Iterative algorithm for minimal norm least squares solution to general linear matrix equations

International Journal of Computer Mathematics

Volumn 87, Issue 11, 2010, Pages 2552-2567

  Li, Zhao Yan ^a   Wang, Yong ^a  

^a HARBIN INSTITUTE OF TECHNOLOGY   (China)

Author keywords

iterative solutions; linear matrix equations; Lyapunov matrix equations; minimal norm least squares solution; Sylvester matrix equations

Indexed keywords

ITERATIVE SOLUTIONS; LEAST SQUARES SOLUTIONS; LINEAR MATRIX EQUATIONS; LYAPUNOV MATRIX EQUATIONS; SYLVESTER MATRIX EQUATIONS;

ALGORITHMS; ITERATIVE METHODS; OPTIMIZATION; PROBLEM SOLVING;

MATRIX ALGEBRA;

EID: 77956507596     PISSN: 00207160     EISSN: 10290265     Source Type: Journal    
DOI: 10.1080/00207160802684459     Document Type: Article

References (28)

1. R.H. Bartels and G.W. Stewart, Solution of the equation AX + XB = C, Comm. A. C. M. 15 (1972), pp. 820-826.
2. S.P. Bhattacharyya and E. de Souza, Pole assignment via Sylvester's equation, Syst. Control Lett. 1(4) (1981), pp. 261-263.
3. J.L. Chen and X.H. Chen, Special Matrices, Tsinghua University Press, Beijing, 2002 (in Chinese).
4. T. Chen and L. Qiu, H∞ design of general multirate sampled-date control systems, Automatica 30 (1994), pp. 1139-1152.
5. D. Chu, H.C. Chan, and D.W.C. Ho, Regularization of singular systems by derivative and proportional output feedback, SIAM J. Matrix Anal. Appl. 19(1) (1998), pp. 21-38.
6. D. Chu, V. Mehrmann, and N.K. Nichols, Minimum norm regularization of descriptor systems by mixed output feedback, Linear Algebra Appl. 296(1-3) (1999), pp. 39-77.
7. F. Ding and T. Chen, Iterative least squares solutions of coupled Sylvester matrix equations, Syst. Control Lett. 54(2) (2005), pp. 95-107.
8. F. Ding andT. Chen,Oniterative solutions of general coupled matrix equations,SIAMJ. Control Optim. 44(6) (2006), pp. 2269-2284.
9. T. Gu, X. Liu, Z. Mo, and X. Chi, Multiple search direction conjugate gradient method I: methods and their propositions, Int. J. Comput. Math. 81(9) (2004), pp. 1133-1143. (Pubitemid 41710373)
10. S.J. Hammarling, Numerical solution of the stable, non-negative definite Lyapunov equation, IMA J. Num. Anal. 2 (1982), pp. 303-323.
11. T. Huang, G. Cheng, D.J. Evans, and X. Cheng, AOR type iterations for solving preconditioned linear systems Int. J. Comput. Math. 82(8) (2005), pp. 969-976.
12. I.E. Kaporin, Explicitly preconditioned conjugate gradient method for the solution of unsymmetric linear systems, Int. J. Comput. Math. 44(1-4) (1992), pp. 169-187.
13. C.T. Kelley, Iterative Methods for Linear and Nonlinear Equations, SIAM, Philadelphia, 1995.
14. J. Lam and W.Y. Yan, Pole assignment with optimal spectral conditioning, Syst. Control Lett. 29(5) (1997), pp. 241-253.
15. J. Lam, W. Yan, and T. Hu, Pole assignment with eigenvalue and stability robustness, Internat. J. Control 72(13) (1999), pp. 1165-1174. (Pubitemid 30510912)
16. L. Qiu and T. Chen, Unitary dilation approach to contractive matrix completion, Linear Algebra Appl. 379 (2004), pp. 345-352.
17. R.S. Varga, Matrix Iterative Analysis, 2nd ed., Series in Computational Mathematics, Springer-Verlag, New York, 2000.
18. S.G.Wang and Z.H.Yang, The Generalized Inverse of Matrix and its Applications, Beijing University of Technology Press, Beijing, 1996 (in Chinese).
19. Q.Wang, J. Lam,Y.Wei, and T. Chen, Iterative solutions of coupled discrete Markovian jump Lyapunov equations, Comput. Math. Appl. 55(4) (2008), pp. 843-850.
20. G.F. Zhang, Q. Zhang, T. Chen, and Y. Lin, On Lyapunov theorems for descriptor systems, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms 10(5) (2003), pp. 709-726.
21. L. Zheng C. Li, and D. J. Evans, Chebyshev acceleration for SOR-like method, Int. J. Comput. Math. 82(5) (2005), pp. 583-593.
22. B. Zhou and G. Duan, An explicit solution to the matrix equation AX - XF = BY, Linear Algebra Appl. 402 (2005), pp. 345-366.
23. B. Zhou and G.R. Duan, A new solution to the generalized Sylvester matrix equation AV - EV F = BW, Syst. Control Lett. 55(3) (2006), pp. 193-198. (Pubitemid 43107554)
24. B. Zhou and G. Duan, Solutions to generalized Sylvester matrix equation by Schur decomposition, Int. J. Syst. Sci. 38(5) (2007), pp. 369-375.
25. B. Zhou and G. Duan, Parametric solutions to the generalized Sylvester matrix equation AX - XF = BY and the regulator equation AX - XF = BY + R, Asian J. Control 9(4) (2007), pp. 475-483.
26. B. Zhou and G.R. Duan, On the generalized Sylvester mapping and matrix equations, Syst. Control Lett. 57(3) (2008), pp. 200-208. (Pubitemid 351625360)
27. K. Zhou, J. Doyle, and K. Glover, Robust and Optimal Control, Prentice-Hall, New Jersey, 1996.
28. B. Zhou, J. Lam, and G. Duan, Gradient-based maximal convergence rate iterative method for solving linear matrix equations, Int. J. Comput. Math. 87(3) (2010), pp. 515-527.