Extended Seventh Order Derivative Free Family of Methods for Solving Nonlinear Equations

A plethora of applications from Computational Sciences can be identified for a system of nonlinear equations in an abstract space. These equations are mostly solved with an iterative method because an analytical method does not exist for such problems. The convergence of the method is established by sufficient conditions. Recently, there has been a surge in the development of high convergence order methods. Local convergence results reveal the degree of difficulty when choosing the initial points. However, these methods may converge even in cases not guaranteed by the conditions. Moreover, it is not known in advance how many iterations should be carried out to reach a certain error tolerance. Furthermore, no computable information is provided about the isolation of the solution in a certain region containing it. The aforementioned concerns constitute the motivation for writing this article. The novelty of the works is the expansion of the applicability of the method under ω − continuity conditions considered for the involved operator. The technique is demonstrated using a derivative-free seventh convergence three-step method. However, it was found that it can be used with the same effectiveness as other methods containing inverses of linear operators. The technique also uses information about the operators appearing in this method. This is in contrast to earlier works utilizing derivatives or divided differences not on the method which may not even exist for the problem at hand. The numerical experiments complement the theory.