Iterative Method for Least Squares Reflexive Solution of General Linear Matrix Equation

张凯院; 田小红; 郑凤芹

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Iterative Method for Least Squares Reflexive Solution of General Linear Matrix Equation

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- Iterative Method for Least Squares Reflexive Solution of General Linear Matrix Equation
- 求线性矩阵方程自反最小二乘解的迭代方法
- Vol. 31, Issue 6, Pages: 548-553(2010)
- 作者机构：
  
  西北工业大学应用数学系
- 作者简介：
- 基金信息：
- DOI：
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- Published：2010
- 稿件说明：
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张凯院, 田小红, 郑凤芹. Iterative Method for Least Squares Reflexive Solution of General Linear Matrix Equation[J]. 2010, 31(6): 548-553.
DOI：

张凯院, 田小红, 郑凤芹. Iterative Method for Least Squares Reflexive Solution of General Linear Matrix Equation[J]. 2010, 31(6): 548-553. DOI：

摘要

基于求线性代数方程组的共轭梯度法的思想

建立了求一般线性矩阵方程的自反最小二乘解的迭代算法

并证明了迭代算法的收敛性.不考虑舍入误差时

迭代算法能够在有限步计算之后得到矩阵方程的自反最小二乘解;选取特殊的初始矩阵时

可求得极小范数自反最小二乘解.同时

也能够给出指定矩阵的最佳逼近自反矩阵.最后

用数值算例对有关结果进行了验证.

Abstract

On the base of conjugate gradient method of solving linear algebraic equations

an iterative method is presented to find the least squares reflexive solution of the general linear matrix equation and its convergence is proved.By the iterative method

the least squares reflexive solution can be obtained within finite iterative steps in the absence of round off errors.And the least squares solution with minimal norm can be obtained by choosing a special initial reflexive 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的参数迭代算法[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 Northwest University,2006,36(1):13-16(in Chinese)

尚丽娜,张凯院.一类Lyapunov矩阵方程对称最小二乘解的迭代算法[J].中北大学学报(自然科学版),2008,29(4):301-307.Shang Lina,Zhang Kaiyuan.An iterative method for the least squares symmetric solution of the Lyapunov matrixequation[J].Journal of North University of China(Natural Science Edition),2008,29(4):301-307.(in Chinese)

尚丽娜,张凯院,陈梅枝.求矩阵方程AXB=C的双对称最小二乘解的迭代算法[J].数值计算与计算机应用,2008,29(2):126-135.Shang Lina,Zhang Kaiyuan,Chen Meizhi.An iterative method for the least squares bisymmetric solution of thematrix equation AXB=C[J].Journal on Numerical Methods and Computer Applications,2008,29(2):126-135.(in Chinese)

Chu K W E.Symmetric solution of linear matrix equations matrix decompositions[J].Linear Algebra Appl,1989,119:35-50.

Dai H.On the symmetric solutions of linear equations[J].Linear Algebra Appl,1990,131:1-7.

Hou Jinjun,Peng Zhenyun,Zhang Xianglin.An iterative method for the least squares symmetric solution 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册.西安:西北工业大学出版社,2006.

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