Abstract
The principal objective of this work is to propose a fourth, eighth and sixteenth order scheme for solving a nonlinear equation. In terms of computational cost, per iteration, the fourth order method uses two evaluations of the function and one evaluation of the first derivative; the eighth order method uses three evaluations of the function and one evaluation of the first derivative; and sixteenth order method uses four evaluations of the function and one evaluation of the first derivative. So these all the methods have satisfied the Kung-Traub optimality conjecture. In addition, the theoretical convergence properties of our schemes are fully explored with the help of the main theorem that demonstrates the convergence order. The performance and effectiveness of our optimal iteration functions are compared with the existing competitors on some standard academic problems. The conjugacy maps of the presented method and other existing eighth order methods are discussed, and their basins of attraction are also given to demonstrate their dynamical behavior in the complex plane. We apply the new scheme to find the optimal launch angle in a projectile motion problem and Planck’s radiation law problem as an application.
Highlights
One of the most frequent problems in engineering, scientific computing and applied mathematics, in general, is the problem of solving a nonlinear equation f ( x ) = 0
The study of dynamical behavior of the rational function associated to an iterative method gives important information about convergence and stability of the method
We note that a point z0 belongs to the Julia set if and only if the dynamics in a neighborhood of z0 displays sensitive dependence on the initial conditions, so that nearby initial conditions lead to wildly different behavior after a number of iterations
Summary
The local order of convergence of Newton’s method is two and it is an optimal method with two function evaluations per iterative step. Mathematics 2019, 7, 322 that the order of convergence of any multi-point without memory method with d function evaluations cannot exceed the bound 2d−1 , the optimal order. Some fourth and eighth order optimal iterative methods have been developed (see [3,4,5,6,7,8,9,10,11,12,13,14] and references therein). The value of p is called the order of convergence of the method.
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