In this article, we construct an optimal family of iterative methods for finding the single root and then extend this family for determining all the distinct as well as multiple roots of single-variable nonlinear equations simultaneously. Convergence analysis is presented for both the cases to show that the optimal order of convergence is 4 in the case of single root finding methods and 6 for simultaneous determination of all distinct as well as multiple roots of a nonlinear equation. The computational cost, basins of attraction, efficiency, log of residual, and numerical test examples show that the newly constructed methods are more efficient as compared to the existing methods in the literature.

中文翻译：

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非线性方程迭代技术的动力学及其在工程中的应用

在本文中，我们构建了一个最优的迭代方法族，以查找单个根，然后扩展了该族，以便同时确定单变量非线性方程的所有不同根以及多个根。两种情况都进行了收敛分析，以表明在单根查找方法的情况下，收敛的最佳阶数为4，而在同时确定非线性方程的所有不同根和多根时，收敛的最佳阶数为6。计算成本，吸引盆，效率，残差的对数以及数值测试示例表明，与文献中的现有方法相比，新构建的方法效率更高。