An efficient algorithm for solving general coupled matrix equations and its application

Mathematical and Computer Modelling

Volumn 51, Issue 9-10, 2010, Pages 1118-1134

  Dehghan, Mehdi ^a   Hajarian, Masoud ^a  

^a AMIRKABIR UNIVERSITY OF TECHNOLOGY   (Iran)

Author keywords

(R, S) skew symmetric; (R, S) symmetric; Iterative algorithm; Least Frobenius norm solution group; Optimal approximation solution group; The general coupled matrix equations

Indexed keywords

COUPLED MATRIX EQUATIONS; ITERATIVE ALGORITHM; LEAST FROBENIUS NORMS; OPTIMAL APPROXIMATION; SKEW-SYMMETRIC;

APPROXIMATION ALGORITHMS; APPROXIMATION THEORY; CONJUGATE GRADIENT METHOD; OPTIMIZATION;

ITERATIVE METHODS;

EID: 77649337553     PISSN: 08957177     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.mcm.2009.12.022     Document Type: Article

References (70)

1. Aliev F.A., and Larin V.B. Optimization of Linear Control Systems: Analytical Methods and Computational Algorithms. Stability and Control: Theory, Methods and Applications vol. 8 (1998), Gordon and Breach
2. Calvetti D., and Reichel L. Application of ADI iterative methods to the restoration of noisy images. SIAM J. Matrix Anal. Appl. 17 (1996) 165-186
3. Dieci L., Osborne M.R., and Russel R.D. A Riccati transformation method for solving linear bvps. I: Theoretical aspects. SIAM J. Numer. Anal. 25 5 (1988) 1055-1073
4. Enright W.H. Improving the efficiency of matrix operations in the numerical solution of stiff ordinary differential equations. ACM Trans. Math. Softw. 4 (1978) 127-136
5. Epton M.A. Methods for the solution of A X D - B X C = E and its applications in the numerical solution of implicit ordinary differential equations. BIT 20 (1980) 341-345
6. Higham N.J. Perturbation theory and backward error analysis for A X - X B = C. BIT 33 (1993) 124-136
7. Hu Q., and Cheng D. The polynomial solution to the Sylvester matrix equation. Appl. Math. Lett. 19 (2006) 859-864
8. Gantmacher F.R. Theory of Matrices (1959), Chelsea, New York
9. Müller P.C. Stability of linear mechanical systems with holonomic constraints. Appl. Mech. Rev. 46 (1993) 160-164
10. Lancaster P., and Rodman L. The Algebraic Riccati Equation (1995), Oxford University Press, Oxford
11. Laub A.J., Heath M.T., Paige C.C., and Ward R.C. Computation of system balancing transformations and other applications of simultaneous diagonalization algorithms. IEEE Trans. Automat. Control 32 (1987) 115-122
12. Dehghan M., and Hajarian M. On the reflexive solutions of the matrix equation A X B + C Y D = E. Bull. Korean Math. Soc. 46 (2009) 511-519
13. Dehghan M., and Hajarian M. A lower bound for the product of eigenvalues of solutions to matrix equations. Appl. Math. Lett. 22 (2009) 1786-1788
14. M. Dehghan, M. Hajarian, The reflexive and anti-reflexive solutions of a linear matrix equation and systems of matrix equations, Rocky Mountain J. Math. (in press)
15. Peng Y.X., Hu X.Y., and Zhang L. An iteration method for the symmetric solutions and the optimal appromation solution of the matrix equation A X B = C. Appl. Math. Comput. 160 (2005) 763-777
16. 2. Appl. Math. Comput. 181 (2006) 988-999
17. Peng Z.Y., and Peng Y.X. An efficient iterative method for solving the matrix equation A X B + C Y D = E. Numer. Linear Algebra Appl. 13 (2006) 473-485
18. Shi Y., Ding F., and Chen T. Multirate crosstalk identification in xDSL systems. IEEE Trans. Commun. 54 (2006) 1878-1886
19. Shi Y., Ding F., and Chen T. 2-norm based recursive design of transmultiplexers with designable filter length. Circuits Systems Signal Process. 25 (2006) 447-462
20. Shi Y., Ding F., and Chen T. Multirate crosstalk identification in xDSL systems. IEEE Transactions on Communication 54 (2006) 1878-1886
21. Shi Y., and Yu B. Output feedback stabilization of networked control systems with random delays modeled by Markov chains. IEEE Trans. Automat. Control 54 (2009) 1668-1674
22. Shi Y., Fang H., and Yan M. Kalman filter-based adaptive control for networked systems with unknown parameters and randomly missing outputs. Int. J. Robust Nonlinear Control 19 (2009) 1976-1992
23. Wang Q.W., Zhang H.S., and Song G.J. A new solvable condition for a pair of generalized Sylvester equations, Electron. J. Linear Algebra 18 (2009) 289-301
24. Zhou B., and Duan G.R. An explicit solution to the matrix equation A X - X F = B Y. Linear Algebra Appl. 402 (2005) 345-366
25. Zhou B., and Yan Z.B. Solutions to right coprime factorizations and generalized Sylvester matrix equations. Trans. Inst. Meas. Control 30 (2008) 397-426
26. Zhou B., Duan G.R., and Li Z.Y. Gradient based iterative algorithm for solving coupled matrix equations. Systems Control Lett. 58 (2009) 327-333
27. Zhou B., Li Z.Y., Duan G.R., and Wang Y. Solutions to a family of matrix equations by using the Kronecker matrix polynomials. Appl. Math. Comput. 212 (2009) 327-336
28. Guennouni A.E., Jbilou K., and Riquet A.J. Block Krylov subspace methods for solving large Sylvester equations. Numer. Algorithms 29 (2002) 75-96
29. Robbé M., and Sadkane M. Use of near-breakdowns in the block Arnoldi method for solving large Sylvester equations. Appl. Numer. Math. 58 (2008) 486-498
30. Starke G., and Niethammer W. SOR for A X - X B = C. Linear Algebra Appl. 154 (1991) 355-375
31. Lin Y. Minimal residual methods augmented with eigenvectors for solving Sylvester equations and generalized Sylvester equations. Appl. Math. Comput. 181 (2006) 487-499
32. Bao L., Lin Y., and Wei Y. A new projection method for solving large Sylvester equations. Appl. Numer. Math. 57 (2007) 521-532
33. Kagström B., and Westin L. Generalized Schur methods with condition estimators for solving the generalized Sylvester equation. IEEE Trans. Automat. Control 34 (1989) 745-751
34. Kagström B., and Poromaa P. LAPACK-style algorithms and software for solving the generalized Sylvester equation and estimating the separation between regular matrix pairs. ACM Trans. Math. Software 22 (1996) 78-103
35. Wang Q.W., Chang H.X., and Ning Q. The common solution to six quaternion matrix equations with applications. Appl. Math. Comput. 198 (2008) 209-226
36. Wang Q.W., and Zhang F. The reflexive re-nonnegative definite solution to a quaternion matrix equation. Electron. J. Linear Algebra 17 (2008) 88-101
37. Wang Q.W., Zhang H.S., and Yu S.W. On solutions to the quaternion matrix equation A X B + C Y D = E. Electron. J. Linear Algebra 17 (2008) 343-358
38. Wang Q.W., Woude J.W., and Chang H.X. A system of real quaternion matrix equations with applications. Linear Algebra Appl. 431 (2009) 2291-2303
39. Wang Q.W., and Li C.K. Ranks and the least-norm of the general solution to a system of quaternion matrix equations. Linear Algebra Appl. 430 (2009) 1626-1640
40. Wang Q.W. Bisymmetric and centrosymmetric solutions to systems of real quaternion matrix equations. Comput. Math. Appl. 49 (2005) 641-650
41. Wang Q.W. The general solution to a system of real quaternion matrix equations. Comput. Math. Appl. 49 (2005) 665-675
42. Smith R.A. Matrix equation, X A + B X = C. SIAM J. Appl. Math. 16 (1968) 198-201
43. Roberts J.D. Linear model reduction and solution of the algebraic Riccati equation by use of the sign function. Internat. J. Control 32 (1980) 677-687
44. Wachspress E. Iterative solution of the Lyapunov matrix equation. Appl. Math. Lett. 1 (1988) 87-90
45. Penzl T. A cyclic low-rank Smith method for large sparse Lyapunov equations. SIAM J. Sci. Comput. 21 (2000) 1401-1418
46. Gugercin S., Sorensen D.C., and Antoulas A.C. A modified low-rank Smith method for large-scale Lyapunov equations. Numer. Algor. 32 (2003) 27-55
47. 2 - A V F - C V = B W. IET Control Theory Appl. 3 (2009) 1401-1408
48. Zhou B., and Duan G.R. A new solution to the generalized Sylvester matrix equation A V - E V F = B W. Systems Control Lett. 55 (2006) 193-198
49. Zhou B., and Duan G.R. Solutions to generalized Sylvester matrix equation by Schur decomposition. Internat. J. Systems Sci. 38 (2007) 369-375
50. Zhou B., Li Z.Y., Duan G.R., and Wang Y. Weighted least squares solutions to general coupled Sylvester matrix equations. J. Comput. Appl. Math. 224 (2009) 759-776
51. Zhou B., and Duan G.R. On the generalized Sylvester mapping and matrix equations. Systems Control Lett. 57 (2008) 200-208
52. Ding F., and Chen T. Gradient based iterative algorithms for solving a class of matrix equations. IEEE Trans. Automat. Control 50 (2005) 1216-1221
53. Ding F., and Chen T. Iterative least squares solutions of coupled Sylvester matrix equations. Systems Control Lett. 54 (2005) 95-107
54. Ding F., and Chen T. On iterative solutions of general coupled matrix equations. SIAM J. Control Optim. 44 (2006) 2269-2284
55. Ding F., Liu P.X., and Ding J. Iterative solutions of the generalized Sylvester matrix equations by using the hierarchical identification principle. Appl. Math. Comput. 197 (2008) 41-50
56. Ding F., and Chen T. Hierarchical gradient-based identification of multivariable discrete-time systems. Automatica 41 (2005) 315-325
57. Ding F., and Chen T. Hierarchical least squares identification methods for multivariable systems. IEEE Trans. Autom. Contr. 50 (2005) 397-402
58. Dehghan M., and Hajarian M. An iterative algorithm for the reflexive solutions of the generalized coupled Sylvester matrix equations and its optimal approximation. Appl. Math. Comput. 202 (2008) 571-588
59. Dehghan M., and Hajarian M. An iterative algorithm for solving a pair of matrix equations A Y B = E, C Y D = F over generalized centro-symmetric matrices. Comput. Math. Appl. 56 (2008) 3246-3260
60. 2 = C. Math. Comput. Model. 49 (2009) 1937-1959
61. Dehghan M., and Hajarian M. An iterative method for solving the generalized coupled Sylvester matrix equations over generalized bisymmetric matrices. Appl. Math. Modelling 34 (2010) 639-654
62. M. Dehghan, M. Hajarian, On the reflexive and anti-reflexive solutions of the generalized coupled Sylvester matrix equations, Int. J. Systems Sci. (in press)
63. Dehghan M., and Hajarian M. The general coupled matrix equations over generalized bisymmetric matrices. Linear Algebra Appl. 432 (2010) 1531-1552
64. Reid J.K. On the method of conjugate gradients for the solution of large sparse systems of linear equations. In: Reid J.K. (Ed). Large Sparse Sets of Linear Equations (1971), Academic Press, New York
65. Trench W.F. Minimization problems for (R, S)-symmetric and (R, S)-skew symmetric matrices. Linear Algebra Appl. 389 (2004) 23-31
66. Trench W.F. Characterization and properties of (R, S)-symmetric, (R, S)-skew symmetric, and (R, S)-conjugate matrices. SIAM J. Matrix Anal Appl. 26 (2005) 748-757
67. Chen H.C. Generalized reflexive matrices: Special properties and applications. SIAM J. Matrix Anal. Appl. 19 (1998) 140-153
68. Liao A., and Xie D.X. Least-squares solution of a class of inverse eigenvalue problems for bisymmetric nonnegative definite matrices. Math. Numer. Sinica 2 (2001) 209-218
69. Woodgate K.G. Least-squares solution of F = P G over positive semidefinite symmetric P. Linear Algebra Appl. 245 (1996) 171-190
70. Xie D.X., and Zhang L. Least-squares solutions of inverse problems for anti-symmetric matrices. J. Engrg. Math. 4 (1993) 25-34