Abstract

A new optimal two-parameter class of derivative-free iterative methods with the application to the Hansen–Patrick type iterations is developed. Using self-accelerating parameters, new higher order methods with memory are obtained. Exact analytical formulas for the optimal values of the parameters are found for the first time. The convergence order is increased from four to seven without any additional computations. Thus, the proposed methods with memory have a high computational efficiency. Numerical examples and comparison with some other available methods confirm the theoretical results and high computational efficiency.

中文翻译：

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求解非线性方程的两点迭代法中参数的最优选择

摘要

开发了一种新的最优两参数类无导数迭代方法，并将其应用于Hansen-Patrick类型迭代。使用自加速参数，可以获得具有存储器的新的高阶方法。首次找到参数最佳值的精确解析公式。收敛阶数从四个增加到七个，而无需任何其他计算。因此，所提出的带有存储器的方法具有很高的计算效率。数值算例和与其他可用方法的比较证实了理论结果和较高的计算效率。