无导数的大范围平方收敛迭代算法及其数值分析

应用迭代法的困难之处在于：一是迭代公式是否存在导数运算；二是初始值选定是否影响迭代公式的收敛性；三是收敛快速和达到需要精度等问题。我们获得了求方程根不用计算导数的平方收敛迭代公式，并设计了求根的大范围收敛算法，编写了C^ 语言程序，进行了算法和数值分析。与其它算法比较，该算法具有无导数计算、初值任意选定、平方快速收敛、大范围收敛和双精度控制(根的精度和函数值的精度控制)等优点。

A DERIVATIVE-FREE ITERATIVE ALGORITHM WITH GLOBAL SECOND-ORDER CONVERGENCE AND ITS NUMERICAL ANALYSES

The difficulties in using iterative algorithms lay as the following: At first, derivative feasibility; Secondly, sensitivity of iterative convergenc e to initial values; thirdly, the convergent rate and the iterative precision. T he paper presents a derivative-free iterative formula with second-order converge nt rate. Based upon it, a global convergent algorithm for the roots of equatio ns is designed using C ++ programming. The numerical analysis show that i t has several features comparing with conventional iterative algorithms, such as de rivative-free, robustness to the selection of the initial values, high convergen t rate, the global optimization and double precision control (root and function al precisions).