Newton's iterative method to solve a nonlinear matrix equation

被引：6
作者：
Peng, Jingjing [1 ]
Liao, Anping [2 ]
Peng, Zhenyun [3 ]
Chen, Zhencheng [4 ]
机构：
[1] Shanghai Univ, Coll Sci, Shanghai, Peoples R China
[2] Hunan Univ, Coll Math & Econometr, Changsha, Hunan, Peoples R China
[3] Guilin Univ Elect Technol, Sch Math & Comp Sci, Guangxi Coll & Univ Key Lab Data Anal & Computat, Guilin 541004, Peoples R China
[4] Guilin Univ Elect Technol, Sch Life & Environm Sci, Guilin, Peoples R China
来源：
LINEAR & MULTILINEAR ALGEBRA | 2019年 / 67卷 / 09期
基金：
中国国家自然科学基金;
关键词：
Nonlinear matrix equation; iterative method; Newton's iterative method; fixed point iterative method; POSITIVE-DEFINITE SOLUTIONS; EXTREME SOLUTIONS; A-ASTERISK-X(-1)A; EXISTENCE; X-2;
D O I：
10.1080/03081087.2018.1472736
中图分类号：
O1 [数学];
学科分类号：
0701 ; 070101 ;
摘要：
In this paper, Newton's iterative method to solve the nonlinear matrix equation x + A* X-n A = Q is studied. For the given initial matrix Q, the main results that the matrix sequence generated by the iterative method is contained in a fixed open ball, and that the matrix sequence generated by the iterative method converges to the only solution of the nonlinear matrix equation in a fixed closed ball are proved. In addition, the error estimate of the approximate solution in the closed ball and a numerical example to illustrate the convergence results are given.
引用
收藏
页码：1867 / 1878
页数：12
相关论文
共 40 条
[1]   LADDER NETWORKS, FIXPOINTS, AND THE GEOMETRIC MEAN [J].
ANDERSON, WN ;
MORLEY, TD ;
TRAPP, GE .
CIRCUITS SYSTEMS AND SIGNAL PROCESSING, 1983, 2 (03) :259-268
[2]   POSITIVE SOLUTIONS TO X = A-BX-1B-STAR [J].
ANDERSON, WN ;
MORLEY, TD ;
TRAPP, GE .
LINEAR ALGEBRA AND ITS APPLICATIONS, 1990, 134 :53-62
[3]   LIMIT OF ITERATES OF CASCADE ADDITION OF MATRICES [J].
ANDO, T .
NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION, 1980, 2 (7-8) :579-589
[4]  
Bucy, 1972, P 6 BERK S MATH STAT, P645
[5]   Positive definite solutions of operator equations Xm+A*X-n A=I [J].
Du, SP ;
Hou, JC .
LINEAR & MULTILINEAR ALGEBRA, 2003, 51 (02) :163-173
[6]   A new inversion free iteration for solving the equation X+A*X-1 A=Q [J].
El-Sayed, SM ;
Al-Dbiban, AM .
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS, 2005, 181 (01) :148-156
[7]   On positive definite solutions of the nonlinear matrix equation X+A☆X-n A = I [J].
El-Sayed, SM ;
Al-Dbiban, AM .
APPLIED MATHEMATICS AND COMPUTATION, 2004, 151 (02) :533-541
[8]   A two-sided iterative method for computing positive definite solutions of a nonlinear matrix equation [J].
El-Sayed, SM .
ANZIAM JOURNAL, 2003, 45 :145-152
[9]   Some properties for the existence of a positive definite solution of matrix equation X+A☆X-2m A = I [J].
El-Sayed, SM ;
El-Alem, M .
APPLIED MATHEMATICS AND COMPUTATION, 2002, 128 (01) :99-108
[10]  
ELSAYED SM, 2005, LINEAR ALGEBRA APPL, V403, P45
1234