A new fifth-order iterative method free from second derivative for solving nonlinear equations

In this recent work, a new two-step iterative method for solving nonlinear equations that have a fifth-order convergence is suggested and analyzed. This new iterative method is free from second derivative of functions and based on Halley's method and Taylor's expansion together by using Hermite orthogonal polynomials basis to implement a suitable approximation of second derivative of functions. In addition, the order of convergence and the corresponding error equations of the new method are proved. Finally, some numerical examples are given to show the efficiency and the performance of the new method as well as a comparison with the original well-known Newton's method and some other relevant methods are illustrated.

Automated summary

In this recent work, a new two-step iterative method with fifth-order convergence for solving nonlinear equations is suggested and analyzed. The method is free from second derivatives of functions, based on Halley's method and Taylor's expansion using Hermite orthogonal polynomials basis for approximating second derivatives. The convergence order and error equations of the method are proven, and numerical examples show its efficiency compared to Newton's method and other relevant methods.