A Sixth and seventh-order derivative-free parameter-based iterative method for solving non-linear equations

In this research, for solving non-linear equations we suggest a two-second derivative-free parameter-based continuation method. As per the Kung-Traub conjecture, both of the proposed schemes are optimal. Additionally, two main theorems that demonstrates the order of convergence and asymptotic error constant, as well as the theoretical and computational characteristics of the suggested schemes, are thoroughly studied. The order of convergence of proposed method is seven for α = 0 and six for α = 1. The performance and effectiveness of the suggested optimal approaches are compared with their closest competitors on a concrete variety of nonlinear equations with the help of Mathematica-11.3, by using its high precision compatibility. Finally, based on the results obtained, our suggested approaches are shown to be more effective than comparable robust procedures of the same order.