Extending the Applicability of a Seventh Order Method Without Inverses of Derivatives Under Weak Conditions

Abstract

The novelty of this article, lies in the fact that we extend the applicability of an efficient seventh convergence order method for solving equations. The convergence of the method was shown in earlier works using derivatives up to order eight, and only in the $$k-$$dimensional Euclidean space setting. Moreover, no computable error estimates or uniqueness of the solution results were given. Hence, the applicability of method is very limited. We address all these problems, and in the more general setting of a Banach space. In particular, we only use hypotheses on the first derivative. Furthermore, computable error bounds and uniqueness results are also presented. Finally, numerical examples are used to show that our results apply to solve equations in cases not possible before.