Iterative algorithms for the generalized centro-symmetric and central anti-symmetric solutions of general coupled matrix equations

Engineering Computations (Swansea, Wales)

Volumn 29, Issue 5, 2012, Pages 528-560

  Dehghan, Mehdi ^a   Hajarian, Masoud ^b  

^a AMIRKABIR UNIVERSITY OF TECHNOLOGY   (Iran)

^b SHAHID BEHESHTI UNIVERSITY   (Iran)

Author keywords

Generalized centro symmetric solution group; Iterative methods; Mathematics; Optimal approximation generalized centro symmetric solution group; The general coupled matrix equations

Indexed keywords

ANTI-SYMMETRIC; ANTISYMMETRIC MATRIX; CENTROSYMMETRIC; CONJUGATE GRADIENT; CONTINUOUS TIME; CONVERGENCE RESULTS; COUPLED MATRIX EQUATIONS; DESIGN/METHODOLOGY/APPROACH; GENERALIZED CENTRO-SYMMETRIC SOLUTION GROUP; ITERATIVE ALGORITHM; LEAST FROBENIUS NORMS; LYAPUNOV EQUATION; MATRIX EQUATIONS; NUMERICAL EXAMPLE; OPTIMAL APPROXIMATION; THEORETICAL RESULT;

ALGEBRA; ALGORITHMS; CONJUGATE GRADIENT METHOD; ITERATIVE METHODS; LYAPUNOV FUNCTIONS; MATHEMATICAL TECHNIQUES; OPTIMIZATION;

MATRIX ALGEBRA;

EID: 84863817689     PISSN: 02644401     EISSN: None     Source Type: Journal    
DOI: 10.1108/02644401211235870     Document Type: Article

References (47)

1. Baksalary, J.K. and Kala, R. (1979), "The matrix equation AX 2 YB=C", Linear Algebra Appl., Vol. 25, pp. 41-3.
2. Baksalary, J.K. and Kala, R. (1980), "The matrix equation AXB+CYD=E", Linear Algebra Appl., Vol. 30, pp. 141-7.
3. Costa, O.L.V. and Fragoso, M.D. (1993), "Stability results for discrete-time linear systems with Markovian jumping parameters", J. Math. Anal. Appl., Vol. 179, pp. 154-78.
4. Dai, H. (1990), "On the symmetric solutions of a linear matrix equations", Linear Algebra Appl., Vol. 131, pp. 1-7.
5. Dehghan, M. and Hajarian, M. (2008a), "An iterative algorithm for solving a pair of matrix equations AYB=E, CYD=F over generalized centro-symmetric matrices", Comput. Math. Appl., Vol. 56, pp. 3246-60.
6. Dehghan, M. and Hajarian, M. (2008b), "An iterative algorithm for the reflexive solutions of the generalized coupled Sylvester matrix equations and its optimal approximation", Appl. Math. Comput., Vol. 202, pp. 571-88.
7. Dehghan, M. and Hajarian, M. (2009a), "A lower bound for the product of eigenvalues of solutions to matrix equations", Appl. Math. Lett., Vol. 22, pp. 1786-8.
8. Dehghan, M. and Hajarian, M. (2009b), "Efficient iterative method for solving the second-order Sylvester matrix equation EVF 2 2 AVF 2 CV=BW", IET Control Theory Appl., Vol. 3, pp. 1401-8.
9. Dehghan, M. and Hajarian, M. (2009c), "Finite iterative algorithms for the reflexive and anti-reflexive solutions of the matrix equation A1X1B1+A2X2B2=C", Math. Comput. Model., Vol. 49, pp. 1937-59.
10. Dehghan, M. and Hajarian, M. (2010a), "An iterative method for solving the generalized coupled Sylvester matrix equations over generalized bisymmetric matrices", Appl. Math. Model., Vol. 34, pp. 639-54.
11. Dehghan, M. and Hajarian, M. (2010b), "The general coupled matrix equations over generalized bisymmetric matrices", Linear Algebra Appl., Vol. 432, pp. 1531-52.
12. Dehghan, M. and Hajarian, M. (2010c), "The reflexive and anti-reflexive solutions of a linear matrix equation and systems of matrix equations", Rocky Mountain J. Math., Vol. 40, pp. 825-48.
13. Ding, F. and Chen, T. (2005a), "Gradient based iterative algorithms for solving a class of matrix equations", IEEE Trans. Autom. Contr., Vol. 50, pp. 1216-21. (Pubitemid 41410122)
14. Ding, F. and Chen, T. (2005b), "Hierarchical gradient-based identification of multivariable discrete-time systems", Automatica, Vol. 41, pp. 315-25. (Pubitemid 41536582)
15. Ding, F. and Chen, T. (2005c), "Hierarchical least squares identification methods for multivariable systems", IEEE Trans. Autom. Contr., Vol. 50, pp. 397-402. (Pubitemid 40448592)
16. Ding, F. and Chen, T. (2005d), "Iterative least squares solutions of coupled Sylvester matrix equations", Systems & Control Letters, Vol. 54, pp. 95-107.
17. Ding, F. and Chen, T. (2006), "On iterative solutions of general coupled matrix equations", SIAM J. Control Optim., Vol. 44, pp. 2269-84. (Pubitemid 44775259)
18. Ding, F., Liu, P.X. andDing, J. (2008), "Iterative solutions of the generalized Sylvester matrix equations by using the hierarchical identification principle", Appl. Math. Comput., Vol. 197, pp. 41-50. (Pubitemid 351215477)
19. Feng, Y.T. (2006), "On the discrete dynamic nature of the conjugate gradient method", J. Comput. Physics, Vol. 211, pp. 91-8. (Pubitemid 41363976)
20. Feng, Y.T. and Owen, D.R.J. (1996), "Conjugate gradient methods for solving the smallest eigenpair of large symmetric eigenvalue problems", Int. J. Numer. Eng., Vol. 39, pp. 2209-29. (Pubitemid 126648023)
21. Golub, G.H. and Van Loan, C.F. (1996), Matrix Computations, 3rd ed., Johns Hopkins University Press, Baltimore, MD.
22. Hestenes, M.R. and Stiefel, E.L. (1952), "Methods of conjugate gradients for solving linear systems", J. Natl. Bur. Stand, Vol. 49, pp. 409-36.
23. Kågström, B. and Westin, L. (1989), "Generalized Schur methods with condition estimators for solving the generalized Sylvester equation", IEEE Trans. Autom. Contr., Vol. 34, pp. 745-51.
24. Mori, H., Tanaka, H. and Kanno, J. (1996), "A preconditioned fast decoupled power flow method for contingency screening", IEEE Trans. Power Syst., Vol. 11, pp. 357-63. (Pubitemid 126781165)
25. Navarra, A., Odell, P.L. and Young, D.M. (2001), "A representation of the general common solution to the matrix equations A1XB1=C1 and A2XB2=C2 with applications", Comput. Math. Appl., Vol. 41, pp. 929-35.
26. Polak, E. (1971), Computational Methods in Optimization: A Unified Approach, Academic Press, New York, NY.
27. Sheng, X. and Chen, G. (2007), "A finite iterative method for solving a pair of linear matrix equations (AXB, CXD)=(E, F)", Appl. Math. Comput., Vol. 189, pp. 1350-8. (Pubitemid 46802751)
28. Sonneveld, P. (1989), "CGS: A fast Lanczos-type solver for nonsymmetric linear systems", SIAM J. Sci. Stat. Comp., Vol. 10, pp. 36-52.
29. Stykel, T. (2006), "Solving projected generalized Lyapunov equations with SLICOT", Proceedings of the IEEE CCA/CACSD/ISIC Conference 2006, Munich, Germany, October 4-6, pp. 14-18.
30. van der Vorst, H. (1992), "Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear equations",SIAMJ. Sci.Stat.Comp.,Vol. 13, pp. 631-44.
31. Wang, Q.W. (2004), "A system of matrix equations and a linear matrix equation over arbitrary regular rings with identity", Linear Algebra Appl., Vol. 384, pp. 43-54.
32. Wang, Q.W. (2005a), "A system of four matrix equations over von Neumann regular rings and its applications", Acta Math. Sinica, Ser. A, Vol. 21, pp. 323-34. (Pubitemid 40550352)
33. Wang, Q.W. (2005b), "Bisymmetric and centrosymmetric solutions to systems of real quaternion matrix equations", Comput. Math. Appl., Vol. 49, pp. 641-50.
34. Wang, Q.W. and Li, C.K. (2009), "Ranks and the least-norm of the general solution to a system of quaternion matrix equations", Linear Algebra Appl., Vol. 430, pp. 1626-40.
35. Wang, Q.W., Chang, H.X. and Ning, Q. (2008), "The common solution to six quaternion matrix equations with applications", Appl. Math. Comput., Vol. 198, pp. 209-26. (Pubitemid 351329381)
36. Wang, Q.W., Woude, J.W. and Chang, H.X. (2009a), "A system of real quaternion matrix equations with applications", Linear Algebra Appl., Vol. 431, pp. 2291-303.
37. Wang, Q.W., Zhang, H.S. and Song, G.J. (2009b), "A new solvable condition for a pair of generalized Sylvester equations", Electron. J. Linear Algebra, Vol. 18, pp. 289-301.
38. Wolfe, M.A. (1978), Numerical Methods for Unconstrained Optimization: An Introduction, Van Nostrand Reinhold, Berkshire.
39. Zhou, B. and Duan, G.R. (2006), "A new solution to the generalized Sylvester matrix equation AV 2 EVF=BW", Systems & Control Letters, Vol. 55, pp. 193-8. (Pubitemid 43107554)
40. Zhou, B. and Duan, G.R. (2007), "Solutions to generalized Sylvester matrix equation by Schur decomposition", Internat. J. Systems Sci., Vol. 38, pp. 369-75. (Pubitemid 46782041)
41. Zhou, B. and Duan, G.R. (2008), "On the generalized Sylvester mapping and matrix equations", Systems & Control Letters, Vol. 57, pp. 200-8. (Pubitemid 351625360)
42. Zhou, B. and Yan, Z.B. (2008), "Solutions to right coprime factorizations and generalized Sylvester matrix equations", Transactions of the Institute of Measurement and Control, Vol. 30, pp. 397-426.
43. Zhou, B., Duan, G.R. and Li, Z.Y. (2009a), "Gradient based iterative algorithm for solving coupled matrix equations", Systems & Control Letters, Vol. 58, pp. 327-33.
44. Zhou, F.Z., Hu, X.Y. and Zhang, L. (2003), "The solvability conditions for the inverse eigenvalue problemof generalized centro- symmetricmatrices", Linear AlgebraAppl.,Vol. 364, pp. 147-60.
45. Zhou, B., Lam, J. and Duan, G.R. (2009b), "On Smith-type iterative algorithms for the Stein matrix equation", Appl. Math. Letters, Vol. 22, pp. 1038-44.
46. Zhou, B., Li, Z.Y., Duan, G.R. and Wang, Y. (2009c), "Solutions to a family of matrix equations by using the Kronecker matrix polynomials", Appl. Math. Comput., Vol. 212, pp. 327-36.
47. Zhou, B., Li, Z.Y., Duan, G.R. and Wang, Y. (2009d), "Weighted least squares solutions to general coupled Sylvester matrix equations", J. Comput. Appl. Math., Vol. 224, pp. 759-76.