high-order method, local convergence, nonlinear equation, chebyshev-type method, radius of convergence, Mathematics, QA1-939, Applied mathematics. Quantitative methods, T57-57.97

In this paper, the local convergence of a high-order Chebyshev-type method without the second derivative is studied. We study the convergence under $ \omega $-continuity conditions based on the first derivative. The uniqueness of the solution and the radii of convergence domains are obtained. In contrast to the conditions used in previous studies, the new conditions of convergence are weaker. In addition, the attractive basins of the family with different parameters are studied, which can show the different stability of the family. Finally, in numerical experiments, the iterative method is used to solve different nonlinear models, including vertical stresses, civil engineering problem, blood rheology model, and so on. Theoretical results of convergence criteria are verified.