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Preconditioned Positive-Definite and Skew-Hermitian Splitting Iteration Methods for Continuous Sylvester Equations AX + XB = C

Part of: Numerical linear algebra

Published online by Cambridge University Press:  31 January 2017

Rong Zhou ,
Xiang Wang  and
Xiao-Bin Tang
Rong Zhou
Affiliation:
Department of Mathematics, School of Sciences, Nanchang University, Nanchang 330031, China
Xiang Wang*
Affiliation:
Department of Mathematics, School of Sciences, Nanchang University, Nanchang 330031, China Numerical Simulation and High-Performance Computing Laboratory, School of Sciences, Nanchang University, Nanchang 330031, China
Xiao-Bin Tang
Affiliation:
School of Statistics, University of International Business and Economics, Beijing 100029, China
*
*Corresponding author. Email address:[email protected] (X. Wang)
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Abstract

In this paper, we present a preconditioned positive-definite and skew-Hermitian splitting (PPSS) iteration method for continuous Sylvester equations AX + XB = C with positive definite/semi-definite matrices. The analysis shows that the PPSS iteration method will converge under certain assumptions. An inexact variant of the PPSS iteration method (IPPSS) has been presented and the analysis of its convergence property in detail has been discussed. Numerical results show that this new method is more efficient and robust than the existing ones.

Keywords

PPSS iteration methodIPPSS iteration methodSylvester equationsconvergence

MSC classification

Secondary: 65F10: Iterative methods for linear systems 65F50: Sparse matrices
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 7 , Issue 1 , February 2017 , pp. 55 - 69
Copyright
Copyright © Global-Science Press 2017 

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References

[1] Anderson, B.D.O., Agathoklis, P., Jury, E.I., Mansour, M.. Stability and the matrix Lyapunov equation for discrete 2-dimensional systems, IEEE Trans. Circuits Syst. 33 (1986) 261–267.CrossRefGoogle Scholar
[2] Bhatia, R., Rosenthal, P.. How and why to solve the operator equation AX – X B = Y, Bull. Lond. Math. Soc. 29 (1997) 1–21.CrossRefGoogle Scholar
[3] Benner, P., Li, R.-C., Truhar, N.. On the ADI method for Sylvester equations, J. Comput. Appl. Math. 233 (2009) 1035–1045.CrossRefGoogle Scholar
[4] Bai, Z.-Z., Golub, G. H., Ng, M.K.. On successive-overrelaxation acceleration of the Hermitian and skew-Hermitian splitting iterations, Numer. Linear Algebra Appl. 14 (2007) 319–335.CrossRefGoogle Scholar
[5] Bai, Z.-Z., On Hermitian and skew-Hermitian spliting iteration methods for continuous Sylvester equations, J. Comput. Math. 29 (2011) 185–198.CrossRefGoogle Scholar
[6] 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 (2005) 844–863.CrossRefGoogle Scholar
[7] 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 (2003) 603–626.CrossRefGoogle Scholar
[8] Bai, Z.-Z., Deng, Y.-B., Gao, Y.-H.. Iterative orthogonal direction methods for Hermitian minimum norm solutions of two consistent matrix equations, Numer. Linear Algebra Appl., 13 (2006) 801–823.Google Scholar
[9] Bai, Z.-Z., A class of two-stage iterative methods for systmes of weakly nonlinear equations, Numer. Algorithms, 14 (1997) 295–319.CrossRefGoogle Scholar
[10] Bai, Z.-Z., Benzi, M., Chen, F.. Modified HSS iteration methods for a class of complex symmetric linear systems, Computing, 87 (2010) 93–111.CrossRefGoogle Scholar
[11] Bai, Z.-Z., Benzi, M., Chen, F.. On preconditioned MHSS iteration methods for complex symmetric linear systems, Numer. Algorithms, 56 (2011) 297–317.CrossRefGoogle Scholar
[12] Bai, Z.-Z., Guo, X.-P.. On Newton-HSS methods for systems of nonlinear equations with positive definite Jacobian matrices, J. Comput. Math., 28 (2010) 235–260.Google Scholar
[13] Bai, Z.-Z., Several splittings for non-Hermitian linear systems, Sci. China (Ser. A: Math), 51(2008) 1339–1348.CrossRefGoogle Scholar
[14] Bai, Z.-Z., Golub, G. H., Li, C.-K., Convergence properties of preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite matrices, Math. Comput., 76 (2007) 287–298.CrossRefGoogle Scholar
[15] Bai, Z.-Z., Golub, G. H., Ng, M. K., On inexact Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, Linear Algebra Appl., 428 (2008) 413–440.CrossRefGoogle Scholar
[16] Bai, Z.-Z., Huang, Y.-M., Ng, M. K.. On preconditioned iterative methods for Burgers equations, SIAM J. Sci. Comput., 29 (2007) 415–439.CrossRefGoogle Scholar
[17] Bai, Z.-Z., Ng, M. K.. Preconditioners for nonsymmetric block Toeplitz-like-plus-diagonal linear systems, Numer. Math., 96 (2003) 197–220.CrossRefGoogle Scholar
[18] Bai, Z.-Z., Yin, J.-F., Su, Y.-F.. A shift-splitting preconditioner for non-Hermitian positive definite matrices, J. Comput. Math., 24 (2006) 539–552.Google Scholar
[19] Calvetti, D., Reichel, L.. Application of ADI iterative methods to the restoration of noisy images, SIAM J. Matrix Anal. Appl. 17(1996) 165–186.CrossRefGoogle Scholar
[20] Dong, Y.-X., Gu, C.-Q.. On PMHSS iteration methods for continuous Sylvester equations, J. Comput. Math., 2016, to appear.Google Scholar
[21] Druskin, V., Knizhnerman, L., Simoncini, V.. Analysis of the rational Krylov subspace and ADI methods for solving the Lyapunov equation, SIAM J. Numer. Anal. 49 (2011) 1875–1898.CrossRefGoogle Scholar
[22] Friswell, M. I., Mottershead, J. E.. Finite Element Model Updating in Structural Dynamics, Kluwer Academic Publishers, Dordrecht, Boston and London, 1995.CrossRefGoogle Scholar
[23] Golub, G. H., Loan, C.F.V.. Matrix Computations, Third edition, The Johns Hopkins University Press, Baltimore and London, MD, 1996.Google Scholar
[24] Golub, G. H., Nash, S. G., Loan, C. F. V.. A Hessenberg-Schur method for the problem AX + X B = C, IEEE Trans. Automat. Control 24 (1979) 909–913.CrossRefGoogle Scholar
[25] Hu, D.-Y., Reichel, L.. Krylov-subspace methods for the Sylvester equation, Linear Algebra Appl. 172 (1992) 283–313.CrossRefGoogle Scholar
[26] Halanay, A., aˇsvan, V. R.. Applications of Lyapunov Methods in Stability, Kluwer Academic Publishers, Dordrecht, Boston and London, 1993.CrossRefGoogle Scholar
[27] Ilic, M.D.. New approaches to voltage monitoring and control, IEEE Control Syst. Mag. 9 (1989) 5–11.CrossRefGoogle Scholar
[28] Jbilou, K.. Low rank approximate solutions to large Sylvester matrix equations, Appl. Math. Comput. 177 (2006) 365–376.Google Scholar
[29] Lewis, F. L., Mertzios, V. G., Vachtsevanos, G., Christodoulou, M. A.. Analysis of bilinear systems using walsh functions, IEEE Trans. Automat. Control 35 (1990) 119–123.CrossRefGoogle Scholar
[30] Liao, A.-P., Bai, Z.-Z., Lei, Y.. Best approximate solution of matrix equation AX B + CY D = E, SIAM J. Matrix Anal. Appl. 27 (2005) 675–688.CrossRefGoogle Scholar
[31] Obinata, G., Anderson, B.D.O.. Model Reduction For Control System Design, Springer-Verlag, London, 2001.CrossRefGoogle Scholar
[32] van der Schaft, A.. L2-Gain and Passivity Techniques in Nonlinear Control, 2nd edition, Springer-Verlag, London, 2000.CrossRefGoogle Scholar
[33] Simoncini, V.. A new iterative method for solving large-scale Lyapunov matrix equations, SIAM J. Sci. Comput. 29 (2007) 1268–1288.CrossRefGoogle Scholar
[34] Zhou, R., Wang, X., Tang, X.-B.. A generalization of the Hermitian and skew-Hermitian splitting iteration method for solving Sylvester equations, Appl. Math. Comput. 271 (2015) 609–617.Google Scholar
[35] Wang, X., Dai, L., Liao, D.. A modified gradient based algorithm for solving Sylvester equations, Appl. Math. Comput. 218 (2012) 5620–5628.Google Scholar
[36] Niu, Q., Wang, X., Lu, L.-Z.. A relaxed gradient based algorithm for solving Sylvester equations, Asian J. Control 13 (2011) 461–464.CrossRefGoogle Scholar
[37] Wang, X., Li, W.-W., Mao, L.-Z.. On positive-definite and skew-Hermitian splitting iteration methods for continuous Sylvester equation AX + X B = C, Comput. Math. Appl. 66 (2013) 2352–2361.CrossRefGoogle Scholar
[38] Zhou, R., Wang, X., Zhou, P.. A modified HSS iteration method for solving the complex linear matrix equation AX B = C, J. Comput. Math. 34 (2016) 437–450.CrossRefGoogle Scholar
[39] Wang, X., Li, W.-W.. A modified GPSS method for non-Hermitian positive definite linear systems, Appl. Math. Comput. 234 (2014) 253–259.Google Scholar
[40] Wang, X., Li, Y., Dai, L.. On Hermitian and skew-Hermitian splitting iteration methods for the linear matrix equation AX B = C, Comput. Math. Appl. 234 (2014) 253–259.Google Scholar
[41] Zheng, Q.-Q., Ma, C.-F.. On normal and skew-Hermitian splitting iteration methods for large sparse continuous Sylvester equations, J. Comput. Appl. Math. 268 (2014) 145–154.CrossRefGoogle Scholar