非线性最小二乘问题的一种正则同伦迭代解法  

A regularization homotopy iterative method for solving nonlinear least square problem

作　　者：唐利民[1,2] 

机构地区：[1]中南大学信息物理工程学院,长沙410083 [2]长沙理工大学公路工程学院,长沙410076

出　　处：《工程勘察》2009年第10期66-70,共5页Geotechnical Investigation & Surveying

摘　　要：当非线性最小二乘问题的数值迭代解算方法其Jacobian矩阵是秩亏或者严重病态时,诸多方法如高斯-牛顿法、修正高斯-牛顿法等将会失效。本文结合同伦延拓和正则化方法,构造了正则同伦函数min(α‖f(x)-L‖2+(1-α)‖x-x0‖2)来解算Jacobian矩阵是秩亏或者严重病态的非线性最小二乘问题。采用将f(x)线性化的策略,建立了非线性最小二乘问题正则同伦方法迭代公式,对其迭代过程进行了详细的推导,给出了其连续性和收敛性的条件。对两个非线性最小二乘问题和一个非线性秩亏自由网平差实例进行了解算,结果表明本文所提出方法是正确和适用的。When the Jacobian matrix is rank-deficient or very ill-conditioned in numerical iterative process to solve the nonlinear least square problem,more methods with standard approaches such as the gauss-newton method or modified gauss-newton method will be in failure.Combined with homotopy continuation and regularization method,this paper constructs a regularization homotopy function： min（α‖f（x）-L‖^2＋（1-α）‖x-x_0‖^2） to solve this problem,in which Jacobian matrix is rank-deficient or very ill-conditioned.A regularization homotopy iterative formula is established for nonlinear least square problem by linearized f（x） in the paper. The iterative process is derived in detial. The continuity and convergence conditions are also given. The calculation results of two nonlinear least square problems and one nonlinear adjustment of free network with rank deficiency example show that the method proposed in this paper is correct and applicable.

关 键 词：非线性最小二乘问题 数值迭代 正则同伦 病态 非线性秩亏自由网平差 

分 类 号：P207.2[天文地球—测绘科学与技术]