{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,14]],"date-time":"2025-10-14T00:47:45Z","timestamp":1760402865342,"version":"build-2065373602"},"reference-count":19,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2020,1,15]],"date-time":"2020-01-15T00:00:00Z","timestamp":1579046400000},"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 discuss the local convergence of a derivative-free eighth order method in a Banach space setting. The present study provides the radius of convergence and bounds on errors under the hypothesis based on the first Fr\u00e9chet-derivative only. The approaches of using Taylor expansions, containing higher order derivatives, do not provide such estimates since the derivatives may be nonexistent or costly to compute. By using only first derivative, the method can be applied to a wider class of functions and hence its applications are expanded. Numerical experiments show that the present results are applicable to the cases wherein previous results cannot be applied.<\/jats:p>","DOI":"10.3390\/a13010025","type":"journal-article","created":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T04:14:41Z","timestamp":1579234481000},"page":"25","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Local Convergence of an Efficient Multipoint Iterative Method in Banach Space"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4627-2795","authenticated-orcid":false,"given":"Janak Raj","family":"Sharma","sequence":"first","affiliation":[{"name":"Department of Mathematics, Sant Longowal Institute of Engineering &amp; Technology, Punjab 148106, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sunil","family":"Kumar","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Sant Longowal Institute of Engineering &amp; Technology, Punjab 148106, 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"}]}],"member":"1968","published-online":{"date-parts":[[2020,1,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Rheinbolt, W.C. (1978). An Adaptive Continuation Process for Solving System of Nonlinear Equations, Banach Center Publications.","DOI":"10.4064\/-3-1-129-142"},{"key":"ref_2","unstructured":"Chui, C.K., and Wuytack, L. (2007). Computational Theory of Iterative Methods, Elsevier Publ. Co."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1016\/S0377-0427(99)00347-7","article-title":"Modification of Kantrovich assumptions for semilocal convergence of Chebyshev method","volume":"126","author":"Salanova","year":"2000","journal-title":"J. Comput. Appl. Math."},{"key":"ref_4","first-page":"452","article-title":"A note on local convergence of iterative methods based on Adomian decomposition method and 3-note quadrature rule","volume":"200","author":"Babajee","year":"2008","journal-title":"Appl. Math. 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Iterative Methods and Their Dynamics with Applications, CRC Press.","DOI":"10.1201\/9781315153469"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Sharma, J.R., Argyros, I.K., and Kumar, S. (2018). Ball convergence of an efficient eighth order iterative method under weak conditions. Mathematics, 6.","DOI":"10.3390\/math6110260"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"56","DOI":"10.1016\/j.jco.2018.07.005","article-title":"A fast and efficient composite Newton\u2013Chebyshev method for systems of nonlinear equations","volume":"49","author":"Sharma","year":"2018","journal-title":"J. Complex."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Kumar, D., Argyros, I.K., and Sharma, J.R. (2019). Convergence ball and complex geometry of an iteration function of higher order. Mathematics, 7.","DOI":"10.3390\/math7010028"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Sharma, J.R., Argyros, I.K., and Kumar, S. (2019). Convergence analysis of weighted-Newton methods of optimal eighth order in Banach spaces. Mathematics, 7.","DOI":"10.3390\/math7020198"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Sharma, J.R., Kumar, S., and Argyros, I.K. (2019). Generalized Kung-Traub method and its multi-step iteration in Banach spaces. J. Complex.","DOI":"10.1016\/j.jco.2019.02.003"},{"key":"ref_14","first-page":"630","article-title":"A third-order Newton-type method to solve systems of nonlinear equations","volume":"187","author":"Darvishi","year":"2007","journal-title":"Appl. Math. Comput."},{"key":"ref_15","first-page":"257","article-title":"A fourth-order method from quadrature formulae to solve systems of nonlinear equations","volume":"188","author":"Darvishi","year":"2007","journal-title":"Appl. Math. 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