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ARTICLE
New Optimal Newton-Householder Methods for Solving Nonlinear Equations and Their Dynamics
Syahmi Afandi Sariman1, Ishak Hashim1, *
1 Department of Mathematical Sciences, Faculty of Science & Technology, Universiti Kebangsaan Malaysia, Bangi, 43600, Malaysia.
* Corresponding Author: Ishak Hashim. Email: .
Computers, Materials & Continua 2020, 65(1), 69-85. https://doi.org/10.32604/cmc.2020.010836
Received 31 March 2020; Accepted 29 May 2020; Issue published 23 July 2020
Abstract
The classical iterative methods for finding roots of nonlinear equations, like
the Newton method, Halley method, and Chebyshev method, have been modified
previously to achieve optimal convergence order. However, the Householder method has
so far not been modified to become optimal. In this study, we shall develop two new
optimal Newton-Householder methods without memory. The key idea in the
development of the new methods is the avoidance of the need to evaluate the second
derivative. The methods fulfill the Kung-Traub conjecture by achieving optimal
convergence order four with three functional evaluations and order eight with four
functional evaluations. The efficiency indices of the methods show that methods perform
better than the classical Householder’s method. With the aid of convergence analysis and
numerical analysis, the efficiency of the schemes formulated in this paper has been
demonstrated. The dynamical analysis exhibits the stability of the schemes in solving
nonlinear equations. Some comparisons with other optimal methods have been conducted
to verify the effectiveness, convergence speed, and capability of the suggested methods.
Keywords
Cite This Article
S. Afandi Sariman and I. Hashim, "New optimal newton-householder methods for solving nonlinear equations and their dynamics,"
Computers, Materials & Continua, vol. 65, no.1, pp. 69–85, 2020. https://doi.org/10.32604/cmc.2020.010836
Citations