{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:24:48Z","timestamp":1760239488771,"version":"build-2065373602"},"reference-count":29,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2020,11,28]],"date-time":"2020-11-28T00:00:00Z","timestamp":1606521600000},"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, which use derivatives in the algorithm, have been proposed in literature. But contrarily, derivative free optimal order techniques for multiple root are almost nonexistent. By this as an inspirational factor, here we present a family of optimal fourth order derivative-free techniques for computing multiple roots of nonlinear equations. At the beginning the convergence analysis is executed for particular values of multiplicity afterwards it concludes in general form. Behl et. al derivative-free method is seen as special case of the family. Moreover, the applicability and comparison is demonstrated on different nonlinear problems that certifies the efficient convergent nature of the new methods. Finally, we conclude that our new methods consume the lowest CPU time as compared to the existing ones. This illuminates the theoretical outcomes to a great extent of this study.<\/jats:p>","DOI":"10.3390\/sym12121969","type":"journal-article","created":{"date-parts":[[2020,11,29]],"date-time":"2020-11-29T21:55:31Z","timestamp":1606686931000},"page":"1969","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["A Family of Derivative Free Optimal Fourth Order Methods for Computing Multiple Roots"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8471-5139","authenticated-orcid":false,"given":"Sunil","family":"Kumar","sequence":"first","affiliation":[{"name":"Department of Mathematics, Sant Longowal Institute of Engineering and Technology, Longowal, Sangrur 148106, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3500-3087","authenticated-orcid":false,"given":"Deepak","family":"Kumar","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Chandigarh University, 140413, Mohali, NH-95 Chandigarh-Ludhiana Highway, Punjab 140413, India"}]},{"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"}]},{"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":[[2020,11,28]]},"reference":[{"key":"ref_1","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. 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