一种新的迭代收敛阶数的证明与推广

A New Proof of Convergence Order for an Iterative Method and Generlization

迭代方法是求解非线性方程近似根的重要方法.本文基于隐函数存在定理,提出了一种新的迭代方法收敛性和收敛阶数的证明方法,并分别对牛顿(Newton)和柯西(Cauchy)迭代方法迭代收敛性和收敛阶数进行了证明.最后,利用本文提出的证明方法,证明了基于三次泰勒(Taylor)展式构成的迭代格式是收敛的,收敛阶数至少为4,并提出猜想,基于n次泰勒展式构成的迭代格式是收敛的,收敛阶数至少为(n+1).

Iterative method is an important technique to solve the approximate root for nonlinear equations. In this article, with the implicit function theorem, we present a new method for proving the convergence and deriving the convergence order for an iterative method. With these new method, we demonstrate that the Newton iteration method and the Cauchy iteration method are convergent and their convergent order are at least 2 and 3 respectively. Then based on the Taylor expansion of degree of 3, we present a local convergent iteration method, which has the convergence order at least 4. At last, we propose a guess that the iteration method, based on the Taylor expansion of degree of n, is convergent and its convergence order is at least （n＋ 1）.