{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,18]],"date-time":"2026-05-18T13:42:50Z","timestamp":1779111770733,"version":"3.51.4"},"reference-count":13,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2015,12,31]],"date-time":"2015-12-31T00:00:00Z","timestamp":1451520000000},"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>The singularity of Jacobian happens when we are looking for a root, with multiplicity greater than one, of a system of nonlinear equations. The purpose of this article is two-fold. Firstly, we will present a modification of an existing method that computes roots with known multiplicities. Secondly, will propose the generalization of a family of methods for solving nonlinear equations with unknown multiplicities, to the system of nonlinear equations. The inclusion of a nonzero multi-variable auxiliary function is the key idea. Different choices of the auxiliary function give different families of the iterative method to find roots with unknown multiplicities. Few illustrative numerical experiments and a critical discussion end the paper.<\/jats:p>","DOI":"10.3390\/a9010005","type":"journal-article","created":{"date-parts":[[2016,1,5]],"date-time":"2016-01-05T02:29:07Z","timestamp":1451960947000},"page":"5","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["A Family of Iterative Methods for Solving Systems of Nonlinear Equations Having Unknown Multiplicity"],"prefix":"10.3390","volume":"9","author":[{"given":"Fayyaz","family":"Ahmad","sequence":"first","affiliation":[{"name":"Dipartimento di Scienza e Alta Tecnologia, Universita dell\u2019Insubria, Via Valleggio 11, Como 22100, Italy"},{"name":"Departament de F\u00edsica i Enginyeria Nuclear, Universitat Polit\u00e8cnica de Catalunya, Comte d\u2019Urgell 187, 08036 Barcelona, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"S.","family":"Serra-Capizzano","sequence":"additional","affiliation":[{"name":"Dipartimento di Scienza e Alta Tecnologia, Universita dell\u2019Insubria, Via Valleggio 11, Como 22100, Italy"},{"name":"Department of Information Technology, Uppsala University, Box 337, SE-751 05 Uppsala, Sweden"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Malik","family":"Ullah","sequence":"additional","affiliation":[{"name":"Dipartimento di Scienza e Alta Tecnologia, Universita dell\u2019Insubria, Via Valleggio 11, Como 22100, Italy"},{"name":"Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"A.","family":"Al-Fhaid","sequence":"additional","affiliation":[{"name":"Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2015,12,31]]},"reference":[{"key":"ref_1","unstructured":"Traub, J.F. (1964). Iterative Methods for the Solution of Equations, Prentice-Hall."},{"key":"ref_2","unstructured":"Ortega, J.M., and Rheinbodt, W.C. (1970). Iterative Solution of Nonlinear Equations in Several Variables, Academic Press."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Ahmad, F., Tohidi, E., and Carrasco, J.A. (2015). A parameterized multi-step Newton method for solving systems of nonlinear equations. Numer. Algorithms.","DOI":"10.1007\/s11075-015-0013-7"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1016\/j.amc.2014.10.103","article-title":"An efficient multi-step iterative method for computing the numerical solution of systems of nonlinear equations associated with ODEs","volume":"250","author":"Ullah","year":"2015","journal-title":"Appl. Math. Comput."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"624","DOI":"10.1016\/j.camwa.2015.05.012","article-title":"Higher order multi-step Jarratt-like method for solving systems of nonlinear equations: Application to PDEs and ODEs","volume":"70","author":"Ahmad","year":"2015","journal-title":"Comput. Math. Appl."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Alaidarous, E.S., Ullah, M.Z., Ahmad, F., and Al-Fhaid, A.S. (2013). An Efficient Higher-Order Quasilinearization Method for Solving Nonlinear BVPs. J. Appl. Math., 2013.","DOI":"10.1155\/2013\/259371"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1007\/s11075-013-9784-x","article-title":"Numerical solution of nonlinear systems by a general class of iterative methods with application to nonlinear PDEs","volume":"67","author":"Ullah","year":"2014","journal-title":"Numer. Algorithms"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Montazeri, H., Soleymani, F., Shateyi, S., and Motsa, S.S. (2012). On a New Method for Computing the Numerical Solution of Systems of Nonlinear Equations. J. Appl. Math., 2012.","DOI":"10.1155\/2012\/751975"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1007\/s11075-009-9359-z","article-title":"A modified Newton-Jarratt\u00e2\u0102\u0179s composition","volume":"55","author":"Cordero","year":"2010","journal-title":"Numer. Algorithms"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1016\/j.cam.2008.04.013","article-title":"Modified Newton\u2019s method for systems of nonlinear equations with singular Jacobian","volume":"224","author":"Hueso","year":"2009","journal-title":"J. Comput. Appl. Math."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1476","DOI":"10.1016\/j.amc.2006.12.035","article-title":"Note on the improvement of Newton\u2019s method for systems of nonlinear equations","volume":"189","author":"Wu","year":"2007","journal-title":"Appl. Math. Comput."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"2367","DOI":"10.12785\/amis\/080532","article-title":"A Family of Iterative Schemes for Finding Zeros of Nonlinear Equations having Unknown Multiplicity","volume":"8","author":"Noor","year":"2014","journal-title":"Appl. Math. Inf. Sci."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"815","DOI":"10.1137\/0721054","article-title":"Tensor methods for nonlinear equations","volume":"21","author":"Schnabel","year":"1984","journal-title":"SIAM J. Numer. 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