{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:25:39Z","timestamp":1760243139715,"version":"build-2065373602"},"reference-count":16,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2015,11,20]],"date-time":"2015-11-20T00:00:00Z","timestamp":1447977600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>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 \ufb01rst order derivative appears in the proposed scheme. In order to overcomethis problem, we proposed the hypotheses up to only the \ufb01rst 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.<\/jats:p>","DOI":"10.3390\/a8041076","type":"journal-article","created":{"date-parts":[[2015,11,24]],"date-time":"2015-11-24T01:57:02Z","timestamp":1448330222000},"page":"1076-1087","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Local Convergence of an Ef\ufb01cient High Convergence Order Method Using Hypothesis Only on the First Derivative"],"prefix":"10.3390","volume":"8","author":[{"given":"Ioannis","family":"Argyros","sequence":"first","affiliation":[{"name":"Department of Mathematics Sciences, 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, Scottsville 3209, Pietermaritzburg, South Africa"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"S.S.","family":"Motsa","sequence":"additional","affiliation":[{"name":"School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X01, Scottsville 3209, Pietermaritzburg, South Africa"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2015,11,20]]},"reference":[{"key":"ref_1","unstructured":"Sharma, J.R., and Arora, H. A new family of optimal eighth order methods with dynamics for nonlinear equations. Appl. Math. Comput., submitted."},{"key":"ref_2","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_3","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_4","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_5","unstructured":"Argyros, I.K. (2008). Convergence and Application of Newton-Type Iterations, Springer."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Argyros, I.K., and Hilout, S. (2013). Numerical Methods in Nonlinear Analysis, World Scientific Publishing Company.","DOI":"10.1142\/8475"},{"key":"ref_7","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_8","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_9","doi-asserted-by":"crossref","first-page":"472","DOI":"10.1016\/j.aml.2009.12.006","article-title":"On some computational orders of convergence","volume":"23","author":"Noguera","year":"2010","journal-title":"Appl. Math. Lett."},{"key":"ref_10","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. Appl."},{"key":"ref_11","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\u2013finding methods","volume":"233","year":"2014","journal-title":"Appl. Math. Comput."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1016\/j.amc.2014.09.061","article-title":"A new tool to study real dynamics: The convergence plane","volume":"248","year":"2014","journal-title":"Appl. Math. Comput."},{"key":"ref_13","unstructured":"Petkovic, M.S., Neta, B., Petkovic, L., and D\u017euni\u010d, J. (2013). 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Lett."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/8\/4\/1076\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T20:52:32Z","timestamp":1760215952000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/8\/4\/1076"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,11,20]]},"references-count":16,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2015,12]]}},"alternative-id":["a8041076"],"URL":"https:\/\/doi.org\/10.3390\/a8041076","relation":{},"ISSN":["1999-4893"],"issn-type":[{"type":"electronic","value":"1999-4893"}],"subject":[],"published":{"date-parts":[[2015,11,20]]}}}