• 论文 •

### 矩阵极分解新的数值方法

1. 华南师范大学数学科学学院, 广州 510631
• 收稿日期:2015-12-08 出版日期:2017-02-15 发布日期:2017-02-17
• 基金资助:

广东省自然科学基金（S2013010012530）资助和华南师范大学研究生科研创新基金（2015lkxm21）和国家自然科学基金（11671158）资助.

Wen Chaotao, Chen Xiaoshan. NEW NUMERICAL METHODS FOR COMPUTING THE POLAR DECOMPOSITION[J]. Mathematica Numerica Sinica, 2017, 39(1): 23-32.

### NEW NUMERICAL METHODS FOR COMPUTING THE POLAR DECOMPOSITION

Wen Chaotao, Chen Xiaoshan

1. School of Mathematics, South China Normal University, Guangzhou 510631, China
• Received:2015-12-08 Online:2017-02-15 Published:2017-02-17
p是大于1的偶数.本文基于方程xp-1=0的Newton和Halley求根公式给出计算非奇异矩阵酉极因子的数值方法，并证明算法的收敛性.用数值列子说明算法的有效性.
Let p > 1 be an even number.In this paper,based on solving the equation xp-1=0 we present the Newton iteration and Halley iteration to compute the polar decomposition of a nonsingular matrix,and prove their convergence properties.Numerical examples show that these algorithms are valid.

MR(2010)主题分类:

()
 [1] 张贤达. 矩阵分析与应用[M]. 北京:清华大学出版社, 2004.[2] 邹财盛, 陈小山. 用割线法求矩阵的极分解[J]. 计算数学, 2014, 36(3):225-230.[3] Gander W. Gander W. Algorithms for the polar decomposition[J]. SIAM Journal on Scientific and Statistical Computing, 1990, 11(6):1102-1115.[4] Golub G H, Van Loan C F. Matrix computations. JHU Press, 2012.[5] Guo C H. On Newton s method and Halley s method for the principal pth root of a matrix[J]. Linear Algebra and its Applications, 2010, 432(8):1905-1922.[6] Horn R A, Johnson C R. Matrix analysis. Cambridge university press, 2012.[7] Higham N J. Functions of matrices:Theory and Computation. Society for Industrial and Applied Mathematics, USA:Philadelphia, 2008.[8] Higham N J. Computing the polar decomposition-with applications[J]. SIAM Journal on Scientific and Statistical Computing, 1986, 7(4):1160-1174.[9] Iannazzo B. On the Newton method for the matrix pth root[J]. SIAM Journal on Matrix Analysis and Applications, 2006, 28(2):503-523.[10] Marcus M, Minc H. Introduction to linear algebra. Courier Corporation, 1988.[11] Nakatsukasa Y, Bai Z, Gygi F. Optimizing Halley's iteration for computing the matrix polar decomposition[J]. SIAM Journal on Matrix Analysis and Applications, 2010, 31(5):2700-2720.
 [1] 蔡文银, 徐玲玲. 核范数和谱范数下广义Sylvester方程最小二乘问题的一类改进算法[J]. 计算数学, 2018, 40(4): 387-401. [2] 李姣芬, 宋丹丹, 李涛, 黎稳. 核范数和谱范数下广义Sylvester方程最小二乘问题的有效算法[J]. 计算数学, 2017, 39(2): 129-150. [3] 裕静静, 江平, 刘植. 两类五阶解非线性方程组的迭代算法[J]. 计算数学, 2017, 39(2): 151-166. [4] 刘晴, 檀结庆, 张旭. 一种基于Chebyshev迭代解非线性方程组的方法[J]. 计算数学, 2015, 37(1): 14-20. [5] 陈小山. 用矩阵符号函数解(广义)周期Sylvester方程[J]. 计算数学, 2012, 34(2): 153-162. [6] 陈小山,黎稳. 关于矩阵方程X+A~*X~(-1)A=P的解及其扰动分析[J]. 计算数学, 2005, 27(3): 303-310.