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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2023, Vol. 40 ›› Issue (2): 332-340.doi: 10.3969/j.issn.1005-3085.2023.02.012

• • 上一篇    

实方阵的Moore-Penrose广义逆的MCG算法探究

陈世军   

  1. 阳光学院基础教研部，福建 福州 350003
  • 收稿日期:2020-11-10 接受日期:2021-07-23 出版日期:2023-04-15 发布日期:2023-06-20
  • 基金资助:
    福建省教育厅中青年教师教育科研项目 (JAT210584).

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

CHEN Shijun   

  1. Department of Basic Teaching and Research, Yango University, Fuzhou, Fujian 350003
  • Received:2020-11-10 Accepted:2021-07-23 Online:2023-04-15 Published:2023-06-20
  • Supported by:
    The Education and Scientific Research Foundation for Young Teachers in Fujian Province (JAT210584).

PDF (PC)

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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。

关键词: Moore-Penrose广义逆, 修正共轭梯度算法, 线性方程组

Abstract:

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.

Key words: Moore-Penrose generalized inverse, MCG algorithm, linear equations

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