基于拟牛顿方程的非线性最小二乘的新算法  

A NEW NON-LINEAR LEAST SQUARE ALGORITHM BASED ON QUASI-NEWTON TYPE'S EQUATION

作　　者：孙风建[1] 肖伟[2] 

机构地区：[1]南京师范大学附属中学,南京210003 [2]南京理工大学理学院,南京210094

出　　处：《高等学校计算数学学报》2010年第4期333-344,共12页Numerical Mathematics A Journal of Chinese Universities

摘　　要：我们知道,非线性最小二乘问题： minf（x）=1/2R（x）^TR（x）=1/2m∑i=1 [ri（x）]^2,（1）其中x∈R^n称为决策变量,R（x）=（r1（x）,r2（x）,…,rm（x））^T称为在点x的残向量,目标函数f（x）的梯度和海森矩阵分别为：This paper employs the new quasi-Newton equation proposed by Zhang in 2001 with the tensor method,which is a Huang＇s quasi-Newton.With the combination of the dual principle and the structure principle of the secant method, Chen＇s algorithm is generalized to the whole Broyden＇s and two corresponding algorithms are presented in the first part.It is proved in the second part that the algorithms,invariant under an orthogonal matrix transformation on variables, possess local and super-linear convergence.Numerical experiments show that our algorithms are superior to Chen＇s.In addition,improvements of the algorithms are also given to enable them to be immune of the influence of the initial point and each iteration point,which effectively expand the application scope of such algorithms.

关 键 词：非线性最小二乘问题 拟牛顿方程 算法 决策变量 目标函数 R^N 向量 矩阵 

分 类 号：O241[理学—计算数学] O242[理学—数学]