Maximally efficient damped composed Newton-type methods to solve nonlinear systems of equations

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.