{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,13]],"date-time":"2026-03-13T19:44:01Z","timestamp":1773431041596,"version":"3.50.1"},"reference-count":40,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2022,9,8]],"date-time":"2022-09-08T00:00:00Z","timestamp":1662595200000},"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>In this paper, we describe iterative derivative-free algorithms for multiple roots of a nonlinear equation. Many researchers have evaluated the multiple roots of a nonlinear equation using the first- or second-order derivative of functions. However, calculating the function\u2019s derivative at each iteration is laborious. So, taking this as motivation, we develop second-order algorithms without using the derivatives. The convergence analysis is first carried out for particular values of multiple roots before coming to a general conclusion. According to the Kung\u2013Traub hypothesis, the new algorithms will have optimal convergence since only two functions need to be evaluated at every step. The order of convergence is investigated using Taylor\u2019s series expansion. Moreover, the applicability and comparisons with existing methods are demonstrated on three real-life problems (e.g., Kepler\u2019s, Van der Waals, and continuous-stirred tank reactor problems) and three standard academic problems that contain the root clustering and complex root problems. Finally, we see from the computational outcomes that our approaches use the least amount of processing time compared with the ones already in use. This effectively displays the theoretical conclusions of this study.<\/jats:p>","DOI":"10.3390\/sym14091881","type":"journal-article","created":{"date-parts":[[2022,9,8]],"date-time":"2022-09-08T09:51:09Z","timestamp":1662630669000},"page":"1881","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Optimal Derivative-Free One-Point Algorithms for Computing Multiple Zeros of Nonlinear Equations"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8471-5139","authenticated-orcid":false,"given":"Sunil","family":"Kumar","sequence":"first","affiliation":[{"name":"Department of Mathematics, Amrita School of Engineering, Amrita Vishwa Vidyapeetham, Channai 601103, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6941-1364","authenticated-orcid":false,"given":"Jai","family":"Bhagwan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Government Post Graduate College, Rohtak 124001, 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":[[2022,9,8]]},"reference":[{"key":"ref_1","unstructured":"Traub, J.F. 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