一类矩阵方程组双对称解的修正共轭梯度法

郑凤芹; 张凯院

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一类矩阵方程组双对称解的修正共轭梯度法

应用基础研究 | 更新时间：2026-09-14

- 一类矩阵方程组双对称解的修正共轭梯度法
- The Modified Conjugate Gradient Method for the Bisymmetric Solutions of a Class Matrix Equations
- 中北大学学报（自然科学版） 2011年32卷第2期页码：128-134
- 作者机构：
  
  西北工业大学应用数学系
- 作者简介：
- 基金信息：
  
  陕西省自然科学基金资助项目(2006A05)
- DOI：
  中图分类号：
- 纸质出版：2011
- 稿件说明：
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郑凤芹, 张凯院. 一类矩阵方程组双对称解的修正共轭梯度法[J]. 中北大学学报（自然科学版）, 2011,32(2):128-134.

郑凤芹, 张凯院. The Modified Conjugate Gradient Method for the Bisymmetric Solutions of a Class Matrix Equations[J]. 2011, 32(2): 128-134.
郑凤芹, 张凯院. 一类矩阵方程组双对称解的修正共轭梯度法[J]. 中北大学学报（自然科学版）, 2011,32(2):128-134. DOI：

郑凤芹, 张凯院. The Modified Conjugate Gradient Method for the Bisymmetric Solutions of a Class Matrix Equations[J]. 2011, 32(2): 128-134. DOI：

摘要

建立了求多变量线性矩阵方程组双对称解的迭代算法.利用该算法不仅可以判断矩阵方程组是否存在双对称解

而且在双对称解存在时选取特殊的初始矩阵

能够在有限步迭代计算之后得到矩阵方程组的极小范数双对称解;同时

能够在矩阵方程组的双对称解集合中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.

Abstract

An iterative method was presented to find the bisymmetric solutions of the multivariable linear matrix equations.With this method

the existence of the bisymmetric solutions can be determined

and the minimal-norm bisymmetric solutions can be got by choosing special initial bisymmetric matrices.In addition

the optimal approximation matrices to the given matrices can be obtained in the solutions set of the matrix equations.Numerical experiments proved the effectiveness of the algorithm.

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Keywords

references

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.(inChinese)

Li Fanliang,Hu Xiyan,Zhang Lei.Least-squares mirrosymmetric solution for matrix equations(AX=B,XC=D)[J].Numerical Mathematics:A Journal of Chinese Universities,2006,15(3):217-226.

Peng Yaxin,Hu Xiyan,Zhang Lei.An iterative method for symmetric solutions and optimal approximate solution ofthe system of matrix equations A1XB1=C1,A2XB2=C2[J].Applied Mathematics and Computation,2006,183:1127-1137.

Navarra A,Odell P L,Young D M.A representation of the general common solution to the equations A1XB1=C1,A2XB2=C2with applications[J].Computers and Mathematics with Applications,2001,41(7-8):929-935.

Mehdi Dehghan,Masoud Hajarian.Finite iterative algorithms for the reflexive and anti-reflexive solutions of thematrix equation A1X1B1+A2X2B2=C[J].Mathematical and Computer Modelling,2009,49:1937-1959.

袁仕芳,廖安平,雷渊.矩阵方程AXB+CYD=E的对称极小范数最小二乘解[J].计算数学,2007,29(2):203-216Yuan Shifang,Liao Anping,Lei Yuan.Least squares symmetric solution of the matrix equation AXB+CYD=E withleast norm[J].Mathematica Numerica Sinica,2007,29(2):203-216.(in Chinese)

Peng Zhuohua,Hu Xiyan,Zhang Lei.The bisymmetric solutions of the matrix equation A1X1B1+A2X2B2+…+AlXlBl=C and its optimal approximation[J].Linear Algebra and its Applications,2007,426:583-595.

张凯院,徐仲.数值代数[M].第2版.北京:科学出版社,2006:25-27.

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