摘要
在共轭梯度思想的启发下,本文给出了迭代算法求解约束矩阵方程AXB+CXD=F的对称解及其最佳逼近.应用迭代算法,矩阵方程AXB+CXD=F的相容性可以在迭代过程中自动判断.当矩阵方程AXB+CXD=F有对称解时,在有限的误差范围内,对任意初始对称矩阵X_1,运用迭代算法,经过有限步可得到矩阵方程的对称解;选取合适的初始迭代矩阵,还可以迭代出极小范数对称解.而且,对任意给定的矩阵X_0,矩阵方程AXB+CXD=F的最佳逼近对称解可以通过迭代求解新的矩阵方程AXB+CXD=F的极小范数对称解得到.文中的数值例子证实了该算法的有效性.
Motivated by the conjugate gradient method, an iterative algorithm is presented to solve the linear matrix equation AXB ＋ CXD = F over symmetric matrix X and its optimal approximation. By this method, the solvability of the equation AXB ＋ CXD = F over symmetric X can be determined automatically. When the equation AXB ＋ CXD = F is consistent over symmetric X, its solution can be obtained within finite iteration steps in the absence of round off errors for any initial symmetric matrix X1, and its least-norm symmetric solution can be derived by choosing a suitable initial iterative matrix. Furthermore, its optimal approximation to the given matrix X0 can be obtained by choosing the least-norm symmetric solution of a new matrix equation AX^-B ＋ CX^-D =F^-. Some numerical examples verify the efficiency of the algorithm.
出处
《计算数学》
CSCD
北大核心
2010年第4期413-422,共10页
Mathematica Numerica Sinica
关键词
矩阵方程
迭代算法
对称解
极小范数解
最佳逼近
matrix equation
iterative algorithm
symmetric solution
least-norm solution
optimal approximation
