{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,25]],"date-time":"2026-04-25T09:15:47Z","timestamp":1777108547617,"version":"3.51.4"},"reference-count":22,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2020,3,29]],"date-time":"2020-03-29T00:00:00Z","timestamp":1585440000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["EP\/P018912\/1"],"award-info":[{"award-number":["EP\/P018912\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["EP\/R010811\/1"],"award-info":[{"award-number":["EP\/R010811\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>Finding roots of equations is at the heart of most computational science. A well-known and widely used iterative algorithm is Newton\u2019s method. However, its convergence depends heavily on the initial guess, with poor choices often leading to slow convergence or even divergence. In this short note, we seek to enlarge the basin of attraction of the classical Newton\u2019s method. The key idea is to develop a relatively simple multiplicative transform of the original equations, which leads to a reduction in nonlinearity, thereby alleviating the limitation of Newton\u2019s method. Based on this idea, we derive a new class of iterative methods and rediscover Halley\u2019s method as the limit case. We present the application of these methods to several mathematical functions (real, complex, and vector equations). Across all examples, our numerical experiments suggest that the new methods converge for a significantly wider range of initial guesses. For scalar equations, the increase in computational cost per iteration is minimal. For vector functions, more extensive analysis is needed to compare the increase in cost per iteration and the improvement in convergence of specific problems.<\/jats:p>","DOI":"10.3390\/a13040078","type":"journal-article","created":{"date-parts":[[2020,3,31]],"date-time":"2020-03-31T13:27:19Z","timestamp":1585661239000},"page":"78","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Beyond Newton: A New Root-Finding Fixed-Point Iteration for Nonlinear Equations"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1755-8807","authenticated-orcid":false,"given":"Ankush","family":"Aggarwal","sequence":"first","affiliation":[{"name":"Glasgow Computational Engineering Centre, James Watt School of Engineering, University of Glasgow, Glasgow G12 8QQ, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2081-308X","authenticated-orcid":false,"given":"Sanjay","family":"Pant","sequence":"additional","affiliation":[{"name":"Zienkiewicz Centre for Computational Engineering, College of Engineering, Swansea University, Swansea SA1 8EN, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,3,29]]},"reference":[{"key":"ref_1","unstructured":"Apostol, T.M. 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