On Hermitian and skew-Hermitian splitting iteration methods for continuous Sylvester equations

Journal of Computational Mathematics

Volumn 29, Issue 2, 2011, Pages 185-198

  Bai, Zhong Zhi ^a  

^a CHINESE ACADEMY OF SCIENCES   (China)

Author keywords

Continuous Sylvester equation; Convergence; HSS iteration method; Inexact iteration

Indexed keywords

EID: 79955132775     PISSN: 02549409     EISSN: None     Source Type: Journal    
DOI: 10.4208/jcm.1009-m3152     Document Type: Article

References (41)

1. B.D.O. Anderson, P. Agathoklis, E.I. Jury and M. Mansour, Stability and the matrix Lyapunov equation for discrete 2-dimensional systems, IEEE Trans. Circuits Systems, 33 (1986), 261-267.
2. O. Axelsson, Z.-Z. Bai and S.-X. Qiu, A class of nested iteration schemes for linear systems with a coefficient matrix with a dominant positive definite symmetric part, Numer. Algorithms, 35 (2004), 351-372.
3. Z.-Z. Bai, A class of two-stage iterative methods for systems of weakly nonlinear equations, Numer. Algorithms, 14 (1997), 295-319.
4. Z.-Z. Bai, Splitting iteration methods for non-Hermitian positive definite systems of linear equations, Hokkaido Math. J., 36 (2007), 801-814.
5. Z.-Z. Bai, G.H. Golub and C.-K. Li, Convergence properties of preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite matrices, Math. Comput., 76 (2007), 287-298.
6. 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.
7. Z.-Z. Bai, X.-X. Guo and S.-F. Xu, Alternately linearized implicit iteration methods for the minimal nonnegative solutions of nonsymmetric algebraic Riccati equations, Numer. Linear Algebra Appl, 13 (2006), 655-674.
8. Z.-Z. Bai, Y.-M. Huang and M.K. Ng, On preconditioned iterative methods for Burgers equations, SIAM J. Sci. Comput, 29 (2007), 415-439.
9. Z.-Z. Bai and M.K. Ng, Preconditioners for nonsymmetric block Toeplitz-like-plus-diagonal linear systems, Numer. Math., 96 (2003), 197-220.
10. R.H. Bartels and G.W. Stewart, Solution of the matrix equation AX + XB = C: Algorithm 432, Commun. ACM, 15 (1972), 820-826.
11. D. Calvetti and L. Reichel, Application of ADI iterative methods to the restoration of noisy images, SIAM J. Matrix Anal Appl, 17 (1996), 165-186.
12. D.J. Evans and E. Galligani, A parallel additive preconditioner for conjugate gradient method for AX + XB = C, Parallel Comput, 20 (1994), 1055-1064.
13. M.I. Friswell and J.E. Mottershead, Finite Element Model Updating in Structural Dynamics, Kluwer Academic Publishers, Dordrecht, Boston and London, 1995.
14. F.R. Gantmacher, The Theory of Matrices, Chelsea, New York, 1959.
15. G.H. Golub, S.G. Nash and C.F. Van Loan, A Hessenberg-Schur method for the problem AX + XB = C, IEEE Trans. Automat. Control, 24 (1979), 909-913.
16. G.H. Golub and C.F. Van Loan, Matrix Computations, 3rd Edition, The Johns Hopkins University Press, Baltimore and London, MD, 1996.
17. C.-Q. Gu and H.-Y. Xue, A shift-splitting hierarchical identification method for solving Lyapunov matrix equations, Linear Algebra Appl, 430 (2009), 1517-1530.
18. C.-H. Guo, Nonsymmetric algebraic Riccati equations and Wiener-Hopf factorization for M- matrices, SIAM J. Matrix Anal Appl, 23 (2001), 225-242.
19. C.-H. Guo and A.J. Laub, On the iterative solution of a class of nonsymmetric algebraic Riccati equations, SIAM J. Matrix Anal. Appl, 22 (2000), 376-391.
20. X.-X. Guo and Z.-Z. Bai, On the minimal nonnegative solution of nonsymmetric algebraic Riccati equation, J. Comput. Math., 23 (2005), 305-320.
21. W.M. Haddad and D.S. Bernstein, The optimal projection equations for reduced-order, discrete-time state estimation for linear systems with multiplicative white noise, Systems Control Lett., 8 (1987), 381-388.
22. A. Halanay and V. Rǎsvan, Applications of Lyapunov Methods in Stability, Kluwer Academic Publishers, Dordrecht, Boston and London, 1993.
23. Y. Halevi, The optimal reduced-order estimator for systems with singular measurement noise, IEEE Trans. Automat. Control, 34 (1989), 777-781.
24. D.-Y. Hu and L. Reichel, Krylov-subspace methods for the Sylvester equation, Linear Algebra Appl, 172 (1992), 283-313.
25. M.D. Ilic, New approaches to voltage monitoring and control, IEEE Control Systems Mag., 9 (1989), 5-11.
26. D.J. Inman and J. Daniel, Vibration with Control, Measurement, and Stability, Prentice-Hall, Englewood Cliffs, New Jersey, 1989.
27. L. Jódar, An algorithm for solving generalized algebraic Lyapunov equations in Hilbert space, applications to boundary value problems, Proc. Edinburgh Math. Society, 31 (1988), 99-105.
28. D. Kreßne, V. Mehrmann and T. Penzl, CTLEX-A Collection of Benchmark Examples for Continuous-Time Lyapunov Equations, SLICOT Working Note 1999-6, May 1999.
29. P. Lancaster, Explicit solutions of linear matrix equations, SIAM Rev., 12 (1970), 544-566.
30. P. Lancaster and L. Rodman, Algebraic Riccati Equations, The Clarendon Press, Oxford and New York, 1995.
31. P. Lancaster and M. Tismenetsky, The Theory of Matrices, 2nd Edition, Academic Press, Orlando, 1985.
32. F.L. Lewis, V.G. Mertzios, G. Vachtsevanos and M.A. Christodoulou, Analysis of bilinear systems using Walsh functions, IEEE Trans. Automat. Control, 35 (1990), 119-123.
33. N. Levenberg and L. Reichel, A generalized ADI iterative method, Numer. Math., 66 (1993), 215-233.
34. A.-P. Liao, Z.-Z. Bai and Y. Lei, Best approximate solution of matrix equation AXB + CYD = E, SIAM J. Matrix Anal. Appl, 27 (2005), 675-688.
35. G. Obinata and B.D.O. Anderson, Model Reduction for Control System Design, Springer-Verlag, Berlin and London, 2001.
36. ∞-optimization: A design method based on the algebraic Riccati equation, IEEE Trans. Automat Control, 32 (1987), 427-429.
37. R.A. Smith, Matrix equation XA + BX = C, SIAM J. Appl. Math., 16 (1968), 198-201.
38. G. Starke and W. Niethammer, SOR for AX - XB = C, Linear Algebra Appl., 154-156 (1991), 355-375.
39. 2-Gain and Passivity Techniques in Nonlinear Control, 2nd Edition, Springer-Verlag, London, 2000.
40. E.L. Wachspress, Iterative solution of the Lyapunov matrix equation, Appl. Math. Lett., 1 (1988), 87-90.
41. G.-P. Xu, M.-S. Wei and D.-S. Zheng, On solutions of matrix equation AXB + CYD = F, Linear Algebra Appl, 279 (1998), 93-109.