非线性最小二乘数值求解中Hesse矩阵的快速计算方法

A Rapid Calculating Method of Hesse Matrix in the Numerical Calculation Process of Nonlinear Least-Square

非线性最小二乘数值求解过程中,残差函数非线性程度较高时,必须计算Hesse矩阵,而其工作量往往很大的。通常的作法是使用拟牛顿法,用一阶导数信息来逼近二阶导数,不仅不能获得准确的数值,而且计算繁琐。通过对Hesse矩阵结构的分析,采用符号运算求导,编译执行求值和向量化传递参数,达到了快速计算Hesse矩阵的目的,使对于大数据量直接使用牛顿法进行非线性最小二乘数值求解成为可能。数值试验的结果表明了该方法的可行性。

In the numerical calculation process of nonlinear least-square, Hesse matrix should be calculated when residual function has more nonlinear degree,but its workload is huge. Usually, Quasi-Newton method is adopted in the situation, using first derivative to approximate second derivative , and its result is not only inexact, but also difficult to calculate. Based on analysis of structure of the matrix, a new method is adopted by using symbol calculation to get derivative, complied executing to get value and transferring parameters in forms of vector, the rapid of this method make it possible to solve problem of nonlinear least-square with huge data by using Newton methd directly. The numerical experiment show its feasibility.