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

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.


Introduction
The solution of a nonlinear equation () = 0 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][2][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 2  ,  + 1 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  =  1/ , 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.
where  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 lim →∞ (  /  ) = , with  being a nonzero constant, we write   = (  ) or   ∼   .The notation and techniques used in the proofs of the results can be found in [9][10][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.

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,  ̸ = 0.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  :  ⊆ R → R is a sufficiently differentiable function in an open interval  and  * ∈  is a simple root of () = 0.If the initial approximation  0 is close enough to  * , then the iterative scheme (2) has optimal fourthorder convergence when  = 1 and  =  − 1 for all nonzero , being in this case the error equation where   =   −  * ,  = 0, 1, . .., and Proof.By expanding in Taylor series (  ), we get Therefore, 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  = 1, third-order is achieved and, then, the error equation is Again, setting  =  − 1, we get order four and Let us remark not only that the fourth-order has been proven but also that factor (1 +   ( * )) 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  = 4 1/3 ≈ 1.587.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.

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  = −1/  ( * ), 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  = −1/  ( * ) 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   = −1/  ( * ),  = 1, 2, . ... We are going to consider different approximations of   ( * ).

Complexity
Then, where  4 (  ,  −1 ) indicates that the sum of the exponents of  −1 and   in the rejected terms of the development is at least 4.
The error at the next step is and, 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  −1 are 4 and 2, respectively, so the polynomial whose real roots give the -order of the method is  2 − 4 + 2 and the order of the method is, at least, 2 + √ 6.
, 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  :  ⊆ R → R in an open interval .If  0 is close enough to  * and  0 is given, then the -order of family MM2 is at least 5. ( , and then The algorithm denoted by MM3() can be presented in the following way: where   =   −  * ,  , =   −  * , and  , =   −  * .Let {  } be the sequence generated by the algorithm MM3.If {  } converges to  * with -order , then such that lim →∞  , =   , with this limit being the asymptotic error constant of the method.So, Let us suppose that {  } has -order  1 , and then Analogously, if sequence {  } has -order  2 , we obtain On the other hand, we obtain the error equation of the different sequences without memory where Therefore, the corresponding error relations with memory are and, by using expression (18) and simple algebraic manipulations, we obtain If we compare the exponents of  −1 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:  1 = 2,  2 = 3, and  = 6.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.

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  2 − 1,  2 , and  2 + 1.

Fundamentals on Real Dynamics.
Let  : R → R be a rational function.The orbit of a point  0 ∈ R is defined as the set { 0 ,  ( 0 ) ,  2 ( 0 ) , . . .,   ( 0 ) , . ..} . (30) A point  0 is a fixed point,   0 , of  if (  0 ) =   0 .The multiplier |  (  0 )| 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 () = 1/(1/) [15].The point ∞ is a fixed point of  if and only if  = 0 is a fixed point of .In this case, the multiplier of ∞ is |  (0)|.When this value is zero, the point ∞ is a superattracting point.
The basin of attraction of an attracting fixed point  * , A( * ), is defined as the set of preimages of any order such that The Fatou set, F(), includes the points whose orbits tend to an attracting point  * .The Julia set, J(), 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) is where  =  + () and  =  − ()/[, ].  .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.Therefore,   3 has both behaviors, depending on the value of  and .For assuring the convergence of the method to  * 1,2 , white areas must be avoided.
Computing    () = 0, 16 critical points can be found.As expected,   1,2 =  * 1,2 .  3,4 = (−1 ± )/ are preimages of  * 1,2 .  5−16 are the roots of a polynomial of degree 12, that depend on the parameters  and .Six of these roots are real (  5−10 ) while the other ones are complex (  11−16 ).A classical result (attributed to Fatou and Julia) establishes that there is at least one critical point associated with each invariant Fatou component, so it is interesting to analyze the behavior of each free critical point (critical point different to the roots of the polynomial) used as initial estimation for all the elements of the family.
Just as the convergence plane [4] represents the final state of the orbits of every starting point in a single-parametric family of methods, we use the parameters plane to guess the final state of the orbit of one critical point in a biparametric family of methods, in a similar manner as [16].Therefore, the parameters plane represents the family of methods   (), where each point (, ) stands for an individual method.If the orbit of the free critical point   tends to −1, 1, or ∞, the point is colored in red, green, or blue, respectively.If   does not converge to any of those points, the point is colored in black.Figure 2 shows the parameter planes of   5−10 .The unified parameter plane represented in Figure 3 gathers in one image the main information of Figure 2. It is composed of the superposition of the black regions.In this way, the election of a point in the white region guarantees that the corresponding scheme tends to ±1 or ∞, while the set of methods in the black region do not.
In order to visualize the behavior of each method, a new representation concept is introduced: the dynamical line.Analogous to the dynamical plane for complex variable, the dynamical line represents the basins of attraction, plotting in different colors where the orbit of each initial estimation tends.In coherence to Figure 2, the red color is assigned to those initial approximations that tend to  * 1 = −1.Analogously, the green color is assigned to  * 2 = 1, the blue color is assigned to  * = ∞, and the black one is assigned to those initial guesses that tend to a point different than the previous three.The final orbit of 1001 initial values of  ∈ [−2, 2] has been calculated, and the convergence to a point is set when its distance is lower than 10 −3 .
Figure 4 represents the dynamical line of a set of methods whose orbits tend to −1, 1, or ∞, that is, methods that belong to the white area of the unified parameter plane.On the contrary, Figure 5 shows methods whose points also tend to another fixed point.
Note that every point in Figure 4 tends to one of the attracting points.However, when a value of (, ) of the black region of Figure 3 is chosen, some points tend to a different point.For instance, these points can be found in Figures 5(a)-5(d) for  0 = −0.2, 0 = 0.2,  0 = −0.28,and  0 = 0.28, respectively.
As expected from the attracting area of   3 and the unified parameter plane, the election of methods with  > 0 is capital, as proved in Figures 4 and 5. 2 .Applying (32) over () =  2 , the fixed point operator is

𝑓(𝑥) = 𝑥
The real fixed points of the operator are   1 = 0 and   2 (, ), while the other two fixed points are complex.Both real fixed points have a different dynamical behavior depending on the parameter .  1 is attracting for every value of  but the interval −3.25 <  < −2.875.  2 is repelling for every value of  ∈ (−3.25, −0.098).The stability planes of Figure 6   x In a similar fashion as the case () =  2 − 1, the infinity is a parabolic fixed point.
The zero values of the derivative of (33) are the critical points, which correspond to the roots of polynomial The polynomial has four real roots and four complex ones.No one of the critical points agrees with the roots of the function  2 , so every one is free.Figure 7 shows the parameters planes of each real critical point.When the critical point tends to  * 1 ,  * 2 , or  * ∞ , the points are plotted in red, green, or blue, respectively, while black is kept for convergence to any other point.Note in Figures 7(b), 7(c), and 7(d) there is a thin black region, where the critical points have periodical orbits, so they do not converge to any one of the fixed points.
For gathering the information of the parameters planes, the unified parameter plane is shown in Figure 8.
Finally, the dynamical behavior is analyzed with the dynamical line.The representation of the methods that converge to any of the roots is covered in Figure 9, while Figure 10 illustrates the methods that also converge to any other point or have a periodical orbit.Color map is held.
The fixed points are the roots of () and the roots of an eighth-degree polynomial.Both groups of roots are complex, so they are left out of the purpose of the current study.
Regarding the critical points, an analogous comment can be made.The roots of () are critical points.The rest of the critical points are   3,4 = −1/± and the roots of a polynomial of degree 12.No critical point is real, so the real dynamics can not be analyzed.

Memory Dynamics
Introducing memory in the iterative expression of a method can improve its order of convergence.Below some fundamentals about methods with memory are introduced [11,17].

Basics on Iterative Methods with Memory.
The standard form of an iterative method with memory of first order is x  where  0 and  1 are the initial estimations.Fixed points must satisfy two conditions:  +1 =   and  −1 =   , so the fixed point function Φ can be defined by means of So, and, consequently,  +1 =   and   =  −1 .Therefore, the discrete dynamical system Φ : R 2 → R 2 , such that where  is the operator of the iterative scheme with memory, has been defined.Below some definitions related to iterative methods with memory are recalled.
The drawing tool to represent the basins of attraction is the dynamical plane [16,19].This tool is based on the application to complex dynamics, where horizontal and vertical axes stand for the real and imaginary parts of a complex value, respectively.For real dynamics with memory, the horizontal axis is devoted to the current iteration   while the vertical one represents the last iteration  −1 .

Stability of Family MM1(𝛼).
As we have seen in Section 3, the memory is introduced in the iterative expression by replacing parameter  by −1/  ( * ), where   ( * ) denotes any approximation of   ( * ).The first family of iterative methods with memory designed in Section 3 is expressed as where  is a free parameter.We are going to analyze the dynamical behavior of this class on the quadratic polynomials  2 − 1,  2 , and  2 + 1.  Figure 11 represents the dynamical plane of the method when (, )   2 is attracting (a) and when it is not (b).White stars plot the attracting points.Orange points represent the set of initial points that converge to (, ) * 1 = (−1, −1), while green points are the (, ) * 3 = (1, 1) corresponding ones.Blue is devoted to initial estimations that finally converge to (, ) * 2 = (0, 0).The initial points are a mesh of 501 × 501 points of values of  and  from −2 to 2. Every initial point has converged to one of the attracting fixed points in less than 100 iterations.Further information about the representation can be found in [16]. 2 .The application of (39) over () =  2 results in the following fixed point operator:
Every fixed point whose value is known is repelling, except the cases collected in Table 2. Related to Theorem 5, only the point (0, 0) is an attracting fixed point for  = −0.381966; the rest of the points are nonhyperbolic.
The dynamical planes of the cases in Table 2 are represented in Figures 13(a)-13(d).In this case, white stars represent the fixed points.Only for the particular case of  = −0.381966 is there an attracting fixed point that attracts a few points of the memory plane, as can be observed in Figure 13.
Since the usual behavior is the nonconvergence to any of the roots, the orbit of different initial points have been obtained and represented in Figures 13(e
The memory dynamics have improved the order of convergence of the original method.For polynomials with real roots, the iterative family of methods has a broad stable behavior.When the family is applied over polynomials with complex roots, many fixed points with repelling behavior appear, and some periodical orbits can be found.

𝐹 3 −
10 (, ) are the roots of an 8th-degree polynomial:   3,4 ∈ R and   5−10 ∈ C. Since our purpose is the study of the real dynamics,   5−10 are rejected.The evaluation of |   (  3,4 )| establishes the behavior of these points.For different values of  and ,   3,4 have different dynamical features, as the stability plane of Figure 1 represents.A mesh of 100 × 100 points covers the values of  ∈ [−2, 2] and  ∈ [−2, 2]

Theorem 2. Let 𝑥 * be a simple zero of a sufficiently differen
iii)   =   − (  )/[  ,   ]. tiable function  :  ⊆ R → R in an open interval .If  0 is close enough to  * and  0 is given, then the -order of family MM1() is at least 2+ √ 6 ≈ 4.45 that corresponds to the positive root of polynomial  2 − 4 − 2.

Table 2 :
Fixed points without repelling behavior.