An efficient algorithm for solving general coupled matrix equations and its application
An efficient algorithm for solving general coupled matrix equations and its application
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DOI：
10.1016/j.mcm.2009.12.022
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发表时间：
2010-05
期刊：
影响因子：
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通讯作者：
M. Dehghan;M. Hajarian
中科院分区：
文献类型：
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作者：
M. Dehghan;M. Hajarian
一般耦合矩阵方程（包括作为特例的广义耦合西尔维斯特矩阵方程）在控制和系统理论的各个分支中都有很好的应用。本文通过推广共轭梯度法的思想，提出了求解一般耦合矩阵方程组的一种有效的迭代算法（Ⅰ）。当矩阵方程（Ⅰ）是相容的时，对于任何初始矩阵组，在没有舍入误差的情况下，可以在有限的迭代步内得到解组。当选取适当的初始矩阵组时，可导出一般耦合矩阵方程的最小Frobenius范数解组。我们可以使用所提出的算法在矩阵方程（I）的解群集合内找到给定矩阵群（X <$1，X <$2，...，X <$1）在Frobenius范数下的最佳逼近解群。数值算例表明了算法的有效性.最后，给出了该算法在求解（R，S）-对称和（R，S）-斜对称矩阵方程组中的应用.
The general coupled matrix equations (including the generalized coupled Sylvester matrix equations as special cases) have nice applications in various branches of control and system theory. In this paper, by extending the idea of conjugate gradient method, we propose an efficient iterative algorithm to solve the general coupled matrix equations (I). When the matrix equations (I) are consistent, for any initial matrix group, a solution group can be obtained within finite iteration steps in the absence of roundoff errors. The least Frobenius norm solution group of the general coupled matrix equations can be derived when a suitable initial matrix group is chosen. We can use the proposed algorithm to find the optimal approximation solution group to a given matrix group (X̂1,X̂2,…,X̂l) in a Frobenius norm within the solution group set of the matrix equations (I). Also several numerical examples are given to illustrate that the algorithm is effective. Furthermore, the application of the proposed algorithm for solving the system of matrix equations over (R,S)-symmetric and (R,S)-skew symmetric matrices is highlighted.