Finite iterative algorithms for solving generalized coupled Sylvester systems-Part II: Two-sided and generalized coupled Sylvester matrix equations over reflexive solutions

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• 作者：Feng Yin^a ; ^{fyin@suse.edu.cn} ; Guang-Xin Huang^b ; ^{huangx@cdut.edu.cn} ; De-Qin Chen^a
• 关键词：Iterative method ; Generalized reflexive solution ; Reflexive solution ; Generalized coupled Sylvester matrix equations ; Optimal approximate solution
• 刊名：Applied Mathematical Modelling
• 出版年：2012
• 出版时间：April, 2012
• 年：2012
• 卷：36
• 期：4
• 页码：1604-1614
• 全文大小：378 K

文摘

In Part I of this article, we proposed a finite iterative algorithm for the one-sided and generalized coupled Sylvester matrix equations (AY ?#xA0;ZB, CY ?#xA0;ZD) = (E, F) and its optimal approximation problem over generalized reflexive matrices solutions. In Part II, an iterative algorithm is constructed to solve the two-sided and generalized coupled Sylvester matrix equations (AXB ?#xA0;CYD, EXF ?#xA0;GYH) = (M, N), which include Sylvester and Lyapunov matrix equations as special cases, over reflexive matrices X and Y. When the matrix equations are consistent, for any initial reflexive matrix pair [X₁, Y₁], the 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 reflexive solutions can be obtained by choosing a special kind of initial matrix pair. The unique optimal approximation reflexive solution pair to a given matrix pair [X₀, Y₀] in Frobenius norm can be derived by finding the least-norm reflexive solution pair of a new corresponding generalized coupled Sylvester matrix equations , where . Several numerical examples are given to show the effectiveness of the presented iterative algorithm.