线性子空间上求解AXB+CXD=F的最小二乘问题的迭代算法  

An iterative algorithm to the least squares problem of AXB+CXD=F over linear subspace

作　　者：周海林[1] ZHOU Hai-lin(Taizhou Institute of Sci.&Tech.,NJUST.,Taizhou 225300,China)

机构地区：[1]南京理工大学泰州科技学院,江苏泰州225300

出　　处：《高校应用数学学报（A辑）》2022年第3期350-364,共15页Applied Mathematics A Journal of Chinese Universities(Ser.A)

基　　金：江苏高校“青蓝工程”(2020)。

摘　　要：应用共轭梯度方法和线性投影算子,给出了求解线性矩阵方程AXB+CXD=F在任意线性子空间上的最小二乘解问题的迭代算法.在不考虑舍入误差的情况下,理论上可以证明,所给迭代算法经过有限步迭代可得到矩阵方程AXB+CXD=F的最小二乘解,极小范数解及其最佳逼近.该算法可以应用于任何线性子空间,包括由对称矩阵,中心对称矩阵等构成的线性子空间.文中的数值例子证实了该算法的有效性.Applying the conjugate gradient method and linear projection operator,an iterative algorithm is presented to solve the least squares problem of linear matrix equation AXB+CXD=F over any linear subspace.It is proved in theory that the least squares solution,the least-norm solution and the optimal approximation of the matrix equation AXB+CXD=F can be obtained in finite iteration steps by the method without considering rounding errors.The algorithm can be applied to any linear subspaces,including linear subspaces composed of symmetric matrices,central symmetric matrices and so on.The numerical examples verify the efficiency of the algorithm.

关 键 词：线性子空间 共轭梯度 投影算子 最小二乘解 最佳逼近 

分 类 号：O151.21[理学—数学]