Local Convergence of an Efficient High Convergence Order Method Using Hypothesis Only on the First Derivative

We present a local convergence analysis of an eighth order three step methodin order to approximate a locally unique solution of nonlinear equation in a Banach spacesetting. In an earlier study by Sharma and Arora (2015), the order of convergence wasshown using Taylor series expansions and hypothese...

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Main Authors: Ioannis K. Argyros, Ramandeep Behl, S.S. Motsa
Format: Article
Language:English
Published: MDPI AG 2015-11-01
Series:Algorithms
Subjects:
Online Access:http://www.mdpi.com/1999-4893/8/4/1076
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spelling doaj-fe8d3aa35146477792c263051aecc5582020-11-24T21:53:01ZengMDPI AGAlgorithms1999-48932015-11-01841076108710.3390/a8041076a8041076Local Convergence of an Efficient High Convergence Order Method Using Hypothesis Only on the First DerivativeIoannis K. Argyros0Ramandeep Behl1S.S. Motsa2Department of Mathematics Sciences, Cameron University, Lawton, OK 73505, USASchool of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X01, Scottsville 3209, Pietermaritzburg, South AfricaSchool of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X01, Scottsville 3209, Pietermaritzburg, South AfricaWe present a local convergence analysis of an eighth order three step methodin order to approximate a locally unique solution of nonlinear equation in a Banach spacesetting. In an earlier study by Sharma and Arora (2015), the order of convergence wasshown using Taylor series expansions and hypotheses up to the fourth order derivative oreven higher of the function involved which restrict the applicability of the proposed scheme. However, only first order derivative appears in the proposed scheme. In order to overcomethis problem, we proposed the hypotheses up to only the first order derivative. In this way,we not only expand the applicability of the methods but also propose convergence domain. Finally, where earlier studies cannot be applied, a variety of concrete numerical examplesare proposed to obtain the solutions of nonlinear equations. Our study does not exhibit thistype of problem/restriction.http://www.mdpi.com/1999-4893/8/4/1076Newton-like methodlocal convergenceefficiency indexoptimum method
collection DOAJ
language English
format Article
sources DOAJ
author Ioannis K. Argyros
Ramandeep Behl
S.S. Motsa
spellingShingle Ioannis K. Argyros
Ramandeep Behl
S.S. Motsa
Local Convergence of an Efficient High Convergence Order Method Using Hypothesis Only on the First Derivative
Algorithms
Newton-like method
local convergence
efficiency index
optimum method
author_facet Ioannis K. Argyros
Ramandeep Behl
S.S. Motsa
author_sort Ioannis K. Argyros
title Local Convergence of an Efficient High Convergence Order Method Using Hypothesis Only on the First Derivative
title_short Local Convergence of an Efficient High Convergence Order Method Using Hypothesis Only on the First Derivative
title_full Local Convergence of an Efficient High Convergence Order Method Using Hypothesis Only on the First Derivative
title_fullStr Local Convergence of an Efficient High Convergence Order Method Using Hypothesis Only on the First Derivative
title_full_unstemmed Local Convergence of an Efficient High Convergence Order Method Using Hypothesis Only on the First Derivative
title_sort local convergence of an efficient high convergence order method using hypothesis only on the first derivative
publisher MDPI AG
series Algorithms
issn 1999-4893
publishDate 2015-11-01
description We present a local convergence analysis of an eighth order three step methodin order to approximate a locally unique solution of nonlinear equation in a Banach spacesetting. In an earlier study by Sharma and Arora (2015), the order of convergence wasshown using Taylor series expansions and hypotheses up to the fourth order derivative oreven higher of the function involved which restrict the applicability of the proposed scheme. However, only first order derivative appears in the proposed scheme. In order to overcomethis problem, we proposed the hypotheses up to only the first order derivative. In this way,we not only expand the applicability of the methods but also propose convergence domain. Finally, where earlier studies cannot be applied, a variety of concrete numerical examplesare proposed to obtain the solutions of nonlinear equations. Our study does not exhibit thistype of problem/restriction.
topic Newton-like method
local convergence
efficiency index
optimum method
url http://www.mdpi.com/1999-4893/8/4/1076
work_keys_str_mv AT ioanniskargyros localconvergenceofanefficienthighconvergenceordermethodusinghypothesisonlyonthefirstderivative
AT ramandeepbehl localconvergenceofanefficienthighconvergenceordermethodusinghypothesisonlyonthefirstderivative
AT ssmotsa localconvergenceofanefficienthighconvergenceordermethodusinghypothesisonlyonthefirstderivative
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