广义约束条件下矩阵方程AXB+CXTD=E最佳逼近解的迭代算法

doi:10.13438/j.cnki.jdzk.2025.01.001

吉首大学学报（自然科学版） ›› 2025, Vol. 46 ›› Issue (1): 1-11.DOI: 10.13438/j.cnki.jdzk.2025.01.001

• 数学 • 下一篇

广义约束条件下矩阵方程AXB+CX^TD=E最佳逼近解的迭代算法

杨家稳，万鹏，梁金荣

（滁州职业技术学院基础教学部,安徽滁州 239000）

出版日期:2025-01-01 发布日期:2025-01-20
作者简介:杨家稳（1972—），男，安徽滁州人，滁州职业技术学院基础教学部教授，硕士，主要从事最优化理论与算法研究.
基金资助:
安徽省高校自然科学重点项目基金（2023AH053087，KJ2018A0839）

Iterative Algorithm for the Optimal Approximation Solution of Matrix Equations AXB+CX^TD=E with Generalized Constraint

YANG Jiawen,WAN Peng,LIANG Jinrong

（Department of Basic Course,Chuzhou Polytechnic,Chuzhou 239000,Anhui China）

Online:2025-01-01 Published:2025-01-20

摘要/Abstract

摘要：为了计算广义约束条件GX=H下矩阵方程AXB+CX^TD=E的最佳逼近解,设计了一种基于梯度投影与搜索方向正交的迭代算法.证明了任意给定一个满足广义约束条件的特殊初始矩阵，通过有限次迭代算法，能够获得广义约束条件下矩阵方程的极小范数最小二乘解，并利用该极小范数最小二乘解计算出最佳逼近解.极

关键词： 小范数最小二乘解, 最佳逼近解, 迭代算法, 梯度投影

Abstract: An iterative algorithm is presented to calculate the optimal approximation solution of the matrix equations AXB+CX^TD=E with constraint conditions GX=H.It is proved that given a special initial matrix satisfying the generalized constraints,the minimal norm least squares solution of the matrix equation under the generalized constraints can be obtained by the finite-order iterative algorithm,and the optimal approximation solution can be calculated by using the minimal norm least squares solution.

Key words: minimal norm least squares solution, optimal approximation solution, iterative algorithm, gradient projection

杨家稳, 万鹏, 梁金荣. 广义约束条件下矩阵方程AXB+CX^TD=E最佳逼近解的迭代算法[J]. 吉首大学学报（自然科学版）, 2025, 46(1): 1-11.

YANG Jiawen, WAN Peng, LIANG Jinrong. Iterative Algorithm for the Optimal Approximation Solution of Matrix Equations AXB+CX^TD=E with Generalized Constraint [J]. Journal of Jishou University(Natural Sciences Edition), 2025, 46(1): 1-11.

参考文献

[1] CHEN HSIN-CHU.Generalized Reflexive Matrices:Special Properties and Applications[J].SIAM Journal on Matrix Analysis and Applications,1998,19(1):140-153.
[2] NICHOLAS J HIGHAM,NATASA STRABIC.Anderson Acceleration of the Alternating Projections Method for Computing the Nearest Correlation Matrix[J].Numerical Algorithms,2016,72(4):1021-1042．
[3] HAJARIAN MASOUD.Generalized Conjugate Direction Algorithm for Solving the General Coupled Matrix Equations over Symmetric Matrices[J].Numerical Algorithms,2016,73:591-609.
[4] ZHANG Huamin,YIN Hongcai.Conjugate Gradient Least Squares Algorithm for Solving the Generalized Coupled Sylvester Matrix Equations[J].Computers & Mathematics with Applications,2017,73(12):2529-2547.
[5] HUANG Baohua,MA Changfeng.Gradient-Based Iterative Algorithms for Generalized Coupled Sylvester-Conjugate Matrix Equations[J].Computers & Mathematics with Applications,2018,75(7):2295-2310.
[6] HUANG Baohua,MA Changfeng.An Iterative Algorithm for the Least Frobenius Norm Hermitian and Generalized Skew Hamiltonian Solutions of the Generalized Coupled Sylvester-Conjugate Matrix Equations[J].Numerical Algorithms,2018,78(4):1271-1301.
[7] HUANG Baohua,MA Changfeng.The Least Squares Solution of a Class of Generalized Sylvester-Transpose Matrix Equations with the Norm Inequality Constraint[J].Journal of Global Optimization,2019,73(1):193- 221.
[8] QU Hongli,XIE Dongxiu,XU Jie.A Numerical Method on the Mixed Solution of Matrix Equation ∑ti=1AiXiBi=E with Sub-Matrix Constraints and Its Application[J].Applied Mathematics and Computation,2021,411 (15):126460-126481.
[9] YUAN Yongxin,ZHANG Huiting,LIU Lina.The Re-nnd and Re-pd Solutions to the Matrix Equations AX=C,XB=D[J].Linear and Multilinear Algebra,2022,70(18):3543-3552.〖ZK）〗
[10] YAN Tongxin,MA Changfeng.An Iterative Algorithm for Generalized Hamiltonian Solution of a Class of Generalized Coupled Sylvester-Conjugate Matrix Equations[J].Applied Mathematics and Computation,2021,411(15):126491-126514.
[11] ZHANG Huiting,LIU Lina,LIU Hao,et al.The Solution of the Matrix Equation AXB=D and the System of Matrix Equations AX=C,XB=D with X*X=Ip[J].Applied Mathematics and Computation,2022,418:126789-126797.
[12] PIAO Fengxian,ZHANG Qingling,WANG Zhefeng.The Solution to Matrix Equation AX+XTC=B[J].Journal of the Franklin Institute,2007,344:1056-1062.
[13] XIE Li,LIU Yanjun,YANG Huizhong.Gradient Based and Least Squares Based Iterative Algorithms for Matrix Equation AXB+CXTD=E[J].Applied Mathematics and Computation,2010,217:2191-2199．
[14] WANG Minghui,CHENG Xuehan,WEI Musheng.Iterative Algorithms for Solving the Matrix Equation AXB+CXTD=E[J].Applied Mathematics and Computation,2007,187(2):622-629.
[15] TERN FERNANDO DE,DOPICO FROILN M.The Solution of the Equation XA+AXT=O and Its Applications to the Theory of Orbits[J].Linear Algebra and Its Applications,2011,434:44-67．
[16] 赵琳琳.矩阵方程AXB+CXTD=E的可解性[J].山东大学学报（理学版），2012,47（10）：45-48.
[17] 盛兴平,苏友峰,陈果良.矩阵方程ATXB+BTXTA=D的极小范数最小二乘解的迭代算法[J].高等学校计算数学学报,2008,30(4):352-362.
[18] MEHDI DEHGHAN,MASOUD HAJARIAN.Finite Iterative Algorithms for the Reflexive and Anti-Reflexive Solutions of the Matrix Equations A1X1B1=C,A2X2B2=D[J].Mathematical and Computer Modeling,2009,49:1937-1959.
[19] MAZIAR DEHGHAN,MASOUD HAJARIAN.On the Generalized Bisymmetric and Skew-Symmetric Solutions of the System of Generalized Sylvester Matrix Equations[J].Linear and Multilinear Algebra,2011,59(11):1281-1309.
[20] PENG Zhuohua,XIN Huimin.The Reflexive Least Squares Solutions of the General Coupled Matrix Equations with a Submatrix Constraint[J].Applied Mathematics and Computation,2013,225:425-445.
[21] LIANG Maolin,DAI Lifang,YANG Yafang.The (P,Q,k+1)-Reflexive Solution of Matrix Equation AXB=C[J].Applied Mathematics and Computing,2013,42(1/2):339-350.
[22] 杨家稳，孙合明.矩阵方程A1Z+ZB1=C1的广义自反最佳逼近的迭代算法[J].数学杂志，2014,34(5):968-976.
[23] ROGER A HORN,CHARLES R JOHNSON.Topics in Matrix Analysis[M].London:Cambridge University Press,1991:259-260.
[24] PENG Zhuohua,HU Xiyan,ZHANG Lei.An Efficient Algorithm for the Least-Squares Reflexive Solution of the Matrix Equation A1XB1=C1,A2XB2=C2[J].Applied Mathematics and Computation,2006,181:988-999.

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广义约束条件下矩阵方程AXB+CX^TD=E最佳逼近解的迭代算法

Iterative Algorithm for the Optimal Approximation Solution of Matrix Equations AXB+CX^TD=E with Generalized Constraint

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广义约束条件下矩阵方程AXB+CXTD=E最佳逼近解的迭代算法

Iterative Algorithm for the Optimal Approximation Solution of Matrix Equations AXB+CXTD=E with Generalized Constraint

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广义约束条件下矩阵方程AXB+CX^TD=E最佳逼近解的迭代算法

Iterative Algorithm for the Optimal Approximation Solution of Matrix Equations AXB+CX^TD=E with Generalized Constraint