{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:18:17Z","timestamp":1760242697842,"version":"build-2065373602"},"reference-count":28,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2016,3,11]],"date-time":"2016-03-11T00:00:00Z","timestamp":1457654400000},"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 homotopy method is presented for the construction of frozen Jacobian iterative methods. The frozen Jacobian iterative methods are attractive because the inversion of the Jacobian is performed in terms of LUfactorization only once, for a single instance of the iterative method. We embedded parameters in the iterative methods with the help of the homotopy method: the values of the parameters are determined in such a way that a better convergence rate is achieved. The proposed homotopy technique is general and has the ability to construct different families of iterative methods, for solving weakly nonlinear systems of equations. Further iterative methods are also proposed for solving general systems of nonlinear equations.<\/jats:p>","DOI":"10.3390\/a9010018","type":"journal-article","created":{"date-parts":[[2016,3,14]],"date-time":"2016-03-14T11:03:45Z","timestamp":1457953425000},"page":"18","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Constructing Frozen Jacobian Iterative Methods for Solving Systems of Nonlinear Equations, Associated with ODEs and PDEs Using the Homotopy Method"],"prefix":"10.3390","volume":"9","author":[{"given":"Uswah","family":"Qasim","sequence":"first","affiliation":[{"name":"Department of Mathematics, Riphah International University, Main Satyana Road, Faisalabad 44000, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zulifqar","family":"Ali","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Riphah International University, Main Satyana Road, Faisalabad 44000, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Fayyaz","family":"Ahmad","sequence":"additional","affiliation":[{"name":"Dipartimento di Scienza e Alta Tecnologia, Universit\u00e0 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, Barcelona 08036, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Stefano","family":"Serra-Capizzano","sequence":"additional","affiliation":[{"name":"Dipartimento di Scienza e Alta Tecnologia, Universit\u00e0 dell\u2019Insubria, Via Valleggio 11, Como 22100, Italy"},{"name":"Department of Information Technology, Uppsala University, Box 337, Uppsala SE-751 05, Sweden"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Malik","family":"Zaka Ullah","sequence":"additional","affiliation":[{"name":"Dipartimento di Scienza e Alta Tecnologia, Universit\u00e0 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":"Mir","family":"Asma","sequence":"additional","affiliation":[{"name":"Department of Computer Science, Virtual University, Lahore 5400, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2016,3,11]]},"reference":[{"key":"ref_1","unstructured":"Ortega, J.M., and Rheinbodt, W.C. 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