{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,8]],"date-time":"2026-05-08T11:02:32Z","timestamp":1778238152996,"version":"3.51.4"},"reference-count":34,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2020,6,21]],"date-time":"2020-06-21T00:00:00Z","timestamp":1592697600000},"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>A plethora of higher order iterative methods, involving derivatives in algorithms, are available in the literature for finding multiple roots. Contrary to this fact, the higher order methods without derivatives in the iteration are difficult to construct, and hence, such methods are almost non-existent. This motivated us to explore a derivative-free iterative scheme with optimal fourth order convergence. The applicability of the new scheme is shown by testing on different functions, which illustrates the excellent convergence. Moreover, the comparison of the performance shows that the new technique is a good competitor to existing optimal fourth order Newton-like techniques.<\/jats:p>","DOI":"10.3390\/sym12061038","type":"journal-article","created":{"date-parts":[[2020,6,24]],"date-time":"2020-06-24T07:14:19Z","timestamp":1592982859000},"page":"1038","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":41,"title":["An Optimal Fourth Order Derivative-Free Numerical Algorithm for Multiple Roots"],"prefix":"10.3390","volume":"12","author":[{"given":"Sunil","family":"Kumar","sequence":"first","affiliation":[{"name":"Department of Mathematics, Sant Longowal Institute of Engineering and Technology, Longowal, Sangrur 148106, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Deepak","family":"Kumar","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Chandigarh University, Gharuan, Mohali 140413, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4627-2795","authenticated-orcid":false,"given":"Janak Raj","family":"Sharma","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Sant Longowal Institute of Engineering and Technology, Longowal, Sangrur 148106, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Clemente","family":"Cesarano","sequence":"additional","affiliation":[{"name":"Section of Mathematics, International Telematic University UNINETTUNO, Corso Vittorio Emanuele II, 39, 00186 Roma, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Praveen","family":"Agarwal","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Anand International College of Engineering, Jaipur 303012, Rajasthan, India"},{"name":"International Center for Basic and Applied Sciences, Jaipur 302029, India"},{"name":"Department of Mathematics, Harish-Chandra Research Institute, Allahabad 211 019, India"},{"name":"Department of Mathematics, Netaji Subhas University of Technology Dwarka Sector-3, Dwarka, Delhi 110078, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yu-Ming","family":"Chu","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Huzhou University, Huzhou 313000, China"},{"name":"Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, Changsha University of Science &amp; Technology, Changsha 410114, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,6,21]]},"reference":[{"key":"ref_1","unstructured":"Brezinski, C., and Wuytack, L. (2001). Historical Developments in Convergence Analysis for Newton\u2019s and Newton-Like Methods. Numerical Analysis: Historical Developments in the 20th Century, Elsevier."},{"key":"ref_2","unstructured":"Kantorovich, L.V. (1949). On Newton\u2019s Method. Collected Works on Approximation Analysis of the Leningrad Branch of the Institute, Academy of Sciences of the Soviet Union. Trudy Mat. Inst. Steklov."},{"key":"ref_3","unstructured":"Argyros, I.K. (2008). Convergence and Applications of Newton-Type Iterations, Springer."},{"key":"ref_4","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_5","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_6","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_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","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1016\/j.cam.2017.10.033","article-title":"Constructing a family of optimal eighth-order modified Newton-type multiple-zero finders along with the dynamics behind their purely imaginary extraneous fixed points","volume":"333","author":"Geum","year":"2018","journal-title":"J. Comp. Appl. Math."},{"key":"ref_15","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_16","doi-asserted-by":"crossref","unstructured":"Behl, R., Alsolami, A.J., Pansera, B.A., Al-Hamdan, W.M., Salimi, M., and Ferrara, M. (2019). A new optimal family of Schrder\u2019s method for multiple zeros. Mathematics, 7.","DOI":"10.3390\/math7111076"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Behl, R., Salimi, M., Ferrara, M., Sharifi, S., and Alharbi, S.K. (2019). Some real-life applications of a newly constructed derivative free iterative scheme. Symmetry, 11.","DOI":"10.3390\/sym11020239"},{"key":"ref_18","unstructured":"Traub, J.F. (1982). Iterative Methods for the Solution of Equations, Chelsea Publishing Company."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Kumar, D., Sharma, J.R., and Argyros, I.K. (2020). Optimal one-point iterative function free from derivatives for multiple roots. Mathematics, 8.","DOI":"10.3390\/math8050709"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Sharma, J.R., Kumar, S., and J\u00e4ntschi, L. (2019). On a class of optimal fourth order multiple root solvers without using derivatives. Symmetry, 11.","DOI":"10.3390\/sym11121452"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Sharma, J.R., Kumar, S., and Argyros, I.K. (2019). Development of optimal eighth order derivative-free methods for multiple roots of nonlinear equations. Symmetry, 11.","DOI":"10.3390\/sym11060766"},{"key":"ref_22","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_23","unstructured":"Wolfram, S. (2003). The Mathematica Book, Wolfram Media. [5th ed.]."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/BF01401018","article-title":"Extraneous fixed points, basin boundaries and chaotic dynamics for Schr\u00f6der and K\u00f6nig rational iteration functions","volume":"52","author":"Vrscay","year":"1988","journal-title":"Numer. Math."},{"key":"ref_25","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_26","unstructured":"J\u00e9z\u00e9quel, F. (2005). Dynamical Control of Approximation Methods, Habilitationa diriger des recherches, Universit Pierre et Marie Curie."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1007\/s00009-019-1350-x","article-title":"Optimal fourth-order Weerakoon-Fernando-type methods for multiple roots and their dynamics","volume":"16","author":"Chand","year":"2019","journal-title":"Mediterr. J. Math."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"608","DOI":"10.1016\/j.cam.2018.06.006","article-title":"A study of dynamics via M\u00f6bius conjugacy map on a family of sixth-order modified Newton-like multiple-zero finders with bivariate polynomial weight functions","volume":"344","author":"Geum","year":"2018","journal-title":"J. Comput. Appl. Math."},{"key":"ref_29","first-page":"520","article-title":"On developing fourth-order optimal families of methods for multiple roots and their dynamics","volume":"265","author":"Behl","year":"2018","journal-title":"Appl. Math. Comput."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Sidorov, N., Sidorov, D., and Li, Y. (2019). Basins of attraction and stability of nonlinear systems\u2019 equilibrium points. Differ. Equ. Dyn. Syst.","DOI":"10.1007\/s12591-019-00511-w"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"5043","DOI":"10.1016\/j.amc.2011.10.071","article-title":"Basin attractors for various methods for multiple roots","volume":"218","author":"Neta","year":"2012","journal-title":"App. Math. Comp."},{"key":"ref_32","first-page":"686","article-title":"Variants of Newtons method using fifth order quadrature formulas","volume":"190","author":"Cordero","year":"2007","journal-title":"Appl. Math. Comput."},{"key":"ref_33","unstructured":"Bradie, B. (2006). A Friendly Introduction to Numerical Analysis, Pearson Education Inc."},{"key":"ref_34","unstructured":"Hoffman, J.D. (1992). Numerical Methods for Engineers and Scientists, McGraw-Hill Book Company."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/6\/1038\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:41:22Z","timestamp":1760175682000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/6\/1038"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,6,21]]},"references-count":34,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2020,6]]}},"alternative-id":["sym12061038"],"URL":"https:\/\/doi.org\/10.3390\/sym12061038","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,6,21]]}}}