{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,18]],"date-time":"2026-03-18T17:53:01Z","timestamp":1773856381013,"version":"3.50.1"},"reference-count":14,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2015,10,9]],"date-time":"2015-10-09T00:00:00Z","timestamp":1444348800000},"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>In this work, we have developed a fourth order Newton-like method based on harmonic mean and its multi-step version for solving system of nonlinear equations. The new fourth order method requires evaluation of one function and two first order Fr\u00e9chet derivatives for each iteration. The multi-step version requires one more function evaluation for each iteration. The proposed new scheme does not require the evaluation of second or higher order Fr\u00e9chet derivatives and still reaches fourth order convergence. The multi-step version converges with order 2r+4, where r is a positive integer and r \u2265 1. We have proved that the root \u03b1 is a point of attraction for a general iterative function, whereas the proposed new schemes also satisfy this result. Numerical experiments including an application to 1-D Bratu problem are given to illustrate the efficiency of the new methods. Also, the new methods are compared with some existing methods.<\/jats:p>","DOI":"10.3390\/a8040895","type":"journal-article","created":{"date-parts":[[2015,10,9]],"date-time":"2015-10-09T11:37:33Z","timestamp":1444390653000},"page":"895-909","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":17,"title":["On Some Improved Harmonic Mean Newton-Like Methods for Solving Systems of Nonlinear Equations"],"prefix":"10.3390","volume":"8","author":[{"given":"Diyashvir","family":"Babajee","sequence":"first","affiliation":[{"name":"African Network for Policy Research & Advocacy for Sustainability (ANPRAS), Midlands, Curepipe 52501, Mauritius"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9328-2996","authenticated-orcid":false,"given":"Kalyanasundaram","family":"Madhu","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Pondicherry Engineering College, Pondicherry 605014, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8814-3249","authenticated-orcid":false,"given":"Jayakumar","family":"Jayaraman","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Pondicherry Engineering College, Pondicherry 605014, India"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2015,10,9]]},"reference":[{"key":"ref_1","unstructured":"Ostrowski, A.M. (1960). Solutions of Equations and System of Equations, Academic Press."},{"key":"ref_2","first-page":"425","article-title":"On Newton-type methods with cubic convergence","volume":"176","author":"Homeier","year":"2005","journal-title":"Comp. Appl. Math."},{"key":"ref_3","unstructured":"Babajee, D.K.R. (2010). Analysis of Higher Order Variants of Newton\u2019s Method and Their Applications to Differential and Integral Equations and in Ocean Acidification. [Ph.D. Thesis, University of Mauritius]."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1016\/j.camwa.2008.10.067","article-title":"Some iterative methods for solving a system of nonlinear equations","volume":"57","author":"Noor","year":"2009","journal-title":"Comput. Math. 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Mat."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"380","DOI":"10.1002\/num.10055","article-title":"Investigations of nonstandard Mickens-type finite-difference schemes for singular boundary value problems in cylindrical or spherical coordinates","volume":"19","author":"Buckmire","year":"2003","journal-title":"Num. Meth. P. Diff. Eqns."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Babajee, D.K.R., Kalyanasundaram, M., and Jayakumar, J. (2015). A family of higher order multi-point iterative methods based on power mean for solving nonlinear equations. Afr. Mat.","DOI":"10.1007\/s13370-015-0380-1"},{"key":"ref_10","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. Algor."},{"key":"ref_11","unstructured":"Ortega, J.M., and Rheinbolt, W.C. (1970). Iterative Solution of Nonlinear Equations in Several Variables, Academic Press."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"686","DOI":"10.1016\/j.amc.2007.01.062","article-title":"Variants of Newton\u2019s method using fifth-order quadrature formulas","volume":"190","author":"Cordero","year":"2007","journal-title":"Appl. Math. Comp."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"771","DOI":"10.1016\/S0096-3003(03)00178-4","article-title":"Third-order methods from Quadrature Formulae for solving systems of nonlinear equations","volume":"140","author":"Frontini","year":"2004","journal-title":"Appl. Math. Comp."},{"key":"ref_14","first-page":"49","article-title":"A note on two dimensional Bratu problem","volume":"29","author":"Odejide","year":"2006","journal-title":"Kragujevac J. Math."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/8\/4\/895\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T20:49:48Z","timestamp":1760215788000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/8\/4\/895"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,10,9]]},"references-count":14,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2015,12]]}},"alternative-id":["a8040895"],"URL":"https:\/\/doi.org\/10.3390\/a8040895","relation":{},"ISSN":["1999-4893"],"issn-type":[{"value":"1999-4893","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,10,9]]}}}