A correction note on "Three-step iterative methods for nonlinear equations" and generalization of method

TL;DR

This paper corrects and clarifies the mathematical derivation of a three-step iterative method for solving nonlinear equations, providing a valid proof of its convergence order and generalizing the method.

Contribution

It rectifies errors in the original derivation, offers a correct proof of convergence order, and extends the method to a more general form.

Findings

Corrected proof of cubic convergence order

Validated the convergence through computational results

Generalized the iterative method

Abstract

In the paper [Muhammad Aslam Noor, Khalida Inayat Noor, Three-step iterative methods for nonlinear equations, Applied Mathematics and Computation, 183 (2006), pp. 322-327 ], Authors presented an algorithm (\textbf{Algorithm 2.3}) and stated a theorem (\textbf{Theorem 2.3}) to prove the cubic order of convergence but the given proof does not show cubic order of convergence. Actually, the mathematical derivation steps to develop the \textbf{Algorithm 2.3} are wrong. In this note, we present the correct mathematical developments and finally provide computational order of convergence in the favor of our claim and provide the generalization of the method.