求解奇异非线性方程组的牛顿不精确最小二乘算法

Inexact Newton-least Squares Methods for Singular Nonlinear System of Equations

对运用M-P逆建立的Newton迭代法做近似,构造不精确的算法.取Newton方程组的最小二乘解的近似解推导构造不精确的算法,结果可得到不精确Gauss-Newton算法和不精确Levenberg-Marquardt算法;用一迭代法计算雅可比矩阵的Moore-Penrose逆,截取它的一个近似矩阵构造不精确的算法,给出了近似程度的控制条件,证明了其收敛性;用雅可比矩阵的局部信息代替其全部信息构造不精确的算法,证明了算法的收敛性.数值例子也表明了不精确算法在求解大型方程组问题上的优越性.

Inexact methods of Newton＇s method constructed with Moore-Penrose inverse are given.First,inexact Gauss-Newton method and inexact Levenberg-Marquardt method are deduced by taking an approximate solution of the least squares solution of Newton equations.Second,the inexact method is constructed by taking an approximate matrix of Moore-Penrose inverse of Jacobian matrix,and its convergence is proved.Third,the inexact method is constructed by using local information instead of the whole information of the Jacobian matrix,and the convergence is proved.The numerical example also indicates its superiority in solving large system of equations.