Abstract
In this paper, we want to construct a new high-order and efficient iterative technique for solving a system of nonlinear equations. For this purpose, we extend the earlier scalar scheme [16] to a system of nonlinear equations preserving the same convergence order. Moreover, by adding one more additional step, we obtain minimum 5th-order convergence for every value of a free parameter, , and for , the method reaches maximum 6-order convergence. We present an extensive convergence analysis of our scheme. The analytical discussion of the work is upheld by performing numerical experiments on some applied science problems and a large system of nonlinear equations. Furthermore, numerical results demonstrate the validity and reliability of the suggested methods.
1. Introduction
Solution of nonlinear equations of the following form:where , is a more challenging and broader problem as compared to the scalar equations. Several issues like neurophysiology, kinematics syntheses, transport theory, chemical equilibrium, reactor and steering combustion, and economics modeling problems can be phrased in the terms of (1) and details can be found in [1–9].
There are different ways to construct iterative schemes for nonlinear systems. We center in the natural way methods constructed for scalar equations and then extend to the multidimensional case. Some researchers like Cordero et al. [10], Abad et al. [11], Cordero et al. [12], and Wang and Zhang [13] adopted this way to construct schemes for nonlinear systems. There is no doubt that some scalar schemes can be extended to a system of nonlinear equations with the help of some transformations or substitutions. But, the most important and crucial task is to retain the same convergence order and lower computational cost.
Therefore, researchers also tried some other approaches and procedures in order to construct new and higher-order iterative methods for systems of nonlinear equations. Recently, Sharma et al. [14] proposed fourth-order and six-order iterative methods based on weighted-Newton iteration. Very recently, Artidiello et al. [15] provided fourth-order methods based on the weight function approach. Some researchers have also used the approaches like quadrature formulae, Adomian polynomial, divided difference approach for constructing iterative schemes to solve nonlinear systems. For the details of the other approaches, one can refer some standard text books [16, 17, 18].
In this paper, our main objective is to construct a new high-order family of iterative methods for a system of nonlinear equations. The development of our scheme is based on the scalar scheme proposed by Kou et al. [19]. First of all, we extend their scheme for a system of nonlinear equations retaining the same convergence order. Then, we increase the convergence order up to 6 by adding one additional substep to the same scheme. The efficiency of the suggested iterative techniques is demonstrated on several applied science problems and academic numerical examples. We found that our proposed methods perform better than the existing ones in terms of residual error, difference between two consecutive iterations, and asymptotic error constants.
2. Higher-Order Scheme for Multidimensional Case
Here, we consider a higher-order scheme for simple zeros on univariate function proposed by Kou et al. [19], which is defined as follows:where is a free disposable parameter. Expression (2) has at least cubic convergence for every θ except . It has maximum fourth-order convergence for . It is a linear combination of well-known Potra-Pták and Newton–Steffensen methods. More details can be found in Kou et al.’s study [19]. We can easily obtain Potra-Pták [20] and Newton–Steffensen [21], for and , respectively, in scheme (2). We can rewrite scheme (2) in the following way:orwhere is the well-known divided difference of first order. But, scheme (2) does not work for a system of nonlinear equations. So, our foremost aim is to extend this scheme for a system of nonlinear equations and retain the same convergence order. Therefore, we rewrite expression (4) in the following way:where θ is a free disposable parameter and is a finite difference of order one.
Now, by adding one extra substep, we have following higher-order scheme:
In Theorem 1, we demonstrate that the minimum convergence orders of methods (5) and (6) are three and five, respectively. In the proof of this result, we use the tools and procedure introduced in [10] which we recall briefly.
Let be sufficiently differentiable in D. The qth derivative of F at , , is the q-linear function such that . It is easy to observe that(1).(2), for all permutation σ of .
From the above properties, we can use the following notation:(a)(b)
On the other hand, for lying in a neighborhood of a solution ξ of , we can apply Taylor’s expansion, and assuming that the Jacobian matrix is nonsingular, we havewhere , . We observe that since and .
In addition, we can express aswhere I is the identity matrix. Therefore, . From (8), we obtainwhere
We denote as the error in the jth-iteration in the multidimensional case. The equation , where M is a p-linear function, , is called the error equation and p is the order of convergence. Let us observe that is .
Theorem 1. Let be a sufficiently differentiable function in an open neighborhood D of its zero ξ. Let us consider that is continuous and nonsingular in the neighborhood of ξ. In addition, we assume the initial guess is close enough to ξ for the guaranteed convergence. Then, the iterative schemes (5) and (6) have minimum third and fifth order of convergence, respectively.
Proof. Let be the error of the jth iteration. Developing and in a neighborhood of ξ, we writeandwhere I is the identity matrix of size and , .
With the help of expression (12), we havewhere , given byFrom expressions (11) and (2), we yieldwhere , i.e.,By inserting expression (15) in the first substep of (5), we obtainwhich further producesandBy using expressions (11)–(19) in scheme (5), we havewhere , i.e.,By using the values of and from (15) in and , we haveand adopting (22) in (21), we yieldFrom expression (23), it clear that scheme (5) has the least third-order convergence for every θ.
Now, expanding the in a neighborhood of ξ, we havewhich further produces, with the help of expression (18),By using equations (24) and (25) in the final substep of (6), we obtain
It is clear from expression (26) that we have minimum fifth-order convergence for every . In addition, we have maximum sixth order of convergence for . Hence, schemes (5) and (6) have minimum third and fifth order convergence, respectively, for every .
3. Numerical Experiments
Here, we checked the efficiency and effectiveness of our methods on 4 real life problems. Some of them have been taken from the paper of R. Behl et al. (see [22]). Additionally, we also consider a large system of equations of to verify the theoretical results. The details of all numerical problems can be seen in examples (1–5). Furthermore, we also mentioned the starting points and solutions of the considered problems in the examples. Now, we employ our fourth and sixth-order schemes (5) and (6), called and , respectively, for in order to verify the computational performance of them with the existing methods.
Now, we compare (2) with sixth-order methods suggested by Mona et al. [23] and Lotfi et al. [24]. We consider methods (27) and (5), respectively, called and . In addition, we also compared our scheme with sixth-order family of iterative methods proposed by Sharma and Arora [25]. From them, we choose method (13), called . Finally, we compare (2) with a sixth-order family of iterative method that is recently designed by Abbasbandy et al. [26] and Hueso et al. [27]. From them, we choose their best expressions (8) and (14, 15) , respectively, denoted by and .
In Tables 1–5, we depicted the iteration indexes , residual error of the corresponding function , error between the two consecutive iterations , approximate computational order of convergence , (where p is either 4 or 6), and η is the last calculated value.
During the current numerical experiments with programming language Mathematica (Version 9), all computations have been performed with multiple precision arithmetic with 1000 digits of mantissa, which minimize round-off errors. The meaning of is in all tables. We adopted the command “AbsoluteTiming” in order to calculate the CPU time. We run our programs three times and depicted the average CPU time in Table 6. Also, one can observe the times used for each iterative method and each problem in Figure 1, where we want to point out that for big size problems, the method uses the minimum time, so it is being very competitive. The configuration of the used computer is given as follows: Processor: Intel(R) Core(TM) i7-4790 CPU 3.60 GHz Made of: HP RAM: 8:00 GB System type: 64-bit-Operating System, x64-based processor
Example 1. Let us consider a boundary value problem described in the book of Ortega and Rheinbolt [16], which is defined as follows:In addition, we consider the following partition of the interval , which is given byMoreover, we also consider the notation, , for . We discretize problem (27) with the help of following numerical approximation for first and second derivativesHence, we yield a nonlinear system of , which is given as follows:Let us consider initial approximation . In particular, we will solve this problem for so that we can obtain a system of nonlinear equations. The solution of this problem isThe numerical results appeared in Table 1. Moreover, we represent in Figure 2 the difference between two consecutive iterations but in logarithmic scale and taking absolute value, , so the biggest value corresponds to the minimum distance error, by the properties of logarithmic function, that is, the iterative method presents the best results.
Example 2. Consider another typical nonlinear problem, that is, Fisher’s equation [28] with homogeneous Neumann’s boundary conditions and D the diffusion coefficient:Again using a finite difference discretization, equation (32) reduces to a system of nonlinear equations. Consider to be its approximate solution at the grid points of the mesh. Let M and N be the number of steps in x and t directions, and h and k be the respective step size. Apply central difference to , backward difference for , and forward difference for . For the solution of the system, we consider and which reduces to a nonlinear system of size 25, with the initial vector convergence towards the following solution:and the numerical results of comparing the behavior different methods are shown in Table 2.
Example 3. The 2D Bratu problem [29, 30] is defined in asThe approximated solution of a nonlinear partial differential equation can be found using finite difference discretization which reduces to solving a system of nonlinear equations. Let be its approximate solution at the grid points of the mesh. Let M and N be the number of steps in x and t directions and h and k be the respective step size. To solve the given PDE, apply central difference to and , i.e., . We look for the solution of the system for M = 11 and N = 11 of size 100, with the initial sinusoidal approximation ; by using different methods, one can see the obtained result in Table 3. The solution of this problem can be seen in Figure 3.
Example 4. In this example, we consider the following Hammerstein integral equation (see [pp. 19, 20]) Ortega to check the effectiveness and applicability of our proposed methods, and compared with other existing methods, the nonlinear integral equation is given bywhere and the kernel F can be written asTo transform the above equation into a finite-dimensional problem by using Gauss Legendre quadrature formula given as where the abscissas and the weights are determined for . Denoting the approximations of by , one gets the system of nonlinear equations:where the abscissas and the weights are known and depicted in Table 7 when .
The different methods converge towards the solution and different conditions of convergence can be checked in Table 1. The initial guess .
Example 5. Finally, let us consider the following system of nonlinear equations (chosen from Grau-Sánchez et al. [31]):We choose in order to check the theoretical results mentioned with a large size system. We take the initial guess for this problem. The obtained solution of this problem isThe obtained results can be observed in Table 5.
4. Concluding Remarks
In this paper, we proposed a new high-order and efficient family of iterative techniques for a system of nonlinear equations. The suggested methods are the extension of the study by Kou et al. [19] for multidimensional case. In addition, our scheme has minimum 5th-order convergence by adding additional substep and for particular value , it reached the maximum 6th order. From the obtained numerical results in Tables 1–5, it is confirmed that the lowest residual error, difference between two consecutive iterations and asymptotic error constant, belongs to our method . Finally, our schemes and , respectively, also consume the first and second lowest average CPU time as compared to the other existing methods in examples 1–5. In the future work, we will try to propose higher-order version of the same or new methods with lower computational cost and that employ more accurate solution.
Data Availability
In this paper, we want to construct a new high-order and efficient iterative technique for solving a system of nonlinear equations. For this purpose, we extend an earlier scalar scheme to a system of nonlinear equations preserving the same convergence order. Moreover, by adding one additional step, we obtain minimum 5th-order convergence for every value of a free parameter, and in some specific cases, the method reaches maximum 6-order convergence. We present an extensive convergence analysis of our scheme. The analytical discussion of the work is upheld by performing numerical experiments on some applied science problems and a large system of nonlinear equations. Further, numerical results demonstrate the validity and reliability of the suggested methods.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant no. (D-349-130-1441). The authors, therefore, acknowledge with thanks, DSR technical and financial support.