矩阵方程AXB+CXD=F的最小二乘解的多步迭代算法  

Multi-step iterative algorithm for solving the least squares solution of the matrix equation AXB+CXD=F

作　　者：王杰 彭振赟[1] 李涛[1] WANG Jie;PENG Zhenyun;LI Tao(School of Mathematics and Computing Science,Guilin University of Electronic Technology,Guilin 541004,China)

机构地区：[1]桂林电子科技大学数学与计算科学学院,广西桂林541004

出　　处：《桂林电子科技大学学报》2022年第2期138-142,共5页Journal of Guilin University of Electronic Technology

基　　金：国家自然科学基金(11961012);广西自然科学基金(2018GXNSFBA281192,2017GXNSFAA198248);桂林电子科技大学研究生教育创新计划(GDYCS201458)。

摘　　要：为了求解矩阵方程AXB+CXD=F的最小二乘解及其最佳逼近解,提出了一种多步迭代算法。证明了由多步迭代算法产生的矩阵序列收敛于矩阵方程AXB+CXD=F的最小二乘问题的最小Frobenius范数解;通过修改系数矩阵F,证明了由多步迭代算法产生的矩阵序列收敛于矩阵方程AXB+CXD=F的最小二乘问题的最佳逼近解,同时给出了多步迭代算法与不动点迭代算法和共轭梯度算法的数值比较。实验结果证明了多步迭代算法比共轭梯度算法和不动点迭代算法更有效。In order to solve the least square solution of the matrix equation AXB+CXD=F and its best approximate solution,a multi-step iterative algorithm was designed.The minimum Frobenius norm solution of the matrix sequence generated by the multi-step iterative algorithm converges to the least squares problem of AXB+CXD=F is proved.By modifying the coefficient matrix F,the optimal approximation solution of the least square problem of the matrix sequence generated by the multi-step iterative algorithm converges to the matrix equation AXB+CXD=F is proved.At the same time,the numerical comparison of multi-step iterative algorithm with fixed point algorithm and conjugate gradient algorithm is given.The experimental results show that the multi-step iterative algorithm is effective.

关 键 词：矩阵方程 最小二乘问题 多步迭代算法 不动点迭代算法 共轭梯度算法 

分 类 号：O241.2[理学—计算数学]