Three-Step Derivative-Free Method of Order Six

Derivative-free iterative methods are useful to approximate the numerical solutions when given function lacks explicit derivative information or derivatives too expensive compute. Exploring convergence properties of such is crucial in their development. The behavior approaches and determining practical applicability require conducting local as well semi-local analysis. In this study, we explore a sixth-order derivative-free method. Previous studies assumed existence high order even method itself was not utilizing any derivatives. These assumptions imposed limitations on its applicability. paper, extend analysis by providing estimates for error bounds Consequently, expands across broader range problems. Moreover, more important challenging investigated earlier also developed. Additionally, survey recent advancements field. outcomes presented paper can be proved valuable practitioners researchers engaged development algorithms. Numerical tests illuminate validate further theoretical results.