Wang, Li; Song, Yongzhong Preconditioned AOR iterative methods for \(M\)-matrices. (English) Zbl 1175.65043 J. Comput. Appl. Math. 226, No. 1, 114-124 (2009). The authors present general preconditioners for solving linear systems with a non-singular \(M\)-matrix. Additionally, they prove that the proposed preconditioners increase the convergence rate of accelerated overrelaxation (AOR) iterative methods. Finally, the authors demonstrate the applicability by solving a convection-diffusion equation and numerical results for the restarted generalized minimal residual (GMRES) method and precondiotioned restarted GMRES method are presented, using MATLAB. Reviewer: George A. Gravvanis (Xanthi) Cited in 1 ReviewCited in 18 Documents MSC: 65F10 Iterative numerical methods for linear systems 65F35 Numerical computation of matrix norms, conditioning, scaling 35J25 Boundary value problems for second-order elliptic equations 65N06 Finite difference methods for boundary value problems involving PDEs Keywords:linear system; M-matrix; preconditioner; convergence; accelerated overrelaxation (AOR) iterative methods; restarted generalized minimal residual (GMRES) method; convection-diffusion equation; numerical results; MATLAB Software:Matlab PDFBibTeX XMLCite \textit{L. Wang} and \textit{Y. Song}, J. Comput. Appl. Math. 226, No. 1, 114--124 (2009; Zbl 1175.65043) Full Text: DOI References: [1] Bai, Z.-Z., On the comparisons of the multisplitting unsymmetric AOR methods for \(M\)-matrices, Calcolo, 32, 207-220 (1995) · Zbl 0882.65018 [2] Bai, Z.-Z., The generalized Stein-Rosenberg type theorem for the PDAOR-method, Math. Numer. Sin., 19, 329-335 (1997), (in Chinese) · Zbl 0906.65029 [3] 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 [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.; 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.; Sun, J.-C.; Wang, D.-R., A unified framework for the construction of various matrix multisplitting iterative methods for large sparse system of linear equations, Comput. Math. Appl., 32, 51-76 (1996) · Zbl 0870.65025 [7] Berman, A.; Plemmons, R. J., Nonnegative Matrices in the Mathematical Sciences (1979), Academic Press: Academic Press New York, SIAM, Philadelphia, PA, 1994 · Zbl 0484.15016 [8] Evans, D. J.; Martins, M. M.; Trigo, M. E., The AOR iterative method for new preconditioned linear systems, J. Comput. Appl. Math., 132, 461-466 (2001) · Zbl 0992.65022 [9] Frommer, A.; Szyld, D. B., H-splitting and two-stage iterative methods, Numer. Math., 63, 345-356 (1992) · Zbl 0764.65018 [10] Gunawardena, A. D.; Jain, S. K.; Snyder, L., Modified iterative methods for consistent linear systems, Linear Algebra Appl., 154-156, 123-143 (1991) · Zbl 0731.65016 [11] Hadjidimos, A., Accelerated overrelaxation method, Math. Comput., 32, 149-157 (1978) · Zbl 0382.65015 [12] Hadjidimos, A.; Noutsos, D.; Tzoumas, M., More on modifications and improvements of classical iterative schemes for \(M\)-matrices, Linear Algebra Appl., 364, 253-279 (2003) · Zbl 1023.65022 [13] Kohno, T.; Kotakemori, H., Improving the modified Gauss-Seidel method for \(Z\)-matrices, Linear Algebra Appl., 267, 113-123 (1997) · Zbl 0886.65030 [14] Kotakemori, H.; Niki, H.; Okamoto, N., Accelerated iterative method for \(Z\)-matrices, J. Comput. Appl. Math., 75, 87-97 (1996) · Zbl 0872.65027 [15] Li, W., A note on the preconditioned Gauss-Seidel (GS) method for linear systems, J. Comput. Appl. Math., 182, 81-90 (2005) · Zbl 1072.65042 [16] Li, W.; Sun, W. W., Modified Gauss-Seidel type methods and Jacobi type methods for \(Z\)-matrices, Linear Algebra Appl., 317, 227-240 (2000) · Zbl 0966.65032 [17] Ortega, J. M.; Rheinboldt, W. C., Iterative Solution of Nonlinear Equations in Several Variables (1970), Academic Press: Academic Press New York, London · Zbl 0241.65046 [18] Rheinboldt, W. C.; Vandergraft, J. S., A simple approach to the Perron-Frobenius theory for positive operators on general partially ordered finite dimensional linear spaces, Math. Comp., 27, 139-145 (1973) · Zbl 0255.15017 [19] Varga, R. S., (Matrix Iterative Analysis (1962), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ). (Springer Series in Computational Mathematics, vol. 27 (2000), Springer-Verlag: Springer-Verlag Berlin) [20] Wang, L.; Bai, Z. Z., Skew-Hermitian triangular splitting iteration methods for non-Hermitian positive definite linear systems of strong skew-Hermitian parts, BIT Numer. Math., 44, 363-386 (2004) · Zbl 1069.65031 [21] Wang, L., On a class of row preconditioners for solving linear systems, Inter. J. Comput. Math., 83, 939-949 (2006) · Zbl 1126.65037 [22] Wu, M.; Wang, L.; Song, Y., Preconditioned AOR iterative method for linear systems, Appl. Numer. Math., 57, 672-685 (2007) · Zbl 1127.65020 [23] Young, D. M., Iterative Solution of Large Linear Systems (1971), Academic Press: Academic Press New York, London · Zbl 0204.48102 [24] Zhang, Y.; Huang, T. Z.; Liu, X. P., Modified iterative methods for nonnegative matrices and \(M\)-matrices linear systems, Comput. Math. Appl., 50, 1587-1602 (2005) · Zbl 1087.65031 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.