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Algorithms 2016, 9(1), 18; doi:10.3390/a9010018

Constructing Frozen Jacobian Iterative Methods for Solving Systems of Nonlinear Equations, Associated with ODEs and PDEs Using the Homotopy Method

1
Department of Mathematics, Riphah International University, Main Satyana Road, Faisalabad 44000, Pakistan
2
Dipartimento di Scienza e Alta Tecnologia, Università dell’Insubria, Via Valleggio 11, Como 22100, Italy
3
Departament de Física i Enginyeria Nuclear, Universitat Politècnica de Catalunya, Comte d’Urgell 187, Barcelona 08036, Spain
4
Department of Information Technology, Uppsala University, Box 337, Uppsala SE-751 05, Sweden
5
Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia
6
Department of Computer Science, Virtual University, Lahore 5400, Pakistan
*
Author to whom correspondence should be addressed.
Academic Editors: Alicia Cordero, Juan R. Torregrosa and Francisco I. Chicharro
Received: 25 December 2015 / Revised: 3 March 2016 / Accepted: 4 March 2016 / Published: 11 March 2016
(This article belongs to the Special Issue Numerical Algorithms for Solving Nonlinear Equations and Systems)
View Full-Text   |   Download PDF [263 KB, uploaded 11 March 2016]

Abstract

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. View Full-Text
Keywords: systems of nonlinear equations; frozen Jacobian; ordinary differential equations; partial differential equations; homotopy method systems of nonlinear equations; frozen Jacobian; ordinary differential equations; partial differential equations; homotopy method
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Qasim, U.; Ali, Z.; Ahmad, F.; Serra-Capizzano, S.; Zaka Ullah, M.; Asma, M. Constructing Frozen Jacobian Iterative Methods for Solving Systems of Nonlinear Equations, Associated with ODEs and PDEs Using the Homotopy Method. Algorithms 2016, 9, 18.

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