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

  • Rong Zhou (a1), Xiang Wang (a1) (a2) and Xiao-Bin Tang (a3)
Abstract
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.

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*Corresponding author. Email address: wangxiang49@ncu.edu.cn (X. Wang)
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[1] B.D.O. Anderson , P. Agathoklis , E.I. Jury , M. Mansour . Stability and the matrix Lyapunov equation for discrete 2-dimensional systems, IEEE Trans. Circuits Syst. 33 (1986) 261267.

[2] R. Bhatia , P. Rosenthal . How and why to solve the operator equation AX – X B = Y, Bull. Lond. Math. Soc. 29 (1997) 121.

[3] P. Benner , R.-C. Li , N. Truhar . On the ADI method for Sylvester equations, J. Comput. Appl. Math. 233 (2009) 10351045.

[4] Z.-Z. Bai , G. H. Golub , M.K. Ng . On successive-overrelaxation acceleration of the Hermitian and skew-Hermitian splitting iterations, Numer. Linear Algebra Appl. 14 (2007) 319335.

[6] Z.-Z. Bai , G. H. Golub , L.-Z. Lu , J.-F. Yin . Block triangular and skew-Hermitian splitting methods for positive-definite linear systems, SIAM J. Sci. Comput. 26 (2005) 844863.

[7] Z.-Z. Bai , G. H. Golub , M. K. Ng . Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, SIAM J. Matrix Anal. Appl., 24 (2003) 603626.

[9] Z.-Z. Bai , A class of two-stage iterative methods for systmes of weakly nonlinear equations, Numer. Algorithms, 14 (1997) 295319.

[10] Z.-Z. Bai , M. Benzi , F. Chen . Modified HSS iteration methods for a class of complex symmetric linear systems, Computing, 87 (2010) 93111.

[11] Z.-Z. Bai , M. Benzi , F. Chen . On preconditioned MHSS iteration methods for complex symmetric linear systems, Numer. Algorithms, 56 (2011) 297317.

[13] Z.-Z. Bai , Several splittings for non-Hermitian linear systems, Sci. China (Ser. A: Math), 51(2008) 13391348.

[14] Z.-Z. Bai , G. H. Golub , C.-K. Li , Convergence properties of preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite matrices, Math. Comput., 76 (2007) 287298.

[15] Z.-Z. Bai , G. H. Golub , M. K. Ng , On inexact Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, Linear Algebra Appl., 428 (2008) 413440.

[16] Z.-Z. Bai , Y.-M. Huang , M. K. Ng . On preconditioned iterative methods for Burgers equations, SIAM J. Sci. Comput., 29 (2007) 415439.

[17] Z.-Z. Bai , M. K. Ng . Preconditioners for nonsymmetric block Toeplitz-like-plus-diagonal linear systems, Numer. Math., 96 (2003) 197220.

[19] D. Calvetti , L. Reichel . Application of ADI iterative methods to the restoration of noisy images, SIAM J. Matrix Anal. Appl. 17(1996) 165186.

[21] V. Druskin , L. Knizhnerman , V. Simoncini . Analysis of the rational Krylov subspace and ADI methods for solving the Lyapunov equation, SIAM J. Numer. Anal. 49 (2011) 18751898.

[22] M. I. Friswell , J. E. Mottershead . Finite Element Model Updating in Structural Dynamics, Kluwer Academic Publishers, Dordrecht, Boston and London, 1995.

[24] G. H. Golub , S. G. Nash , C. F. V. Loan . A Hessenberg-Schur method for the problem AX + X B = C, IEEE Trans. Automat. Control 24 (1979) 909913.

[25] D.-Y. Hu , L. Reichel . Krylov-subspace methods for the Sylvester equation, Linear Algebra Appl. 172 (1992) 283313.

[26] A. Halanay , V. R. aˇsvan . Applications of Lyapunov Methods in Stability, Kluwer Academic Publishers, Dordrecht, Boston and London, 1993.

[27] M.D. Ilic . New approaches to voltage monitoring and control, IEEE Control Syst. Mag. 9 (1989) 511.

[29] F. L. Lewis , V. G. Mertzios , G. Vachtsevanos , M. A. Christodoulou . Analysis of bilinear systems using walsh functions, IEEE Trans. Automat. Control 35 (1990) 119123.

[30] A.-P. Liao , Z.-Z. Bai , Y. Lei . Best approximate solution of matrix equation AX B + CY D = E, SIAM J. Matrix Anal. Appl. 27 (2005) 675688.

[31] G. Obinata , B.D.O. Anderson . Model Reduction For Control System Design, Springer-Verlag, London, 2001.

[32] A. van der Schaft . L2-Gain and Passivity Techniques in Nonlinear Control, 2nd edition, Springer-Verlag, London, 2000.

[33] V. Simoncini . A new iterative method for solving large-scale Lyapunov matrix equations, SIAM J. Sci. Comput. 29 (2007) 12681288.

[36] Q. Niu , X. Wang , L.-Z. Lu . A relaxed gradient based algorithm for solving Sylvester equations, Asian J. Control 13 (2011) 461464.

[37] X. Wang , W.-W. Li , L.-Z. Mao . On positive-definite and skew-Hermitian splitting iteration methods for continuous Sylvester equation AX + X B = C, Comput. Math. Appl. 66 (2013) 23522361.

[38] R. Zhou , X. Wang , P. Zhou . A modified HSS iteration method for solving the complex linear matrix equation AX B = C, J. Comput. Math. 34 (2016) 437450.

[41] Q.-Q. Zheng , C.-F. Ma . On normal and skew-Hermitian splitting iteration methods for large sparse continuous Sylvester equations, J. Comput. Appl. Math. 268 (2014) 145154.

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East Asian Journal on Applied Mathematics
  • ISSN: 2079-7362
  • EISSN: 2079-7370
  • URL: /core/journals/east-asian-journal-on-applied-mathematics
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