{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:37:05Z","timestamp":1760240225511,"version":"build-2065373602"},"reference-count":19,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2019,4,11]],"date-time":"2019-04-11T00:00:00Z","timestamp":1554940800000},"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>Here, we propose optimal fourth-order iterative methods for approximating multiple zeros of univariate functions. The proposed family is composed of two stages and requires 3 functional values at each iteration. We also suggest an extensive convergence analysis that demonstrated the establishment of fourth-order convergence of the developed methods. It is interesting to note that some existing schemes are found to be the special cases of our proposed scheme. Numerical experiments have been performed on a good number of problems arising from different disciplines such as the fractional conversion problem of a chemical reactor, continuous stirred tank reactor problem, and Planck\u2019s radiation law problem. Computational results demonstrates that suggested methods are better and efficient than their existing counterparts.<\/jats:p>","DOI":"10.3390\/sym11040526","type":"journal-article","created":{"date-parts":[[2019,4,12]],"date-time":"2019-04-12T03:46:37Z","timestamp":1555040797000},"page":"526","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Modified Optimal Class of Newton-Like Fourth-Order Methods for Multiple Roots"],"prefix":"10.3390","volume":"11","author":[{"given":"Munish","family":"Kansal","sequence":"first","affiliation":[{"name":"School of Mathematics, Thapar Institute of Engineering and Technology, Patiala 147004, India"}]},{"given":"Ramandeep","family":"Behl","sequence":"additional","affiliation":[{"name":"Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia"}]},{"given":"Mohammed Ali A.","family":"Mahnashi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia"}]},{"given":"Fouad Othman","family":"Mallawi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2019,4,11]]},"reference":[{"key":"ref_1","unstructured":"Ostrowski, A.M. (1960). Solution of Equations and Systems of Equations, Academic Press."},{"key":"ref_2","unstructured":"Traub, J.F. (1964). Iterative Methods for the Solution of Equations, Prentice-Hall."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"564","DOI":"10.1016\/j.amc.2013.06.097","article-title":"New optimal class of higher-order methods for multiple roots, permitting f\u2032(xn)=0","volume":"222","author":"Kanwar","year":"2013","journal-title":"Appl. Math. Comput."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"520","DOI":"10.1016\/j.amc.2015.05.004","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_5","doi-asserted-by":"crossref","first-page":"775","DOI":"10.1007\/s11075-015-0023-5","article-title":"An optimal fourth-order family of methods for multiple roots and its dynamics","volume":"71","author":"Behl","year":"2016","journal-title":"Numer. Algor."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1288","DOI":"10.1016\/j.amc.2009.06.065","article-title":"A new fourth-order iterative method for finding multiple roots of nonlinear equations","volume":"215","author":"Li","year":"2009","journal-title":"Appl. Math. Comput."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"358","DOI":"10.1016\/j.amc.2013.08.077","article-title":"On the development of iterative methods for multiple roots","volume":"224","author":"Neta","year":"2013","journal-title":"Appl. Math. Comput."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"878","DOI":"10.1016\/j.amc.2010.06.031","article-title":"Modified Jarratt method for computing multiple roots","volume":"217","author":"Sharma","year":"2010","journal-title":"Appl. Math. Comput."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"4199","DOI":"10.1016\/j.cam.2011.03.014","article-title":"Constructing higher-order methods for obtaining the multiple roots of nonlinear equations","volume":"235","author":"Zhou","year":"2011","journal-title":"Comput. Appl. Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"8436759","DOI":"10.1155\/2016\/8436759","article-title":"A triparametric family of optimal fourth-order multiple-root finders and their Dynamics","volume":"2016","author":"Kim","year":"2016","journal-title":"Discret. Dyn. Nat. Soc."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"126","DOI":"10.1016\/j.camwa.2009.08.066","article-title":"Some fourth-order nonlinear solvers with closed formulae for multiple roots","volume":"59","author":"Li","year":"2010","journal-title":"Comput. Math. Appl."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1023","DOI":"10.1080\/00207160802272263","article-title":"Extension of Murakami\u2019s high-order non-linear solver to multiple roots","volume":"87","author":"Neta","year":"2010","journal-title":"Int. J. Comput. Math."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"6030","DOI":"10.1016\/j.amc.2012.12.041","article-title":"Families of third and fourth order methods for multiple roots of nonlinear equations","volume":"219","author":"Zhou","year":"2013","journal-title":"Appl. Math. Comput."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"564","DOI":"10.1016\/j.amc.2017.08.005","article-title":"On the dynamics of tri-parametric family of optimal fourth-order multiple-zero finders with a weight function of the principal mth root of a function-function ratio","volume":"315","author":"Lee","year":"2017","journal-title":"Appl. Math. Comput."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Zafar, F., Cordero, A., and Torregrosa, J.R. (2018). Stability analysis of a family of optimal fourth-order methods for multiple roots. Numer. Algor.","DOI":"10.1007\/s11075-018-0577-0"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1455","DOI":"10.1002\/nme.1620230805","article-title":"Numerical solution of constrained nonlinear algebraic equations","volume":"23","author":"Shacham","year":"1986","journal-title":"Int. J. Numer. Method Eng."},{"key":"ref_17","unstructured":"Constantinides, A. (1999). , Mostoufi, N. Numerical Methods for Chemical Engineers with MATLAB Applications, Prentice Hall PTR."},{"key":"ref_18","unstructured":"Douglas, J.M. (1972). Process Dynamics and Control, Prentice Hall."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1072","DOI":"10.1080\/00207160.2012.746677","article-title":"Families of Newton-like method with fourth-order convergence","volume":"90","author":"Jain","year":"2013","journal-title":"Int. J. Comput. Math."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/4\/526\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:44:45Z","timestamp":1760186685000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/4\/526"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,4,11]]},"references-count":19,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2019,4]]}},"alternative-id":["sym11040526"],"URL":"https:\/\/doi.org\/10.3390\/sym11040526","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2019,4,11]]}}}