Gradient based iterative solutions for general linear matrix equations

In this paper, we present a gradient based iterative algorithm for solving general linear matrix equations by extending the Jacobi iteration and by applying the hierarchical identification principle. Convergence analysis indicates that the iterative solutions always converge fast to the exact solutions for any initial values and small condition numbers of the associated matrices. Two numerical examples are provided to show that the proposed algorithm is effective.

[1]  G. Duan,et al.  An explicit solution to the matrix equation AX − XF = BY , 2005 .

[2]  Feng Ding,et al.  Reconstruction of continuous-time systems from their non-uniformly sampled discrete-time systems , 2009, Autom..

[3]  Feng Ding,et al.  Hierarchical gradient-based identification of multivariable discrete-time systems , 2005, Autom..

[4]  Adem Kiliçman,et al.  Some new connections between matrix products for partitioned and non-partitioned matrices , 2007, Comput. Math. Appl..

[5]  Chuanqing Gu,et al.  A numerical algorithm for Lyapunov equations , 2008, Appl. Math. Comput..

[6]  Feng Ding,et al.  Gradient Based Iterative Algorithms for Solving a Class of Matrix Equations , 2005, IEEE Trans. Autom. Control..

[7]  Guang-Ren Duan,et al.  Solutions to generalized Sylvester matrix equation by Schur decomposition , 2007, Int. J. Syst. Sci..

[8]  Feng Ding,et al.  Performance analysis of multi-innovation gradient type identification methods , 2007, Autom..

[9]  Mehdi Dehghan,et al.  An iterative algorithm for the reflexive solutions of the generalized coupled Sylvester matrix equations and its optimal approximation , 2008, Appl. Math. Comput..

[10]  Feng Ding,et al.  Auxiliary model based multi-innovation extended stochastic gradient parameter estimation with colored measurement noises , 2009, Signal Process..

[11]  Feng Ding,et al.  Performance analysis of stochastic gradient algorithms under weak conditions , 2008, Science in China Series F: Information Sciences.

[12]  Feng Ding,et al.  Iterative solutions of the generalized Sylvester matrix equations by using the hierarchical identification principle , 2008, Appl. Math. Comput..

[13]  Tongwen Chen,et al.  Hierarchical least squares identification methods for multivariable systems , 2005, IEEE Transactions on Automatic Control.

[14]  Feng Ding,et al.  On Iterative Solutions of General Coupled Matrix Equations , 2006, SIAM J. Control. Optim..

[15]  Guang-Ren Duan,et al.  On the generalized Sylvester mapping and matrix equations , 2008, Syst. Control. Lett..

[16]  Gene H. Golub,et al.  Matrix computations , 1983 .

[17]  Guang-Ren Duan,et al.  PARAMETRIC SOLUTIONS TO THE GENERALIZED SYLVESTER MATRIX EQUATION AX ‐ XF = BY AND THE REGULATOR EQUATION AX ‐ XF = BY + R , 2010 .

[18]  Bin Zhou,et al.  A new solution to the generalized Sylvester matrix equation AV-EVF=BW , 2006, Syst. Control. Lett..

[19]  Feng Ding,et al.  Parameter Identification and Intersample Output Estimation for Dual-Rate Systems , 2008, IEEE Transactions on Systems, Man, and Cybernetics - Part A: Systems and Humans.

[20]  Feng Ding,et al.  Iterative least-squares solutions of coupled Sylvester matrix equations , 2005, Syst. Control. Lett..

[21]  Zeyad Abdel Aziz Al Zhour,et al.  Vector least-squares solutions for coupled singular matrix equations , 2007 .

[22]  Toru Yamamoto,et al.  A Numerical Algorithm for Finding Solution of Cross-Coupled Algebraic Riccati Equations , 2008, IEICE Trans. Fundam. Electron. Commun. Comput. Sci..