非精确修正牛顿法

Inexact Modified Newton's Methods

牛顿法是求解非线性方程组的经典的高阶算法 .当xk 远离解x 时 ,实际上不必花费庞大的工作量以求解大型线性方程组 (牛顿方程组 )F′(xk)sk=-F(xk)的精确解 .类似地 ,F′(xk)也可以被某些简便的近似值所替代 .因此 ,本文讨论非精确修正牛顿法 ,在自然合理的条件下 。

Newton's method is a classical algorithm with a high order of convergence for solving systems of nonlinear equations. However,it is not justified to find an exact solution to a large system of linear equations F′(x k)s k=-F(x k) (Newton equations) with massive work especially when x k is far from root x *. Similarly, F′(x k ) should be replaced by some simpler approximation, too. On this point, inexact modified Newton's methods are discussed in this paper. Linear convergence for inexact Newton's methods and inexact modified Newton's methods are proved under natural and reasonable conditions respectively.