{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:58:07Z","timestamp":1760237887632,"version":"build-2065373602"},"reference-count":16,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2020,7,1]],"date-time":"2020-07-01T00:00:00Z","timestamp":1593561600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Solving equations in abstract spaces is important since many problems from diverse disciplines require it. The solutions of these equations cannot be obtained in a form closed. That difficulty forces us to develop ever improving iterative methods. In this paper we improve the applicability of such methods. Our technique is very general and can be used to expand the applicability of other methods. We use two methods of linear interpolation namely the Secant as well as the Kurchatov method. The investigation of Kurchatov\u2019s method is done under rather strict conditions. In this work, using the majorant principle of Kantorovich and our new idea of the restricted convergence domain, we present an improved semilocal convergence of these methods. We determine the quadratical order of convergence of the Kurchatov method and order      1 +  5   2     for the Secant method. We find improved a priori and a posteriori estimations of the method\u2019s error.<\/jats:p>","DOI":"10.3390\/sym12071093","type":"journal-article","created":{"date-parts":[[2020,7,2]],"date-time":"2020-07-02T05:00:08Z","timestamp":1593666008000},"page":"1093","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Extending the Convergence Domain of Methods of Linear Interpolation for the Solution of Nonlinear Equations"],"prefix":"10.3390","volume":"12","author":[{"given":"Ioannis K.","family":"Argyros","sequence":"first","affiliation":[{"name":"Department of Mathematics, Cameron University, Lawton, OK 73505, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3845-6260","authenticated-orcid":false,"given":"Stepan","family":"Shakhno","sequence":"additional","affiliation":[{"name":"Department of Theory of Optimal Processes, Ivan Franko National University of Lviv, Universitetska Str. 1, 79000 Lviv, Ukraine"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Halyna","family":"Yarmola","sequence":"additional","affiliation":[{"name":"Department of Computational Mathematics, Ivan Franko National University of Lviv, Universitetska Str. 1, 79000 Lviv, Ukraine"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,7,1]]},"reference":[{"unstructured":"Shakhno, S.M. (2004, January 24\u201328). Nonlinear majorants for investigation of methods of linear interpolation for the solution of nonlinear equations. Proceedings of the ECCOMAS 2004\u2014European Congress on Computational Methods in Applied Sciences and Engineering, Jyv\u00e4skyl\u00e4, Finland. Available online: http:\/\/www.mit.jyu.fi\/eccomas2004\/proceedings\/pdf\/424.pdf.","key":"ref_1"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1153","DOI":"10.1080\/00207160412331284123","article-title":"On the local convergence of secant-type methods","volume":"81","author":"Amat","year":"2004","journal-title":"Int. J. Comput. Math."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1016\/S0893-9659(01)00150-1","article-title":"The Secant method for nondifferentiable operators","volume":"15","author":"Hernandez","year":"2002","journal-title":"Appl. Math. Lett."},{"unstructured":"Arkeryd, L., Bergh, J., Brenner, P., and Pettersson, R. (1999). Iterative-Difference Methods for Solving Nonlinear Least-Squares Problem. Progress in Industrial Mathematics at ECMI 98, Verlag B. G. Teubner GMBH.","key":"ref_4"},{"key":"ref_5","first-page":"433","article-title":"Algorithms of the generalized Steffensen method","volume":"14","year":"1965","journal-title":"Izv. Akad. Nauk ESSR Ser. Fiz.-Mat."},{"key":"ref_6","first-page":"13","article-title":"On generalized divided differences I, II","volume":"16","year":"1967","journal-title":"Izv. Akad. Nauk ESSR Ser. Fiz.-Mat."},{"unstructured":"Kantorovich, L.V., and Akilov, G.P. (1982). Functional Analysis, Pergamon Press.","key":"ref_7"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1080\/01630568508816182","article-title":"On an iterative algorithm of order 1.839... for solving nonlinear operator equations","volume":"7","author":"Potra","year":"1985","journal-title":"Numer. Funct. Anal. Optim."},{"key":"ref_9","first-page":"55","article-title":"Iteration methods with divided differences of the second order","volume":"158","year":"1964","journal-title":"Doklady Akademii Nauk SSSR"},{"unstructured":"Schwetlick, H. (1979). Numerische L\u00f6sung Nichtlinearer Gleichungen, VEB Deutscher Verlag der Wissenschaften.","key":"ref_10"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1016\/j.jmaa.2006.09.075","article-title":"A Kantorovich-type analysis for a fast iterative method for solving nonlinear equations","volume":"332","author":"Argyros","year":"2007","journal-title":"J. Math. Anal. Appl."},{"doi-asserted-by":"crossref","unstructured":"Argyros, I.K., and George, S. (2019). On a two-step Kurchatov-type method in Banach space. Mediterr. J. Math., 16.","key":"ref_12","DOI":"10.1007\/s00009-018-1285-7"},{"doi-asserted-by":"crossref","unstructured":"Argyros, I.K., and Magre\u00f1\u00e1n, A.A. (2017). Iterative Methods and Their Dynamics with Applications: A Contemporary Study, CRC Press.","key":"ref_13","DOI":"10.1201\/9781315153469"},{"key":"ref_14","first-page":"524","article-title":"On a method of linear interpolation for the solution of functional equations","volume":"198","author":"Kurchatov","year":"1971","journal-title":"Dokl. Akad. Nauk SSSR"},{"key":"ref_15","first-page":"235","article-title":"Kurchatov method of linear interpolation under generalized Lipschitz conditions for divided differences of first and second order","volume":"77","author":"Shakhno","year":"2012","journal-title":"Visnyk Lviv. Univ. Ser. Mech. Math."},{"key":"ref_16","first-page":"105","article-title":"On the difference method with quadratic convergence for solving nonlinear operator equations","volume":"26","author":"Shakhno","year":"2006","journal-title":"Matematychni Studii"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/7\/1093\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:46:00Z","timestamp":1760175960000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/7\/1093"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,7,1]]},"references-count":16,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2020,7]]}},"alternative-id":["sym12071093"],"URL":"https:\/\/doi.org\/10.3390\/sym12071093","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2020,7,1]]}}}