Derivative-Free Iterative Methods with Some Kurchatov-Type Accelerating Parameters for Solving Nonlinear Systems

: Some Kurchatov-type accelerating parameters are used to construct some derivative-free iterative methods with memory for solving nonlinear systems. New iterative methods are developed from an initial scheme without memory with order of convergence three. New methods have the convergence order 2 + √ 5 ≈ 4.236 and 5, respectively. The application of new methods can solve standard nonlinear systems and nonlinear ordinary differential equations (ODEs) in numerical experiments. Numerical results support the theoretical results.


Introduction
Many real-world problems that arise in various scientific fields are modeled by mathematically interesting nonlinear systems F(x) = 0. Symmetries and conservation laws are powerful tools for studying explicit solutions of nonlinear systems. Finding the solution of nonlinear systems is an important problem in the area of mathematics, where F : D ⊆ R n → R n . Iterative method is a kind of efficient method for solving nonlinear systems. Optimization and acceleration of iterative methods can be achieved by applying symmetries. Newton's method [1] is the oldest method for solving nonlinear systems, which is quadratically convergent assuming that initial approximation is close enough to the root. Based on Newton's method, some high-order iterative method have been proposed in the literature. For example, Torres-Hernandez et al. [2], Gdawiec et al. [3], Akgül et al. [4] and Cordero et al. [5] developed some variants of Newton's method by using fractional derivatives. Behl et al. [6] and Geum et al. [7] proposed some high-order iterative methods and their dynamics are investigated. Schwandt [8] proposed a symmetric iterative method for solving nonlinear systems. Barco et al. [9] obtained the local solutions of partial differential equations by symmetry approach. Derivative-free method is a kind of variant of Newton's method, which can solve the solution of non-differentiable nonlinear systems. One of the celebrated derivative-free iterative methods is Traub's method [10], which is given by where s (j) = z (j) + B F(z (j) ), B is a nonzero arbitrary parameter and [s (j) , z (j) ; F] −1 is the inverse of the first-order divided difference operator [s (j) , z (j) ; F]. The first-order divided difference operator [., .; F] : D × D ⊂ R n × R n → L(R n ) is an n × n matrix, which is defined by [11]: [z + h, z; F] = 1 0 F (z + th)dt, ∀(z, h) ∈ R n × R n , where h = s − z. Developing the Taylor's expansion of F (z + th) on the point z, we obtain 1 0 F (z + th)dt = F (z) In the process of computation, the first-order divided difference operator [s (j) , z (j) ; F] is calculated by [11] [s (j) , Based on Traub's method, many derivative-free methods have been studied in the literature [12][13][14][15][16][17][18][19]. Derivative-free methods can be divided into two groups: iterative method with memory and iterative method without memory. Iterative method with memory is superior to iterative method without memory in terms of computational efficiency and stability. To date, very few derivative-free methods with memory for solving nonlinear systems have been proposed in the literature. Recently, Petković and Sharma [12] designed the following derivative-free method with memory for solving nonlinear systems by using the variable parameter method where ) and c ∈ R − 0. Method (5) has the convergence order 2 + √ 5 ≈ 4.236, when the parameter a = 3. Using the same variable parameter B (j) as method (5), Ahmad et al. [13] and Kansal et al. [14] proposed some high order iterative methods with memory for solving nonlinear systems. Using the Kurchatov's divided difference operator [15], Chicharro et al. [16] designed two derivative-free methods with memory for solving nonlinear systems. Firstly, they constructed the following third-order iterative method without memory which satisfies the following error equation where ε (j) = z (j) − ζ, ζ is the zero of nonlinear function F and A j = 1 j! F (ζ) −1 F (j) (ζ), (j = 1, 2, · · · , n). Replacing the constant parameter B with B (j) = −[2z (j) − z (j−1) , z (j−1) ; F] −1 in method (6), they obtained the following fourth-order method with memory where the first-order divided difference operator [2z (j) − z (j−1) , z (j−1) ; F] is called Kurchatov's divided difference operator. Using the Kurchatov's divided difference operator to design the variable parameter, Cordero et al. [17], Argyros et al. [18] and Candela et al. [19] proposed some efficient Kurchatov-type methods. Variable parameters can be designed by different schemes, which usually uses iterative sequences from the current and previous steps.
In this paper, we design some new variable parameters by using some new Kurchatovtype divided difference operators. This paper is organized as follows. Some new Kurchatovtype divided difference operators are used to construct iterative schemes with memory for the numerical solution of nonlinear systems in Section 2. The main advantage of the new Kurchatov-type divided difference operators is that it has less errors than the Kurchatov's first-order operator [2z (j) − z (j−1) , z (j−1) ; F]. The order of the basic method (6) is increased from 3 to (2+ √ 5) ≈ 4.236 and 5, respectively. The application of new methods to solve standard nonlinear systems and nonlinear ordinary differential equations (ODEs) is made in numerical experiments. Numerical experiments are made in Section 3. A short summary is given in Section 4.

Some New Iterative Schemes with Memory
If I + BF (ζ) = 0 in (7), the convergence order of method (6) is three. Letting B = −F (ζ) −1 , the order of convergence of method (6) can be improved. However, F (ζ) is unknown in practice. In order to improve the order of convergence of method (6), we could choose a variable parameter B (j) to replace constant parameter B. The variable parameter where iterative sequence {z (j) } → ζ as j → ∞.
To obtain some more effective iterative methods, we design some new Kurchatov-type first-order divided difference operators to construct the accelerating parameter B (j) . If j → ∞, then iterative sequences generated by iterative method (7) satisfy {t (j) } → ζ, {z (j) } → ζ and {s (j) } → ζ. Using t (j) , z (j) and s (j) , we can design some first-order divided difference operators to approach F (ζ). Scheme 1. Using t (j−1) and z (j) , we design the first-order divided difference operator [2t (j−1) − z (j) , z (j) ; F] and obtain the following variable parameter Using (2) and (3), we have Using Taylor's expansion around ζ and taking into account F(ζ) = 0, we have and where ε (j) = z (j) − ζ.
where ε From (16), we obtain This means that the first order divided difference operator [2t (j−1) − z (j) , z (j) ; F] can be used to construct the variable parameter B (j) .

Numerical Results
Our methods (39)-(45) are compared with Petković's method (5) and Chicharro's method (8) for solving nonlinear systems and ODEs. For numerical experiments, Maple 14 with 2048 digits is used. The stopping criterion ||z (j) − z (j−1) || < 10 −100 is used in numerical algorithms. The initial parameter B (0) is the identity matrix. Tables 1-4 give the numerical results and the following information: NI means the number of iterations, EF means function values at the last step, EV represents the error values of ||z (j) − z (j−1) ||, e − Time represents the CPU time (in second) and ACOC [20] is the approximated computational order of convergence. Figures 1-4 show the iterative processes of different methods for solving nonlinear systems.
The solution ζ ≈ {0.2074, · · · ,0.2074} T is obtained by the initial guess z (0) = {0.038, · · · 0.038} T . The iterative processes of different methods for solving Example 1 are shown by Figure 1. Figure 1 shows that our method (44) has higher computational accuracy than other methods.
For n = 6, the solution {0.07748, 0.12494, 0.14093, . . . , 0.07748} T is founded by the initial value is z (0) = (0.3, · · · 0.3) T . The numerical results are displayed in Table 3. The iterative processes of different methods for solving Example 3 are shown by Figure 3. Figure 3 show that our method (44) has higher computational accuracy than other methods. Methods (39), (40) and (41) have the similar convergence behavior for Example 3.
For n = 8, the solution {0.0846, 0.1767, 0.2776, . . . , 0.8159} T is founded by the initial guess z (0) = (0.97, · · · 0.97) T . Table 4 shows the numerical results.   Figures 1-4 show that our methods (44) have higher computational accuracy than other methods. The iterative processes of different methods for solving Example 4 are shown by Figure 4. Figure 4 shows that our method (44) has higher computational accuracy than other methods.

Conclusions
In this paper, we proposed four new Kurchatov-type first-order divided operators. Using these new Kurchatov-type first-order divided operators, we designed some new accelerating parameters and constructed seven derivative-free iterative methods with memory for solving nonlinear systems. The local convergence order of Chicharro's method without memory (6) was improved from 3 to 2+ √ 5 ≈ 4.236 and 5, respectively. Numerical results support the theoretical results. We should note that the main objective of this paper was to develop a high-order method and prove the local convergence order of new methods. The initial approximation must be close enough to zero of the nonlinear function. If the initial approximation is far from the zero of nonlinear function, then the iterative sequence generated by iterative method converges slowly or diverges. Therefore, the choice of good initial approximations is very important to iterative methods. Some strategies for finding sufficiently good initial approximation have been proposed [23][24][25]. Finding good initial approximation for multipoint iterative method needs further research.