A generalized preconditioned HSS method for non-Hermitian positive definite linear systems. (English) Zbl 1198.65065

The paper presents a generalized preconditioned HSS (Hermitian and skew-Hermitian) spliting method for a large sparse non-Hermitian positive definite linear system.
The first section concerns iterative methods called HSS and preconditioned HSS(PHSS)methods based on the Hermitian/skew-Hermitian spliting, presenting also the algorithm of the new generalized preconditioned HSS method (or simply GPHSS method).
The second section focuses on the study of the convergence rate of the GPHSS iteration. This new two-parameter two-step iterative method can be generalized to the two-step splitting iterative framework. Also, for the upper bound of the spectral radius of the iteration matrix, the optimal parameters for the GPHSS method are provided.
In the third section the authors introduce an efficient preconditioner based on the incremental unknowns method. The efficiency of the GPHSS method is numerically tested. A comparison with the HSS and PHSS methods shows that the new method is more efficient.
In the last section, a concluding remark is given.


65F10 Iterative numerical methods for linear systems
65F08 Preconditioners for iterative methods
Full Text: DOI


[1] Bai, Z.-Z.; Golub, G.H.; Li, C.-K., Convergence properties of pre-conditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite matrices, Math. comput., 76, 287-298, (2007) · Zbl 1114.65034
[2] 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
[3] 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) · Zbl 1056.65025
[4] Chen, M.; Temam, R., Incremental unknowns in finite differences: condition number of the matrix, SIAM J. matrix anal. appl., 14, 432-455, (1993) · Zbl 0773.65080
[5] Eiermann, M.; Niethammer, W.; Varga, R.S., Acceleration of relaxation methods for non-Hermitian linear systems, SIAM J. matrix anal. appl., 13, 979-991, (1992) · Zbl 0757.65032
[6] Garcia, S., Algebraic conditioning analysis of the incremental unknowns preconditioner, Appl. math. model., 22, 351-366, (1998)
[7] Golub, G.H.; Van Loan, C., Matrix computations, (1996), The Johns Hopkins University Press Baltimore · Zbl 0865.65009
[8] Li, L.; Huang, T.-Z.; Liu, X.-P., Modified Hermitian and skew-Hermitian splitting methods for non-Hermitian positive-definite linear systems, Numer. linear algebra appl., 14, 217-235, (2007) · Zbl 1199.65109
[9] Thomas, J.W., Numerical partial differential equations: finite difference methods, (1995), Springer-Verlag New York · Zbl 0831.65087
[10] Wang, C.-L.; Bai, Z.-Z., Sufficient conditions for the convergent splittings of non-Hermitian positive definite matrices, Linear algebra appl., 330, 215-218, (2001) · Zbl 0983.65044
[11] Wu, Y.-J.; Yang, A.-L., Incremental unknowns for the heat equation with time-dependent coefficients: semi-implicit \(\theta\)-schemes and their stability, J. comput. math., 25, 573-582, (2007) · Zbl 1142.65406
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.