Broadening the convergence domain of Seventh-order method satisfying Lipschitz and H\"{o}lder conditions

TL;DR

This paper extends the convergence analysis of a seventh-order method for solving nonlinear equations, relaxing traditional assumptions by using only first-order derivatives and covering Lipschitz and Hölder conditions.

Contribution

It provides a novel convergence analysis that avoids Taylor expansion, broadening the applicability of the method under less restrictive derivative conditions.

Findings

Derived radii of convergence balls

Established error bounds in terms of distances

Confirmed convergence through numerical examples

Abstract

In this paper, the local convergence analysis of the multi-step seventh order method is presented for solving nonlinear equations assuming that the first-order Fr\'echet derivative belongs to the Lipschitz class. The significance of our work is that it avoids the standard practice of Taylor expansion thereby, extends the applicability of the scheme by applying the technique based on the first-order derivative only. Also, this study provides radii of balls of convergence, the error bounds in terms of distances in addition to the uniqueness of the solution. Furthermore, generalization of this analysis satisfying H\"{o}lder continuity condition is provided since it is more relaxed than Lipschitz continuity condition. We have considered some numerical examples and computed the radii of the convergence balls.