{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,30]],"date-time":"2025-12-30T08:53:13Z","timestamp":1767084793451,"version":"build-2065373602"},"reference-count":24,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2017,1,18]],"date-time":"2017-01-18T00:00:00Z","timestamp":1484697600000},"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>A modification to an existing iterative method for computing zeros with unknown multiplicities of nonlinear equations or a system of nonlinear equations is presented. We introduce preconditioners to nonlinear equations or a system of nonlinear equations and their corresponding Jacobians. The inclusion of preconditioners provides numerical stability and accuracy. The different selection of preconditioner offers a family of iterative methods. We modified an existing method in a way that we do not alter its inherited quadratic convergence. Numerical simulations confirm the quadratic convergence of the preconditioned iterative method. The influence of preconditioners is clearly reflected in the numerically achieved accuracy of computed solutions.<\/jats:p>","DOI":"10.3390\/a10010017","type":"journal-article","created":{"date-parts":[[2017,1,18]],"date-time":"2017-01-18T10:00:47Z","timestamp":1484733647000},"page":"17","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["A Preconditioned Iterative Method for Solving Systems of Nonlinear Equations Having Unknown Multiplicity"],"prefix":"10.3390","volume":"10","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, Eduard Maristanu 10, Barcelona 08019, Spain"},{"name":"UCERD (Pvt) Ltd, Islamabad 44000, Pakistan"}]},{"given":"Toseef","family":"Bhutta","sequence":"additional","affiliation":[{"name":"Department of Computer Science, University of Gujrat, Gujrat 50700, Pakistan"}]},{"given":"Umar","family":"Shoaib","sequence":"additional","affiliation":[{"name":"Department of Computer Science, University of Gujrat, Gujrat 50700, Pakistan"}]},{"given":"Malik","family":"Zaka 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"}]},{"given":"Ali","family":"Alshomrani","sequence":"additional","affiliation":[{"name":"Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia"}]},{"given":"Shamshad","family":"Ahmad","sequence":"additional","affiliation":[{"name":"Heat and Mass Transfer Technological Center, Technical University of Catalonia, Colom 11, Terrassa 08222, Spain"}]},{"given":"Shahid","family":"Ahmad","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Government College University Lahore, Lahore 54000, Pakistan"}]}],"member":"1968","published-online":{"date-parts":[[2017,1,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"631","DOI":"10.1007\/s11075-015-0013-7","article-title":"A parameterized multi-step Newton method for solving systems of nonlinear equations","volume":"71","author":"Ahmad","year":"2016","journal-title":"Numer. 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Math."},{"key":"ref_5","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_6","doi-asserted-by":"crossref","first-page":"751975","DOI":"10.1155\/2012\/751975","article-title":"On a New Method for Computing the Numerical Solution of Systems of Nonlinear Equations","volume":"2012","author":"Montazeri","year":"2012","journal-title":"J. Appl. Math."},{"key":"ref_7","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. 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