{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:35:59Z","timestamp":1760060159183,"version":"build-2065373602"},"reference-count":22,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2025,8,4]],"date-time":"2025-08-04T00:00:00Z","timestamp":1754265600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Institute of Technology Karnataka, India"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>A multi-step method introduced by Raziyeh and Masoud for solving nonlinear systems with convergence order five has been considered in this paper. The convergence of the method was studied using Taylor series expansion, which requires the function to be six times differentiable. However, our convergence study does not depend on the Taylor series. We use the derivative of F up to two only in our convergence analysis, which is presented in a more general Banach space setting. Semi-local analysis is also discussed, which was not given in earlier studies. Unlike in earlier studies (where two sets of assumptions were used), we used the same set of assumptions for semi-local analysis and local convergence analysis. We discussed the dynamics of the method and also gave some numerical examples to illustrate theoretical findings.<\/jats:p>","DOI":"10.3390\/a18080483","type":"journal-article","created":{"date-parts":[[2025,8,5]],"date-time":"2025-08-05T07:49:58Z","timestamp":1754380198000},"page":"483","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["An Improved Convergence Analysis of a Multi-Step Method with High-Efficiency Indices"],"prefix":"10.3390","volume":"18","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3530-5539","authenticated-orcid":false,"given":"Santhosh","family":"George","sequence":"first","affiliation":[{"name":"Department of Mathematical and Computational Sciences, National Institute of Technology Karnataka, Mangalore 575025, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0007-1093-4404","authenticated-orcid":false,"given":"Manjusree","family":"Gopal","sequence":"additional","affiliation":[{"name":"Department of Mathematical and Computational Sciences, National Institute of Technology Karnataka, Mangalore 575025, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0005-7170-6670","authenticated-orcid":false,"given":"Samhitha","family":"Bhide","sequence":"additional","affiliation":[{"name":"Department of Mathematical and Computational Sciences, National Institute of Technology Karnataka, Mangalore 575025, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9189-9298","authenticated-orcid":false,"given":"Ioannis K.","family":"Argyros","sequence":"additional","affiliation":[{"name":"Department of Computing and Mathematical Sciences, Cameron University, Lawton, OK 73505, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,8,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1007\/s11075-009-9342-8","article-title":"On new iterative method for solving systems of nonlinear equations","volume":"54","author":"Awawdeh","year":"2010","journal-title":"Numer. 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A Contemporary Study of Iterative Methods: Convergence, Dynamics and Applications, Academic Press.","DOI":"10.1016\/B978-0-12-809214-9.00023-1"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1007\/s10092-013-0097-1","article-title":"Efficient Jarratt-like methods for solving systems of nonlinear equations","volume":"51","author":"Sharma","year":"2014","journal-title":"Calcolo"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1007\/s11075-009-9359-z","article-title":"A modified Newton-Jarratt\u2019s composition","volume":"55","author":"Cordero","year":"2010","journal-title":"Numer. Algorithms"},{"key":"ref_8","first-page":"129401","article-title":"First optimal vectorial eighth-order iterative scheme for solving non-linear systems","volume":"498","author":"Cordero","year":"2025","journal-title":"Appl. Math. 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