In this paper, we present a new optimal derivative free scheme of eighth-order methods without memory in a general way. The advantage of our scheme over the earlier iteration functions, it is applicable to every optimal fourth-order derivative free scheme whose first sub step should be Steffensen’s type method to develop more advanced optimal iteration techniques of order eight. In addition, the theoretical convergence properties of our schemes are fully explored with the help of main theorem that demonstrate the convergence order. Each member of the proposed scheme satisfies the classical Kung and Traub conjecture which is related to multi-point iterative methods without memory. On the basis of average number of iterations required per point and the number of points requiring 40 iterations, we confirmed that our methods are more effective and comparable to the existing robust optimal eighth-order derivative free methods. Further, the dynamical study of these methods also supports the theoretical aspects.

中文翻译：

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一类非线性方程的最优八阶无导数方法

在本文中，我们以一般的方式提出了一种新的无记忆八阶方法的最优无导数方案。我们的方案优于较早的迭代函数，它适用于每一个最优四阶无导数方案，其第一个子步骤应该是 Steffensen 类型的方法，以开发更先进的八阶最优迭代技术。此外，在证明收敛顺序的主定理的帮助下，我们充分探索了我们方案的理论收敛特性。所提出方案的每个成员都满足经典的 Kung 和 Traub 猜想，该猜想与无记忆的多点迭代方法有关。在平均每个点所需的迭代次数和需要40次迭代的点数的基础上，我们证实我们的方法更有效，并且与现有的鲁棒最优八阶无导数方法相比具有可比性。此外，这些方法的动态研究也支持理论方面。