4阶收敛的史蒂芬森型迭代格式(英文)

A Steffensen Type Method with Fourth-order Convergence

针对非线性方程求根问题,提出了一种4阶收敛的史蒂芬森型方法.在迭代过程中新方法不需要计算任何导数,仅仅需要计算3个函数值,就可达到4阶收敛.该方法的计算效率为1.587.依据Kung与Traub提出的假设,即若一个迭代法在迭代过程中需要计算n个函数值,则该方法能达到最优收敛阶为2n-1.可知当n=3时,新方法是最优的.数值试验进一步证明了该方法的收敛性.

In this paper, a Steffensen type method of fourth-order convergence for solving nonlinear equations is suggested. The derivative-free method only uses three evaluations of the function per iteration to achieve fourth- order convergence. Therefore, the new method has an efficiency index equal to 1. 587. Kung and Traub conjectured that a multipoint iteration without memory based on n evaluations could achieve optimal convergence order 2^n- 1 The new method agrees with Kung-Traub conjecture for n = 3. Numerical comparisons are made to show the performance of the presented method, as shown in the illustration examples.