实方阵的Moore-Penrose广义逆的MCG算法探究  

Research on MCG Algorithm of Moore-Penrose Generalized Inverse of Real Matrix

作　　者：陈世军 CHEN Shijun(Department of Basic Teaching and Research,Yango University,Fuzhou,Fujian 350003)

机构地区：[1]阳光学院基础教研部,福建福州350003

出　　处：《工程数学学报》2023年第2期332-340,共9页Chinese Journal of Engineering Mathematics

基　　金：福建省教育厅中青年教师教育科研项目(JAT210584).

摘　　要：证明了矩阵Moore-Penrose逆的唯一性以及建立了求矩阵Moore-Penrose逆的算法。首先将求矩阵的Moore-Penrose逆转为求解含有三个矩阵变量的矩阵方程组,其次建立求该矩阵方程组的修正共轭梯度算法(MCG算法),给出了MCG算法的性质和收敛性证明,对于任意给定的初始矩阵该算法能在有限步迭代计算后得到矩阵的Moore-Penrose逆。最后给出数值算例,证明MCG算法在求解矩阵Moore-Penrose逆中具有很高的计算效率。In this paper,the uniqueness of the Moore-Penrose inverse of a matrix is proved and an algorithm for solving the Moore-Penrose inverse of a matrix is proposed.First of all,the Moore-Penrose inverse of the matrix is reversed to the solution of a matrix equations with three matrix variables.Then a modified conjugate gradient algorithm(MCG algorithm)is established to solve the matrix equations.Moreover,the properties and convergence of the MCG algorithm are proved.For any given initial matrix,we can obtain the Moore-Penrose inverse of the matrix after finite iterative steps.Finally,several numerical examples are given to prove that MCG algorithm has high computational efficiency for solving the Moore-Penrose inverse of a matrix.

关 键 词：MOORE-PENROSE广义逆 修正共轭梯度算法 线性方程组 

分 类 号：O241.6[理学—计算数学]