Study of Local Convergence and Dynamics of a King-Like Two-Step Method with Applications

In this paper, we present the local results of the convergence of the two-step King-like method to approximate the solution of nonlinear equations. In this study, we only apply conditions to the first derivative, because we only need this condition to guarantee convergence. As a result, the applicability of the method is expanded. We also use different convergence planes to show family behavior. Finally, the new results are used to solve some applications related to chemistry.


Introduction
In this study, inspired in previous papers such us [1], we want to approximate a locally unique solution z * of equation where G : I ⊂→ T, T = R or T = C, I is convex and G is a differentiable function. As society changes, the way in which we understand science also changes, as can be seen in [2][3][4][5][6]. In the case of Mathematical Modelling [2][3][4][5] it is important to comment that it enables us to solve problems with the form of the Equation (1). Based on the results, researchers are working to use the same assumptions to increase the convergence domain of the iterative method because it is usually small, the final purpose is to estimate: z n+1 − z n and z n − z * .
The dynamic behavior of iterative methods provides key information about its reliability and stability. In the past ten years, some studies related to dynamic characteristics have been presented [3,4,[6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]. The main problem of this research is related to the fact that these families or methods have only one parameter, or even no parameters. In our case, we have more free parameters, so we need another tool to study it. The tools in [19,21], arevery useful for studying the dynamic behavior of families with two different parameters or the actual dynamics in other interactive applications. In this article, we will study the convergence of: G(z n ) + βG(w n ) G(z n ) + (β − 2)G(w n ) , Includes computable convergence radius and error estimates. In addition, we will show the behaviors related to the family by using the convergence planes.
We recognize local, semi-local and global convergence. In the case of local (convergence ball is centered at the solution) and semi-local (convergence ball is centered at the starting point x 0 ) the convergence is faster, and less expensive than global convergence. But global convergence provides all roots inside a domain. It is worth noticing that loocal convergence results are important because they demonstrate the degree of difficulty in choosing starters x 0 . There is a plethora of literature on solvers. See for example [26][27][28][29][30] and the references therein. In these references other techniques are used on polynomials (or not) considering proper intervals containing a root or all roots. These references do not provide computable bounds on x n − x * or uniqueness of the solution x * results. But we do. Moreover, the starter x 0 really a shot in the dark, whereas in our case it is picked up from a predetermined ball of convergence.
Consider the motivational example for T = R, Then, we have x * = 0, and Then, obviously, function f (t) is unbounded on I. Hence, earlier results using the third (or higher order, that do not appear on (2), b ut used to compute the order of convergence) derivatives cannot guarantee convergence (2) to z * , The convergence studies so far have shown convergence using high order derivatives that do not appear on the method. Hence, limiting the applicability of the method. But we show convergence using only the first derivative that actually appears on the method. Hence, we expand the applicability of the method. A radius of convergence, estimates on x n − x * and uniqueness results not given before are also provided. Our technique can be used to expand the applicability of other methods along the same lines. The convergence order is determined using (COC) orr (ACOC) to be precised in Remark 2.
All this work is organized into different chapters: Next section, number 2, we display the method and the theorems that guarantee the local convergence of the method. In the next section, number 3, the authors study the convergence planes of the method of the previous section applied to a quadratic polynomial. In the next section we introduce some chemical applications that reinforce the applicability of the method and in the last section, number 5, we write the conclusions that complete the article.

Local Convergence Analysis
We start to introduce in this section the method we will study throughout the article. We study the local convergence analysis of the two-step King-like method defined as follow for each n = 0, 1, . . . by where z 0 is an initial point, α ∈ T and β ∈ T are parameters. Some well-known methods, such as our King's family methods or Traub-Ostrowski method, are special cases of this family. We consider N ≥ 1, L 0 , L > 0 and α, β ∈ T. We now want to demonstrate the local convergence of the method (2) for which we are going to use the following auxiliary functions and parameters defined as: The first auxiliary function that we are going to define is called k 1 on the interval [0, 1 L 0 ) by and Next, taking into account the definition of N and α, we can write Based on the above, we obtain as a consequence Now consider the interval [0, 1 L 0 ) and defining a new auxiliary functions and other auxiliary function It clearly follows that Using now the Intermediate Value Theorem, it is clear that there exist at least one root of p 0 located in the interval (0, 1 L 0 ), we will call r 0 the smallest zero in that interval. Now we can perform the same technique, for this we define on the interval [0, r 0 ) a new auxiliary functions Similarly, suppose it is easy to see that So, it is clear that there exist at least one root of p 2 located in the interval (0, r 0 ). We will call r 2 as the smallest zero in the interval.
We also define the value r as the minimum of the values between r 1 and r 2 r = min{r 1 , r 2 }.
For t ∈ [0, r), the following conditions are fulfilled by definition of the functions and From now on, we will also consider U(v, ρ) as the open ball in T with center v ∈ T and radius ρ > 0, asŪ(v, ρ) the closed one and I 0 = I ∩ U(Z * , 1 L 0 ), where L 0 is positive therefore we can present the following result. Theorem 1. Let G : I ⊂ T → T be a differentiable function. Suppose that there exists z * ∈ I, L 0 , L > 0, N ≥ 1 and α, β ∈ T such that the following requirements are true Then, the method (2) generate a well defined sequence {z n } for z 0 ∈ U(z * , r) \ {z * }, and {z n } remains in U(z * , r) for each n = 0, 1, 2, . . . and moreover, the sequence converges to z * .
Furthermore, the inequalities written below are also fulfilled and Moreover, if we take R in the form the limit point z * is the only solution of Equation (1) in the domain

Remark 1.
• Moreover for the error bounds in practice we can use the computational order of convergence (COC) [31] using for each n = 0, 1, 2, . . .
• or the approximate computational order of convergence (ACOC) [31] using iterations instead of the exact solution for each n = 1, 2, . . .

Remark 2.
(a) Condition (11) and estimates (12)- (14) are natural Lipschitz-type conditions which are standard in the study of iterative methods. In fact (12)- (14) can be condensed in one condition. But used this way is better for reasons of precision. (b) Concerning the second and the thirs conditions in Theorem 1, these were left as uncluttered as possible.
One can easily see that there are infinitely many pair of values (α, β) satisfying these inequalities (but not as weak). Indeed, suppose that Then, the second inequality is satisfied provided that Then, the original inequalities certainly hold, if (27) and (28) hold. Moreover, (27) (in view of (28)) can ever be replaced by However as noted earlier condtions (27) and (28) or (28) and (29) are stronger than the two inequalities appearing in the statemetn of Theorem 1 (and can certainly replace them). Either way, as claimed above there are infinitely many choices of (α, β).

The Convergence Planes Applied to the Quadratic Polynomial p(z) = z 2 − 1 Using Method (2)
Once the convergence of the method has been demonstrated in the previous section, we will study the convergence planes. We should first study the parameter planes of the method (2), but the details of these parameter planes can be seen in [24]. We will show the convergence planes applied to p(z) = z 2 − 1 a polynomial of degree two, taking different values of the method parameters α and β. In this article, to draw the points, we iterate a maximum of 200 times and a tolerance of 10 −3 . We will use different colors to differentiate the numerical behavior of the points when they are iterated: Consequently, we are only interested in points painted in cyan or magenta since it means that the selection of parameters α and β for that initial point z 0 in numerical terms is a good one.
In Figure 1 we see the convergence plane associated to the method (2) applied to the two order polynomial p(z) = z 2 − 1 with the initial point z 0 = 0.5 and in the region of the plane (α, β) ∈ [−5, 5] × [ −5, 5]. and in Figure 2

Application Examples
To show the applicability of the theorems and results presented in this article, let's look at the Planck's radiation law problem, which can be obtained from the reference [32]: which estimates the density of within an isothermal black body. If we use a variable change: The expression which estimates the density of within an isothermal black body can be defined in the following terms: And if we now define the function h(x) as: As a result, if we are able to find the roots of Equation (32) can provide us the maximum wavelength of radiation (λ) through the expression: Let's consider the interval D = [4,6] and the solution x * = 4.965114 . . .. Then, we will introduce the applicability of the three special cases introduced in previous Section 3. Moreover, we can conclude the following conditions are satisfied. Then, due to the definition of the "k" functions we can derive r 0 = 9.71092 . . . , r 1 = 9.23282 . . . and r 2 = 2.64413 . . . .
Therefore, if we take the alpha and beta values as α = 0.9, β = 2.5 and r = r 2 = 2.64413 . . . we we can assure that the method described in (2) converges to the unique solution x * of the function h(x) by Theorem 1. Moreover, we can conclude the following conditions  Moreover, we can conclude the following conditions are satisfied. Now, applying the definition of the "k" functions we get r 0 = 9.71092 . . . , r 1 = 9.23282 . . . and r 2 = 2.64413 . . . .
So,taking the values: α = 0.9, β = 2.5 and r = r 2 = 2.64413 . . . we can assure the convergence of the method described in (2) to the solution x * of f (x) by Theorem 1.

Conclusions
In this paper, we have studied the local convergence of the two-step King-like method of Equation (2). The local convergence of the method has been proved giving the theorem that assure it, in addition to its proof. The advantage of this research is that it is only necessary to apply conditions to the first derivative to ensure local convergence, which can avoid the study of advanced derivatives that may not be simple, thus expanding the applicability of the method. In addition, the dynamics of the method for quadratic polynomials has been studied to prove the performance of the method. We finish by showing several examples using the new proposed method where we can assure the convergence.
Author Contributions: All authors have contributed equally, read and agreed to the published version of the manuscript.

Conflicts of Interest:
The authors declare no conflict of interest.