Newton-like Normal S-iteration under Weak Conditions

In the present paper, we introduced a quadratically convergent Newton-like normal S-iteration method free from the second derivative for the solution of nonlinear equations permitting f′(x)=0 at some points in the neighborhood of the root. Our proposed method works well when the Newton method fails and performs even better than some higher-order converging methods. Numerical results verified that the Newton-like normal S-iteration method converges faster than Fang et al.’s method. We studied different aspects of the normal S-iteration method regarding the faster convergence to the root. Lastly, the dynamic results support the numerical results and explain the convergence, divergence, and stability of the proposed method.