Abstract
In this paper, using the idea of weight functions on the Potra–Pták method, an optimal fourth order method, a non optimal sixth order method, and a family of optimal eighth order methods are proposed. These methods are tested on some numerical examples, and the results are compared with some known methods of the corresponding order. It is proved that the results obtained from the proposed methods are compatible with other methods. The proposed methods are tested on some problems related to engineering and science. Furthermore, applying these methods on quadratic and cubic polynomials, their stability is analyzed by means of their basins of attraction.
Highlights
IntroductionThe efficiency index as defined by Ostroswki in [1], which relates the order of convergence of a method p with the number of function evaluations per iteration d, is given by the expression p1/d
For solving nonlinear equations iteratively, the Newton’s method given by x n +1 = x n − f f 0is one of the most commonly used methods
We analyze the dynamical behavior of the methods that we have developed in this paper
Summary
The efficiency index as defined by Ostroswki in [1], which relates the order of convergence of a method p with the number of function evaluations per iteration d, is given by the expression p1/d. Potra and Pták [9], as an attempt to improve Newton’s method, gave the method yn x n +1 This method is cubically convergent but is not optimal, as it requires three function evaluations per iteration. In this sense, the study of the rational function resulting from the application of the methods to several nonlinear functions is developed, and their basins of attraction are represented.
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