Complexity

Volume 2017, Article ID 2713145, 15 pages

https://doi.org/10.1155/2017/2713145

## King-Type Derivative-Free Iterative Families: Real and Memory Dynamics

^{1}Instituto Universitario de Matemática Multidisciplinar, Universitat Politècnica de València, València, Spain^{2}Instituto Tecnológico de Santo Domingo (INTEC), Santo Domingo, Dominican Republic

Correspondence should be addressed to M. P. Vassileva; od.ude.cetni@avoknep.airam

Received 29 June 2017; Accepted 4 October 2017; Published 31 October 2017

Academic Editor: Guido Caldarelli

Copyright © 2017 F. I. Chicharro et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A biparametric family of derivative-free optimal iterative methods of order four, for solving nonlinear equations, is presented. From the error equation of this class, different families of iterative schemes with memory can be designed increasing the order of convergence up to six. The real stability analysis of the biparametric family without memory is made on quadratic polynomials, finding areas in the parametric plane with good performance. Moreover, in order to study the real behavior of the parametric class with memory, we associate it with a discrete multidimensional dynamical system. By analyzing the fixed and critical points of its vectorial rational function, we can select those methods with best stability properties.

#### 1. Introduction

The solution of a nonlinear equation is a common problem in topics related to several disciplines of science. However, most of these equations can not be solved analytically. In this case, iterative methods acquire prominence, because they can reach an approximate solution of . Newton’s scheme is the most-applied iterative method, but the evaluation of the derivative of is its most remarkable inconvenience.

Several authors have made great efforts to study the complex dynamics of some derivative-free iterative methods [1–3], but the analysis of the real dynamics is not as usual as the complex one [4, 5].

During the last years, many researchers have focused on the construction of optimal multipoint methods without memory based on the Kung and Traub’s conjecture [6], which claims that to create an optimal method without memory with optimal order , functional evaluations are required. Construction and development of multipoint methods with memory have more advantages in comparison with methods without memory. In other words, they are able to increase the convergence order without any new functional evaluation. Due to this basic fact, they can also get higher efficiency index (the efficiency index was defined by Ostrowski [7] as , where is the order of convergence and is the number of functional evaluations required per iteration). As another plus, they generate more stable computations than optimal methods without memory, even with higher convergence orders.

The basic idea for the construction of multipoint methods with memory was introduced by Traub [8], who presented a version with memory from Steffensen’s method. Recently, based on this method, some schemes with memory have been developed by several authors. We can see interesting overviews in [9, 10].

In this paper, we deeply analyze the real dynamics of a version with memory of a King-type derivative-free iterative family. Two new tools are introduced in the real dynamics section: the unified parameter plane and the dynamical line. The former gathers the information of different parameters planes into one plane. The latter becomes a comfortable tool to visualize the dynamical behavior of a method for a set of initial points in the real line.

Starting from King’s family of fourth-order schemeswhere is a disposable parameter, we design a derivative-free family of fourth-order methods. The order of convergence can be increased introducing memory, as can be deduced from the error equation.

In the manuscript, we use symbols and in the following way: if , with being a nonzero constant, we write or . The notation and techniques used in the proofs of the results can be found in [9–11].

The rest of this paper is organized as follows. In Section 2, we introduce the derivative-free parametric family and its error equation. Section 3 addresses the increase of the order of convergence with the introduction of the memory in the iterative schemes. In Section 4, the real dynamics of the biparametric family is studied for different quadratic polynomials, while Section 5 covers the dynamical study of the methods with memory. Finally, Section 6 collects some conclusions about the obtained results.

#### 2. Parametric Families of Iterative Schemes

Following the structure of King’s family and replacing the derivative by a first-order divided difference we present the following schemes:where , with , , , and being real parameters, .

The order of convergence of methods (2) is established in the following result. The proof only requires the development in Taylor series of the elements of the iterative expression and some algebraic manipulations.

Theorem 1. *Let us suppose that is a sufficiently differentiable function in an open interval and is a simple root of . If the initial approximation is close enough to , then the iterative scheme (2) has optimal fourth-order convergence when and for all nonzero , being in this case the error equation where , , and .*

*Proof. *By expanding in Taylor series , we getTherefore, the error at the first step is By obtaining the Taylor expansion of and , we get the following expression for the error equation: It is clear that, imposing , third-order is achieved and, then, the error equation is Again, setting , we get order four andLet us remark not only that the fourth-order has been proven but also that factor appears also in the term of order five.

Let us observe that this family of derivative-free methods (2) supports Kung-Traub’s conjecture [6], having optimal efficiency index . On the other hand, by observing the expressions of the error equation (8) we can choose different values of the free disposable parameters in order to obtain iterative methods with memory, increasing the order of convergence.

#### 3. Iterative Methods with Memory

We are going to design derivative-free schemes with memory based on the proposed methods (2).

From (8) we can assure that the order of convergence of family (2) increases up to six if , but the value of is not available in practice and such acceleration is not possible. However, we can use an approximation , calculated by using known information. Therefore, by setting we can increase the convergence order without using new functional evaluations. The main idea in constructing methods with memory consists of the calculation of the parameter as the iteration proceeds by , . We are going to consider different approximations of .

() Let be Newton’s interpolation polynomial of first degree through two available approximations , ; that is , so and the algorithm denoted by MM1() can be presented in the following way: (i) are given.(ii).(iii).(iv),

where and is a free parameter.

The order of convergence of this algorithm is analyzed in the following result.

Theorem 2. *Let be a simple zero of a sufficiently differentiable function in an open interval . If is close enough to and is given, then the -order of family MM1() is at least that corresponds to the positive root of polynomial .*

*Proof. *By expanding in Taylor series and , we get Then,where indicates that the sum of the exponents of and in the rejected terms of the development is at least 4.

The error at the next step isand, then,Moreover,Finally, by combining the previous expansions from (12)–(14), Since the lower term of the error equation is , applying Theorem 9.2.9 of [12], the powers of and are 4 and 2, respectively, so the polynomial whose real roots give the -order of the method is and the order of the method is, at least, .

() Let (can be also used as ) be Newton’s interpolation polynomial of second degree; that is, , and therefore and a similar algorithm to the previous one, denoted by MM2(), can be presented. The convergence of the method is established in the following result, whose proof is similar to that of the last proposed method. This is the reason why it is avoided.

Theorem 3. *Let be a simple zero of a sufficiently differentiable function in an open interval . If is close enough to and is given, then the -order of family MM2 is at least 5.*

() Let be Newton’s interpolation polynomial of third degree: , and then The algorithm denoted by MM3() can be presented in the following way: (i), are given.(ii).(iii).(iv),

where and is a free parameter.

Theorem 4. *Let be a simple zero of a sufficiently differentiable function in an open interval . If is close enough to and is given, then the -order of family MM3 is at least 6.*

*Proof. *By using Taylor expansion around of the different elements of , we prove thatwhere , , and .

Let be the sequence generated by the algorithm MM3. If converges to with -order , then such that , with this limit being the asymptotic error constant of the method. So,Let us suppose that has -order , and thenAnalogously, if sequence has -order , we obtainOn the other hand, we obtain the error equation of the different sequences without memorywhere andTherefore, the corresponding error relations with memory are and, by using expression (18) and simple algebraic manipulations, we obtainIf we compare the exponents of between “(20), (21), (22), and (26)), ((21), (22), (26), and (27)” and “(22), (26), (27), and (28),” we obtain the following nonlinear system:It is easy to obtain the unique real solution of this system: , , and . Therefore, the -order of algorithm MM3 is 6 for any value of parameter .

Many other approximations of are possible, but they either are of sixth order (with more computational cost) or are lower than six.

#### 4. Real Dynamics

In this section, the real dynamics of the iterative family are analyzed. After an introduction of real dynamics fundamentals [13, 14], the features of the family are described when it is applied over the nonlinear polynomials , , and .

##### 4.1. Fundamentals on Real Dynamics

Let be a rational function. The orbit of a point is defined as the set A point is a fixed point, , of if . The multiplier classifies the fixed points as attracting, repelling, or neutral if its value is lower than, greater than, or equal to 1, respectively. When the multiplier is zero, the fixed point is superattracting. Both attracting and superattracting fixed points are denoted as .

The character of the infinity is set by the study of [15]. The point is a fixed point of if and only if is a fixed point of . In this case, the multiplier of is . When this value is zero, the point is a superattracting point.

The basin of attraction of an attracting fixed point , , is defined as the set of preimages of any order such that

The Fatou set, , includes the points whose orbits tend to an attracting point . The Julia set, , is its complementary. It covers the repelling points and sets the borders between the basins of attraction.

The fixed point operator of the biparametric family (2) iswhere and .

##### 4.2. Application on Quadratic Polynomials

We are going to analyze the dynamical behavior of the rational function obtained when family (32) is applied on .

###### 4.2.1.

In this case, expression (32) is , where and are polynomials of degrees 11 and 10, respectively, depending on and on parameters and .

There are 10 fixed points of . are superattracting points, while are the roots of an 8th-degree polynomial: and . Since our purpose is the study of the real dynamics, are rejected. The evaluation of establishes the behavior of these points. For different values of and , have different dynamical features, as the stability plane of Figure 1 represents. A mesh of points covers the values of . The white area represents where the multiplier is lower than 1, while the corresponding black represents where the multiplier is greater than 1. Its dynamical meaning is immediate, since white and black regions represent attracting and repelling behavior, respectively.