{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:22:45Z","timestamp":1760059365504,"version":"build-2065373602"},"reference-count":29,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2025,6,9]],"date-time":"2025-06-09T00:00:00Z","timestamp":1749427200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"PAID-11-24 Vicerrectorado de Investigaci\u00f3n de la Universitat Polit\u00e8cnica de Val\u00e8ncia, (UPV)"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>In this work, using the weight function technique, we introduce a new family of fourth-order iterative methods optimal in the sense of Kung and Traub for scalar equations, generalizing Jarratt\u2019s method. Through Taylor series expansions, we confirm that all members of this family achieve fourth-order convergence when derivatives up to the fourth order are bounded. Additionally, a stability analysis is performed on quadratic polynomials using complex discrete dynamics, enabling differentiation among the methods based on their stability. To demonstrate practical applicability, a numerical example illustrates the effectiveness of the proposed family. Extending our findings to Banach spaces, we conduct local convergence analyses on a specific subfamily containing Jarratt\u2019s method, requiring only boundedness of the first derivative. This significantly broadens the method\u2019s applicability to more general spaces and reduces constraints on higher-order derivatives. Finally, additional examples validate the existence and uniqueness of approximate solutions in Banach spaces, provided the initial estimate lies within the locally determined convergence radius obtained using majorizing functions.<\/jats:p>","DOI":"10.3390\/computation13060142","type":"journal-article","created":{"date-parts":[[2025,6,9]],"date-time":"2025-06-09T09:13:01Z","timestamp":1749460381000},"page":"142","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Stability Analysis and Local Convergence of a New Fourth-Order Optimal Jarratt-Type Iterative Scheme"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2869-4334","authenticated-orcid":false,"given":"Eulalia","family":"Mart\u00ednez","sequence":"first","affiliation":[{"name":"Multidisciplinary Mathematics Institute, Universitat Polit\u00e8cnica de Val\u00e8ncia (UPV), 46022 Valencia, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9397-7571","authenticated-orcid":false,"given":"Jos\u00e9 A.","family":"Reyes","sequence":"additional","affiliation":[{"name":"Departamento de Ciencias B\u00e1sicas, Instituto Tecnol\u00f3gico de Santo Domingo (INTEC), Santo Domingo 10602, Dominican Republic"},{"name":"Escuela de Matem\u00e1tica, Universidad Aut\u00f3noma de Santo Domingo (UASD), Alma M\u00e1ter, Santo Domingo 10105, Dominican Republic"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7462-9173","authenticated-orcid":false,"given":"Alicia","family":"Cordero","sequence":"additional","affiliation":[{"name":"Multidisciplinary Mathematics Institute, Universitat Polit\u00e8cnica de Val\u00e8ncia (UPV), 46022 Valencia, Spain"}],"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":"Multidisciplinary Mathematics Institute, Universitat Polit\u00e8cnica de Val\u00e8ncia (UPV), 46022 Valencia, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,6,9]]},"reference":[{"key":"ref_1","unstructured":"Danby, J.M.A. (1992). Fundamentals of Celestial Mechanics, Willmann-Bell."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Yaseen, S., Zafar, F., and Alsulami, H.H. (2023). An Efficient Jarratt-Type Iterative Method for Solving Nonlinear Global Positioning System Problems. Axioms, 12.","DOI":"10.3390\/axioms12060562"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Yaseen, S., Zafar, F., and Chicharro, F.I. (2023). A Seventh Order Family of Jarratt Type Iterative Method for Electrical Power Systems. Fractal Fract., 7.","DOI":"10.3390\/fractalfract7040317"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Bonilla-Correa, D.M., Coronado-Hern\u00e1ndez, O.E., Fuertes-Miquel, V.S., Besharat, M., and Ramos, H.M. (2023). Application of Newton-Raphson Method for Computing the Final Air-Water Interface Location in a Pipe Water Filling. Water, 15.","DOI":"10.3390\/w15071304"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"643","DOI":"10.1145\/321850.321860","article-title":"Optimal Order of One-Point and Multipoint Iteration Methods for Solving Nonlinear Equations","volume":"21","author":"Kung","year":"1974","journal-title":"J. Assoc. Comput. Mach."},{"key":"ref_6","unstructured":"Streater, J. (1687). Philosophi\u00e6 Naturalis Principia Mathematica, Royal Society Press."},{"key":"ref_7","unstructured":"Blanchard, P. (1993, January 9\u201313). The Dynamics of Newton\u2019s Method. Proceedings of the Symposium in Applied Mathematics, Vancouver, BC, Canada."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"358","DOI":"10.1016\/j.cam.2014.09.020","article-title":"Study of Iterative Methods through the Cayley Quadratic Test","volume":"291","author":"Babajee","year":"2016","journal-title":"J. Comput. Appl. Math."},{"key":"ref_9","first-page":"261","article-title":"On the Stability of a Two-Step Method for a Fourth-Degree Family by Computer Designs Along with Applications","volume":"14","author":"Moccari","year":"2023","journal-title":"Int. J. Nonlinear Anal. Appl."},{"key":"ref_10","first-page":"3","article-title":"Review of some iterative root-finding methods from a dynamical point of view","volume":"10","author":"Amat","year":"2004","journal-title":"Scientia"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"110","DOI":"10.1016\/j.cam.2018.06.042","article-title":"Highly Efficient Family of Iterative Methods for Solving Nonlinear Models","volume":"346","author":"Behl","year":"2019","journal-title":"J. Comput. Appl. Math."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Chicharro, F.I., Contreras, R.A., and Garrido, N. (2020). A Family of Multiple-Root Finding Iterative Methods Based on Weight Functions. Mathematics, 8.","DOI":"10.3390\/math8122194"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"971","DOI":"10.29020\/nybg.ejpam.v15i3.4397","article-title":"Convergence and Stability of Optimal Two-Step Fourth-Order and Its Expanding to Sixth Order for Solving Nonlinear Equations","volume":"15","author":"Khirallah","year":"2022","journal-title":"Eur. J. Pure Appl. Math."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Cordero, A., Reyes, J.A., Torregrosa, J.R., and Vassileva, M.P. (2024). Stability Analysis of a New Fourth-Order Optimal Iterative Scheme for Nonlinear Equations. Axioms, 13.","DOI":"10.3390\/axioms13010034"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"876","DOI":"10.1137\/0710072","article-title":"A family of fourth-order methods for nonlinear equations","volume":"10","author":"King","year":"1973","journal-title":"SIAM J. Numer. Anal."},{"key":"ref_16","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_17","unstructured":"Argyros, I.K., George, S., and Thapa, N. (2018). Mathematical Modeling for the Solution of Equations and Systems of Equations with Applications, Nova Science Publishers. Available online: https:\/\/novapublishers.com\/shop\/mathematical-modeling-for-the-solution-of-equations-and-systems-of-equations-with-applications-volume-i\/."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Qureshi, S., Chicharro, F.I., Argyros, I.K., Soomro, A., Alahmadi, J., and Hincal, E. (2024). A New Optimal Numerical Root-Solver for Solving Systems of Nonlinear Equations Using Local, Semi-Local, and Stability Analysis. Axioms, 13.","DOI":"10.3390\/axioms13060341"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"65","DOI":"10.4067\/S0719-06462018000300065","article-title":"Ball Comparison between Jarratt\u2019s and Other Fourth Order Methods for Solving Equations","volume":"20","author":"Argyros","year":"2018","journal-title":"CUBO A Math. J."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1007\/s00211-006-0025-2","article-title":"Construction of Newton-Like Iterative Methods for Solving Nonlinear Equations","volume":"104","author":"Chun","year":"2006","journal-title":"Numer. Math."},{"key":"ref_21","unstructured":"Ostrowski, A.M. (1973). Solution of Equations in Euclidean and Banach Spaces, Academic Press."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Devaney, R.L. (2020). A First Course in Chaotic Dynamical Systems: Theory and Experiment, Taylor & Francis Group. [2nd ed.].","DOI":"10.1201\/9780429280665"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1090\/S0273-0979-1984-15240-6","article-title":"Complex Analytic Dynamics on the Riemann Sphere","volume":"11","author":"Blanchard","year":"1984","journal-title":"Bull. Am. Math. Soc."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"161","DOI":"10.24033\/bsmf.998","article-title":"Sur les \u00e9quations fonctionnelles","volume":"47","author":"Fatou","year":"1919","journal-title":"Bull. Soc. Math. Fr."},{"key":"ref_25","first-page":"47","article-title":"M\u00e9moire sur l\u2019it\u00e9ration des fonctions rationnelles","volume":"8","author":"Julia","year":"1918","journal-title":"J. Math. Pures Appl."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"780153","DOI":"10.1155\/2013\/780153","article-title":"Drawing Dynamical and Parameter Planes of Iterative Families and Methods","volume":"2013","author":"Chicharro","year":"2013","journal-title":"Sci. World J."},{"key":"ref_27","first-page":"686","article-title":"Variants of Newton\u2019s Methods Using Fifth-Order Quadrature Formulas","volume":"190","author":"Cordero","year":"2007","journal-title":"Appl. Math. Comput."},{"key":"ref_28","unstructured":"Taylor, A.E., and Lay, D.C. (1980). Introduction to Functional Analysis, Wiley. [2nd ed.]."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Petkovi\u0107, M.S., Neta, B., Petkovi\u0107, L., and D\u017euni\u0107, J. (2013). Multipoint Methods for Solving Nonlinear Equations, Elsevier.","DOI":"10.1016\/B978-0-12-397013-8.00002-9"}],"container-title":["Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2079-3197\/13\/6\/142\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:48:55Z","timestamp":1760032135000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2079-3197\/13\/6\/142"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,6,9]]},"references-count":29,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2025,6]]}},"alternative-id":["computation13060142"],"URL":"https:\/\/doi.org\/10.3390\/computation13060142","relation":{},"ISSN":["2079-3197"],"issn-type":[{"type":"electronic","value":"2079-3197"}],"subject":[],"published":{"date-parts":[[2025,6,9]]}}}