多矩阵变量线性矩阵方程的广义自反解的迭代算法  

AN ITERATIVE ALGORITHM FOR THE GENERALIZED REFLEXIVE SOLUTION OF THE MULTI-MATRIX-VARIABLE LINEAR MATRIX EQUATION

作　　者：王娇[1] 张凯院[1] 李书连[1] 

机构地区：[1]西北工业大学应用数学系,西安710072

出　　处：《数值计算与计算机应用》2013年第1期9-19,共11页Journal on Numerical Methods and Computer Applications

基　　金：国家自然科学基金(11071196)

摘　　要：基于求线性矩阵方程约束解的修正共轭梯度法的思想方法,通过修改某些矩阵的结构,建立了求特殊类型的多矩阵变量线性矩阵方程的广义自反解的迭代算法,证明了迭代算法的收敛性,解决了给定矩阵在该矩阵方程的广义自反解集合中的最佳逼近计算问题.当矩阵方程相容时,该算法可以在有限步计算后得到其一组广义自反解;选取特殊的初始矩阵,能够求得其极小范数广义自反解.数值算例表明,迭代算法是有效的.Based on the method of the modified conjugate gradient to the linear matrix equation over constrained matrices, and by modifying the construction of some matrices, an iterative algorithm is presented to find the generalized reflexive solution of the matrix equation which is a special type with several matrix variables. The convergence of the iterative algorithm is proved. And the problem of the optimal approximation to the given matrix is solved in the generalized reflexive solution set of this matrix equation. When this matrix equation is consistent, its generalized reflexive solution can be obtained within finite iterative steps. And its least-norm generalized reflexive solution can be got by choosing the special initial matrices. The numerical example shows that the iterative algorithm is quite efficient.

关 键 词：线性矩阵方程 广义自反矩阵 极小范数解 迭代算法 最佳逼近 

分 类 号：O151.21[理学—数学]