{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,4]],"date-time":"2025-11-04T16:07:02Z","timestamp":1762272422041,"version":"build-2065373602"},"reference-count":13,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2019,9,2]],"date-time":"2019-09-02T00:00:00Z","timestamp":1567382400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100014440","name":"Ministerio de Ciencia, Innovaci\u00f3n y Universidades","doi-asserted-by":"publisher","award":["PGC2018-095896-B-C22"],"award-info":[{"award-number":["PGC2018-095896-B-C22"]}],"id":[{"id":"10.13039\/100014440","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003359","name":"Generalitat Valenciana","doi-asserted-by":"publisher","award":["PROMETEO 2016\/089"],"award-info":[{"award-number":["PROMETEO 2016\/089"]}],"id":[{"id":"10.13039\/501100003359","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Several authors have designed variants of Newton\u2019s method for solving nonlinear equations by using different means. This technique involves a symmetry in the corresponding fixed-point operator. In this paper, some known results about mean-based variants of Newton\u2019s method (MBN) are re-analyzed from the point of view of convex combinations. A new test is developed to study the order of convergence of general MBN. Furthermore, a generalization of the Lehmer mean is proposed and discussed. Numerical tests are provided to support the theoretical results obtained and to compare the different methods employed. Some dynamical planes of the analyzed methods on several equations are presented, revealing the great difference between the MBN when it comes to determining the set of starting points that ensure convergence and observing their symmetry in the complex plane.<\/jats:p>","DOI":"10.3390\/sym11091106","type":"journal-article","created":{"date-parts":[[2019,9,3]],"date-time":"2019-09-03T03:06:14Z","timestamp":1567479974000},"page":"1106","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["A Convex Combination Approach for Mean-Based Variants of Newton\u2019s Method"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7462-9173","authenticated-orcid":false,"given":"Alicia","family":"Cordero","sequence":"first","affiliation":[{"name":"Institute of Multidisciplinary Mathematics, Universitat Polit\u00e8cnica de Val\u00e8ncia, Camino de Vera, s\/n, 46022-Valencia, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jonathan","family":"Franceschi","sequence":"additional","affiliation":[{"name":"Institute of Multidisciplinary Mathematics, Universitat Polit\u00e8cnica de Val\u00e8ncia, Camino de Vera, s\/n, 46022-Valencia, Spain"},{"name":"Universit\u00e1 di Ferrara, via Ludovico Ariosto, 35, 44121 Ferrara, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9893-0761","authenticated-orcid":false,"given":"Juan R.","family":"Torregrosa","sequence":"additional","affiliation":[{"name":"Institute of Multidisciplinary Mathematics, Universitat Polit\u00e8cnica de Val\u00e8ncia, Camino de Vera, s\/n, 46022-Valencia, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Anna C.","family":"Zagati","sequence":"additional","affiliation":[{"name":"Institute of Multidisciplinary Mathematics, Universitat Polit\u00e8cnica de Val\u00e8ncia, Camino de Vera, s\/n, 46022-Valencia, Spain"},{"name":"Universit\u00e1 di Ferrara, via Ludovico Ariosto, 35, 44121 Ferrara, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,9,2]]},"reference":[{"key":"ref_1","unstructured":"Traub, J.F. (1964). Iterative Methods for the Solution of Equations, Prentice-Hall."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Petkovi\u0107, M.S., Neta, B., Petkovi\u0107, L.D., and D\u017euni\u0107, J. (2013). Multipoint Methods for Solving Nonlinear Equations, Academic Press.","DOI":"10.1016\/B978-0-12-397013-8.00002-9"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Amat, S., and Busquier, S. (2016). Advances in Iterative Methods for Nonlinear Equations, Springer.","DOI":"10.1007\/978-3-319-39228-8"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1016\/S0893-9659(00)00100-2","article-title":"A variant of Newton\u2019s method with accelerated third-order convergence","volume":"13","author":"Weerakoon","year":"2000","journal-title":"Appl. Math. Lett."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"677","DOI":"10.1016\/S0893-9659(04)90104-8","article-title":"Some new variants of Newton\u2019s method","volume":"17","year":"2004","journal-title":"Appl. Math. Lett."},{"key":"ref_6","first-page":"1071","article-title":"New Newton\u2019s method with third order convergence for solving nonlinear equations","volume":"61","author":"Ababneh","year":"2012","journal-title":"World Acad. Sci. Eng. Technol."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1026","DOI":"10.1016\/j.aml.2006.09.010","article-title":"A class of Newton\u2019s methods with third-order convergence","volume":"20","author":"Xiaojian","year":"2007","journal-title":"Appl. Math. Lett."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"174","DOI":"10.14445\/22315373\/IJMTT-V49P524","article-title":"A new-mean type variant of Newton\u2019s method for simple and multiple roots","volume":"49","author":"Singh","year":"2017","journal-title":"Int. J. Math. Trends Technol."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"126","DOI":"10.1504\/IJCSM.2016.076403","article-title":"On the centroidal mean Newton\u2019s method for simple and multiple roots of nonlinear equations","volume":"7","author":"Verma","year":"2016","journal-title":"Int. J. Comput. Sci. Math."},{"key":"ref_10","first-page":"43","article-title":"A generalized family of quadrature based iterative methods","volume":"18","author":"Zafar","year":"2010","journal-title":"Gen. Math."},{"key":"ref_11","first-page":"686","article-title":"Variants of Newton\u2019s method using fifth-order quadrature formulas","volume":"190","author":"Cordero","year":"2007","journal-title":"Appl. Math. Comput."},{"key":"ref_12","unstructured":"Ostrowski, A.M. (1966). Solution of Equations and Systems of Equatiuons, Academic Press."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"780153","DOI":"10.1155\/2013\/780153","article-title":"Drawing dynamical and pa- rameters planes of iterative families and methods","volume":"2013","author":"Chicharro","year":"2013","journal-title":"Sci. World J."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/9\/1106\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:16:10Z","timestamp":1760188570000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/9\/1106"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,9,2]]},"references-count":13,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2019,9]]}},"alternative-id":["sym11091106"],"URL":"https:\/\/doi.org\/10.3390\/sym11091106","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2019,9,2]]}}}