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Journal of Applied Mathematics
Volume 2015, Article ID 562529, 12 pages
Research Article

LSMR Iterative Method for General Coupled Matrix Equations

1Department of Applied Mathematics, School of Mathematical Sciences, Ferdowsi University of Mashhad, Iran
2The Center of Excellence on Modelling and Control Systems, Ferdowsi University of Mashhad, Iran
3Faculty of Mathematical Sciences, University of Guilan, Iran

Received 18 March 2014; Accepted 28 July 2014

Academic Editor: D. R. Sahu

Copyright © 2015 F. Toutounian et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


By extending the idea of LSMR method, we present an iterative method to solve the general coupled matrix equations , , (including the generalized (coupled) Lyapunov and Sylvester matrix equations as special cases) over some constrained matrix groups , such as symmetric, generalized bisymmetric, and -symmetric matrix groups. By this iterative method, for any initial matrix group , a solution group can be obtained within finite iteration steps in absence of round-off errors, and the minimum Frobenius norm solution or the minimum Frobenius norm least-squares solution group can be derived when an appropriate initial iterative matrix group is chosen. In addition, the optimal approximation solution group to a given matrix group in the Frobenius norm can be obtained by finding the least Frobenius norm solution group of new general coupled matrix equations. Finally, numerical examples are given to illustrate the effectiveness of the presented method.