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Algorithms 2015, 8(4), 895-909; doi:10.3390/a8040895

On Some Improved Harmonic Mean Newton-Like Methods for Solving Systems of Nonlinear Equations

1
African Network for Policy Research & Advocacy for Sustainability (ANPRAS), Midlands, Curepipe 52501, Mauritius
2
Department of Mathematics, Pondicherry Engineering College, Pondicherry 605014, India
*
Author to whom correspondence should be addressed.
Academic Editors: Alicia Cordero, Juan R. Torregrosa and Francisco I. Chicharro
Received: 5 September 2015 / Revised: 23 September 2015 / Accepted: 24 September 2015 / Published: 9 October 2015
(This article belongs to the Special Issue Numerical Algorithms for Solving Nonlinear Equations and Systems)
View Full-Text   |   Download PDF [342 KB, uploaded 9 October 2015]   |  

Abstract

In this work, we have developed a fourth order Newton-like method based on harmonic mean and its multi-step version for solving system of nonlinear equations. The new fourth order method requires evaluation of one function and two first order Fréchet derivatives for each iteration. The multi-step version requires one more function evaluation for each iteration. The proposed new scheme does not require the evaluation of second or higher order Fréchet derivatives and still reaches fourth order convergence. The multi-step version converges with order 2r+4, where r is a positive integer and r ≥ 1. We have proved that the root α is a point of attraction for a general iterative function, whereas the proposed new schemes also satisfy this result. Numerical experiments including an application to 1-D Bratu problem are given to illustrate the efficiency of the new methods. Also, the new methods are compared with some existing methods. View Full-Text
Keywords: system of nonlinear equation; Newton’s method; order of convergence; point of attraction system of nonlinear equation; Newton’s method; order of convergence; point of attraction
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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Babajee, D.K.R.; Madhu, K.; Jayaraman, J. On Some Improved Harmonic Mean Newton-Like Methods for Solving Systems of Nonlinear Equations. Algorithms 2015, 8, 895-909.

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