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Modified HSS iteration methods for a class of non-Hermitian positive-definite linear systems. (English) Zbl 1246.65056

Summary: We consider the numerical solution of a class of non-Hermitian positive-definite linear systems by the modified Hermitian and skew-Hermitian splitting (MHSS) iteration method. We show that the MHSS iteration method converges unconditionally even when the real and the imaginary parts of the coefficient matrix are nonsymmetric and positive semidefinite and, at least, one of them is positive definite. At each step the MHSS iteration method requires to solve two linear sub-systems with real nonsymmetric positive definite coefficient matrices. We propose to use inner iteration methods to compute approximate solutions of these linear sub-systems. We illustrate the performance of the MHSS method and its inexact variant by two numerical examples.

MSC:

65F10 Iterative numerical methods for linear systems
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[1] Axelsson, O.; Bai, Z.-Z.; Qiu, S.-X., A class of nested iteration schemes for linear systems with a coefficient matrix with a dominant positive definite symmetric part, Numer. Algor., 35, 351-372 (2004) · Zbl 1054.65028
[2] Bai, Z.-Z., Splitting iteration methods for non-Hermitian positive definite systems of linear equations, Hokkaido Math. J., 36, 801-814 (2007) · Zbl 1138.65027
[3] Bai, Z.-Z.; Benzi, M.; Chen, F., Modified HSS iteration methods for a class of complex symmetric linear systems, Computing, 87, 93-111 (2010) · Zbl 1210.65074
[4] Bai, Z.-Z.; Golub, G. H., Accelerated Hermitian and skew-Hermitian splitting iteration methods for saddle-point problems, SIAM J. Numer. Anal., 27, 1-23 (2007) · Zbl 1134.65022
[5] 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) · Zbl 1116.65039
[6] Bai, Z.-Z.; Golub, G. H.; Li, C.-K., Convergence properties of preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite matrices, Math. Comput., 76, 287-298 (2007) · Zbl 1114.65034
[7] 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) · Zbl 1079.65028
[8] 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) · Zbl 1036.65032
[9] 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) · Zbl 1135.65016
[10] Cao, Y.; Jiang, M.-Q.; Zheng, Y.-L., A splitting preconditioner for saddle point problems, Numer. Linear Algebra Appl., 18, 875-895 (2011) · Zbl 1249.65065
[11] Bertaccini, D.; Golub, G. H.; Stefano, S.-C., Spectral analysis of a preconditioned iterative method for the convection-diffusion equation, SIAM J. Matrix Anal. Appl., 29, 260-278 (2006) · Zbl 1140.65024
[12] Zhang, G.-F.; Ren, Z.-R.; Zhou, Y.-Y., On HSS-based constraint preconditioners for generalized saddle-point problems, Numer. Algor., 57, 273-287 (2011) · Zbl 1220.65038
[13] Chen, F.; Jiang, Y.-L.; Zheng, B., On contraction and semi-condition factors of GSOR method for augmented linear systems, J. Comput. Math., 28, 901-912 (2010) · Zbl 1240.65106
[14] Chen, F.; Jiang, Y.-L., On HSS and AHSS iteration methods for nonsymmetric positive definite Toeplitz systems, J. Comput. Appl. Math., 234, 2432-2440 (2010) · Zbl 1191.65027
[15] Hamilton, S.; Benzi, M.; Haber, E., New multigrid smoothers for the Oseen problem, Numer. Linear Algebra Appl., 17, 557-576 (2010) · Zbl 1240.76003
[16] Benzi, M., A generalization of the Hermitian and skew Hermitian splitting iteration, SIAM J. Matrix Anal. Appl., 31, 360-374 (2009) · Zbl 1191.65025
[17] Russo, A.; Possio, C. T., Preconditioned Hermitian and skew-Hermitian splitting method for finite element approximations of convection-diffusion equations, SIAM J. Matrix Anal. Appl., 31, 997-1018 (2009) · Zbl 1202.65156
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