Journal of Applied Mathematics
Volume 2012 (2012), Article ID 152805, 28 pages
http://dx.doi.org/10.1155/2012/152805
Research Article

An Iterative Algorithm for the Generalized Reflexive Solutions of the Generalized Coupled Sylvester Matrix Equations

Feng Yin1 and Guang-Xin Huang2

1School of Science, Sichuan University of Science and Engineering, Zigong 643000, China
2College of Management Science, Geomathematics Key Laboratory of Sichuan Province, Chengdu University of Technology, Chengdu 610059, China

Received 14 November 2011; Revised 13 March 2012; Accepted 25 March 2012

Academic Editor: Alain Miranville

Copyright © 2012 Feng Yin and Guang-Xin Huang. 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.

Abstract

An iterative algorithm is constructed to solve the generalized coupled Sylvester matrix equations ( 𝐴 𝑋 𝐵 − 𝐶 𝑌 𝐷 , 𝐸 𝑋 𝐹 − 𝐺 𝑌 𝐻 ) = ( 𝑀 , 𝑁 ) , which includes Sylvester and Lyapunov matrix equations as special cases, over generalized reflexive matrices 𝑋 and 𝑌 . When the matrix equations are consistent, for any initial generalized reflexive matrix pair [ 𝑋 1 , 𝑌 1 ] , the generalized reflexive solutions can be obtained by the iterative algorithm within finite iterative steps in the absence of round-off errors, and the least Frobenius norm generalized reflexive solutions can be obtained by choosing a special kind of initial matrix pair. The unique optimal approximation generalized reflexive solution pair [ 𝑋 , 𝑌 ] to a given matrix pair [ 𝑋 0 , 𝑌 0 ] in Frobenius norm can be derived by finding the least-norm generalized reflexive solution pair [ 𝑋 ∗ , 𝑌 ∗ ] of a new corresponding generalized coupled Sylvester matrix equation pair ( 𝐴 𝑋 𝐵 − 𝐶 𝑌 𝐷 , 𝐸 𝑋 𝐹 − 𝐺 𝑌 𝐻 ) = ( 𝑀 , 𝑁 ) , where 𝑀 = 𝑀 − 𝐴 𝑋 0 𝐵 + 𝐶 𝑌 0 𝐷 , 𝑁 = 𝑁 − 𝐸 𝑋 0 𝐹 + 𝐺 𝑌 0 𝐻 . Several numerical examples are given to show the effectiveness of the presented iterative algorithm.