Gradient-based iterative algorithm is suggested for solving a coupled complex conjugate and transpose matrix equations. Using the hierarchical identification principle and the real representation of a complex matrix, a convergence proof is offered. The necessary and sufficient conditions for the optimal convergence factor are determined. A numerical example is offered to validate the efficacy of the suggested algorithm.
References
[1]
Young, D.M. and Young, M.M. (2017) A General Hermitian Nonnegative-Definite Solution to the Matrix Equation . Advances in Linear Algebra & Matrix Theory, 7, 7-17. https://doi.org/10.4236/alamt.2017.71002
[2]
Gao, D.J. (2017) Iterative Methods for Solving the Nonlinear Matrix Equation . Advances in Linear Algebra & Matrix Theory, 7, 72-78.
[3]
Wu, A.G., Lv, L.L. and Duan, G.R. (2011) Iterative Algorithms for Solving a Class of Complex Conjugate and Transpose Matrix Equations. Applied Mathematics and Computation, 217, 8343-8353. https://doi.org/10.1016/j.amc.2011.02.113
[4]
Gu, C.Q. and Xue, H.Y. (2009) A Shift-Splitting Hierarchical Identification Method for Solving Lyapunov Matrix Equations. Linear Algebra and its Applications, 430, 1517-1530. https://doi.org/10.1016/j.laa.2008.01.010
[5]
Zhou, B., Li, Z.Y., Duan, G.R. and Wang, Y. (2009) Weighted Least Squares Solutions to General Coupled Sylvester Matrix Equations. Journal of Computational and Applied Mathematics, 224, 759-776. https://doi.org/10.1016/j.cam.2008.06.014
[6]
Zhou, B., Lam, J. and Duan, G.R. (2010) Gradient-Based Maximal Convergence Rate Iterative Method for Solving Linear Matrix Equations. International Journal of Computer Mathematics, 87, 515-527. https://doi.org/10.1080/00207160802123458
[7]
Zhou, B., Lam J. and Duan, G.R. (2008) Convergence of Gradient-Based Iterative Solution of the Coupled Markovian Jump Lyapunov Equations. Computers & Mathematics with Applications, 56, 3070-3078. https://doi.org/10.1016/j.camwa.2008.07.037
[8]
Zhou, B., Duan, G.R. and Li, Z.Y. (2009) Gradient Based Iterative Algorithm for Solving Coupled Matrix Equations. Systems & Control Letters, 58, 327-333. https://doi.org/10.1016/j.sysconle.2008.12.004
[9]
Li, Z.Y., Wang, Y., Zhou, B. and Duan, G.R. (2010) Least Squares Solution with the Minimum-Norm to General Matrix Equations via Iteration. Applied Mathematics and Computation, 215, 3547-3562. https://doi.org/10.1016/j.amc.2009.10.052
[10]
Jian, T.S. and Wei, M.S. (2003) On Solutions of the Matrix Equations and . Linear Algebra and its Applications, 367, 225-233. https://doi.org/10.1016/S0024-3795(02)00633-X
[11]
Zhou, B., Lam, J. and Duan, G.R. (2011) Toward Solution of Matrix Equations . Linear Algebra and its Applications, 435, 1370-1398.
[12]
Ding, F., Liu, X.P. and Ding, J. (2008) Iterative Solutions of the Generalized Sylvester Matrix Equations by Using the Hierarchical Identification Principle. Applied Mathematics and Computation, 197, 41-50. https://doi.org/10.1016/j.amc.2007.07.040
