{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:20:08Z","timestamp":1760145608800,"version":"build-2065373602"},"reference-count":43,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2024,8,14]],"date-time":"2024-08-14T00:00:00Z","timestamp":1723593600000},"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>We present a new family of optimal eighth-order numerical methods for finding the multiple zeros of nonlinear functions. The methodology used for constructing the iterative scheme is based on the approach called the \u2018weight factor approach\u2019. This approach ingeniously combines weight functions to enhance convergence properties and stability. An extensive convergence analysis is conducted to prove that the proposed scheme achieves optimal eighth-order convergence, providing a significant improvement in efficiency over lower-order methods. Furthermore, the applicability of these novel methods to some real-world problems is demonstrated, showcasing their superior performance in terms of speed and accuracy. This is illustrated through a series of three examples involving basins of attraction with reflection symmetry, confirming the dominance of the new methods over existing counterparts. The examples highlight not only the robustness and precision of the proposed methods but also their practical utility in solving the complex nonlinear equations encountered in various scientific and engineering domains. Consequently, these eighth-order methods hold great promise for advancing computational techniques in fields that require the resolution of multiple roots with high precision.<\/jats:p>","DOI":"10.3390\/sym16081045","type":"journal-article","created":{"date-parts":[[2024,8,14]],"date-time":"2024-08-14T09:20:51Z","timestamp":1723627251000},"page":"1045","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["An Optimal Family of Eighth-Order Methods for Multiple-Roots and Their Complex Dynamics"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8471-5139","authenticated-orcid":false,"given":"Sunil","family":"Kumar","sequence":"first","affiliation":[{"name":"Department of Mathematics, University Centre for Research and Development, Chandigarh University, Mohali 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 Technology, Longowal 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"}]}],"member":"1968","published-online":{"date-parts":[[2024,8,14]]},"reference":[{"key":"ref_1","unstructured":"Traub, J.F. 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