矩阵方程AXB+CYD=E的M对称解的迭代算法

AN ITERATIVE ALGORITHM FOR M SYMMETRIC SOLUTIONS OF THE MATRIX EQUATION AXB+CYD=E

在共轭梯度思想的启发下,结合线性投影算子,给出迭代算法求解了线性矩阵方程AXB+CYD=E的M对称解[X,Y]及其最佳逼近.当矩阵方程AXB+CYD=E有M对称解时,应用迭代算法,在有限的误差范围内,对任意初始M对称矩阵对[X_,Y_1],经过有限步迭代可得到矩阵方程的M对称解;选取合适的初始迭代矩阵,还可得到极小范数M对称解.而且,对任意给定的矩阵对[X,Y],矩阵方程AXB+CYD=E的最佳逼近可以通过迭代求解新的矩阵方程AXB+CYD=E的极小范数M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, combined with the linear projection operator, an iterative algorithm is presented to solve the linear matrix equation AXB＋CYD = E over M symmetric solution [X, Y] and its optimal approximation. When the matrix equation AXB ＋ CYD = E is consistent over M symmetric solution, by this method, its solution can be obtained within finite iteration steps in the absence of round off errors for any initial M symmetric matrix pair [X1, Y1], and its least-norm M symmetric solution can be derived by choosing a suitable initial iterative matrix. Furthermore, for any given matrix pair [X, Y], the optimal approximation of the matrix equation AXB ＋ CYD = E can be obtained by choosing the least-norm M symmetric solution of a new matrix equation AXB ＋ CYD =E. Some numerical examples verify the efficiency of the algorithm.