Fixed-Point Iterative Method with Eighth-Order Constructed by Undetermined Parameter Technique for Solving Nonlinear Systems

In this manuscript, by using undetermined parameter method, an efficient iterative method with eighth-order is designed to solve nonlinear systems. The new method requires one matrix inversion per iteration, which means that computational cost of our method is low. The theoretical efficiency of the proposed method is analyzed, which is superior to other methods. Numerical results show that the proposed method can reduce the computational time, remarkably. New method is applied to solve the numerical solution of nonlinear ordinary differential equations (ODEs) and partial differential equations (PDEs). The nonlinear ODEs and PDEs are discretized by finite difference method. The validity of the new method is verified by comparison with analytic solutions.


Introduction
Symmetry is a fundamental topic in many areas of physics and mathematics. Many problems from engineering and mathematics possess symmetry, which can be transformed into the nonlinear systems F(z) = 0. Some nonlinear ODEs and PDEs can be discretized into nonlinear systems. Using the iterative method to solve these nonlinear systems, we can find the numerical solution of ODEs and PDEs. Newton's method [1,2] is the oldest and famous iterative method for the numerical solution of nonlinear systems, where J j = F (z (j) ) is the Jacobin matrix. Newton's method is one-point iterative method, which is convergent quadratically. Any one-point iterative method which is constructed by F and its first r − 1 derivatives cannot get higher order than r. Researchers try to improve the computational efficiency of one-point method by different ways. One effective way is to increase the iteration step of iterative method. This kind of methods are called multistep or multipoint iterative methods. It has better efficiency than one-point iterative method. For example, based on a quadrature formulae, Darvishi et al. [3] suggested an efficient multipoint method with order four that requires two F, two matrix inversions and three F . Grau-Sánchez et al. [4] proposed a variant of Ostrowski's method, which needs two matrix inversions, three F, two F and one first-order divided difference. Using the pseudocompostion technique, Cordero et al. [5] presented a sixth-order Jarratt-type method that uses three F , two matrix inversions and three F. They [6] also presented a four-step eighth-order method that requires the same computational cost as their sixth-order Jarratttype method. Using the weight function technique, Sharma and Arora [7] suggested a Jarratt-type method with order six, which requires one matrix inversion, three F and two F . Behl and Cordero [8] designed a sixth-order scheme that needs two matrix inversions, two F and one first-order divided difference. Behl and Arora [9] proposed a derivativefree scheme with order seven for solving nonlinear systems, which needs one matrix inversion, five F and two first-order divided differences. Using the interpolation technique, we [10] obtained a seventh-order method that is extendible to solve nonlinear systems. This method requires three matrix inversions, four F and five first-order divided differences. We [11] also obtained another seventh-order fixed-point method that needs one matrix inversion, five F and three first-order divided differences. Sharma and Aroa [12] obtained a seventh-order derivative-free method. This method requires two matrix inversions, four F and five first-order divided differences. Using the undetermined parameter technique, Narang et al. [13] designed a seventh-order method, which needs one matrix inversion, three F and two first-order divided differences per iteration. Many efficient multipoint iterative methods for solving nonlinear equations have been proposed, see [14][15][16][17][18][19][20]. However, not all multipoint iterative methods can be extended to solve nonlinear systems. Therefore, it is an interesting research to construct multipoint iterative method for solving systems of nonlinear equations. Ham and Chun [14] proposed the following fifth-order method for solving nonlinear equations which is called Ham-Chun's method. We generalize Ham-Chun's method to Banach space to solve nonlinear systems and obtain the following iterative scheme where N j = 5F (t (j) ) − J j and J j = F (z (j) ). Method (3) is called HM5 in this paper, which requires two F, two matrix inversions and two F per iteration. Method HM5 is not the fixed-point iterative method, so its computational efficiency is low.
In this paper, we propose an eighth-order fixed-point iterative method for the numerical solution of nonlinear systems. First, we prove the order of convergence of method HM5 in Section 2. Inspired by method HM5, we propose an eighth-order fixed-point method by using the undetermined parameter method in Section 3. The proposed method requires one LU decomposition per iteration, which means that this method has low computational cost. The computational efficiency of iterative method is analyzed in Section 4. The proposed method is used to solve the solution of nonlinear systems, nonlinear ODEs and PDEs in Section 5. Section 6 gives a short conclusion.

Iterative Method with Order Five for Solving Nonlinear Systems
Theorem 1. Let ξ ∈ R m be the zero of F : D ⊂ R m → R m ,F(z) be sufficiently Fréchet differentiable and F (z) be continuous and nonsingular at D. Then, method HM5 converges to ξ with order five, if initial guess z (0) close to ξ.
Based on method HM5, we will give a new fixed-point method with order eight in the following section.

Fixed-Point Iterative Method with Order Eight
Inspired by method HM5, we design a fixed-point iterative method by using the undetermined parameter technique as follows: where M (j) = J −1 j (J j − F (t (j) )), I is an identity matrix and g i , (i = 1, 2) are constant parameters to be determined.

Theorem 2.
Let ξ ∈ R m be the zero of F(z) : D ⊂ R m → R m , F(z) be sufficiently Fréchet differentiable and F (z) be continuous and nonsingular at D. If g 1 = 5 4 , g 2 = 3 2 and initial guess z (0) is close to ξ, then method (13) reaches eighth-order convergence.
Thus, we get an iterative method with eighth-order as follows where (19) is called NM8 in this paper.

Computational Efficiency
The computational efficiency indexes CEI is proposed by Grau-Sánchez et al. [4,16,17]: where In per iteration, C(ν 0 , ν 1 , m) represents the cost of iterative method. Parameter S 0 (m) is the number of the scalar functions used in F and [y, x; F]. Parameter S 1 (m) is the number of scalar functions used in F . S(m, l) is the number of products . The parameter l in (21) is the ratio between the divisions and products. The parameters ν 0 and ν 1 in (21) are the ratios between products and evaluations that are required to express C(ν 0 , ν 1 , m) in terms of products. ρ represents the convergence order. The LU decomposition is used to solve linear systems in the processing of iteration. The following methods are used to compared the computational efficiency: Darvishi et al. fourth-order method [3] (DM4) where ; F] is the first-order divided difference operator.
Cordero et al. sixth-order method [5] where The C and CEI of different iterative methods are given below: The computational efficiencies of different methods are compared by the ratio R A,B [4,10] If R A,B > 1, then method B is less efficient than method A. R A,B = 1 is the boundary between the computational efficiencies, which is an equation of ν 0 as a function of m,l and ν 1 . Parameters in (38) satisfy ν 0 > 0, ν 1 > 0, l ≥ 1 and m ≥ 2, the CEI of different methods is studied in the following result.
Proof. 1. From (26), (27) and (36)-(38), we get the boundary R N M8,DM4 = 1 that is given by (39) Figure 1 shows the boundary R N M8,DM4 = 1, which implies that CEI N M8 > CEI DM4 on the above and CEI N M8 < CEI DM4 on the below of boundary plane. Parameter m ≥ 8, ν 1 is always negative which means that DM4 is less efficient than NM8 for all m ≥ 8 and l ≥ 1. Figure 2 shows the boundary lines in (l, ν 1 )-plane for m = 4, 8 and 9 which means that CEI N M8 > CEI DM4 for m = 8, m = 9 and l ≥ 1. For m < 4, CEI N M8 > CEI DM4 on the above of boundary line and CEI N M8 < CEI DM4 on the below of boundary line.

Numerical Results
New methods HM5 and NM8 are compared with methods DM4, GM6, CM6 and CM8 for solving some nonlinear systems. Numerical algorithms are written by the Maple 14. Table 1 shows an estimate of the cost of functions in product units. The computational cost of elementary functions based on product is showed in Table 1.  Table 1 shows that the computing time of one product is 0.110 milliseconds (ms). The computational cost of division with respect to product is l = 1 and the computational cost of function √ x with respect to product is l = 5. In Tables 2-9, EF represents the errors of function values at the last step, N I represents the number of iterations, CC represents the computational cost, EV represents the errors values of ||z (j) − z (j−1) || , CEI represents the computational efficiency index, ACOC [17] represents the approximated computational order of convergence, e − Time represents the mean CPU time. We choose the tolerance ||z (j) − z (j−1) || < 10 −200 for implementing the methods. Example 1. Nonlinear system [16]: For this example, we can get the computational cost of different methods by using parameters l = 1, m = 4, ν 0 = 110 and ν 1 = 28. The solution ξ ≈ (0.5149, · · · ,0.5149) T is founded by initial guess z (0) = (1.0, · · · 1.0) T . The results of comparisons for this example are displayed in Table 2.
This PDE problem is a Heat Conduction Problems. We transform this problem to the nonlinear systems by using finite differences. The intervals [0, 1] and [0, T] are partitioned into N smaller intervals, and get step size h = 1/N and k = T/N in z and t directions. Let u = u(z, t) be the exact solution and u i,j ≈ u(z i , t j ). Using difference method, we use obtain approximations We get a nonlinear system as follows: . . , N − 1 and j = 1, 2, . . . , N For fixed j, we get some nonlinear systems with size (N − 1) × (N − 1). Choosing different N and T, we solve this problem by different methods. The results are given in Table 9, where N I represents the number of iteration and e − Time represents the CUP time. The approximate solution and absolute value of error of this problem are shown in Figures 7 and 8, when T = 0.01 and N = 50. Figure 9 shows the exact solutions of this problems.  9 show that compared with other methods, method NM8 has lower computational cost and higher computational efficiency. We find that method NM8 costs less computing time than other methods. This advantage is obvious for solving large scale nonlinear systems. Figures 7-9 show that method NM8 can be used to solve nonlinear PDE equations with high accuracy.

Conclusions
We have extended the fifth-order Ham-Chun's method to Banach space and developed an efficient method with eighth-order for solving standard nonlinear systems, nonlinear ODEs and PDEs. Our method requires three functions F, two derivatives F and one matrix inversion F −1 per iteration. Tables 2-9 show that our method NM8 show a highly efficient especially in large scale nonlinear systems. Numerical results verify that our method NM8 is better than other methods in this paper.

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