@Article{cmc.2020.010836,
AUTHOR = {Syahmi Afandi Sariman, Ishak Hashim},
TITLE = {New Optimal Newton-Householder Methods for Solving  Nonlinear Equations and Their Dynamics},
JOURNAL = {Computers, Materials \& Continua},
VOLUME = {65},
YEAR = {2020},
NUMBER = {1},
PAGES = {69--85},
URL = {http://www.techscience.com/cmc/v65n1/39554},
ISSN = {1546-2226},
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.},
DOI = {10.32604/cmc.2020.010836}
}