The order of convergence for Jarratt-type methods in solving nonlinear equations is determined without relying on Taylor expansion. Unlike previous studies, we depend solely on assumptions about the derivatives of the involved operator up to the second order. The proof presented in this paper is independent of the Taylor series expansion, thereby reducing the need for assumptions about derivatives of higher order of the involved operator and enhancing the applicability of these methods. The method’s applicability is broadened by employing the concept of generalized conditions in local convergence analysis and majorizing sequences in semi-local analysis. This study includes numerical examples and basins of attraction for the methods.

The convergence order of Jarratt-type methods for solving nonlinear equations are obtained without using the Taylor expansion. We use assumptions on the derivatives of the involved operator up to second order only contrary to the earlier studies. The proof provided in this paper does not depend on the Taylor series expansion which in turn reduces assumptions on the higher order derivatives of the involved operator and increases the applicability of these methods. The applicability of the method is further extended using the concept of generalized condition in the local convergence and majorizing sequences in the semi-local analysis. Numerical examples and Basins of attractions of the methods are provided in this study.