## On inexact Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems.(English)Zbl 1135.65016

The authors discuss Hermitian/skew-Hermitian splitting methods for solving systems of linear equations $$Ax=b$$, where $$A$$ is a non-Hermitian and positive definite matrix. In each iteration step one has to solve a system of equations with the matrix $$\alpha I + H$$, $$H = (A + A^\ast)/2$$ and the matrix $$\alpha I + S$$, $$S = (A - A^\ast)/2$$. The systems with the matrix $$\alpha I + H$$ are solved by means of the conjugate gradient method and the other systems by means of the Lanczos method or the conjugate gradient method applied to the normal equations. This approach leads to inexact Hermitian/skew-Hermitian splitting methods.
In the paper convergence properties of these methods are studied. It is shown that the contraction factor and the asymptotic convergence rates depend dominantly on the spectrum of the Hermitian part. Furthermore, the computational efficiency of the presented methods is analysed. Optimal choices of the number of inner iteration steps are discussed. The methods are compared by numerical examples.

### MSC:

 65F10 Iterative numerical methods for linear systems 65Y20 Complexity and performance of numerical algorithms
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### References:

 [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. Algorithms, 35, 351-372 (2004) · Zbl 1054.65028 [2] Bai, Z.-Z.; Gao, Z.-F.; Huang, T.-Z., Measure parameters of the effectiveness of the parallel iteration methods, Math. Numer. Sinica, 21, 325-330 (1999), (in Chinese) · Zbl 0933.65054 [3] 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) · Zbl 1134.65022 [4] 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 [5] 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 [6] 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 [7] Bai, Z.-Z.; Qiu, S.-X., Splitting-MINRES methods for linear systems with the coefficient matrix with a dominant indefinite symmetric part, Math. Numer. Sinica, 24, 113-128 (2002), (in Chinese) · Zbl 1495.65031 [8] 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) · Zbl 1120.65054 [9] Bai, Z.-Z.; Zhang, S.-L., A regularized conjugate gradient method for symmetric positive definite system of linear equations, J. Comput. Math., 20, 437-448 (2002) · Zbl 1002.65040 [10] Benzi, M.; Gander, M.; Golub, G. H., Optimization of the Hermitian and skew-Hermitian splitting iteration for saddle-point problems, BIT, 43, 881-900 (2003) · Zbl 1052.65015 [11] Benzi, M.; Golub, G. H., A preconditioner for generalized saddle point problems, SIAM J. Matrix Anal. Appl., 26, 20-41 (2004) · Zbl 1082.65034 [12] Concus, P.; Golub, G. H., A generalized conjugate gradient method for non-symmetric systems of linear equations, (Glowinski, R.; Lions, J. R., Computing Methods in Applied Sciences and Engineering. Computing Methods in Applied Sciences and Engineering, Lecture Notes in Econom. and Math. Systems, vol. 134 (1976), Springer-Verlag: Springer-Verlag Berlin), 56-65 [13] Bertaccini, D.; Golub, G. H.; Serra Capizzano, S.; Tablino Possio, C., Preconditioned HSS method for the solution of non-Hermitian positive definite linear systems and applications to the discrete convection-diffusion equation, Numer. Math., 99, 441-484 (2005) · Zbl 1068.65041 [14] 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 [15] Golub, G. H.; Van Loan, C. F., Matrix Computations (1996), The Johns Hopkins University Press: The Johns Hopkins University Press Baltimore and London · Zbl 0865.65009 [16] Greenbaum, A., Iterative Methods for Solving Linear Systems (1997), SIAM: SIAM Philadelphia · Zbl 0883.65022 [17] 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 [18] Widlund, O. B., A Lanczos method for a class of nonsymmetric systems of linear equations, SIAM J. Numer. Anal., 15, 801-812 (1978) · Zbl 0398.65030
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