{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,18]],"date-time":"2026-05-18T13:43:03Z","timestamp":1779111783857,"version":"3.51.4"},"reference-count":25,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2023,3,12]],"date-time":"2023-03-12T00:00:00Z","timestamp":1678579200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>In this paper, we present a new third-order family of iterative methods in order to compute the multiple roots of nonlinear equations when the multiplicity (m\u22651) is known in advance. There is a plethora of third-order point-to-point methods, available in the literature; but our methods are based on geometric derivation and converge to the required zero even though derivative becomes zero or close to zero in vicinity of the required zero. We use the exponential fitted curve and tangency conditions for the development of our schemes. Well-known Chebyshev, Halley, super-Halley and Chebyshev\u2013Halley are the special members of our schemes for m=1. Complex dynamics techniques allows us to see the relation between the element of the family of iterative schemes and the wideness of the basins of attraction of the simple and multiple roots, on quadratic polynomials. Several applied problems are considered in order to demonstrate the performance of our methods and for comparison with the existing ones. Based on the numerical outcomes, we deduce that our methods illustrate better performance over the earlier methods even though in the case of multiple roots of high multiplicity.<\/jats:p>","DOI":"10.3390\/a16030156","type":"journal-article","created":{"date-parts":[[2023,3,13]],"date-time":"2023-03-13T03:03:57Z","timestamp":1678676637000},"page":"156","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["A New Third-Order Family of Multiple Root-Findings Based on Exponential Fitted Curve"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7923-1324","authenticated-orcid":false,"given":"Vinay","family":"Kanwar","sequence":"first","affiliation":[{"name":"University Institute of Engineering and Technology, Panjab University, Chandigarh 160014, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7462-9173","authenticated-orcid":false,"given":"Alicia","family":"Cordero","sequence":"additional","affiliation":[{"name":"Instituto de Matem\u00e1tica Multidisciplinar, Universitat Polit\u00e8cnica de Val\u00e8ncia, Camino de Vera, s\/n, 46022 Valencia, 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":"Instituto de Matem\u00e1tica Multidisciplinar, Universitat Polit\u00e8cnica de Val\u00e8ncia, Camino de Vera, s\/n, 46022 Valencia, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mithil","family":"Rajput","sequence":"additional","affiliation":[{"name":"Indian Institute of Technology, Kharagpur 721302, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1505-8945","authenticated-orcid":false,"given":"Ramandeep","family":"Behl","sequence":"additional","affiliation":[{"name":"Mathematical Modelling and Applied Computation Research Group (MMAC), Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,3,12]]},"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":"1073","DOI":"10.7546\/CR-2013-66-6-13101331-5","article-title":"Convergence of Schr\u00f6der\u2019s method for polynomial zeros of unknown multiplicity","volume":"66","author":"Proinov","year":"2013","journal-title":"C. R. Acad. Bulg. Sci."},{"key":"ref_3","first-page":"73","article-title":"On the numerical solution of equations","volume":"56","author":"Obreshkov","year":"1963","journal-title":"Annuaire Univ. Sofia Fac. Sci. Phys. Math."},{"key":"ref_4","unstructured":"Traub, J.F. (1964). 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