comput. Math. 24, 539–552 (2006)
Bai, Z.-Z.: Sharp error bounds of some Krylov subspace methods for non-Hermitian linear systems. Appl. Math. Comput. 109, 273–285 (2000)
Bai, Z.-Z.: Motivations and realizations of Krylov subspace methods for large sparse linear systems. J. Comput. Appl. Math. 283, 71–78 (2015)
Chen, C.-R., Ma, C.-F.: A generalized shift-splitting preconditioner for saddle point problems. Appl. Math. Lett. 43, 49–55 (2015)
Chen, C.-R., Ma, C.-F.: A generalized shift-splitting preconditioner for singular saddle point problems. Appl. Math. Comput. 269, 947–955 (2015)
Chen, C.-R., Ma, C.-F.: AOR-Uzawa iterative method for a class of complex symmetric linear system of equations. Comput. Math. Appl. 72, 2462–2472 (2016)
Cao, Y., Du, J., Niu, Q.: Shift-splitting preconditioners for saddle point problems. J. Comput. Appl. Math. 272, 239–250 (2014)
Cao, Y., Miao, S.-X.: On semi-convergence of the generalized shift-splitting iteration method for singular nonsymmetric saddle point problems. Comput. Math. Appl. 71, 1503–1511 (2016)
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)
Frommer, A., Lippert, T., Medeke, B., Schilling, K.: Numerical challenges in lattice quantum chromodynamics. Lect. Notes Comput. Sci. Eng. 15, 66–83 (2000)
Freund, R.W.: Conjugate gradient-type methods for linear systems with complex symmetric coefficient matrices. SIAM J. Sci. Stat. Comput. 13, 425–448 (1992)
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)
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)
Ling, S.-T., Liu, Q.-B.: New local generalized shift-splitting preconditioners for saddle point problems. Appl. Math. Comput. 302, 58–67 (2017)
Li, X., Yang, A.-L., Wu, Y.-J.: Lopsided PMHSS iteration method for a class of complex symmetric linear systems. Numer. Algorithms 66, 555–568 (2014)
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)
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)
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)
Zhang, F.-Z.: Matrix Theory. Springer, New York (2011)
Zeng, M.-L., Ma, C.-F.: A parameterized SHSS iteration method for a class of complex symmetric system of linear equations. Comput. Math. Appl. 71, 2124–2131 (2016)
Zhang, J.-H., Dai, H.: Inexact splitting-based block preconditioners for block two-by-two linear systems. Appl. Math. Lett. 60, 89–95 (2016)
Zeng, M.-L., Zhang, G.-F.: Parameterized rotated block preconditioning techniques for block two-by-two systems with application to complex linear systems. Comput. Math. Appl. 70, 2946–2957 (2015)
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)
Acknowledgments
The authors would like to thank the editor and the anonymous referees for their detailed comments which greatly improve the presentation. This research is supported by National Science Foundation of China (41725017, 41590864), National Basic Research Program of China under grant number 2014CB845906. It is also partially supported by the CAS/CAFEA international partnership Program for creative research teams (No. KZZD-EW-TZ-19 and KZZD-EW-TZ-15), Strategic Priority Research Program of the Chinese Academy of Sciences (XDB18010202) and Fujian Natural Science Foundation (2016J01005).
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Li, CL., Ma, CF. Efficient parameterized rotated shift-splitting preconditioner for a class of complex symmetric linear systems. Numer Algor 80, 337–354 (2019). https://doi.org/10.1007/s11075-018-0487-1
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DOI: https://doi.org/10.1007/s11075-018-0487-1