牛顿法的一种新的迭代格式及其程序实现

A New Iteration of Newton Method and Implementation of the Program

牛顿迭代法是一种非常实用的计算方法，无论是函数零点的确定（它的特例是多项式求根）还是非线性方程组的求解。它在有关逼近一类的数值计算中实际上都有应用。本文提出了一种新的计算格式，用于求解任意次复系数多项式的所有零点。本文提出的这种方法，它的特点在于：我们不是用牛顿迭代法去直接逼近方程的根，而是用牛顿迭代法去逼近方程的二次因子。这样做的最大好处就是可以避免当逼近的根是复根时牛顿法所表现出来的动荡性。因此这种计算格式有很好的计算稳定性。这种方法的另一个特点是：它对迭代计算的初值要求不高。本文给出了两个实系数的情况和一个复系数时的情况。从计算结果看，这三个数值结果非常令人满意。

Newton iterate method is one of the very useful methods in approximating zero points of functions (especial in approximating the roots of polynomials). In this paper a new variant of the iterate method is established, and it can be applied to solution of the roots of polynomials with complex coefficients. The method presented in this paper is not to approximate directly with Newton iterate the roots of a polynomial but to approximate its second order factors . In this way one can avoid the unstable caused if the root approximated is a complex . Moreover, this iterate process needs little demand on the iterate initial data. Using this iterate process, satisfied results are obtain for three examples in this note, two are concerned with polynomials of real coefficients, one is concerned with a polynomial of complex coefficients.