On the Semi-Local Convergence of a Fifth-Order Convergent Method for Solving Equations

Christopher I Argyros,Ioannis K Argyros,Stepan Shakhno,Halyna Yarmola

doi:10.3390/foundations2010008

Abstract

We study the semi-local convergence of a three-step Newton-type method for solving nonlinear equations under the classical Lipschitz conditions for first-order derivatives. To develop a convergence analysis, we use the approach of restricted convergence regions in combination with majorizing scalar sequences and our technique of recurrent functions. Finally, a numerical example is given.

Highlights

Let us consider an equation G(x) = 0. (1)Here, G : Ω ⊂ X → Y is a nonlinear Fréchet-differentiable operator, X and Y are Banach spaces, Ω is an open convex subset of X
To study the multi-step method, it is often required that the operator F be a sufficiently differentiable function in a neighborhood of solutions
That is why we develop a semi-local convergence analysis of Method (2) under classical Lipschitz conditions for first-order derivatives only

Summary

Introduction

To study the multi-step method, it is often required that the operator F be a sufficiently differentiable function in a neighborhood of solutions. That is why we develop a semi-local convergence analysis of Method (2) under classical Lipschitz conditions for first-order derivatives only. There is a plethora of single, two-step, three-step, and multi-step methods whose convergence has been shown using the second or higher-order derivatives or divided differences [1,2,3,5,6,7,8,9,10,12].

Majorizing Sequence

Numerical Example

Conclusions