{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:57:57Z","timestamp":1760147877891,"version":"build-2065373602"},"reference-count":20,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2023,3,8]],"date-time":"2023-03-08T00:00:00Z","timestamp":1678233600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In the present paper, we introduced a quadratically convergent Newton-like normal S-iteration method free from the second derivative for the solution of nonlinear equations permitting f\u2032(x)=0 at some points in the neighborhood of the root. Our proposed method works well when the Newton method fails and performs even better than some higher-order converging methods. Numerical results verified that the Newton-like normal S-iteration method converges faster than Fang et al.\u2019s method. We studied different aspects of the normal S-iteration method regarding the faster convergence to the root. Lastly, the dynamic results support the numerical results and explain the convergence, divergence, and stability of the proposed method.<\/jats:p>","DOI":"10.3390\/axioms12030283","type":"journal-article","created":{"date-parts":[[2023,3,8]],"date-time":"2023-03-08T03:59:32Z","timestamp":1678247972000},"page":"283","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Newton-like Normal S-iteration under Weak Conditions"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3383-2623","authenticated-orcid":false,"given":"Manoj K.","family":"Singh","sequence":"first","affiliation":[{"name":"College of Technology, Sardar Vallabhbhai Patel University of Agriculture and Technology, Meerut 250110, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ioannis K.","family":"Argyros","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, Cameron University, Lawton, OK 73505, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Arvind K.","family":"Singh","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi 221005, India"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,3,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1119","DOI":"10.1007\/s40819-017-0405-6","article-title":"A Family of High Order Numerical Methods for Solving Nonlinear Algebraic Equations with Simple and Multiple Roots","volume":"3","author":"Baccouch","year":"2017","journal-title":"Int. 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Appl."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"135","DOI":"10.37193\/CMI.2016.02.03","article-title":"A qualitative study of Agarwal et al. iteration procedure for fixed points approximation","volume":"25","author":"Ardelean","year":"2016","journal-title":"Creat. Math. Inform."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"88","DOI":"10.1016\/j.amc.2011.05.055","article-title":"A comparison between iterative methods by using the basins of attraction","volume":"218","author":"Ardelean","year":"2011","journal-title":"Appl. Math. Comput."},{"key":"ref_8","first-page":"53","article-title":"Convergence region of Newton iterative power flow method: Numerical studies","volume":"4","author":"Deng","year":"2013","journal-title":"J. Appl. Math."},{"key":"ref_9","unstructured":"Kotarski, W., Gdawiec, K., and Lisowska, A. (2012). International Symposium on Visual Computing, Springer."},{"key":"ref_10","first-page":"3","article-title":"Review of some iterative root-finding methods from a dynamical point of view","volume":"10","author":"Amat","year":"2004","journal-title":"Scientia"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"409","DOI":"10.1016\/j.cam.2007.08.013","article-title":"A cubically convergent Newton-type method under weak conditions","volume":"220","author":"Fang","year":"2008","journal-title":"J. Comput. Appl. Math."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1016\/S0096-3003(99)00049-1","article-title":"A new continuation Newton-like method and its deformation","volume":"112","author":"Wu","year":"2000","journal-title":"Appl. Math. Comput."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"725","DOI":"10.1007\/s10440-009-9470-0","article-title":"Note on a Cubically Convergent Newton\u2013Type Method Under Weak Conditions","volume":"110","author":"Wang","year":"2010","journal-title":"Acta Appl. Math."},{"key":"ref_14","first-page":"187","article-title":"Applications of the S-iteration process to constrained minimization problems and split feasibility problems","volume":"12","author":"Sahu","year":"2011","journal-title":"Fixed Point Theory"},{"key":"ref_15","unstructured":"Sahu, D.R. Strong convergence of a fixed point iteration process with applications. Proceedings of the International Conference on Recent Advances in Mathematical Sciences and Applications, Available online: https:\/\/scholar.google.ca\/scholar?hl=en&as_sdt=0%2C5&q=Strong+convergence+of+a+fixed+point+iteration+process+with+applications&btnG=."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"2584","DOI":"10.1016\/j.amc.2011.07.076","article-title":"Basin attractors for various methods","volume":"218","author":"Scott","year":"2011","journal-title":"Appl. Math. Comput."},{"key":"ref_17","unstructured":"Argyros, I.K., and Magrenan, A.A. (2017). Iterative Methods and Their Dynamics with Applications: A Contemporary Study, CRC Press, Taylor and Francis."},{"key":"ref_18","first-page":"47","article-title":"Memoire sure l\u2019iteration des fonction rationelles","volume":"81","author":"Julia","year":"1918","journal-title":"J. Math. Pures et Appl."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Mandelbrot, B.B. (1983). The Fractal Geometry of Nature, WH Freeman.","DOI":"10.1119\/1.13295"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1084","DOI":"10.1016\/j.amc.2009.06.041","article-title":"Newton\u2019s method\u2019s basins of attraction revisited","volume":"215","author":"Susanto","year":"2009","journal-title":"Appl. Math. 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