An extended study of two multistep higher order convergent methods for solving nonlinear equations

Abstract

The local analysis of convergence for two competing methods of order seven or eight is developed to solve Banach space-valued equations. Previous studies have used the eighth or ninth derivative of the operator involved, which do not appear on the methods, to show the convergence of these methods on the finite-dimensional Euclidean space. In addition, no computable error distances or isolation of the solution results are provided in this study. These problems limit the applicability of this method to solving equations with operators that are at least nine times differentiable. In the current study, only conditions on the first derivative, appearing in these methods, are employed to show the convergence of these methods. Moreover, computable error bounds depend on the distance in the world as well as the isolation of the solution. Results are provided based on generalized continuity conditions on the first derivative. Furthermore, the more interesting semi-local analysis of convergence not given before these methods is presented using a majorizing sequence. Finally, a great deal of impressive numerical results has been shown on real-world problems.