Convergence and Dynamics of a Higher-Order Method

Abstract

Solving problems in various disciplines such as biology, chemistry, economics, medicine, physics, and engineering, to mention a few, reduces to solving an equation. Its solution is one of the greatest challenges. It involves some iterative method generating a sequence approximating the solution. That is why, in this work, we analyze the convergence in a local form for an iterative method with a high order to find the solution of a nonlinear equation. We extend the applicability of previous results using only the first derivative that actually appears in the method. This is in contrast to either works using a derivative higher than one, or ones not in this method. Moreover, we consider the dynamics of some members of the family in order to see the existing differences between them.

One Parameter Optimal Derivative-Free Family to Find the Multiple Roots of Algebraic Nonlinear Equations

“…It provides a wide range of initial guesses to find the roots of an equation, which leads to better understanding of the performance behavior of iterative techniques. Many researchers [31][32][33][34][35][36] have opted for the basin of attraction idea to check the effectiveness of the developed iterative techniques.…”

Purely Iterative Algorithms for Newton’s Maps and General Convergence

“…The following family of iterative functions represents Newton's method, Chebyshev's iterative function, Halley's method, Super-Halley, c-iterative function (considering θ = 0 below) and Chebyshev-Halley family, among others. See for instance [22,[29][30][31][32][33][34][35][36][37][38]. The family of iterative methods given by…”