广义Lyapunov方程A^TX+X^TA=C的一般解及其最佳逼近解  

The General Solution to the Generalized Lyapunov Matrix Equation A^TX+X^TA=C and its Optional Approximation Solution

作　　者：袁艳杰[1] 周富照[1] 

机构地区：[1]长沙理工大学数学与计算科学学院,长沙410114

出　　处：《数学理论与应用》2015年第3期1-8,共8页Mathematical Theory and Applications

基　　金：国家自然科学基金资助项目(11371072)

摘　　要：本文讨论矩阵方程ATX+XTA=C的一般解及其最佳逼近解的正交投影迭代解法.首先,利用矩阵的结构特点及相关性质,并借助矩阵空间的相关理论,给出求该矩阵方程一般解正交投影迭代算法;其次,根据奇异值分解、F-范数正交变换不变性证明算法的收敛性并推导出算法的收敛速率估计式,当方程相容时,该算法收敛于问题的极小范数解,且对该算法稍加修改,就可得到相应最佳逼近解;最后,用数值实例验证算法的有效性.The orthogonal projection iterative method for the matrix equation ATx ＋ XTA = C and its optimal approximation solution is studied. Firstly, the general solution of the orthogonal projection iterative method is constructed by means of the structure features and related properties of the matrix, and the concerned theories of matrix space. Then, by using singular value decomposition and the invariance of the Frobenius norm orthogonal transformation, the convergence of th algorithm is proven and the estimation of the convergent rate is derived. When the matrix equation is consistent, the approximation solution will converge to the least - norm solution of the equation and its optimal approximation solution is obtained only to make slight changes. Finally, the effectiveness of the algorithm is verified by a numerical example.

关 键 词：Lyapunov矩阵方程正交投影迭代法 最佳逼近解收敛速率极小范数解 

分 类 号：O241.6[理学—计算数学]