Local Convergence of an Efficient Multipoint Iterative Method in Banach Space

Janak Raj Sharma,Ioannis K Argyros,Sunil Kumar

doi:10.3390/a13010025

Janak Raj Sharma, Ioannis K Argyros + Show 1 more

Open Access

https://doi.org/10.3390/a13010025

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Abstract

We discuss the local convergence of a derivative-free eighth order method in a Banach space setting. The present study provides the radius of convergence and bounds on errors under the hypothesis based on the first Fréchet-derivative only. The approaches of using Taylor expansions, containing higher order derivatives, do not provide such estimates since the derivatives may be nonexistent or costly to compute. By using only first derivative, the method can be applied to a wider class of functions and hence its applications are expanded. Numerical experiments show that the present results are applicable to the cases wherein previous results cannot be applied.

Highlights

We study local criteria for obtaining a unique solution u∗ of the nonlinear model
For Banach space valued mappings with F : Ω ⊂ B → B, where F is differentiable in the sense of Fréchet [1,2]
The method is of order eight using only divided differences, derivatives up to the order nine and Taylor expansions in the special case when B = Ri and Q(u) = f 1m (u), f 2m (u), · · ·, f im (u), m ≥ 2, where Q : B → B, Q(u∗ ) = F (u∗ ) = 0

Summary

Introduction

The method is of order eight using only divided differences, derivatives up to the order nine and Taylor expansions in the special case when B = Ri and Q(u) = f 1m (u), f 2m (u), · · · , f im (u) , m ≥ 2, where Q : B → B, Q(u∗ ) = F (u∗ ) = 0. We study this method in the more general setting of a Banach space setting: Algorithms 2020, 13, 25; doi:10.3390/a13010025 www.mdpi.com/journal/algorithms

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