{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:13:30Z","timestamp":1760242410973,"version":"build-2065373602"},"reference-count":10,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2017,6,27]],"date-time":"2017-06-27T00:00:00Z","timestamp":1498521600000},"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 investigate the efficiency of multi-step Newton method (the classical Newton method in which the first derivative is re-evaluated periodically after m steps) for solving nonlinear equations,     F  ( x )  = 0 ,  F : D \u2286  R n  \u2192  R n     . We highlight the following property of multi-step Newton method with respect to some other Newton-type method: for a given n, there exist thresholds of m, that is an interval     (  m i  ,  m s  )    , such that for m inside of this interval, the efficiency index of multi-step Newton method is better than that of other Newton-type method. We also search for optimal values of m.<\/jats:p>","DOI":"10.3390\/a10030075","type":"journal-article","created":{"date-parts":[[2017,6,28]],"date-time":"2017-06-28T10:25:56Z","timestamp":1498645556000},"page":"75","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Thresholds of the Inner Steps in Multi-Step Newton Method"],"prefix":"10.3390","volume":"10","author":[{"given":"Stefan","family":"Maruster","sequence":"first","affiliation":[{"name":"Department of Informatics, West University of Timisoara, B-dul V. Parvan No.4, Timisoara 300223, Romania"}]}],"member":"1968","published-online":{"date-parts":[[2017,6,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"6021","DOI":"10.22436\/jnsa.009.12.09","article-title":"Frozen Jacobian iterative method for solving systems of nonlinear equations: Application to nonlinear IVPs and BVPs","volume":"9","author":"Ullah","year":"2016","journal-title":"J. Nonlinear Sci. Appl."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1016\/j.cam.2014.06.008","article-title":"Semilocal convergence by using recurrence relations for a fifth-order method in Banach spaces","volume":"273","author":"Cordero","year":"2015","journal-title":"J. Comput. Appl. Math."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"434","DOI":"10.1090\/S0025-5718-66-99924-8","article-title":"Some fourth order multipoint iterative methods for solving equations","volume":"20","author":"Jarrat","year":"1966","journal-title":"Math. Comput."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"441","DOI":"10.1007\/s11075-010-9438-1","article-title":"Semilocal convergence for a sixth-order Jarratt method in Banach spaces","volume":"57","author":"Wang","year":"2011","journal-title":"Numer. Algorithms"},{"key":"ref_5","unstructured":"Potra, F.A., and Ptak, V. (1984). Nondiscrete Induction and Iterative Proccesses, Pitman."},{"key":"ref_6","first-page":"293","article-title":"Optimal eighth-order simple root-finders free from derivative","volume":"8","author":"Soleymani","year":"2011","journal-title":"WSEAS Trans. Inf. Sci. Appl."},{"key":"ref_7","first-page":"14","article-title":"New modification of Newton method with third order of convergence for solving nonlinear equation of type f(0) = 0","volume":"6","author":"Thukral","year":"2016","journal-title":"Am. J. Comput. Appl. Math."},{"key":"ref_8","unstructured":"Traub, J.F. (1982). Iterative Methods for the Solution of Equations, Chelsea Publishing Company."},{"key":"ref_9","unstructured":"Ortega, J.M., and Rheinboldt, W.C. (1970). Iterative Solution of Nonlinear Equation in Several Variables, Academic Press."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1016\/j.jco.2009.04.001","article-title":"An optimization of Chebyshev\u2019s method","volume":"25","author":"Ezquerro","year":"2009","journal-title":"J. Complex."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/10\/3\/75\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T18:40:33Z","timestamp":1760208033000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/10\/3\/75"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,6,27]]},"references-count":10,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2017,9]]}},"alternative-id":["a10030075"],"URL":"https:\/\/doi.org\/10.3390\/a10030075","relation":{},"ISSN":["1999-4893"],"issn-type":[{"type":"electronic","value":"1999-4893"}],"subject":[],"published":{"date-parts":[[2017,6,27]]}}}