The main contribution of this manuscript is to introduce to the scientific community the concept of maximally efficient damped composed Newton-type method and the design of two schemes of this class of orders four and six. It is obtained from a different and new extension of the vectorial optimal fourth-order Ermakov's Hyperfamily, in the sense of Cordero-Torregrosa conjecture. We call this class biparametric CRTV Hyperfamily, which contains the uniparametric CRTV6Sλ class. We define concepts, such as maximally efficient and least-eval-cost, that allow us to classify new and existing methods in terms of effectiveness, as an alternative to the tools used to date. Finally, we analyze the efficiency of the proposed family and perform numerical tests with big-sized nonlinear problems, up to size 500, comparing with other methods of the same order, low computational cost or maximum efficiency. These tests confirm the validity of the proposed class of methods.

中文翻译：

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用于求解非线性方程组的最大有效阻尼组合牛顿型方法


这份手稿的主要贡献是向科学界介绍了最大效率阻尼组合牛顿型方法的概念以及此类 4 阶和 6 阶的两种方案的设计。它是从向量最优四阶 Ermakov 超家族的不同和新的扩展中获得的，在 Cordero-Torregrosa 猜想的意义上。我们称这个类为双参数 CRTV 超家族，它包含单参数 CRTV6Sλ 类。我们定义了诸如最高效率和最低评估成本等概念，这些概念使我们能够根据有效性对新方法和现有方法进行分类，作为迄今为止使用的工具的替代方案。最后，我们分析了所提出的族的效率，并对最大 500 个大小的大型非线性问题进行了数值测试，与其他同阶方法、低计算成本或最高效率进行了比较。这些测试确认了所建议的方法类的有效性。