High Order Three-Steps Newton Raphson-like Schemes for Solving Nonlinear Equation Systems

Abstract

This study proposes several new 3-steps schemes based on the Newton-Raphson method for solving non-linear equation systems. The proposed schemes are analysed and formulated based on the Newton-Raphson method and the Newtoncotes open form numerical integration method. In general, the schemes can be considered as a predictor and corrector principles. In the first and the third steps, the Newton-Raphson method is applied. Furthermore, Newton-cotes Open Form numerical integration modification is operated in the second step of the proposed schemes. The convergence analysis of the proposed schemes is given. It shows that the proposed scheme provides the 8th order of convergence. The performance of the proposed schemes is compared and assessed with several numerical examples.