[1]张文红,李冲.Gauss-Newton法的半局部收敛性[J].东南大学学报(自然科学版),2001,31(5):135-139.[doi:10.3969/j.issn.1001-0505.2001.05.029]
 Zhang Wenhong,Li Chong.Convergence of Gauss-Newton’s Method[J].Journal of Southeast University (Natural Science Edition),2001,31(5):135-139.[doi:10.3969/j.issn.1001-0505.2001.05.029]
点击复制

Gauss-Newton法的半局部收敛性()
分享到:

《东南大学学报(自然科学版)》[ISSN:1001-0505/CN:32-1178/N]

卷:
31
期数:
2001年第5期
页码:
135-139
栏目:
数学、物理学、力学
出版日期:
2001-09-20

文章信息/Info

Title:
Convergence of Gauss-Newton’s Method
作者:
张文红 李冲
东南大学应用数学系, 南京 210096
Author(s):
Zhang Wenhong Li Chong
Department of Applied Mathematics, Southeast University, Nanjing 210096, China)
关键词:
非线性最小二乘问题 Gauss-Newton法 半局部收敛性
Keywords:
nonlinear least squares problems Gauss-Newton′s method convergence
分类号:
O214.7
DOI:
10.3969/j.issn.1001-0505.2001.05.029
摘要:
设f:Rn→Rm是Frechet可微的,m≥n. 则非线性最小二乘问题可描述为下面的极小化问题:minF(x):=1/2f(x)Tf(x). Gauss-Newton法是求解非线性最小二乘问题的最基本的方法之一,其 n+1 步迭代定义为:xn+1=xn-[f ′(xn)Tf ′(x)]-1f ′(xn)Tf(xn).本文主要研究解非线性最小二乘问题的Gauss-Newton法的半局部收敛性.假设f(x)在B(x0,r)内连续可导且f ′(x0)满秩,若f的导数满足Lipschitz连续=F ′(x)-f ′(x′)=≤γ=x-x′=,x,x′∈B(x0,r). 在一个关于初始点x0的判断准则 c==f(x0)=, β==[f ′T(x0)f ′(x0)]-1f ′(x0)T=, β2cγ<1/10下, Gauss-Newton法产生的序列{xn}收敛到一个驻点x*,从而给出了Gauss-Newton法的半局部收敛性.
Abstract:
Let f:Rn→Rm be a nonlinear Frechet differentiable map, where m>n. The authors investigate the convergence problem of Gauss-Newton’s method xn+1=xn-[f ′(xn)Tf ′(xn)]-1f ′(xn)Tf(xn), n=0,1,2,… for finding the approximation solution of nonlinear least squares problems minF(x)=1/2f(x)Tf(x). Under the hypothesis that f ′(x0)-1 exists and the derivative of f in B(x0,r) satisfies the Lipschitz continuous: =f ′(x)-f ′(x′)=≤γ=x-x′=, x,x′∈B(x0,r). With a criterion on the initial value c==f(x0)=, β==[f ′T(x0)f ′(x0)]-1 f ′(x0)T=, β2cγ<1/10, we judge that the {xn} produced by Gauss-Newton’s method is convergent to x*. Thereby the convergence theorem of Gauss-Newton’s method is obtained.

参考文献/References:

[1] 袁亚湘,孙文渝.最优化理论与方法.北京:科学出版社,1997.375~382
[2] 王国荣.矩阵与算子广义逆.北京:科学出版社,1997.155~164
[3] Stewart G W.On the continuity of the generalized inverse.SIAM J Apple Math,1960,17:33~45
[4] Wedin P A.Perturbation theory for pseudo-inverse.BIT,1973,13:217~232
[5] Ben-Israel A.A Newton-Raphson method for the solution of systems of equations.Journal of Mathematical Analysis and Its Applications,1966,15:243~252

备注/Memo

备注/Memo:
作者简介:张文红,女,1975年生,硕士,助教.
基金项目:国家自然科学基金资助项目(19971013)和江苏省自然科学基金资助项目(BK99001).
更新日期/Last Update: 2001-09-20