关于Newton迭代公式的几个改进

Several variations of Newton iteration method

通过对Newton迭代公式进行改进,本文构造了三种新的迭代公式。迭代公式I是一种单步迭代公式,在单根附近具有二阶收敛速度,且无须求函数的导数值;迭代公式II也是一种单步迭代公式,在单根附近具有三阶收敛速度;迭代公式III是一种两步迭代公式,具有至少三阶收敛速度,虽然该公式形式比较复杂,但是具有计算时不需求函数的导数值的优点。此外,证明了三种新的迭代公式的收敛性。最后,通过数值实验验证了三种迭代公式的有效性。

Based on Newton iterative method,three new kinds of iterative methods are consfructed in this paper. The first iterative formula （Ⅰ） is a one - step iterative formula. It has second order convergence rate at single root, and needn＇ t evaluate derivative of function. The second iterative formula （Ⅱ） is also one - step iterative formula and has three order convergence at single root. The third iterative formula （Ⅲ） is a two -step iterative formula. There is at least three order convergence rate. Although iterative formula （Ⅲ） is complex, it has advantage of free - computing derivative of function. Moreover, the convergence of three kinds of methods is proved. Finally, some numerical experiments are given, and numerical results are satisfied.