M-矩阵代数Riccati方程的一类新的线性迭代法

doi:10.3969/j.issn.0253-2778.2020.12.004

中国科学技术大学学报 ›› 2020, Vol. 50 ›› Issue (12): 1472-1477.DOI: 10.3969/j.issn.0253-2778.2020.12.004

M-矩阵代数Riccati方程的一类新的线性迭代法

1.太原师范学院数学系,山西晋中030619;
2.上海工程技术大学数理与统计学院,上海201620

收稿日期:2020-10-10 接受日期:2020-11-24 出版日期:2020-12-30 发布日期:2021-03-04

A novel linear iteration method for M-matrix algebraic Riccati equations

1. Department of Mathematics, Taiyuan Normal University,Jinzhong 030619, China; 2. School of Mathematics, Physics and Statistics, Shanghai University of Engineering Science,Shanghai 201620 China

Received:2020-10-10 Accepted:2020-11-24 Online:2020-12-30 Published:2021-03-04
Contact: Guan Jinrui ( corresponding author ) is now an associate professor at Departement of Mathematics, Taiyuan Normal University, China. He received his Ph. D. degree in computational mathematics at Xiamen University in 2016. His research interests focus on matrix theory and numerical linear algebra with applications.
Supported by:
The work is supported by National Natural Science Foundation of China (12001363, 12001395), Natural Science Foundation of Shanxi province, China (201901D211423).

摘要/Abstract

摘要：

研究了M-矩阵代数Riccati方程的数值解法,这类方程由于广泛的应用成为近年来的研究热点.提出了一种新的线性迭代法来计算方程的最小非负解,该方法在每步迭代中只需要矩阵乘法.通过适当选取参数,证明了当系数矩阵为非奇异M-矩阵或不可约奇异M-矩阵时新方法的收敛性.理论分析和数值实验表明,新方法是可行的,而且在一定情况下比现有的一些方法更加有效.

关键词: 代数Riccati方程, M-矩阵, 最小非负解, 牛顿法, 加倍算法

Abstract:

Numerical solutions of the M-matrix algebraic Riccati equation (MARE) were studied, which has become a hot topic in recent years due to its broad applications. A novel linear iteration method for computing the minimal nonnegative solution of MARE was proposed, in which only matrix multiplications are needed at each iteration. Convergence of the new method was proved by choosing proper parameters for the MARE associated with a nonsingular M-matrix or an irreducible singular M- matrix. Theoretical analysis and numerical experiments show that the new method is feasible and is effective than some existing methods under certain conditions.

Key words: algebraic Riccati equation, M-matrix, minimal nonnegative solution, Newton method, doubling algorithm

中图分类号:

O241.6

关晋瑞∗, 冯月华 . M-矩阵代数Riccati方程的一类新的线性迭代法 [J]. 中国科学技术大学学报, 2020, 50(12): 1472-1477.

GUAN Jinrui∗, FENG Yuehua. A novel linear iteration method for M-matrix algebraic Riccati equations [J]. Journal of University of Science and Technology of China, 2020, 50(12): 1472-1477.

M-矩阵代数Riccati方程的一类新的线性迭代法

A novel linear iteration method for M-matrix algebraic Riccati equations

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