Abstract We derive two iterative methods with memory for approximating a simple root of any nonlinear equation. For this purpose, we take two optimal methods without memory of order four and eight and convert them into the methods with memory without increasing any further function evaluation. These methods involve a self-accelerator (parameter) that depends upon the iteration index to increase the order of the optimal methods. Consequently, the efficiency of the new methods is considerably high as compared to the methods without memory. Some numerical examples are provided in support of the theoretical results.

中文翻译：

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具有较高收敛 R 阶记忆的导数自由迭代方法

摘要 我们推导出两种具有记忆的迭代方法，用于逼近任何非线性方程的简单根。为此，我们采用了四阶和八阶无记忆的两种最优方法，并将它们转换为有记忆的方法，而不增加任何进一步的函数评估。这些方法涉及一个自加速器（参数），它依赖于迭代索引来增加最优方法的阶数。因此，与没有内存的方法相比，新方法的效率相当高。提供了一些数值例子来支持理论结果。