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Algorithms 2016, 9(1), 5; doi:10.3390/a9010005

A Family of Iterative Methods for Solving Systems of Nonlinear Equations Having Unknown Multiplicity

1
Dipartimento di Scienza e Alta Tecnologia, Universita dell’Insubria, Via Valleggio 11, Como 22100, Italy
2
Departament de Física i Enginyeria Nuclear, Universitat Politècnica de Catalunya, Comte d’Urgell 187, 08036 Barcelona, Spain
3
Department of Information Technology, Uppsala University, Box 337, SE-751 05 Uppsala, Sweden
4
Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia
*
Author to whom correspondence should be addressed.
Academic Editors: Alicia Cordero, Juan R. Torregrosa and Francisco I. Chicharro
Received: 8 December 2015 / Revised: 19 December 2015 / Accepted: 22 December 2015 / Published: 31 December 2015
(This article belongs to the Special Issue Numerical Algorithms for Solving Nonlinear Equations and Systems)
View Full-Text   |   Download PDF [230 KB, uploaded 31 December 2015]

Abstract

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. View Full-Text
Keywords: systems of nonlinear equations; singular Jacobian; roots with multiplicity; auxiliary function systems of nonlinear equations; singular Jacobian; roots with multiplicity; auxiliary function
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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Ahmad, F.; Serra-Capizzano, S.; Ullah, M.Z.; Al-Fhaid, A.S. A Family of Iterative Methods for Solving Systems of Nonlinear Equations Having Unknown Multiplicity. Algorithms 2016, 9, 5.

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