In the recent literature, very few high-order Jacobian-free methods with memory for solving nonlinear systems appear. In this paper, we introduce a new variant of King’s family with order four to solve nonlinear systems along with its convergence analysis. The proposed family requires two divided difference operators and to compute only one inverse of a matrix per iteration. Furthermore, we have extended the proposed scheme up to the sixth-order of convergence with two additional functional evaluations. In addition, these schemes are further extended to methods with memory. We illustrate their applicability by performing numerical experiments on a wide variety of practical problems, even big-sized. It is observed that these methods produce approximations of greater accuracy and are more efficient in practice, compared with the existing methods.

中文翻译：

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国王家族解决非线性系统的新变种中的记忆

在最近的文献中，很少有带有记忆的用于解非线性系统的高阶无雅可比方法。在本文中，我们介绍了King族的一个新变种，其中有四阶可以解决非线性系统及其收敛性分析。所提出的族需要两个分开的差分算子，并且每次迭代仅计算矩阵的一个逆。此外，我们还通过两个附加功能评估将提议的方案扩展到收敛的六阶。另外，这些方案被进一步扩展到具有存储器的方法。我们通过对各种实际问题（甚至大型问题）进行数值实验来说明它们的适用性。可以看出，与现有方法相比，这些方法产生的精度更高，并且在实践中效率更高。