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

尚丽娜; 张凯院

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

更新时间：2026-09-14

- An Iterative Method for the Least Squares Symmetric Solution of the Lyapunov Matrix Equation
- 一类Lyapunov矩阵方程对称最小二乘解的迭代算法
- Issue 4, Pages: 301-307(2008)
- 作者机构：
  
  西北工业大学应用数学系
- 作者简介：
- 基金信息：
- DOI：
  CLC：
- Published：2008
- 稿件说明：
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尚丽娜, 张凯院. An Iterative Method for the Least Squares Symmetric Solution of the Lyapunov Matrix Equation[J]. 2008, (4): 301-307.
DOI：

尚丽娜, 张凯院. An Iterative Method for the Least Squares Symmetric Solution of the Lyapunov Matrix Equation[J]. 2008, (4): 301-307. DOI：

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.

Abstract

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.

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references

张凯院,蔡元虎.矩阵方程AXB+CXD=F的参数迭代解法[J].西北大学学报,2006,36(1):13-16.Zhang Kaiyuan,Cai Yuanhu.On Parameter iterative method for solving matrix equation AXB+CXD=F[J].Journal of Northweast University,2006,36(1):13-16.(in Chinese)

张凯院,徐仲.数值代数[M].第2版.北京:科学出版社,2006:162-189.

彭亚新.求解约束矩阵方程及其最佳逼近的迭代法的研究[D].长沙:湖南大学,2004.

Hou Jinjun,Peng Zhenyun,Zhang Xianglin.An iterative method for the least squares symmetricsolution of matrixequation AXB=C[J].Numer Algor,2006,42:181-192.

程云鹏,张凯院,徐仲.矩阵论[M].第3版.西安:西北工业大学出版社,2006:338-339.

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