矩阵方程AXB+CXD=F的广义中心对称解的算法分析

An Algorithm for the Generalized Center Symmetric Solutions of AXB+CXD=F

应用共轭梯度迭代算法求解方程AXB+CXD=F的广义中心对称解及其最佳逼近.应用此迭代算法,在迭代过程中方程的相容性可以自动地判断.当矩阵方程AXB+CXD=F有解时,在有限的误差范围内,对任意初始广义中心对称矩阵X1,运用迭代算法,方程的广义中心对称解可经过有限步迭代得到;选取适当的初始矩阵,可以迭代出极小范数广义中心对称解.并且,对任意的矩阵X0,矩阵方程AXB+CXD=F的最佳逼近解可以通过迭代求解新的矩阵方程A珘XB+C珘XD=珘F的极小范数广义中心对称解得到.

The conjugate gradient iteration algorithm was presented to find the generalized centrosymmetric solution and its optimal approximation of the constraint matrix equation AXB ＋ CXD = F. By this method, the solvability of the equation can be determined automatically. If the matrix equation AXB ＋ CXD = F is consistent, then its generalized centrosymmetric solution can be obtained within finite iteration steps in the absence of round off errors for any initial symmetric matrix X1 , and generalized centrosymmetric solution with the least norm can be derived by choosing a proper initial matrix. In addition, the optimal approximation solution for a given matrix of the matrix equation AXB ＋ CXD = F can be obtained by choosing the generalized centrosymmetric solution with the least norm of a new matrix equation AXB ＋ CXD =F.