Fixed Point Root-Finding Methods of Fourth-Order of Convergence

Abstract

In this manuscript, by using the weight-function technique, a new class of iterative methods for solving nonlinear problems is constructed, which includes many known schemes that can be obtained by choosing different weight functions. This weight function, depending on two different evaluations of the derivative, is the unique difference between the two steps of each method, which is unusual. As it is proven that all the members of the class are optimal methods in the sense of Kung-Traub's conjecture, the dynamical analysis is a good tool to determine the best elements of the family in terms of stability. Therefore, the dynamical behavior of this class on quadratic polynomials is studied in this work. We analyze the stability of the presented family from the multipliers of the fixed points and critical points, along with their associated parameter planes. In addition, this study enables us to select the members of the class with good stability properties.

On Locating and Counting Satellite Components Born along the Stability Circle in the Parameter Space for a Family of Jarratt-Like Iterative Methods

“…Applications of the above theorem cover a number of existing studies introduced in [11] and references cited therein. For m = 1, by considering the one-point compactification of C, namely, the Riemann sphere C = C ∪ {∞}, Cordero et al [11] have pursued the complex dynamics on C for Jarratt-like uniparametric iterative map (2), with…”

“…Thanks to the basic properties of the dynamical plane D consisting of a union of attractor basins, extensively studied in [11], we in this paper only investigate some interesting properties of the parameter plane P. According to Remark 4, we consider only one branch of the critical points ζ 1 for its typical orbit behavior.…”

“…The lemma in Section 1.2.1 of [10] suffices to deal with Ψ f that is a rational function. Cordero et al [11] have recently introduced a family of optimal iterative root-finding methods for a system of nonlinear equations F(x) = 0 in the following form:…”

Bifurcations along the Boundary Curves of Red Fixed Components in the Parameter Space for Uniparametric, Jarratt-Type Simple-Root Finders

“…Recently, an optimal family of fourth-order simple-root finders locating a zero of f (x) ∈ C m have been studied by Cordero et al [9] in the form:…”

“…Cordero et al [9] have presented some dynamical aspects on the Riemann sphere C for the conjugated map via Möbius conjugacy [12] map M(z) = z−a z−b , (a = b) with f (z) = (z − a)(z − b) in the λ-parameter space, a detailed analysis of which has been strengthened by Geum-Kim [13].…”