An Iterative Scheme of Arbitrary Odd Order and its Basins of Attraction for Nonlinear Systems

In this paper, we propose a fifth-order scheme for solving systems of nonlinear equations. The convergence analysis of the proposed technique is discussed. The proposed method is generalized and extended to be of any odd order of the form 2n 1. The scheme is composed of three steps, of which the first two steps are based on the two-step Homeier’s method with cubic convergence, and the last is a Newton step with an appropriate approximation for the derivative. Every iteration of the presented method requires the evaluation of two functions, two Fréchet derivatives, and three matrix inversions. A comparison between the efficiency index and the computational efficiency index of the presented scheme with existing methods is performed. The basins of attraction of the proposed scheme illustrated and compared to other schemes of the same order. Different test problems including large systems of equations are considered to compare the performance of the proposed method according to other methods of the same order. As an application, we apply the new scheme to some real-life problems, including the mixed Hammerstein integral equation and Burgers’ equation. Comparisons and examples show that the presented method is efficient and comparable to the existing techniques of the same order.


Introduction
One popular research area in mathematics is to find the solution a ¼ ða 1 ; a 2 ; . . . ; a n Þ t of the system of nonlinear equation . . . ; f n x ð ÞÞ t , and X ¼ ðx 1 ; x 2 ; . . . x n Þ t 2 R n . This type of problems occurs in many applied sciences like engineering, physics, biology and chemistry. Many researchers developed iterative methods for solving this kind of systems using different techniques. The most popular iterative method for solving system of nonlinear equations is the well-known Newton's method which has second order of convergence [1]. To improve the order of convergence and increase the accuracy of the solution obtained, many researchers tried to improve Newton's method. Some authors used different forms and modifications based on Adomian decomposition technique for solving systems of nonlinear equations, see for instance [2][3][4][5][6]. Another way to improve some schemes for systems of nonlinear equation is by using homotopy analysis method and homotopy perturbation method, see for example [7,8]. Grau-Sánchez et al. [9] used the harmonic mean of the derivative to improve an iterative scheme for solving systems of nonlinear equations. By applying some quadrature formulas, some researchers implement their techniques to solve systems of nonlinear equations, for instance [10][11][12]. Also, some derivative-free schemes for systems of nonlinear equations were proposed, see for example [13][14][15] and the references therein. One of the well-known modifications of Newton method is Jarratt method of order four. Cordero et al. [16] extended Jarratt method to solve systems of nonlinear equations preserving the same order of convergence. Many variants of Jarratt type methods have been developed, see for example [17][18][19] and the references therein. Many other different orders of convergence schemes for nonlinear modules can be found in the literature, see for example [20,21] and the references therein.
Some techniques to improve the order of convergence of the iterative schemes for systems of nonlinear equations have been proposed, for instance, see [22,23]. In general, obtaining a higher-order iterative method is not the only important thing; as the computational and the time cost are crucial issue also. So, establishing a high order iterative method based on low computational and time cost is very important.
In this paper, we develop a new multi-step scheme of arbitrary odd order for nonlinear equations. The proposed method can be used in the multidimensional case preserving the same order. The convergence analysis of the new scheme is discussed. Several examples are given to show the efficiency of the generalized method and its comparison with other iterative schemes of the same order. To confirm the applicability of the new technique, we apply the new technique to some real-life problems.

The Proposed Method
In this section we will derive the proposed technique for nonlinear modules. We begin by writing the function f x ð Þ as: As we want f x ð Þ ¼ 0, and by using midpoint quadrature formula and writing the equation as an iterative scheme, one gets Now, to write the iterative scheme (2) in explicit form, is the Newton step. So, scheme (2) becomes: The iterative method (3) was proposed by Frontini and Sormani [10]. The multidimensional case of scheme (3) was discussed by Homeier [24] and can be written as: where F 0 ðX n Þ À1 and F 0 ðY n Þ À1 are the inverse of the first Fréchet derivative of F X n ð Þ and F Y n ð Þ respectively. Scheme (4) is of third-order of convergence, and requires at each iteration the evaluation of one function, two Fréchet derivatives and two matrix inversions. In order to increase the convergence order and the computational efficiency of scheme (4) Sharma et al. [20] proposed a new scheme of the fifth-order of convergence by adding one step to scheme (4): W n ¼ X n À F 0 ðY n Þ À1 FðX n Þ; X nþ1 ¼ W n À ð2F 0 ðY n Þ À1 À F 0 ðX n Þ À1 ÞFðW n Þ: Per iteration, scheme (5) requires the evaluations of two functions, two Fréchet derivatives and two matrix inversions. Now, to derive the new scheme for solving systems of nonlinear equations, we start by composing scheme (3) to additional Newton step, that is: Now, to reduce number of functional evaluations at each iteration, we will use divided difference approximation to write the derivative f 0 w n ð Þ using some already computed functions from the previous steps. To do that, one can write f 0 y n ð Þ ¼ f y n ð Þ À f x n ð Þ y n À x n ¼ f y n ; x n ½ %f 0 x n ð Þ; in the same manner, we have f 0 y n ð Þ ¼ f y n ð Þ À f w n ð Þ y n À w n ¼ f y n ; w n ½ %f 0 w n ð Þ; by adding (7) and (8), one easily can conclude that Now, if we substitute (9) in (6), then we will have a new scheme for solving nonlinear equations: To generalize scheme (10) to the multidimensional case to solve systems of nonlinear modules, the scheme becomes: W n ¼ X n À F 0 ðY n Þ À1 FðX n Þ; X nþ1 ¼ W n À ð2F 0 ðY n Þ À F 0 ðX n ÞÞ À1 FðW n Þ: Scheme (11) requires at each iteration the evaluation of two functions, two Fréchet derivatives and three matrix inversions. The proposed scheme is of fifth-order of convergence as we will see in the next section.

Order of Convergence
We will discuss in this section the order of convergence of the proposed schemes (11) and (12). Assume for the next theorems that C j ¼ 1 j! F 0 ðaÞ À1 F j ð Þ a ð Þ, j ! 2, and e n ¼ X n À a.
Theorem 1 Let a be the solution of the system F X ð Þ ¼ 0 where F : D R n ! R n be a sufficiently differentiable function on a neighborhood D of a. Suppose that F 0 X ð Þ is continuous and nonsingular in a. If X 0 2 D is an initial approximation which is close enough to a, then the sequence X n f g n!0 obtained by scheme (11) converges to the root a, and the order of convergence equals 5, with asymptotic equation Proof. By using the Taylor expansion of F X n ð Þ we can write F X n ð Þ ¼ F 0 a ð Þ e n þ C 2 e 2 n þ C 3 e 3 n þ À C 3 e 3 n þ C 4 e 4 n þ . . .Þ. Now, we use the following Mathematica code to show the convergence order of scheme (11) for m ¼ 3: In [ À Á e 5 n . By this, we show that scheme (11) is at least of fifth-order of convergence. Now, we want to discuss the order of convergence of the generalized scheme given by (12).
Theorem 2 Let a be the solution of the system F X ð Þ ¼ 0 where F : D R n ! R n be a sufficiently differentiable function on a neighborhood D of a. Suppose that F 0 X ð Þ is continuous and nonsingular in a. If X 0 2 D is an initial approximation which is close enough to a, then the sequence X n f g n!0 obtained by scheme (12) converges to the root a, and the order of convergence equals 2m À 1, for any integer m ! 3, with asymptotic equation of the form e kþ1 ¼ 1 Proof. We will use the mathematical induction to prove the convergence order of scheme (12).
Firstly, we will prove that scheme (12) is convergent for m ¼ 3, and the convergence order satisfies 2 3 ð Þ À 1 ¼ 5. Note that for m ¼ 3, scheme (12) reduces to scheme (11) which we have been proved that it has the fifth-order of convergence in the previous theorem. Now, to complete the proof using the mathematical induction, suppose that scheme (12) is true and converges for all m r for some positive r > 3 and satisfy the given asymptotic equation. We need to show that the scheme converges for m ¼ r þ 1, and satisfy the given asymptotic equation. To do so, consider the following code of Mathematica: In[2]:= y ¼ e À Series 1 2 ðF 0 ½eÞ À1 F½e; fe; 0; 6g In [5]:= e nþ1 ¼ x½r À ð2F 0 ½y À F 0 ½eÞ −1 F½x½r //FullSimplify Hence, this shows that for m ¼ r þ 1, we have e kþ1 ¼ 1

Computational Efficiency
In this section, we compare the efficiency index of our proposed method with other methods in the literature. Commonly in the literature, the efficiency index EI ¼ p 1 d is used, where p is the order of convergence of the iterative scheme, and d is the number of functions needed to be found per iteration in the iterative scheme. Another common index that can be used in the comparison between iterative scheme is the computational efficiency index CEI ¼ p To find the number of functions required to be found per iteration in an iterative scheme, the following rules applied: Any computation of F X ð Þ needs n evaluations of scalar functions. Any computation of the Jacobian F 0 X ð Þ needs n 2 evaluations of scalar functions. Also, the floating points for obtaining the LU factorization are 2 3 n 3 , and to solve the triangular system we need n 2 floating points operations. Finally, n 2 operations required to find a matrix-vector multiplication, and n 3 operations needed to find a matrixmatrix multiplication.
We compare the efficiency index and the computational efficiency index for the proposed method (PM) (11) to the following iterative schemes: • The third-order Frontini-Sormani method (FS) [10] given by (4).
• The fifth order scheme (CHMT) proposed by Cordero et al. [23], given by • The fifth order scheme (MMK) proposed by Waseem et al. [4], which is given by: • The fifth-order iterative scheme (SG) presented by Sharma et al. [20], which is defined by scheme (5).
A comparison of the number of functional evaluations of the selected iterative schemes is illustrated in Tab. 1. Also, the computational efficiency indices of the selected schemes are compared (for n ¼ 2; 3; 4; 5; 10; 20; 50), see Fig. 1. Note that the proposed scheme does not attain the best efficiency in this comparison, especially for small n. We will see in the next two sections that this issue does not affect the scheme negatively when applied to some numerical tests.      2 illustrates the efficiency indices for the selected methods. Note that CHMT, SG and our proposed method have the same efficiency indices. However, this does not guarantee that they have the same behavior, accuracy and computational time cost.

Basins of Attraction
The concept of basins of attraction is a method to show how different starting points affect the behavior of the function. In this way, we can compare different root-finding schemes depending on the convergence area of the basins of attraction. In this sense, the iterative scheme is better if it has a larger area of convergence. Here, we mean by the area of convergence, the number of convergent points to a root a of f x ð Þ in a selected range.
To check the stability and the area of convergence of our proposed method, we select the case m ¼ 3 of scheme (12). We denote the proposed method by PM 5 . For comparison, we compare PM 5 with the following schemes of the same order of convergence: The scheme CHMT given by Cordero et al. (13), the scheme SG proposed by Sharma and Gupta (5), and the scheme MMK presented by Waseem et al. (14). We choose three test examples to visualize the basins of attraction. All examples are polynomials with roots of multiplicity one. The test polynomials are  Basins of attraction of PM 5 shows that the proposed method is comparable to other methods of the same order, with an area of convergence which is larger or the same as the areas of convergence of the other methods used in the comparison.

Numerical Tests and Applications
In this part, we consider some numerical problems to clarify the computational efficiency and convergence behavior of the proposed scheme. All calculations have been performed using 4000 significant digits on Mathematics 9. For comparisons, we find the number of iterations n needed to satisfy the stopping criterion k X n À X nÀ1 k þ k F X n ð Þ k< 10 À150 for each selected method. Also, we use the approximated computational order of convergence for each iterative scheme, which can be found by ACOC % ln k X nþ1 À x n ð Þk= k X n À X nÀ1 ð Þk ð Þ ln k X n À X nÀ1 ð Þk= k X nÀ1 À X nÀ2 ð Þk ð Þ : Finally, we compare for the selected schemes the distance between two consecutive iterations k X n À X nÀ1 k and the value of k F X n ð Þ k for n ¼ 1; 2; 3. To be consistent in the comparison, we compare the proposed scheme PM 5 defined by scheme (12) for m ¼ 3, to the original method which we derived from, that is, FS scheme given by (4). Also, we use the following fifth-order iterative schemes in the comparison: CHMT 5 method defined by scheme (13), MMK method defined by (14), and SG method defined by (5). To test the efficiency of the extension of our proposed scheme to higher orders schemes, we compare the proposed scheme PM 7 of seventh-order given by scheme (12) for m ¼ 4, to the extension of CHMT 5 to the seventh-order scheme CHMT 7 given by:

Numerical Tests
To be not selective in our examples, we choose most test problems from the same papers which contain the schemes used in the comparisons, see [4,20,23]. Also, we choose two distinct initial guesses for all problems to test the validity and the applicability of iterative schemes. We consider the following test problems and applications: Example 1 Consider the following system of two nonlinear equations: x þ 1 À e y ¼ 0; x þ cos y ð Þ À 2 ¼ 0; & with initial guesses X 0 ¼ f0; 0g t and X 0 ¼ f2; 2g t . The exact solution of this problem is a ¼ f1:3401918575555883401 . . ., 0:8502329164169513268 . . . g t .
Example 3 Consider the following system Assume the following partitioning for the interval 0; 1 ½ : is the step size, m is the system size. Let y i ¼ y u i ð Þ for i ¼ 0; 1; 2; . . . ; m þ 1. We use the finite difference method to solve the problem, in which the second derivative y 00 will be replaced by the central difference y 00 % y iÀ1 À 2y i þ y iþ1 h 2 , i ¼ 1; 2; . . . ; m. By this, we obtain m Â m system given by: . . . ; m: We solve this system for m ¼ 10 by selecting X 0 ¼ À1; À1; . . . ; À1 f g t zfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{ 10Àtimes , and X 0 ¼    Method X 0 n k X 1 À X 0 k k X 2 À X 1 k k X 3 À X 2 k k F X 1 ð Þ k k F X 2 ð Þ k k F X 3 ð Þ k ACOC  Method X 0 n k X 1 À X 0 k k X 2 À X 1 k k X 3 À X 2 k k F X 1 ð Þ k k F X 2 ð Þ k k F X 3 ð Þ k ACOC  Method X 0 n k X 1 À X 0 k k X 2 À X 1 k k X 3 À X 2 k k F X 1 ð Þ k k F X 2 ð Þ k k F X 3 ð Þ k ACOC

Applications
To check the applicability of the proposed scheme on real-life problems, we apply it on the mixed Hammerstein integral equation and Burgers' equation.
Comparisons in Tab. 7 show that the proposed schemes have a good functioning and comparable to the other iterative methods of the same order.

Problem 2
Consider the following Burgers' equation selected from [25]: We use discretization to solve this problem. Let h ¼ b À a N , and k ¼ T M be the spatial and temporal step sizes respectively, where N and M are numbers of subintervals in x and t directions respectively. Therefore, for this problem we select points x i ; t j À Á from a grid of domain 0; 1 . . . ; M and t j ¼ 0 þ jk, j ¼ 0; 1; . . . ; N . Let w ij be the approximate solution at x i ; t j À Á . By applying the central differences to u x and u xx , and the backward difference to u t , we get the following nonlinear system: If we let r ¼ Dk h 2 , and z i ¼ w ij , then we are trying to solve the following system of equations: For the unknowns z 1 ; z 2 ; . . . ; z M . The last term w i;jÀ1 is known from the previous time step. The first and the last equations in the system can be replaced by using the given boundary conditions. In our problem, the first and the last equations in the system are  We find the approximate solution of the problem using the proposed method PM 5 at x ¼ 0:5 and t ¼ 0:2. To check the effect of the temporal step sizes on the solution, we select different values for k, which means a different number of steps to reach the wanted time. Consider for our problem that the diffusion coefficient D ¼ 0:05, a ¼ 5, and b ¼ 4. The exact solution of the given Burgers' equation is given by: The exact solution for this problem is u 0:5; 0:2 ð Þ¼0:2277071734 . . .. Based on spatial step size equals 1 10 , we choose X 0 ¼ 0:6; 0:6; Á Á Á ; 0:6 f g zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{ 10Àtimes . Tab. 8 illustrates the numerical results of this problem. The effect of the selected k is clear. The results become better whenever we have a smaller temporal step size. Based on that, we will compare our proposed schemes to the other schemes for h ¼ 0:1 and k ¼ 0:01. We compare the approximate solutions at x ¼ 0:5 and t ¼ 0:2 for n ¼ 3 and n ¼ 4, that is X 3 and X 4 . Also, we find the norms of the functions F X 3 ð Þ and F X 3 ð Þ. Finally, we find the norm of the difference between the two consecutive iterations X 4 À X 3 for each selected method. Comparisons results are shown in Tab. 9. It is clear that the proposed schemes perform in a good way, and in general, give results which are better than the other selected schemes.

Conclusion
In this study, we have proposed an iterative scheme for systems of nonlinear equations of fifth-order of convergence. We have improved the proposed scheme to a generalized scheme of arbitrary odd order. The proposed method is based on Frontini-Sormani iterative method and developed using additional step with the usage of first derivative approximation. The software Mathematica has been used to show the order of convergence of the proposed method. Different comparisons were used to compare our proposed scheme to the other schemes of the same order, including the efficiency index, computational efficiency index, basins of attractions and several numerical problems. Comparisons show that the efficiency index and the computational efficiency index need not be proper tools for the efficiency of the iterative scheme. As an application, we test the proposed method on the mixed Hammerstein integral equation and Burgers' equation. Comparisons show that the proposed scheme is of excellent performance and overall, it is comparable to the other iterative techniques used in the comparisons regarding the convergence speed, accuracy and the area of convergence in the basins of attraction.
Funding Statement: We are grateful for the financial support from UKM's research Grant GUP-2019-033.

Conflicts of Interest:
The authors declare that they have no conflicts of interest to report regarding the present study.