{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:16:27Z","timestamp":1760217387281,"version":"build-2065373602"},"reference-count":41,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2015,8,19]],"date-time":"2015-08-19T00:00:00Z","timestamp":1439942400000},"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>We study the local convergence of an eighth order Newton-like method to approximate a locally-unique solution of a nonlinear equation. Earlier studies, such as Chen et al. (2015) show convergence under hypotheses on the seventh derivative or even higher, although only the first derivative and the divided difference appear in these methods. The convergence in this study is shown under hypotheses only on the first derivative. Hence, the applicability of the method is expanded. Finally, numerical examples are also provided to show that our results apply to solve equations in cases where earlier studies cannot apply.<\/jats:p>","DOI":"10.3390\/a8030645","type":"journal-article","created":{"date-parts":[[2015,8,19]],"date-time":"2015-08-19T10:51:50Z","timestamp":1439981510000},"page":"645-655","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Local Convergence of an Optimal Eighth Order Method under Weak Conditions"],"prefix":"10.3390","volume":"8","author":[{"given":"Ioannis","family":"Argyros","sequence":"first","affiliation":[{"name":"Cameron University, Department of Mathematics Sciences Lawton, OK 73505, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ramandeep","family":"Behl","sequence":"additional","affiliation":[{"name":"School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X01, Scottsville, Pietermaritzburg 3209, South Africa"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"S.S.","family":"Motsa","sequence":"additional","affiliation":[{"name":"School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X01, Scottsville, Pietermaritzburg 3209, South Africa"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2015,8,19]]},"reference":[{"key":"ref_1","unstructured":"Chen, Y., Wang, Y., and Tan, D. (2015). A family of three-step iterative methods with eighth-order convergence for nonlinear equations. Appl. Math. Comput."},{"key":"ref_2","unstructured":"Argyros, I.K. (2008). Convergence and Application of Newton-Type Iterations, Springer."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Argyros, I.K., and Hilout, S. (2013). Numerical Methods in Nonlinear Analysis, World Scientific Publishing Company.","DOI":"10.1142\/8475"},{"key":"ref_4","unstructured":"Traub, J.F. (1964). Iterative Methods for the Solution of Equations, Prentice-Hall Series in Automatic Computation."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1016\/j.amc.2014.01.037","article-title":"Different anomalies in a Jarratt family of iterative root\u2013finding methods","volume":"233","year":"2014","journal-title":"Appl. Math. Comput."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1016\/j.amc.2014.09.061","article-title":"A new tool to study real dynamics: The convergence plane","volume":"248","year":"2014","journal-title":"Appl. Math. Comput."},{"key":"ref_7","unstructured":"Petkovic, M.S., Neta, B., Petkovic, L., and D\u017euni\u010d, J. (2013). Multipoint Methods for Solving Nonlinear Equations, Elsevier."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"6126","DOI":"10.1016\/j.amc.2012.12.016","article-title":"Means based modifications of Newton\u2019s method for solving nonlinear equations","volume":"219","author":"Herceg","year":"2013","journal-title":"Appl. Math. Comput."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1016\/j.cam.2012.12.008","article-title":"Third-order modifications of Newton\u2019s method based on Stolarsky and Gini means","volume":"245","author":"Herceg","year":"2013","journal-title":"J. Comput. Appl. Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"425","DOI":"10.1016\/j.cam.2004.07.027","article-title":"On Newton-type methods with cubic convergence","volume":"176","author":"Homeier","year":"2005","journal-title":"J. Comput. Appl. Math."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"677","DOI":"10.1016\/S0893-9659(04)90104-8","article-title":"Some new variants of Newton\u2019s method","volume":"17","author":"Ozban","year":"2004","journal-title":"Appl. Math. Lett."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1016\/S0893-9659(00)00100-2","article-title":"A variant of Newton\u2019s method with accelerated third order convergence","volume":"13","author":"Weerakoon","year":"2000","journal-title":"Appl. Math. Lett."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"164","DOI":"10.1016\/j.amc.2008.08.050","article-title":"A modified Chebyshev\u2019s iterative method with at least sixth order of convergence","volume":"206","author":"Amat","year":"2008","journal-title":"Appl. Math. Comput."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1016\/j.amc.2014.02.083","article-title":"An improvement of the Chebyshev\u2013Halley methods free from second derivative","volume":"235","author":"Li","year":"2014","journal-title":"Appl. Math. Comput."},{"key":"ref_15","unstructured":"Rheinboldt, W.C. An Adaptive Continuation Process for Solving Systems of Nonlinear Equations. Available online: https:\/\/eudml.org\/doc\/208686."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"472","DOI":"10.1016\/j.aml.2009.12.006","article-title":"On some computational orders of convergence","volume":"23","author":"Noguera","year":"2010","journal-title":"Appl. Math. Lett."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"212","DOI":"10.1007\/s00010-004-2733-y","article-title":"Dynamics of the King and Jarratt iterations","volume":"69","author":"Amat","year":"2005","journal-title":"Aequationes Math."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"24","DOI":"10.1016\/j.jmaa.2010.01.047","article-title":"Chaotic dynamics of a third-order Newton-type method","volume":"366","author":"Amat","year":"2010","journal-title":"J. Math. Anal. Appl."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1007\/BF02241866","article-title":"Recurrence relations for rational cubic methods I: The Halley method","volume":"44","author":"Candela","year":"1990","journal-title":"Computing"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Chicharro, F., Cordero, A., and Torregrosa, J.R. (2013). Drawing dynamical and parameters planes of iterative families and methods. Sci. World J., 2013.","DOI":"10.1155\/2013\/780153"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1432","DOI":"10.1016\/j.amc.2007.02.023","article-title":"Some improvements of Jarratt\u2019s method with sixth-order convergence","volume":"190","author":"Chun","year":"2007","journal-title":"Appl. Math. Comput."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"842","DOI":"10.1016\/j.aml.2013.03.012","article-title":"Chaos in King\u2019s iterative family","volume":"26","author":"Cordero","year":"2013","journal-title":"Appl. Math. Lett."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"8568","DOI":"10.1016\/j.amc.2013.02.042","article-title":"Dynamics of a family of Chebyshev-Halley type methods","volume":"219","author":"Cordero","year":"2013","journal-title":"Appl. Math. Comput."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"686","DOI":"10.1016\/j.amc.2007.01.062","article-title":"Variants of Newton\u2019s method using fifth-order quadrature formulas","volume":"190","author":"Cordero","year":"2007","journal-title":"Appl. Math. Comput."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"325","DOI":"10.1007\/s10543-009-0226-z","article-title":"New iterations of R-order four with reduced computational cost","volume":"49","author":"Ezquerro","year":"2009","journal-title":"BIT Numer. Math."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"591","DOI":"10.1016\/j.jmaa.2004.08.057","article-title":"On the R-order of the Halley method","volume":"303","author":"Ezquerro","year":"2005","journal-title":"J. Math. Anal. Appl."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0898-1221(98)00168-0","article-title":"Recurrence relations for the super-Halley method","volume":"36","year":"1998","journal-title":"Comput. Math. Appl."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"433","DOI":"10.1016\/S0898-1221(00)00286-8","article-title":"Chebyshev\u2019s approximation algorithms and applications","volume":"41","year":"2001","journal-title":"Comput. Math. Appl."},{"key":"ref_29","first-page":"29","article-title":"Sufficient conditions for semilocal convergence of a fourth order multipoint iterative method for solving equations in Banach spaces","volume":"1","author":"Salanova","year":"1999","journal-title":"Southwest J. Pure Appl. Math."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"244","DOI":"10.1016\/j.cam.2014.02.026","article-title":"Sixth-order modifications of Newton\u2019s method based on Stolarsky and Gini means","volume":"267","author":"Herceg","year":"2014","journal-title":"J. Comput. Appl. Math."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"434","DOI":"10.1090\/S0025-5718-66-99924-8","article-title":"Some fourth order multipoint methods for solving equations","volume":"20","author":"Jarratt","year":"1966","journal-title":"Math. Comput."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"369","DOI":"10.1007\/s11075-011-9519-9","article-title":"Semilocal convergence of a modified multi-point Jarratt method in Banach spaces under general continuity conditions","volume":"60","author":"Kou","year":"2012","journal-title":"Numer. Algor."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"126","DOI":"10.1016\/j.amc.2007.01.011","article-title":"On Chebyshev\u2013Halley methods with sixth-order convergence for solving non-linear equations","volume":"190","author":"Kou","year":"2007","journal-title":"Appl. Math. Comput."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.cam.2006.03.022","article-title":"Third-order modification of Newton\u2019s method","volume":"205","author":"Kou","year":"2007","journal-title":"J. Comput. Appl. Math."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"30","DOI":"10.1016\/j.aml.2007.02.010","article-title":"Geometric mean Newton\u2019s method for simple and multiple roots","volume":"21","year":"2008","journal-title":"Appl. Math. Lett."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1080\/00207167908803166","article-title":"A sixth order family of methods for nonlinear equations","volume":"7","author":"Neta","year":"1979","journal-title":"Int. J. Comput. Math."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"873","DOI":"10.1016\/j.cam.2006.08.027","article-title":"Recurrence relations for a Newton-like method in Banach spaces","volume":"206","author":"Parhi","year":"2007","journal-title":"J. Comput. Appl. Math."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1016\/j.amc.2008.03.037","article-title":"A sixth order method for nonlinear equations","volume":"203","author":"Parhi","year":"2008","journal-title":"Appl. Math. Comput."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"585","DOI":"10.1007\/s11075-009-9302-3","article-title":"New variants of Jarratt\u2019s method with sixth-order convergence","volume":"52","author":"Ren","year":"2009","journal-title":"Numer. Algor."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"441","DOI":"10.1007\/s11075-010-9438-1","article-title":"Semilocal convergence of a sixth-order Jarratt method in Banach spaces","volume":"57","author":"Wang","year":"2011","journal-title":"Numer. Algor."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1026","DOI":"10.1016\/j.aml.2006.09.010","article-title":"A class of Newton\u2019s methods with third-order convergence","volume":"20","author":"Zhou","year":"2007","journal-title":"Appl. Math. Lett."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/8\/3\/645\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T20:47:03Z","timestamp":1760215623000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/8\/3\/645"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,8,19]]},"references-count":41,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2015,9]]}},"alternative-id":["a8030645"],"URL":"https:\/\/doi.org\/10.3390\/a8030645","relation":{},"ISSN":["1999-4893"],"issn-type":[{"type":"electronic","value":"1999-4893"}],"subject":[],"published":{"date-parts":[[2015,8,19]]}}}