{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T20:11:02Z","timestamp":1760213462553,"version":"build-2065373602"},"reference-count":17,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2016,9,29]],"date-time":"2016-09-29T00:00:00Z","timestamp":1475107200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"No funding"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>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.<\/jats:p>","DOI":"10.3390\/a9040065","type":"journal-article","created":{"date-parts":[[2016,9,29]],"date-time":"2016-09-29T09:54:50Z","timestamp":1475142890000},"page":"65","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Local Convergence Analysis of an Eighth Order Scheme Using Hypothesis Only on the First Derivative"],"prefix":"10.3390","volume":"9","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9189-9298","authenticated-orcid":false,"given":"Ioannis","family":"Argyros","sequence":"first","affiliation":[{"name":"Department of Mathematics Sciences Lawton, Cameron University, Lawton, OK 73505, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ramandeep","family":"Behl","sequence":"additional","affiliation":[{"name":"School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X01, Pietermaritzburg 3209, South Africa"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sandile","family":"Motsa","sequence":"additional","affiliation":[{"name":"School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X01, Pietermaritzburg 3209, South Africa"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2016,9,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"212","DOI":"10.1007\/s00010-004-2733-y","article-title":"Dynamics of the King and Jarratt iterations","volume":"69","author":"Amat","year":"2005","journal-title":"Aequ. Math."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"24","DOI":"10.1016\/j.jmaa.2010.01.047","article-title":"Chaotic dynamics of a third-order Newton-type method","volume":"366","author":"Amat","year":"2010","journal-title":"J. Math. Anal. Appl."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"164","DOI":"10.1016\/j.amc.2008.08.050","article-title":"A modified Chebyshev\u2019s iterative method with at least sixth order of convergence","volume":"206","author":"Amat","year":"2008","journal-title":"Appl. Math. Comput."},{"key":"ref_4","unstructured":"Argyros, I.K. (2008). Convergence and Application of Newton-type Iterations, Springer."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Argyros, I.K., and Hilout, S. (2013). Numerical Methods in Nonlinear Analysis, World Scientific Publisher Company.","DOI":"10.1142\/8475"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Behl, R., and Motsa, S.S. (2015). Geometric construction of eighth-order optimal families of Ostrowski\u2019s method. Sci. World J., 2015.","DOI":"10.1155\/2015\/614612"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"325","DOI":"10.1007\/s10543-009-0226-z","article-title":"New iterations of R-order four with reduced computational cost","volume":"49","author":"Ezquerro","year":"2009","journal-title":"BIT Numer. Math."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"4021","DOI":"10.1016\/j.camwa.2011.09.039","article-title":"Simply constructed family of a Ostrowski\u2019s method with optimal order of convergence","volume":"62","author":"Kanwar","year":"2011","journal-title":"Comput. Math. Appli."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1016\/j.amc.2014.01.037","article-title":"Different anomalies in a Jarratt family of iterative root-finding methods","volume":"233","year":"2014","journal-title":"Appl. Math. 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