{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,13]],"date-time":"2026-03-13T19:44:10Z","timestamp":1773431050215,"version":"3.50.1"},"reference-count":24,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2019,11,26]],"date-time":"2019-11-26T00:00:00Z","timestamp":1574726400000},"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 optimal order multiple root techniques involving derivatives have been proposed in literature. On the contrary, optimal order multiple root techniques without derivatives are almost nonexistent. With this as a motivational factor, here we develop a family of optimal fourth-order derivative-free iterative schemes for computing multiple roots. The procedure is based on two steps of which the first is Traub\u2013Steffensen iteration and second is Traub\u2013Steffensen-like iteration. Theoretical results proved for particular cases of the family are symmetric to each other. This feature leads us to prove the general result that shows the fourth-order convergence. Efficacy is demonstrated on different test problems that verifies the efficient convergent nature of the new methods. Moreover, the comparison of performance has proven the presented derivative-free techniques as good competitors to the existing optimal fourth-order methods that use derivatives.<\/jats:p>","DOI":"10.3390\/sym11121452","type":"journal-article","created":{"date-parts":[[2019,11,26]],"date-time":"2019-11-26T10:57:27Z","timestamp":1574765847000},"page":"1452","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":36,"title":["On a Class of Optimal Fourth Order Multiple Root Solvers without Using Derivatives"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4627-2795","authenticated-orcid":false,"given":"Janak Raj","family":"Sharma","sequence":"first","affiliation":[{"name":"Department of Mathematics, Sant Longowal Institute of Engineering and Technology, Longowal, Sangrur 148106, India"}]},{"given":"Sunil","family":"Kumar","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Sant Longowal Institute of Engineering and Technology, Longowal, Sangrur 148106, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8524-743X","authenticated-orcid":false,"given":"Lorentz","family":"J\u00e4ntschi","sequence":"additional","affiliation":[{"name":"Department of Physics and Chemistry, Technical University of Cluj-Napoca, 400114 Cluj-Napoca, Romania"},{"name":"Institute of Doctoral Studies, Babe\u015f-Bolyai University, 400084 Cluj-Napoca, Romania"}]}],"member":"1968","published-online":{"date-parts":[[2019,11,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"317","DOI":"10.1007\/BF01444024","article-title":"\u00dcber unendlich viele Algorithmen zur Aufl\u00f6sung der Gleichungen","volume":"2","year":"1870","journal-title":"Math. Ann."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1007\/BF01396176","article-title":"A family of root finding methods","volume":"27","author":"Hansen","year":"1977","journal-title":"Numer. Math."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"329","DOI":"10.1080\/00207168208803346","article-title":"A higher order method for multiple zeros of nonlinear functions","volume":"12","author":"Victory","year":"1983","journal-title":"Int. J. Comput. Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"363","DOI":"10.1080\/00207168708803576","article-title":"A family of multipoint iterative functions for finding multiple roots of equations","volume":"21","author":"Dong","year":"1987","journal-title":"Int. J. Comput. Math."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1016\/0377-0427(94)00044-1","article-title":"An optimal multiple root-finding method of order three","volume":"51","author":"Osada","year":"1994","journal-title":"J. Comput. Appl. Math."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"162","DOI":"10.1016\/j.amc.2008.01.031","article-title":"New third order nonlinear solvers for multiple roots","volume":"202","author":"Neta","year":"2008","journal-title":"App. Math. Comput."},{"key":"ref_7","first-page":"1288","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_8","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_9","first-page":"878","article-title":"Modified Jarratt method for computing multiple roots","volume":"217","author":"Sharma","year":"2010","journal-title":"Appl. Math. Comput."},{"key":"ref_10","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":"J. Comput. Appl. Math."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"764","DOI":"10.1016\/j.camwa.2011.11.040","article-title":"Finding the solution of nonlinear equations by a class of optimal methods","volume":"63","author":"Sharifi","year":"2012","journal-title":"Comput. Math. Appl."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"346","DOI":"10.1016\/j.joems.2013.03.011","article-title":"On a numerical technique for finding multiple zeros and its dynamics","volume":"21","author":"Soleymani","year":"2013","journal-title":"J. Egypt. Math. Soc."},{"key":"ref_13","first-page":"387","article-title":"A class of two-point sixth-order multiple-zero finders of modified double-Newton type and their dynamics","volume":"270","author":"Geum","year":"2015","journal-title":"Appl. Math. Comput."},{"key":"ref_14","first-page":"349","article-title":"On some optimal multiple root-finding methods and their dynamics","volume":"10","author":"Kansal","year":"2015","journal-title":"Appl. Appl. Math."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1843002","DOI":"10.1142\/S0219876218430028","article-title":"An optimal eighth-order scheme for multiple zeros of unvariate functions","volume":"16","author":"Behl","year":"2019","journal-title":"Int. J. Comput. Meth."},{"key":"ref_16","unstructured":"Traub, J.F. (1982). Iterative Methods for the Solution of Equations, Chelsea Publishing Company."},{"key":"ref_17","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_18","unstructured":"Wolfram, S. (2003). The Mathematica Book, Wolfram Media. [5th ed.]."},{"key":"ref_19","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_20","unstructured":"Bradie, B. (2006). A Friendly Introduction to Numerical Analysis, Pearson Education Inc."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"e25614","DOI":"10.1002\/qua.25614","article-title":"Conformational study of C24 cyclic polyyne clusters","volume":"118","year":"2018","journal-title":"Int. J. Quantum Chem."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Azad, H., Anaya, K., Al-Dweik, A.Y., and Mustafa, M.T. (2018). Invariant solutions of the wave equation on static spherically symmetric spacetimes admitting G7 isometry algebra. Symmetry, 10.","DOI":"10.3390\/sym10120665"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Matko, V., and Brezovec, B. (2018). Improved data center energy efficiency and availability with multilayer node event processing. Energies, 11.","DOI":"10.3390\/en11092478"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"J\u00e4ntschi, L. (2019). The eigenproblem translated for alignment of molecules. Symmetry, 11.","DOI":"10.3390\/sym11081027"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/12\/1452\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:37:40Z","timestamp":1760189860000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/12\/1452"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,11,26]]},"references-count":24,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2019,12]]}},"alternative-id":["sym11121452"],"URL":"https:\/\/doi.org\/10.3390\/sym11121452","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,11,26]]}}}