{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:45:04Z","timestamp":1760237104775,"version":"build-2065373602"},"reference-count":22,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2020,3,5]],"date-time":"2020-03-05T00:00:00Z","timestamp":1583366400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100007801","name":"Fundaci\u00f3n S\u00e9neca","doi-asserted-by":"publisher","award":["20928\/PI\/18"],"award-info":[{"award-number":["20928\/PI\/18"]}],"id":[{"id":"10.13039\/100007801","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003329","name":"Ministerio de Econom\u00eda y Competitividad","doi-asserted-by":"publisher","award":["PGC2018-095896-B-C21"],"award-info":[{"award-number":["PGC2018-095896-B-C21"]}],"id":[{"id":"10.13039\/501100003329","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Solving problems in various disciplines such as biology, chemistry, economics, medicine, physics, and engineering, to mention a few, reduces to solving an equation. Its solution is one of the greatest challenges. It involves some iterative method generating a sequence approximating the solution. That is why, in this work, we analyze the convergence in a local form for an iterative method with a high order to find the solution of a nonlinear equation. We extend the applicability of previous results using only the first derivative that actually appears in the method. This is in contrast to either works using a derivative higher than one, or ones not in this method. Moreover, we consider the dynamics of some members of the family in order to see the existing differences between them.<\/jats:p>","DOI":"10.3390\/sym12030420","type":"journal-article","created":{"date-parts":[[2020,3,6]],"date-time":"2020-03-06T09:26:41Z","timestamp":1583486801000},"page":"420","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Convergence and Dynamics of a Higher-Order Method"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7138-4385","authenticated-orcid":false,"given":"Alejandro","family":"Moysi","sequence":"first","affiliation":[{"name":"Universidad de la Rioja, Av. de la Paz, 93-103, 26006 Logro\u00f1o, La Rioja, Spain"}]},{"given":"Ioannis K.","family":"Argyros","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, Cameron University, Lawton, OK 73505, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0035-1022","authenticated-orcid":false,"given":"Samundra","family":"Regmi","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, Cameron University, Lawton, OK 73505, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5282-7251","authenticated-orcid":false,"given":"Daniel","family":"Gonz\u00e1lez","sequence":"additional","affiliation":[{"name":"Escuela de Ciencias F\u00edsicas y Matem\u00e1ticas, Universidad de las Americas, Avda. de los Granados y Colimes, Quito 170125, Ecuador"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6991-5706","authenticated-orcid":false,"given":"\u00c1. Alberto","family":"Magre\u00f1\u00e1n","sequence":"additional","affiliation":[{"name":"Universidad de la Rioja, Av. de la Paz, 93-103, 26006 Logro\u00f1o, La Rioja, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5962-4147","authenticated-orcid":false,"given":"Juan Antonio","family":"Sicilia","sequence":"additional","affiliation":[{"name":"Universidad Internacional de la Rioja (UNIR), Av. de la Paz 137, 26006 Logro\u00f1o, La Rioja, Spain"}]}],"member":"1968","published-online":{"date-parts":[[2020,3,5]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Petkovi\u0107, M.S., Neta, B., Petkovi\u0107, L.D., and D\u017auni\u0107, J. (2013). Multipoint Methods for Solving Nonlinear Equations: A Survey, Elsevier.","DOI":"10.1016\/B978-0-12-397013-8.00002-9"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"412","DOI":"10.1016\/j.cam.2014.06.010","article-title":"Convergence, efficiency and dynamics of new fourth and sixth order families of iterative methods for nonlinear systems","volume":"275","author":"Hueso","year":"2015","journal-title":"J. Comput. Appl. Math."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"422","DOI":"10.3846\/mma.2019.026","article-title":"Higher-order families of multiple root finding methods suitable for non-convergent cases and their dynamics","volume":"24","author":"Behl","year":"2019","journal-title":"Math. Model. Anal."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1016\/S0377-0427(03)00420-5","article-title":"Geometric constructions of iterative functions to solve nonlinear equations","volume":"157","author":"Amat","year":"2003","journal-title":"J. Comput. Appl. Math."},{"key":"ref_5","unstructured":"Chui, C.K., and Wuytack, L. (2007). Computational Theory of Iterative Methods. Series: Studies in Computational Mathematics, 15, Elsevier Publ. Co."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Argyros, I.K., and Magre\u00f1\u00e1n, \u00c1.A. (2017). Iterative Methods and Their Dynamics with Applications: A Contemporary Study, Taylor & Francis Group.","DOI":"10.1201\/9781315153469"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Argyros, I.K., and Magre\u00f1\u00e1n, \u00c1.A. (2017). A Contemporary Study of Iterative Methods: Convergence, Dynamics and Applications, Elsevier.","DOI":"10.1201\/9781315153469"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Argyros, I.K., and Hilout, S. (2013). Computational Methods in Nonlinear Analysis. Efficient Algorithms, Fixed Point Theory and Applications, World Scientific.","DOI":"10.1142\/8475"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Argyros, I.K., and Hilout, S. (2013). Numerical Methods in Nonlinear Analysis, World Scientific Publ. Comp.","DOI":"10.1142\/8475"},{"key":"ref_10","unstructured":"Kantorovich, L.V., and Akilov, G.P. (1982). Functional Analysis, Pergamon Press."},{"key":"ref_11","unstructured":"Ortega, J.M., and Rheinboldt, W.C. (1970). 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Nonlinear Sci."},{"key":"ref_19","first-page":"29","article-title":"Different anomalies in a Jarratt family of iterative root\u2013finding methods","volume":"233","year":"2014","journal-title":"Appl. Math. Comput."},{"key":"ref_20","first-page":"215","article-title":"A new tool to study real dynamics: The convergence plane","volume":"248","year":"2014","journal-title":"Appl. Math. Comput."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"236","DOI":"10.1016\/j.cam.2015.11.036","article-title":"On the local convergence and the dynamics of Chebyshev-Halley methods with six and eight order of convergence","volume":"298","author":"Argyros","year":"2016","journal-title":"J. Comput. Appl. Math."},{"key":"ref_22","first-page":"347","article-title":"A variant of Steffensen-King\u2019s type family with accelerated sixth-order convergence and high efficiency index: Dynamic study and approach","volume":"252","author":"Lotfi","year":"2015","journal-title":"Appl. Math. Comput."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/3\/420\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:04:35Z","timestamp":1760173475000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/3\/420"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,3,5]]},"references-count":22,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2020,3]]}},"alternative-id":["sym12030420"],"URL":"https:\/\/doi.org\/10.3390\/sym12030420","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2020,3,5]]}}}