Damped Traub’s method: Convergence and stability

Abstract

In this paper, a parametric family including Newton’s and Traub’s iterative schemes is presented. Its local convergence and dynamical behavior on quadratic polynomials are studied. The analysis of fixed and critical points and the associated parameter plane show the dynamical richness of the family and allow us to find members of it with good numerical properties, as well as other ones with very unstable behavior.

Introduction

Real world phenomena are often modeled by means of nonlinear equations $f\left(z\right)=0$; for example, the numerical solution of nonlinear equations is needed in the study of dynamical models of chemical reactors  [11], in radioactive transfer  [17], preliminary orbit determination of satellites  [4] or even the approximation of the eigenvalues of square matrices, which is known to have many applications in areas as image processing, dynamical systems, control theory, etc. (see, for example,  [12]).

For solving these equations, iterative schemes must be used. The best known iterative approach is Newton’s method. In last decades, many researchers have proposed different iterative methods to improve Newton’s scheme (see, for example, the review  [25], and the references therein). These variants of Newton’s method have been designed by means of different techniques, providing in the most of cases multistep schemes. Some of them come from Adomian decomposition (see  [1] and  [7], for example). Another procedure to develop iterative methods is the replacement of the second derivative in Chebyshev-type methods by some approximation: in  [27], Traub presented a family of multi-point methods based on approximating the second derivative that appears in the iterative formula of Chebyshev’s scheme and Babajee et al. in [8] designed two Chebyshev-like methods free from second derivatives. A common way to generate new schemes is the direct composition of known methods with a later treatment to reduce the number of functional evaluations. For example, by composing Newton’s method with itself, holding the derivative “frozen” in the second step, third-order Traub’s method  [27] is obtained.

Recently, the weight-function procedure has been used to increase the order of convergence of known methods [25], [26], allowing to get optimal methods, under the point of view of Kung–Traub’s conjecture  [21]. These authors conjectured that an iterative method without memory which uses $d$ functional evaluations per iteration can reach, at most, order of convergence $2^{d-1}$. When this bound is reached, the scheme is called optimal. Although the aim of many researches in this area is to design optimal high-order methods (see, for example  [14], [18], [20]), it is also known that the higher the order is, the more sensitive the scheme to initial estimations will be  [22]. On the other hand, recent studies on damped Newton’s procedure show (see, for example  [23]) that small damping parameters widen the set of initial guesses that make the method convergent, although the speed of convergence decreases.

In this paper, we present a damped Traub’s-type family of iterative methods. We will prove that it is an uniparametric set of second and third-order iterative schemes to estimate simple roots of nonlinear equations, whose iterative expression is

$$z_{n+1}=y_n-\gamma \frac{f\left(y_n\right)}{f^{\prime }\left(z_n\right)}\text{,}n=0,1,\dots \text{,}$$

where $y_n=z_n-\frac{f\left(z_n\right)}{f^{\prime }\left(z_n\right)}$ and $\gamma$ is the damping parameter. Let us note that, if $\gamma =1$ we get Traub’s scheme and $\gamma =0$ corresponds to Newton’s method.

The application of iterative methods on polynomials gives rise to rational functions whose complex dynamics is not well-known. However, Amat et al. in  [2] studied real dynamics of Traub’s scheme on quadratic and cubic polynomials. From the numerical point of view, the dynamical behavior of the rational function associated with an iterative method gives us important information about its stability and reliability. In these terms, Amat et al. in [3] described the dynamical behavior of several well-known families of iterative methods. More recently, in [15], [16], [19], [5], [24], the authors analyze the qualitative behavior of different known iterative families. The most of these studies show different pathological numerical behavior, such as periodic orbits, attracting fixed points different from the solution of the problem, etc. Indeed, parameter planes associated to a family of methods allow us to understand the behavior of the different members of the family of methods, helping us in the election of a particular one.

The rest of the paper is organized as follows: in Section  2 we introduce the basic concepts on complex dynamics, in Section  3 we study the local convergence of family (1) of iterative methods. Section  4 is devoted to the analysis of the fixed and critical points of the operator $O_{\gamma }\left(z\right)$ and the study, the stability of the fixed points, is shown in Section  5. These regions of stability also appear in the associated parameter spaces (Section  6), as well as the stability region of the attractive 2-periodic orbits, whose analytical expression is found, depending on the damping parameter. We finish the work with some remarks and conclusions.