Local Convergence Analysis of an Eighth Order Scheme Using Hypothesis Only on the First Derivative
In this paper, we propose a local convergence analysis of an eighth order three-step method to approximate a locally unique solution of a nonlinear equation in a Banach space setting. Further, we also study the dynamic behaviour of that scheme. In an earlier study, Sharma and Arora (2015) did not di...
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doaj-35dfb99812ba42a48887c6dbd2c756032020-11-25T00:49:16ZengMDPI AGAlgorithms1999-48932016-09-01946510.3390/a9040065a9040065Local Convergence Analysis of an Eighth Order Scheme Using Hypothesis Only on the First DerivativeIoannis K. Argyros0Ramandeep Behl1Sandile S. Motsa2Department of Mathematics Sciences Lawton, Cameron University, Lawton, OK 73505, USASchool of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X01, Pietermaritzburg 3209, South AfricaSchool of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X01, Pietermaritzburg 3209, South AfricaIn this paper, we propose a local convergence analysis of an eighth order three-step method to approximate a locally unique solution of a nonlinear equation in a Banach space setting. Further, we also study the dynamic behaviour of that scheme. In an earlier study, Sharma and Arora (2015) did not discuss these properties. Furthermore, the order of convergence was shown using Taylor series expansions and hypotheses up to the fourth order derivative or even higher of the function involved which restrict the applicability of the proposed scheme. However, only the first order derivatives appear in the proposed scheme. To overcome this problem, we present the hypotheses for the proposed scheme maximum up to first order derivative. In this way, we not only expand the applicability of the methods but also suggest convergence domain. Finally, a variety of concrete numerical examples are proposed where earlier studies can not be applied to obtain the solutions of nonlinear equations on the other hand our study does not exhibit this type of problem/restriction.http://www.mdpi.com/1999-4893/9/4/65Kung-Traub methodlocal convergencedivided differenceBanach spaceLipschitz constantradius of convergence |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Ioannis K. Argyros Ramandeep Behl Sandile S. Motsa |
spellingShingle |
Ioannis K. Argyros Ramandeep Behl Sandile S. Motsa Local Convergence Analysis of an Eighth Order Scheme Using Hypothesis Only on the First Derivative Algorithms Kung-Traub method local convergence divided difference Banach space Lipschitz constant radius of convergence |
author_facet |
Ioannis K. Argyros Ramandeep Behl Sandile S. Motsa |
author_sort |
Ioannis K. Argyros |
title |
Local Convergence Analysis of an Eighth Order Scheme Using Hypothesis Only on the First Derivative |
title_short |
Local Convergence Analysis of an Eighth Order Scheme Using Hypothesis Only on the First Derivative |
title_full |
Local Convergence Analysis of an Eighth Order Scheme Using Hypothesis Only on the First Derivative |
title_fullStr |
Local Convergence Analysis of an Eighth Order Scheme Using Hypothesis Only on the First Derivative |
title_full_unstemmed |
Local Convergence Analysis of an Eighth Order Scheme Using Hypothesis Only on the First Derivative |
title_sort |
local convergence analysis of an eighth order scheme using hypothesis only on the first derivative |
publisher |
MDPI AG |
series |
Algorithms |
issn |
1999-4893 |
publishDate |
2016-09-01 |
description |
In this paper, we propose a local convergence analysis of an eighth order three-step method to approximate a locally unique solution of a nonlinear equation in a Banach space setting. Further, we also study the dynamic behaviour of that scheme. In an earlier study, Sharma and Arora (2015) did not discuss these properties. Furthermore, the order of convergence was shown using Taylor series expansions and hypotheses up to the fourth order derivative or even higher of the function involved which restrict the applicability of the proposed scheme. However, only the first order derivatives appear in the proposed scheme. To overcome this problem, we present the hypotheses for the proposed scheme maximum up to first order derivative. In this way, we not only expand the applicability of the methods but also suggest convergence domain. Finally, a variety of concrete numerical examples are proposed where earlier studies can not be applied to obtain the solutions of nonlinear equations on the other hand our study does not exhibit this type of problem/restriction. |
topic |
Kung-Traub method local convergence divided difference Banach space Lipschitz constant radius of convergence |
url |
http://www.mdpi.com/1999-4893/9/4/65 |
work_keys_str_mv |
AT ioanniskargyros localconvergenceanalysisofaneighthorderschemeusinghypothesisonlyonthefirstderivative AT ramandeepbehl localconvergenceanalysisofaneighthorderschemeusinghypothesisonlyonthefirstderivative AT sandilesmotsa localconvergenceanalysisofaneighthorderschemeusinghypothesisonlyonthefirstderivative |
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