Abstract
By extending the idea of LSMR method, we present an iterative method to solve the general coupled matrix equations∑k=1qAikXkBik=Ci,i=1,2,…,p, (including the generalized (coupled) Lyapunov and Sylvester matrix equations as special cases) over some constrained matrix groups(X1,X2,…,Xq), such as symmetric, generalized bisymmetric, and(R,S)-symmetric matrix groups. By this iterative method, for any initial matrix group(X1(0),X2(0),…,Xq(0)), a solution group(X1*,X2*,…,Xq*)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(X¯1,X¯2,…,X¯q)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.
Highlights
In control and system theory [1,2,3,4,5,6,7], we often encounter Lyapunov and Sylvester matrix equations which have been playing a fundamental role
The LSMR algorithm uses an algorithm of Golub and Kahan [44], which stated as procedure Bidiag 4, to reduce M to lower bidiagonal form
We will present our matrix iterative method based on the LSMR algorithm, for solving (1) and problem (2)
Summary
In control and system theory [1,2,3,4,5,6,7], we often encounter Lyapunov and Sylvester matrix equations which have been playing a fundamental role. By using the hierarchical identification principle [9,10,11, 28,29,30,31,32], a gradient-based iterative (GI) method was presented to solve the solutions and the least-squares solutions of the general coupled matrix equations. Li and Huang [37] proposed a matrix LSQR iterative method to solve the constrained solutions of the generalized coupled Sylvester matrix equations. We construct a matrix iterative method based on the LSMR algorithm [40] to solve the constrained solutions of the following problems.
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