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A generalization of the positive-definite and skew-Hermitian splitting iteration
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1. | Department of Mathematical Sciences, Xi'an Jiaotong University, Xi'an 710049, China, China, China |
References:
[1] |
K. Arrow, L. Hurwicz and H. Uzawa, "Studies in Nonlinear Programming," Stanford University Press, Stanford, 1958. |
[2] |
Z. Z. Bai, Structured preconditioners for nonsingular matrices of block two-by-two structures, Math. Comput., 75 (2006), 791-815.
doi: 10.1090/S0025-5718-05-01801-6. |
[3] |
Z. Z. Bai and G. H. Golub, Accelerated Hermitian and skew-Hermitian splitting iteration methods for saddle-point problems, IMA J. Numer. Anal., 27 (2007), 1-23.
doi: 10.1093/imanum/drl017. |
[4] |
Z. Z. Bai, G. H. Golub and M. K. Ng, Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, SIAM J. Matrix Anal. Appl., 24 (2003), 603-626.
doi: 10.1137/S0895479801395458. |
[5] |
Z. Z. Bai, G. H. Golub and J. Y. Pan, Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems, Numer. Math., 98 (2004), 1-32. |
[6] |
Z. Z. Bai and G. Q. Li, Restrictively preconditioned conjugate gradient methods for systems of linear equations, IMA J. Numer. Anal., 23 (2003), 561-580.
doi: 10.1093/imanum/23.4.561. |
[7] |
Z. Z. Bai, B. N. Parlett and Z. Q. Wang, On generalized successive overrelaxation methods for augmented linear systems, Numer. Math., 102 (2005), 1-38.
doi: 10.1007/s00211-005-0643-0. |
[8] |
Z. Z. Bai and Z. Q. Wang, On parameterized inexact Uzawa methods for generalized saddle point problems, Linear Algebra Appl., 428 (2008), 2900-2932.
doi: 10.1016/j.laa.2008.01.018. |
[9] |
Z. Z. Bai, L. Wang and J. Y. Yuan, Weak-convergence theory of quasi-nonnegative splittings for singular matrices, Linear Algebra Appl., 47 (2003), 75-89. |
[10] |
M. Benzi and G. H. Golub, A preconditioner for generalized saddle point problems, SIAM J. Matrix Anal. Appl., 26 (2004), 20-41.
doi: 10.1137/S0895479802417106. |
[11] |
J. H. Bramble, J. E. Pasciak and A. T. Vassilev, Analysis of the inexact Uzawa algorithm for saddle point problems, SIAM J. Numer. Anal., 34 (1997), 1072-1092.
doi: 10.1137/S0036142994273343. |
[12] |
H. C. Elman and G. H. Golub, Inexact and preconditioned Uzawa algorithms for saddle point problems, SIAM J. Numer. Anal., 31 (1994), 1645-1661.
doi: 10.1137/0731085. |
[13] |
G. H. Golub, X. Wu and J.-Y. Yuan, SOR-like methods for augmented systems, BIT, 41 (2001), 71-85.
doi: 10.1023/A:1021965717530. |
[14] |
C.-J. Li, Z. Li, D. J. Evans and T. Zhang, A note on an SOR-like method for augmented systems, IMA J. Numer. Anal., 23 (2003), 581-592.
doi: 10.1093/imanum/23.4.581. |
[15] |
Y.-L. Jiang, R. M. M. Chen and O. Wing, Improving convergence performance of relaxation-based transient analysis by matrix splitting in circuit simulation, IEEE Trans. Circuits Systems: Part I, 48 (2001), 769-780.
doi: 10.1109/81.928160. |
[16] |
Y.-L. Jiang, Y.-W. Liu, K.-K. Mei and R. M. M. Chen, A new iterative technique for large and dense linear systems from the MEI method in electromagnetics, Appl. Math. Comput., 139 (2003), 157-163.
doi: 10.1016/S0096-3003(02)00187-X. |
[17] |
D. M. Young, "Iterative Solutions of Large Linear Systmes," Academic Press, New York, 1971. |
[18] |
J.-F. Yin and Z.-Z. Bai, The restrictively preconditioned conjugate gradient methods on normal residual for block two-by-two linear systems, J. Comput. Math., 26 (2008), 240-249. |
[19] |
G.-F. Zhang and Q.-H. Lu, On generalized symmetric SOR method for augmented systems, J. Comput. Appl. Math., 219 (2008), 51-58.
doi: 10.1016/j.cam.2007.07.001. |
[20] |
B. Zheng, Z.-Z. Bai and X. Yang, On semi-convergence of parameterized Uzawa methods for singular saddle point problems, Linear Algebra Appl., 431 (2009), 808-817.
doi: 10.1016/j.laa.2009.03.033. |
[21] |
B. Zheng, K. Wang and Y.-J. Wu, SSOR-like methods for saddle point problems, Intern. J. Comput. Math., 86 (2009), 1405-1423.
doi: 10.1080/00207160701871835. |
show all references
References:
[1] |
K. Arrow, L. Hurwicz and H. Uzawa, "Studies in Nonlinear Programming," Stanford University Press, Stanford, 1958. |
[2] |
Z. Z. Bai, Structured preconditioners for nonsingular matrices of block two-by-two structures, Math. Comput., 75 (2006), 791-815.
doi: 10.1090/S0025-5718-05-01801-6. |
[3] |
Z. Z. Bai and G. H. Golub, Accelerated Hermitian and skew-Hermitian splitting iteration methods for saddle-point problems, IMA J. Numer. Anal., 27 (2007), 1-23.
doi: 10.1093/imanum/drl017. |
[4] |
Z. Z. Bai, G. H. Golub and M. K. Ng, Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, SIAM J. Matrix Anal. Appl., 24 (2003), 603-626.
doi: 10.1137/S0895479801395458. |
[5] |
Z. Z. Bai, G. H. Golub and J. Y. Pan, Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems, Numer. Math., 98 (2004), 1-32. |
[6] |
Z. Z. Bai and G. Q. Li, Restrictively preconditioned conjugate gradient methods for systems of linear equations, IMA J. Numer. Anal., 23 (2003), 561-580.
doi: 10.1093/imanum/23.4.561. |
[7] |
Z. Z. Bai, B. N. Parlett and Z. Q. Wang, On generalized successive overrelaxation methods for augmented linear systems, Numer. Math., 102 (2005), 1-38.
doi: 10.1007/s00211-005-0643-0. |
[8] |
Z. Z. Bai and Z. Q. Wang, On parameterized inexact Uzawa methods for generalized saddle point problems, Linear Algebra Appl., 428 (2008), 2900-2932.
doi: 10.1016/j.laa.2008.01.018. |
[9] |
Z. Z. Bai, L. Wang and J. Y. Yuan, Weak-convergence theory of quasi-nonnegative splittings for singular matrices, Linear Algebra Appl., 47 (2003), 75-89. |
[10] |
M. Benzi and G. H. Golub, A preconditioner for generalized saddle point problems, SIAM J. Matrix Anal. Appl., 26 (2004), 20-41.
doi: 10.1137/S0895479802417106. |
[11] |
J. H. Bramble, J. E. Pasciak and A. T. Vassilev, Analysis of the inexact Uzawa algorithm for saddle point problems, SIAM J. Numer. Anal., 34 (1997), 1072-1092.
doi: 10.1137/S0036142994273343. |
[12] |
H. C. Elman and G. H. Golub, Inexact and preconditioned Uzawa algorithms for saddle point problems, SIAM J. Numer. Anal., 31 (1994), 1645-1661.
doi: 10.1137/0731085. |
[13] |
G. H. Golub, X. Wu and J.-Y. Yuan, SOR-like methods for augmented systems, BIT, 41 (2001), 71-85.
doi: 10.1023/A:1021965717530. |
[14] |
C.-J. Li, Z. Li, D. J. Evans and T. Zhang, A note on an SOR-like method for augmented systems, IMA J. Numer. Anal., 23 (2003), 581-592.
doi: 10.1093/imanum/23.4.581. |
[15] |
Y.-L. Jiang, R. M. M. Chen and O. Wing, Improving convergence performance of relaxation-based transient analysis by matrix splitting in circuit simulation, IEEE Trans. Circuits Systems: Part I, 48 (2001), 769-780.
doi: 10.1109/81.928160. |
[16] |
Y.-L. Jiang, Y.-W. Liu, K.-K. Mei and R. M. M. Chen, A new iterative technique for large and dense linear systems from the MEI method in electromagnetics, Appl. Math. Comput., 139 (2003), 157-163.
doi: 10.1016/S0096-3003(02)00187-X. |
[17] |
D. M. Young, "Iterative Solutions of Large Linear Systmes," Academic Press, New York, 1971. |
[18] |
J.-F. Yin and Z.-Z. Bai, The restrictively preconditioned conjugate gradient methods on normal residual for block two-by-two linear systems, J. Comput. Math., 26 (2008), 240-249. |
[19] |
G.-F. Zhang and Q.-H. Lu, On generalized symmetric SOR method for augmented systems, J. Comput. Appl. Math., 219 (2008), 51-58.
doi: 10.1016/j.cam.2007.07.001. |
[20] |
B. Zheng, Z.-Z. Bai and X. Yang, On semi-convergence of parameterized Uzawa methods for singular saddle point problems, Linear Algebra Appl., 431 (2009), 808-817.
doi: 10.1016/j.laa.2009.03.033. |
[21] |
B. Zheng, K. Wang and Y.-J. Wu, SSOR-like methods for saddle point problems, Intern. J. Comput. Math., 86 (2009), 1405-1423.
doi: 10.1080/00207160701871835. |
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