求多变量线性矩阵方程组自反解的迭代算法  

AN ITERATIVE METHOD FOR THE REFLEXIVE SOLUTIONS OF THE LINEAR MATRIX EQUATIONS WITH SEVERAL VARIABLES

作　　者：郑凤芹[1] 张凯院[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》2010年第1期39-54,共16页Journal on Numerical Methods and Computer Applications

基　　金：陕西省自然科学基金资助项目(2006A05)

摘　　要：利用矩阵分解的方法求多变量线性矩阵方程组的自反解是很困难的.本文建立了一种迭代方法来解决这个问题,利用此迭代方法可以判断多变量线性矩阵方程组的可解性,且当矩阵方程组相容时,可以在有限步迭代后得到其自反解.选取特殊的初始矩阵时,能够求得矩阵方程组的极小范数自反解.进一步,通过求新的线性矩阵方程组的极小范数自反解,能够求得给定矩阵的最佳逼近矩阵.数值算例表明,迭代算法是有效的.The reflexive solutions of the linear matrix equations with several variables are too difficult to be obtained by applying matrices decomposition. In this paper, an iterative method is applied to solve this problem. With it, the solvability of the linear matrix equations with several variables can be determined automatically, when this system of matrix equations is consistent, its reflexive solutions can be obtained within finite iterative steps. And its least-norm reflexive solutions can be obtained by choosing the special kind of initial iterative matrices, furthermore, its optimal approximation solutions to the given matrices can be derived by finding the/east-norm reflexive solutions of the new linear matrix equations. The numerical examples show that the iterative algorithm is quite efficient.

关 键 词：线性矩阵方程组 自反矩阵 极小范数解 迭代算法 最佳逼近 

分 类 号：O241.6[理学—计算数学]