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...
Main Authors: | , , |
---|---|
Format: | Article |
Language: | English |
Published: |
MDPI AG
2015-11-01
|
Series: | Algorithms |
Subjects: | |
Online Access: | http://www.mdpi.com/1999-4893/8/4/1076 |
id |
doaj-fe8d3aa35146477792c263051aecc558 |
---|---|
record_format |
Article |
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 |
_version_ |
1725873338997276672 |