On product-type generalized block AOR method for augmented linear systems

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2012, 2(4): 797-809. doi: 10.3934/naco.2012.2.797

On product-type generalized block AOR method for augmented linear systems

Fang Chen ^1, , Ning Gao ^1, and Yao- Lin Jiang ^1,

1.	Department of Mathematical Sciences, Xi'an Jiaotong University, Xi'an 710049, China, China, China

Received December 2011 Revised August 2012 Published November 2012

Abstract
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The generalized inexact accelerated overrelaxation ( GIAOR) method was presented by Bai, Parlett and Wang (Numer. Math. 102(2005)1-38) for solving the augmented system of linear equations. In this paper, a product-type generalized block AOR ( PGBAOR ) method is proposed, which is a two-step generalization of the GIAOR method. Both convergence and semi-convergence of the PGBAOR method are proved for the nonsingular and the singular augmented linear systems.

Keywords: Augmented linear system, block AOR method, convergence, semi-convergence..

Mathematics Subject Classification: Primary: 65F10, 65F5.

Citation: Fang Chen, Ning Gao, Yao- Lin Jiang. On product-type generalized block AOR method for augmented linear systems. Numerical Algebra, Control & Optimization, 2012, 2 (4) : 797-809. doi: 10.3934/naco.2012.2.797

References:

[1]	K. Arrow, L. Hurwicz and H. Uzawa, "Studies in Nonlinear Programming,", Stanford University Press, (1958).
[2]	Z. Z. Bai, Structured preconditioners for nonsingular matrices of block two-by-two structures,, Math. Comput., 75 (2006), 791. 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. 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. 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.
[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. 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. 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. 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.
[10]	M. Benzi and G. H. Golub, A preconditioner for generalized saddle point problems,, SIAM J. Matrix Anal. Appl., 26 (2004), 20. 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. 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. doi: 10.1137/0731085.
[13]	G. H. Golub, X. Wu and J.-Y. Yuan, SOR-like methods for augmented systems,, BIT, 41 (2001), 71. 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. 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. 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. doi: 10.1016/S0096-3003(02)00187-X.
[17]	D. M. Young, "Iterative Solutions of Large Linear Systmes,", Academic Press, (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.
[19]	G.-F. Zhang and Q.-H. Lu, On generalized symmetric SOR method for augmented systems,, J. Comput. Appl. Math., 219 (2008), 51. 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. 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. doi: 10.1080/00207160701871835.

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References:

[1]	K. Arrow, L. Hurwicz and H. Uzawa, "Studies in Nonlinear Programming,", Stanford University Press, (1958).
[2]	Z. Z. Bai, Structured preconditioners for nonsingular matrices of block two-by-two structures,, Math. Comput., 75 (2006), 791. 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. 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. 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.
[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. 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. 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. 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.
[10]	M. Benzi and G. H. Golub, A preconditioner for generalized saddle point problems,, SIAM J. Matrix Anal. Appl., 26 (2004), 20. 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. 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. doi: 10.1137/0731085.
[13]	G. H. Golub, X. Wu and J.-Y. Yuan, SOR-like methods for augmented systems,, BIT, 41 (2001), 71. 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. 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. 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. doi: 10.1016/S0096-3003(02)00187-X.
[17]	D. M. Young, "Iterative Solutions of Large Linear Systmes,", Academic Press, (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.
[19]	G.-F. Zhang and Q.-H. Lu, On generalized symmetric SOR method for augmented systems,, J. Comput. Appl. Math., 219 (2008), 51. 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. 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. doi: 10.1080/00207160701871835.

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