A Family of Multiple-Root Finding Iterative Methods Based on Weight Functions

Abstract

A straightforward family of one-point multiple-root iterative methods is introduced. The family is generated using the technique of weight functions. The order of convergence of the family is determined in its convergence analysis, which shows the constraints that the weight function must satisfy to achieve order three. In this sense, a family of iterative methods can be obtained with a suitable design of the weight function. That is, an iterative algorithm that depends on one or more parameters is designed. This family of iterative methods, starting with proper initial estimations, generates a sequence of approximations to the solution of a problem. A dynamical analysis is also included in the manuscript to study the long-term behavior of the family depending on the parameter value and the initial guess considered. This analysis reveals the good properties of the family for a wide range of values of the parameter. In addition, a numerical test on academic and engineering multiple-root functions is performed.

Memorizing Schröder’s Method as an Efficient Strategy for Estimating Roots of Unknown Multiplicity

“…The qualitative performance of different iterative schemes designed for solving nonlinear equations with multiple roots has been studied by different authors (see, for example, Reference [17][18][19]). It has been made by using discrete complex dynamics, as all these schemes are without memory.…”

Introducing memory to a family of multi-step multidimensional iterative methods with weight function

“…In this paper, we are inspired precisely by this idea, which is to increase the speed of convergence of Steffensen's method while trying not to deteriorate its other characteristics, such as accessibility and efficiency. For this purpose, we have constructed a parametric family of multi-step iterative methods without derivatives, making use of a weight function H(t (k) ) (see [6,8]), which allow us to explore the different advantages of each of the methods that are part of this family depending on the form of this function. The family is described by the following iterative scheme:…”

Stability Analysis of a New Fourth-Order Optimal Iterative Scheme for Nonlinear Equations

“…It has been observed that iterative schemes stable for such functions tend to perform better when applied to more complicated functions than methods exhibiting pathologies. To this end, the tools of complex discrete dynamics are employed to analyze stability in quadratic polynomials (see for example the work of Amat et al in [13,14], Argyros et al in [15], Behl et al in [16], Chicharro et al in [17], Rafiq et al in [18], Kansal et al in [19], Khirallah et al in [20], and Moccari et al in [21], among others).…”