{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:43:41Z","timestamp":1760240621815,"version":"build-2065373602"},"reference-count":16,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2019,8,2]],"date-time":"2019-08-02T00:00:00Z","timestamp":1564704000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Many problems in diverse disciplines such as applied mathematics, mathematical biology, chemistry, economics, and engineering, to mention a few, reduce to solving a nonlinear equation or a system of nonlinear equations. Then various iterative methods are considered to generate a sequence of approximations converging to a solution of such problems. The goal of this article is two-fold: On the one hand, we present a correct convergence criterion for Newton\u2013Hermitian splitting (NHSS) method under the Kantorovich theory, since the criterion given in Numer. Linear Algebra Appl., 2011, 18, 299\u2013315 is not correct. Indeed, the radius of convergence cannot be defined under the given criterion, since the discriminant of the quadratic polynomial from which this radius is derived is negative (See Remark 1 and the conclusions of the present article for more details). On the other hand, we have extended the corrected convergence criterion using our idea of recurrent functions. Numerical examples involving convection\u2013diffusion equations further validate the theoretical results.<\/jats:p>","DOI":"10.3390\/sym11080981","type":"journal-article","created":{"date-parts":[[2019,8,2]],"date-time":"2019-08-02T11:58:16Z","timestamp":1564747096000},"page":"981","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Extended Convergence Analysis of the Newton\u2013Hermitian and Skew\u2013Hermitian Splitting Method"],"prefix":"10.3390","volume":"11","author":[{"given":"Ioannis K","family":"Argyros","sequence":"first","affiliation":[{"name":"Department of Mathematical Sciences, Cameron University, Lawton, OK 73505, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3530-5539","authenticated-orcid":false,"given":"Santhosh","family":"George","sequence":"additional","affiliation":[{"name":"Department of Mathematical and Computational Sciences, National Institute of Technology, Karnataka 575 025, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Chandhini","family":"Godavarma","sequence":"additional","affiliation":[{"name":"Department of Mathematical and Computational Sciences, National Institute of Technology, Karnataka 575 025, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Alberto A","family":"Magre\u00f1\u00e1n","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1ticas y Computaci\u00f3n, Universidad de la Rioja, 26006 Logro\u00f1o, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,8,2]]},"reference":[{"key":"ref_1","unstructured":"Argyros, I.K., and Szidarovszky, F. (1993). The Theory and Applications of Iteration Methods, CRC Press."},{"key":"ref_2","unstructured":"Argyros, I.K., and Magr\u00e9\u00f1an, A.A. (2018). A Contemporary Study of Iterative Methods, Elsevier (Academic Press)."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1007\/s40324-017-0127-z","article-title":"Local convergence for an almost sixth order method for solving equations under weak conditions","volume":"75","author":"Argyros","year":"2018","journal-title":"SeMA J."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"603","DOI":"10.1137\/S0895479801395458","article-title":"Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems","volume":"24","author":"Bai","year":"2003","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"235","DOI":"10.4208\/jcm.2009.10-m2836","article-title":"The Newton-HSS methods for systems of nonlinear equations with positive-definite Jacobian matrices","volume":"28","author":"Bai","year":"2010","journal-title":"J. Comput. Math."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"400","DOI":"10.1137\/0719025","article-title":"Inexact Newton methods","volume":"19","author":"Dembo","year":"1982","journal-title":"SIAM J. Numer. Anal."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1002\/nla.713","article-title":"Semi-local and global convergence of the Newton-HSS method for systems of nonlinear equations","volume":"18","author":"Guo","year":"2011","journal-title":"Numer. Linear Algebra Appl."},{"key":"ref_8","first-page":"29","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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Eng."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"331","DOI":"10.1007\/BF02684384","article-title":"Avoiding slave points in an adaptive refinement procedure for convection-diffusion problems in 2D","volume":"61","author":"Axelsson","year":"1998","journal-title":"Computing"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/8\/981\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:12:36Z","timestamp":1760188356000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/8\/981"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,8,2]]},"references-count":16,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2019,8]]}},"alternative-id":["sym11080981"],"URL":"https:\/\/doi.org\/10.3390\/sym11080981","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2019,8,2]]}}}