In this paper, a class of efficient iterative methods with increasing order of convergence for solving systems of nonlinear equations is developed and analyzed. The methodology uses well-known third-order Potra–Pták iteration in the first step and Newton-like iterations in the subsequent steps. Novelty of the methods is the increase in convergence order by an amount three per step at the cost of only one additional function evaluation. In addition, the algorithm uses a single inverse operator in each iteration, which makes it computationally more efficient and attractive. Local convergence is studied in the more general setting of a Banach space under suitable assumptions. Theoretical results of convergence and computational efficiency are verified through numerical experimentation. Comparison of numerical results indicates that the developed algorithms outperform the other similar algorithms available in the literature, particularly when applied to solve the large systems of equations. The basins of attraction of some of the existing methods along with the proposed method are given to exhibit their performance.

中文翻译：

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一类非线性方程组的高阶类牛顿法

本文开发和分析了一类有效的收敛阶递增求解非线性方程组的迭代方法。该方法在第一步中使用众所周知的三阶 Potra-Pták 迭代，在后续步骤中使用类牛顿迭代。这些方法的新颖之处在于，收敛顺序每一步增加了三个，而代价是只需要一个额外的函数评估。此外，该算法在每次迭代中使用单个逆算子，这使得它在计算上更加高效和有吸引力。在适当的假设下，在更一般的 Banach 空间设置中研究局部收敛。通过数值实验验证了收敛性和计算效率的理论结果。数值结果的比较表明，所开发的算法优于文献中可用的其他类似算法，特别是在应用于求解大型方程组时。给出了一些现有方法的吸引盆以及所提出的方法以展示它们的性能。