{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T04:37:56Z","timestamp":1773203876615,"version":"3.50.1"},"reference-count":44,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2024,7,7]],"date-time":"2024-07-07T00:00:00Z","timestamp":1720310400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Ayuda a Primeros Proyectos de Investigaci\u00f3n","award":["PAID-06-23"],"award-info":[{"award-number":["PAID-06-23"]}]},{"name":"Vicerrectorado de Investigaci\u00f3n de la Universitat Polit\u00e8cnica de Val\u00e8ncia (UPV)","award":["PAID-06-23"],"award-info":[{"award-number":["PAID-06-23"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this manuscript, we introduce a novel parametric family of multistep iterative methods designed to solve nonlinear equations. This family is derived from a damped Newton\u2019s scheme but includes an additional Newton step with a weight function and a \u201cfrozen\u201d derivative, that is, the same derivative than in the previous step. Initially, we develop a quad-parametric class with a first-order convergence rate. Subsequently, by restricting one of its parameters, we accelerate the convergence to achieve a third-order uni-parametric family. We thoroughly investigate the convergence properties of this final class of iterative methods, assess its stability through dynamical tools, and evaluate its performance on a set of test problems. We conclude that there exists one optimal fourth-order member of this class, in the sense of Kung\u2013Traub\u2019s conjecture. Our analysis includes stability surfaces and dynamical planes, revealing the intricate nature of this family. Notably, our exploration of stability surfaces enables the identification of specific family members suitable for scalar functions with a challenging convergence behavior, as they may exhibit periodical orbits and fixed points with attracting behavior in their corresponding dynamical planes. Furthermore, our dynamical study finds members of the family of iterative methods with exceptional stability. This property allows us to converge to the solution of practical problem-solving applications even from initial estimations very far from the solution. We confirm our findings with various numerical tests, demonstrating the efficiency and reliability of the presented family of iterative methods.<\/jats:p>","DOI":"10.3390\/axioms13070458","type":"journal-article","created":{"date-parts":[[2024,7,8]],"date-time":"2024-07-08T09:01:19Z","timestamp":1720429279000},"page":"458","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Achieving Optimal Order in a Novel Family of Numerical Methods: Insights from Convergence and Dynamical Analysis Results"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5182-3099","authenticated-orcid":false,"given":"Marlon","family":"Moscoso-Mart\u00ednez","sequence":"first","affiliation":[{"name":"Institute for Multidisciplinary Mathematics, Universitat Polit\u00e8cnica de Val\u00e8ncia, Camino de Vera s\/n, 46022 Val\u00e8ncia, Spain"},{"name":"Faculty of Sciences, Escuela Superior Polit\u00e9cnica de Chimborazo (ESPOCH), Panamericana Sur km 1 1\/2, Riobamba 060106, Ecuador"},{"name":"Higher School of Engineering and Technology, Universidad Internacional de la Rioja (UNIR), Avda. de la Paz 137, 26006 Logro\u00f1o, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9116-2870","authenticated-orcid":false,"given":"Francisco I.","family":"Chicharro","sequence":"additional","affiliation":[{"name":"Institute for Multidisciplinary Mathematics, Universitat Polit\u00e8cnica de Val\u00e8ncia, Camino de Vera s\/n, 46022 Val\u00e8ncia, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7462-9173","authenticated-orcid":false,"given":"Alicia","family":"Cordero","sequence":"additional","affiliation":[{"name":"Institute for Multidisciplinary Mathematics, Universitat Polit\u00e8cnica de Val\u00e8ncia, Camino de Vera s\/n, 46022 Val\u00e8ncia, 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":"Institute for Multidisciplinary Mathematics, Universitat Polit\u00e8cnica de Val\u00e8ncia, Camino de Vera s\/n, 46022 Val\u00e8ncia, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0001-5842-7379","authenticated-orcid":false,"given":"Gabriela","family":"Ure\u00f1a-Callay","sequence":"additional","affiliation":[{"name":"Faculty of Sciences, Escuela Superior Polit\u00e9cnica de Chimborazo (ESPOCH), Panamericana Sur km 1 1\/2, Riobamba 060106, Ecuador"},{"name":"Higher School of Engineering and Technology, Universidad Internacional de la Rioja (UNIR), Avda. de la Paz 137, 26006 Logro\u00f1o, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,7,7]]},"reference":[{"key":"ref_1","first-page":"364","article-title":"Gauss meets Newton again: How to make Gauss orbit determination from two position vectors more efficient and robust with Newton\u2013Raphson iterations","volume":"195","author":"Danchick","year":"2008","journal-title":"Appl. Math. Comput."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Tostado-V\u00e9liz, M., Kamel, S., Jurado, F., and Ruiz-Rodriguez, F.J. (2021). On the Applicability of Two Families of Cubic Techniques for Power Flow Analysis. Energies, 14.","DOI":"10.3390\/en14144108"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1802","DOI":"10.1016\/j.mcm.2010.11.063","article-title":"Approximation of artificial satellites\u2019 preliminary orbits: The efficiency challenge","volume":"54","author":"Arroyo","year":"2011","journal-title":"Math. Comput. Model."},{"key":"ref_4","unstructured":"Traub, J.F. (1964). Iterative Methods for the Solution of Equations, Prentice-Hall."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Petkovi\u0107, M., Neta, B., Petkovi\u0107, L., 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_6","doi-asserted-by":"crossref","unstructured":"Amat, S., and Busquier, S. (2017). Advances in Iterative Methods for Nonlinear Equations, Springer.","DOI":"10.1007\/978-3-319-39228-8"},{"key":"ref_7","unstructured":"Ostrowski, A.M. (1973). Solution of Equations in Euclidean and Banach Spaces, Academic Press."},{"key":"ref_8","first-page":"1","article-title":"A new family of optimal fourth-order iterative methods for nonlinear equations","volume":"8","author":"Kaya","year":"2022","journal-title":"Results Control Optim."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Adomian, G. (1994). Solving Frontier Problem of Physics: The Decomposition Method, Kluwer Academic Publishers.","DOI":"10.1007\/978-94-015-8289-6"},{"key":"ref_10","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_11","first-page":"454","article-title":"Some fourth-order iterative methods for solving nonlinear equations","volume":"195","author":"Chun","year":"2008","journal-title":"Appl. Math. Comput."},{"key":"ref_12","unstructured":"Ostrowski, A.M. (1960). Solution of Equations and Systems of Equations, Academic Press."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"643","DOI":"10.1145\/321850.321860","article-title":"Optimal Order of One-Point and Multipoint Iteration","volume":"21","author":"Kung","year":"1974","journal-title":"J. Assoc. Comput. Mach."},{"key":"ref_14","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_15","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_16","doi-asserted-by":"crossref","first-page":"1017","DOI":"10.1007\/s11075-020-00997-4","article-title":"New fourth- and sixth-order classes of iterative methods for solving systems of nonlinear equations and their stability analysis","volume":"87","author":"Kansal","year":"2021","journal-title":"Numer. Algorithms"},{"key":"ref_17","first-page":"398","article-title":"Dynamical analysis of iterative methods for nonlinear systems or how to deal with the dimension?","volume":"244","author":"Cordero","year":"2014","journal-title":"Appl. Math. Comput."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Cordero, A., Moscoso-Mart\u00ednez, M., and Torregrosa, J.R. (2021). Chaos and Stability in a New Iterative Family for Solving Nonlinear Equations. Algorithms, 14.","DOI":"10.3390\/a14040101"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Husain, A., Nanda, M.N., Chowdary, M.S., and Sajid, M. (2022). Fractals: An Eclectic Survey, Part I. Fractal Fract., 6.","DOI":"10.3390\/fractalfract6070379"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Husain, A., Nanda, M.N., Chowdary, M.S., and Sajid, M. (2022). Fractals: An Eclectic Survey, Part II. Fractal Fract., 6.","DOI":"10.3390\/fractalfract6070379"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1007\/BF03025310","article-title":"Graphic and numerical comparison between iterative methods","volume":"24","author":"Varona","year":"2002","journal-title":"Math. Intell."},{"key":"ref_22","first-page":"3","article-title":"Review of some iterative root\u2013finding methods from a dynamical point of view","volume":"10","author":"Amat","year":"2004","journal-title":"SCI. A Math. Sci."},{"key":"ref_23","first-page":"567","article-title":"Basins of attraction for optimal eighth order methods to find simple roots of nonlinear equations","volume":"227","author":"Neta","year":"2014","journal-title":"Appl. Math. Comput."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"842","DOI":"10.1016\/j.aml.2013.03.012","article-title":"Chaos in King\u2019s iterative family","volume":"26","author":"Cordero","year":"2013","journal-title":"Appl. Math. Lett."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Magre\u00f1\u00e1n, A., and Argyros, I. (2018). A Contemporary Study of Iterative Methods, Academic Press.","DOI":"10.1016\/B978-0-12-809214-9.00023-1"},{"key":"ref_26","first-page":"3087","article-title":"Long\u2013term orbit dynamics viewed through the yellow main component in the parameter space of a family of optimal fourth-order multiple-root finders","volume":"25","author":"Geum","year":"2020","journal-title":"Discrete Contin. Dyn. Syst. B"},{"key":"ref_27","first-page":"8568","article-title":"Dynamics of a family of Chebyshev-Halley type method","volume":"219","author":"Cordero","year":"2012","journal-title":"Appl. Math. Comput."},{"key":"ref_28","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_29","unstructured":"Devaney, R. (1989). An Introduction to Chaotic Dynamical Systems, Addison-Wesley Publishing Company."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Beardon, A. (1991). Iteration of Rational Functions, Graduate Texts in Mathematics; Springer.","DOI":"10.1007\/978-1-4612-4422-6"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"161","DOI":"10.24033\/bsmf.998","article-title":"Sur les \u00e9quations fonctionelles","volume":"47","author":"Fatou","year":"1919","journal-title":"Bull. Soc. Mat. Fr."},{"key":"ref_32","first-page":"47","article-title":"M\u00e9moire sur l\u2019iteration des fonctions rationnelles","volume":"8","author":"Julia","year":"1918","journal-title":"Mat. Pur. Appl."},{"key":"ref_33","first-page":"2584","article-title":"Basin attractors for various methods","volume":"218","author":"Scott","year":"2011","journal-title":"Appl. Math. Comput."},{"key":"ref_34","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_35","doi-asserted-by":"crossref","first-page":"1687","DOI":"10.1080\/00207160.2015.1080354","article-title":"Hueso, E.M.; Teruel, C. Multipoint efficient iterative methods and the dynamics of Ostrowski\u2019s method","volume":"96","year":"2019","journal-title":"Int. J. Comput. Math."},{"key":"ref_36","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_37","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_38","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1007\/s00211-006-0025-2","article-title":"Construction of Newton-like iteration methods for solving nonlinear equations","volume":"104","author":"Chun","year":"2006","journal-title":"Numer. Math."},{"key":"ref_39","first-page":"383","article-title":"A fourth order iterative method for solving nonlinear equations","volume":"211","author":"Maheshwari","year":"2009","journal-title":"Appl. Math. Comput."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"38","DOI":"10.1016\/j.cam.2016.12.008","article-title":"A family of second derivative free fourth order continuation method for solving nonlinear equations","volume":"318","author":"Behl","year":"2017","journal-title":"J. Comput. Appl. Math."},{"key":"ref_41","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_42","doi-asserted-by":"crossref","first-page":"2049","DOI":"10.1080\/00207160.2012.748900","article-title":"Local convergence and dynamical analysis of a new family of optimal fourth-order iterative methods","volume":"90","author":"Artidiello","year":"2013","journal-title":"Int. J. Comput. Math."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1016\/j.jksus.2010.07.018","article-title":"A new general fourth-order family of methods for finding simple roots of nonlinear equations","volume":"23","author":"Ghanbari","year":"2011","journal-title":"J. King Saud Univ. Sci."},{"key":"ref_44","first-page":"471","article-title":"A composite fourth-order iterative method for solving non-linear equations","volume":"184","author":"Kou","year":"2007","journal-title":"Appl. Math. Comput."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/7\/458\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T15:11:19Z","timestamp":1760109079000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/7\/458"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,7,7]]},"references-count":44,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2024,7]]}},"alternative-id":["axioms13070458"],"URL":"https:\/\/doi.org\/10.3390\/axioms13070458","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,7,7]]}}}