{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:18:09Z","timestamp":1760059089908,"version":"build-2065373602"},"reference-count":28,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2025,5,24]],"date-time":"2025-05-24T00:00:00Z","timestamp":1748044800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The order of convergence for Jarratt-type methods in solving nonlinear equations is determined without relying on Taylor expansion. Unlike previous studies, we depend solely on assumptions about the derivatives of the involved operator up to the second order. The proof presented in this paper is independent of the Taylor series expansion, thereby reducing the need for assumptions about derivatives of higher order of the involved operator and enhancing the applicability of these methods. The method\u2019s applicability is broadened by employing the concept of generalized conditions in local convergence analysis and majorizing sequences in semi-local analysis. This study includes numerical examples and basins of attraction for the methods.<\/jats:p>","DOI":"10.3390\/axioms14060401","type":"journal-article","created":{"date-parts":[[2025,5,25]],"date-time":"2025-05-25T20:26:50Z","timestamp":1748204810000},"page":"401","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On the Convergence Order of Jarratt-Type Methods for Nonlinear Equations"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6523-0873","authenticated-orcid":false,"given":"Shobha M.","family":"Erappa","sequence":"first","affiliation":[{"name":"Department of Mathematics, Manipal Institute of Technology, Manipal Academy of Higher Education, Manipal 576 104, Udupi, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Suma P.","family":"Bheemaiah","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Manipal Institute of Technology, Manipal Academy of Higher Education, Manipal 576 104, Udupi, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Santhosh","family":"George","sequence":"additional","affiliation":[{"name":"Department of Mathematical and Computational Sciences, National Institute of Technology Karnataka, Surathkal 575 025, Mangaluru, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3599-5488","authenticated-orcid":false,"given":"Kanagaraj","family":"Karuppaiah","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Srinivasa Ramanujan Centre, SASTRA Deemed to Be University, Kumbakonam 612 001, Tamil Nadu, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ioannis K.","family":"Argyros","sequence":"additional","affiliation":[{"name":"Department of Computing and Mathematical Sciences, Cameron University, Lawton, OK 73505, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,5,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"624","DOI":"10.1016\/j.camwa.2015.05.012","article-title":"Higher order multi-step Jarratt-like method for solving systems of nonlinear equations: Application to PDEs and ODEs","volume":"70","author":"Ahmad","year":"2015","journal-title":"Comput. Math. Appl."},{"key":"ref_2","first-page":"249","article-title":"An efficient multi-step iterative method for computing the numerical solution of systems of nonlinear equations associated with ODEs","volume":"250","author":"Ullah","year":"2015","journal-title":"Appl. Math. Comput."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Argyros, I.K. (2022). The Theory and Applications of Iteration Methods, Taylor and Francis Group, CRC Press.","DOI":"10.1201\/9781003128915"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1007\/s11075-013-9784-x","article-title":"Numerical solution of nonlinear systems by a general class of iterative methods with application to nonlinear PDEs","volume":"67","author":"Ullah","year":"2014","journal-title":"Numer. Algorithms"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"7847","DOI":"10.3934\/math.2025360","article-title":"A single parameter fourth-order Jarratt type iterative method for solving nonlinear systems","volume":"10","author":"Yu","year":"2025","journal-title":"AIMS Math."},{"key":"ref_6","first-page":"827","article-title":"Newton\u2019s method in Banach spaces","volume":"6","author":"Bartle","year":"1955","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"243","DOI":"10.1016\/0022-247X(66)90115-6","article-title":"A Newton-Raphson method for the solution of systems of equations","volume":"15","year":"1966","journal-title":"J. Math. Anal. Appl."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"434","DOI":"10.1090\/S0025-5718-66-99924-8","article-title":"Some fourth order multipoint iterative methods for solving equations","volume":"20","author":"Jarratt","year":"1966","journal-title":"Math. Comput."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1269","DOI":"10.1186\/s40064-016-2909-7","article-title":"A new Newton-like method for solving nonlinear equations","volume":"5","author":"Saheya","year":"2016","journal-title":"SpringerPlus"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"585","DOI":"10.1007\/s11075-009-9302-3","article-title":"New variants of Jarratt\u2019s method with sixth-order convergence","volume":"52","author":"Ren","year":"2009","journal-title":"Numer. Algorithms"},{"key":"ref_11","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":"8","author":"Weerakoon","year":"2000","journal-title":"Appl. Math. Lett."},{"key":"ref_12","first-page":"635","article-title":"Multipoint methods for solving nonlinear equations: A survey","volume":"226","author":"Neta","year":"2014","journal-title":"Appl. Math. Comput."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1007\/s11075-009-9359-z","article-title":"A modified Newton-Jarratt\u2019s composition","volume":"55","author":"Cordero","year":"2010","journal-title":"Numer. Algorithms"},{"key":"ref_14","first-page":"520","article-title":"On developing fourth-order optimal families of methods for multiple roots and their dynamics","volume":"265","author":"Behl","year":"2015","journal-title":"Appl. Math. Comput."},{"key":"ref_15","first-page":"29","article-title":"Different anomalies in a Jarratt family of iterative root finding methods","volume":"233","year":"2014","journal-title":"Appl. Math. Comput."},{"key":"ref_16","first-page":"82","article-title":"Convergence analysis of a two step method for the nonlinear squares problem with decomposition of operator","volume":"128","author":"Shakhno","year":"2018","journal-title":"J. Numer. Appl. Math."},{"key":"ref_17","first-page":"253","article-title":"On an iterative algorithm of order 1.839\u2026 for solving nonlinear operator equations","volume":"161","author":"Shakhno","year":"2005","journal-title":"Appl. Math. Appl."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"124381","DOI":"10.1016\/j.jmaa.2020.124381","article-title":"A Newton-type midpoint method with high efficiency index","volume":"491","author":"Castro","year":"2020","journal-title":"J. Math. Anal. Appl."},{"key":"ref_19","unstructured":"Traub, J.F. (1982). Iterative methods for the solution of equations. Am. Math. Soc., 312."},{"key":"ref_20","unstructured":"Ostrowski, A.M. (2016). Solution of Equations and Systems of Equations: Pure and Applied Mathematics, Elsevier."},{"key":"ref_21","first-page":"446","article-title":"New iterative technique for solving a system of nonlinear equations","volume":"271","author":"Noor","year":"2015","journal-title":"Appl. Math. Comput."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Alqahtani, H.F., Behl, R., and Kansal, M. (2019). Higher-Order Iteration Schemes for Solving Nonlinear Systems of Equations. Mathematics, 7.","DOI":"10.3390\/math7100937"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"58","DOI":"10.9790\/5728-0755861","article-title":"Generalized Simpson-Newton\u2019s Method for Solving Nonlinear Equations with Cubic Convergence","volume":"7","author":"Jayakumar","year":"2013","journal-title":"IOSR J. Math."},{"key":"ref_24","first-page":"3","article-title":"Numerical method with order t for solving system nonlinear equations","volume":"30","author":"Iliev","year":"2000","journal-title":"Collect. Sci. Work."},{"key":"ref_25","first-page":"6427","article-title":"On optimal fourth-order iterative methods free from second derivative and their dynamics","volume":"218","author":"Chun","year":"2012","journal-title":"Appl. Math. Comput."},{"key":"ref_26","first-page":"2584","article-title":"Basin attractors for various methods","volume":"218","author":"Scott","year":"2011","journal-title":"Appl. Math. Comput."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Ortega, J.M., and Rheinboldt, W.C. (2000). Iterative solution of nonlinear equations in several variables. Classics in Applied Mathematics, Philadelphia: Society for Industrial and Applied Mathematics, Academic Press.","DOI":"10.1137\/1.9780898719468"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"333","DOI":"10.1007\/BF01397005","article-title":"\u00dcber ein Verfahren der Ordnung 1 + 2 zur Nullstellenbestimmung","volume":"32","author":"Werner","year":"1979","journal-title":"Numer. Math."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/6\/401\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:39:39Z","timestamp":1760031579000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/6\/401"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,5,24]]},"references-count":28,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2025,6]]}},"alternative-id":["axioms14060401"],"URL":"https:\/\/doi.org\/10.3390\/axioms14060401","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,5,24]]}}}