{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,31]],"date-time":"2025-10-31T14:34:14Z","timestamp":1761921254493,"version":"build-2065373602"},"reference-count":35,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2024,4,10]],"date-time":"2024-04-10T00:00:00Z","timestamp":1712707200000},"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>Iterative algorithms requiring the computationally expensive in general inversion of linear operators are difficult to implement. This is the reason why hybrid Newton-like algorithms without inverses are developed in this paper to solve Banach space-valued nonlinear equations. The inverses of the linear operator are exchanged by a finite sum of fixed linear operators. Two types of convergence analysis are presented for these algorithms: the semilocal and the local. The Fr\u00e9chet derivative of the operator on the equation is controlled by a majorant function. The semi-local analysis also relies on majorizing sequences. The celebrated contraction mapping principle is utilized to study the convergence of the Krasnoselskij-like algorithm. The numerical experimentation demonstrates that the new algorithms are essentially as effective but less expensive to implement. Although the new approach is demonstrated for Newton-like algorithms, it can be applied to other single-step, multistep, or multipoint algorithms using inverses of linear operators along the same lines.<\/jats:p>","DOI":"10.3390\/a17040154","type":"journal-article","created":{"date-parts":[[2024,4,10]],"date-time":"2024-04-10T06:07:46Z","timestamp":1712729266000},"page":"154","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Hybrid Newton-like Inverse Free Algorithms for Solving Nonlinear Equations"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9189-9298","authenticated-orcid":false,"given":"Ioannis K.","family":"Argyros","sequence":"first","affiliation":[{"name":"Department of Computing and Mathematical Sciences, Cameron University, Lawton, OK 73505, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3530-5539","authenticated-orcid":false,"given":"Santhosh","family":"George","sequence":"additional","affiliation":[{"name":"Department of Mathematical & Computational Science, National Institute of Technology Karnataka, Surathkal 575025, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0035-1022","authenticated-orcid":false,"given":"Samundra","family":"Regmi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Houston, Houston, TX 77205, USA"}]},{"given":"Christopher I.","family":"Argyros","sequence":"additional","affiliation":[{"name":"School of Computational Science and Engineering, Georgia Institute of Technology, North Avenue Atlanta, Atlanta, GA 30332, USA"}]}],"member":"1968","published-online":{"date-parts":[[2024,4,10]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Driscoll, T.A., and Braun, R.J. 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