Local and semilocal convergence for a family of methods without Jacobian in Hilbert space

For the solution of equations, many derivative-free methods with and without memory are proposed in the literature. In the scalar situation, the convergence order of the methods is determined by the use of Taylor expansions based on the conjectures on higher-order derivatives despite the fact that such derivatives do not appear in the methods. Therefore, the methods are constrained in this way. For this reason, the assessment of local and semi-local convergence of a derivative-free algorithm is presented in this study utilizing just the divided differences of order one, which are present in the method. Additionally, we present computable error distances and the uniqueness of the solution results. The article also includes numerical experiments to support the theoretical findings.