一类Lyapunov矩阵方程对称最小二乘解的迭代算法  

An Iterative Method for the Least Squares Symmetric Solution of the Lyapunov Matrix Equation

作　　者：尚丽娜[1] 张凯院[1] 

机构地区：[1]西北工业大学应用数学系,陕西西安710072

出　　处：《中北大学学报（自然科学版）》2008年第4期301-307,共7页Journal of North University of China(Natural Science Edition)

基　　金：陕西省自然科学基金资助项目(2004CS110002)

摘　　要：基于共轭梯度法,建立了一类Lyapunov矩阵方程的对称最小二乘解的迭代算法.使用该算法不仅可以判断这类矩阵方程的对称解的存在性,而且无论对称解是否存在,都能够在有限步迭代计算之后得到对称最小二乘解.选取特殊的初始矩阵时,可求得极小范数对称最小二乘解,同时也能给出指定矩阵的最佳逼近对称矩阵.最后,利用数值算例对有关结果进行了验证.On the basis of conjugate gradient method, an iterative method was presented to solve the least squares symmetric solution of Lyapunov matrix equation. By the iterative method, the solvability of the equation over symmetric solution can be determined, Whether the matrix equation is consistent or not, the least squares symmetric solution can be obtained automatically within finite iteration steps. And the symmetric solution with least norm can be obtained by choosing a special initial symmetric matrix. In addition, its optimal approximation matrix to a given matrix can be obtained. The given numerical examples show that the iterative method is quite efficient.

关 键 词：矩阵方程 对称矩阵 最小二乘解 极小范数解 迭代算法 最佳逼近 

分 类 号：O241.6[理学—计算数学]