非线性病态问题解算的扰动分析

The Perturbation Analysis of Nonlinear Ill-conditioned Solution

非线性病态问题除了要分析模型本身的病态性外,还要考虑非线性最小二乘算法中初始值选取、正交化过程等造成的扰动对病态分析的影响。在广义条件数定义的基础上,推导非线性病态法方程解的扰动估计式,并以扰动估计式为基础,讨论非线性算法对非线性病态问题判断和分析的影响,产生影响的原因主要包括线性近似时系数矩阵的扰动、附加的截断误差及近似正交过程。通过两个实例进行验证,并对结果加以讨论和分析,分析计算表明:当模型的非线性程度较大时,选取不同初始值及近似正交过程可导致线性近似引起的系数矩阵扰动和截断误差很大,可采用正则化迭代解法。

In addition to the ill-conditioned analysis of the nonlinear model itself,the perturbation stemmed from truncation and orthogonalization process needs to be considered.In this paper,the perturbative estimation inequality of nonlinear ill-conditioned problem was derived from the definition of the generalized condition number.And based on the perturbative estimation inequality,the impacts of judgement and analysis were studied on nonlinear ill-conditioned problem.The impacts stemmed primarily from the disturbance of the coefficient matrix of linear approximation and additional truncation error,also including orthogonal approximation.Therefore,nonlinear ill-conditioned problem was verified and analyzed through two examples.The results of the research show that： when the model nonlinearity is strong,the disturbance of the coefficient matrix and additional truncation error are very significant due to the selection of different approximation values and orthogonal approximation.In this case,iterative regularization method can be used to solve nonlinear ill-conditioned problems.