Finding higher-order optimal derivative-free methods for multiple roots $(m\ge 2)$ of nonlinear expressions is one of the most fascinating and difficult problems in the area of numerical analysis and Computational mathematics. In this study, we introduce a new fourth order optimal family of Ostrowski’s method without derivatives for multiple roots of nonlinear equations. Initially the convergence analysis is performed for particular values of multiple roots—afterwards it concludes in general form. Moreover, the applicability and comparison demonstrated on three real life problems (e.g., Continuous stirred tank reactor (CSTR), Plank’s radiation and Van der Waals equation of state) and two standard academic examples that contain the clustering of roots and higher-order multiplicity $(m=100)$ problems, with existing methods. Finally, we observe from the computational results that our methods consume the lowest CPU timing as compared to the existing ones. This illustrates the theoretical outcomes to a great extent of this study.

中文翻译：

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非线性方程多重根的最优无导数Ostrowski格式

寻找多个根的高阶最优无导数方法 $（米\ge 2）$非线性表达式的求解是数值分析和计算数学领域中最引人入胜和最困难的问题之一。在这项研究中，我们为非线性方程的多个根引入了一种新的无导数的Ostrowski方法的四阶最优族。最初，对多个根的特定值执行收敛分析，然后以一般形式得出结论。此外，在三个现实生活问题（例如，连续搅拌釜反应器（CSTR），普朗克辐射和范德华状态方程）和两个包含根的聚类和高阶多重性的标准学术示例中证明了适用性和比较性$（米=100）$问题，使用现有方法。最后，从计算结果可以看出，与现有方法相比，我们的方法消耗的CPU时间最少。这在很大程度上说明了这项研究的理论成果。