Convergence Analysis and Numerical Study of a Fixed-Point Iterative Method for Solving Systems of Nonlinear Equations

We present a fixed-point iterative method for solving systems of nonlinear equations. The convergence theorem of the proposed method is proved under suitable conditions. In addition, some numerical results are also reported in the paper, which confirm the good theoretical properties of our approach.


Introduction
One of the basic problems in mathematics is how to solve nonlinear equations ( ) = 0. (1) In order to solve these equations, we can use iterative methods such as Newton's method and its variants. Recently, there has been some progress on iterative methods with higher order of convergence using decomposition techniques; see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] and the reference therein. The zeros of a nonlinear equation cannot in general be expressed in closed form; thus we have to use approximate methods. Nowadays, we often use iterative methods to get the approximate solution of the system (1); the best known method is the classical Newton's method. Recently, there has been some progress on solving the system (1), which allows us to get the iterative formula by using essentially Taylor's polynomial (see [16,17]), quadrature formulas (see [7,[9][10][11][12]), homotopy perturbation method (see [8]), and so on.
In this paper, we will present a new fixed point iterative method for solving the system (1) and prove that the method is cubic convergent under suitable conditions. This paper is organized as follows. In Section 2, we introduce new iterative methods to solve (1). In Section 3 we extend these methods to solve systems of nonlinear equations, and we also prove convergence of the proposed method. Some numerical results are reported in Section 4, while the paper is concluded in the last section.

Iterative Methods and Convergence Analysis
We now consider the following nonlinear equation: Assume that * is a simple root of (2); that is, ( * ) = 0. For , ∈ [ , ], using Taylor's formula, we have ( ) = ( ) + ( ) ( − ) Taking = 1 in the above equality, we get The Scientific World Journal If the value of ( + ( − )) in the interval [0, 1] is replaced with its value in = 0, that is, with ( ), then we have By using (5) in (4), we have We can get an iterative method from (6) to solve the system (2); it is the famous Newton's formula The formula (7) has already been proved to be quadratically convergent. Now we begin to deduce a higher order iterative method. In fact, if we estimate ( + ( − )) in the interval [0, 1] by its value in = 1, that is, by ( ), then we have By using (8) in (4), and cutting off the error [ ( )] (still use "="), we have Let +1 be the solution of (9); we can obtain a new iterative method Since the iterative method (10) is implicit-type method, we use classical Newton's formula (7) as predictor and then use the above scheme (10) as corrector; in this way, we can get a workable iterative method.
In this section, we consider the convergence and convergent rate of Algorithm 1. We obtain a convergence theorem as follows.
Proof. By (12) we get Multiplying the above equation by ( ), we can get Let = * in (3) and ( * ) = 0; we have thus, Dividing both sides of the above equation by ( ), we can get Furthermore, let = in (3); we have From (11) we get The Scientific World Journal 3 It follows from the above equation that ) . (21) By applying (18) we have After some manipulations we obtain Now, applying Taylor's expansion for ( ) at the point , we have From (11), (18), and (24), we obtain And after some manipulations we can get By substituting (17), (23), and (26) into (15) we have Note that Then, it follows from (27) that This proves the conclusion of the theorem and the proof is completed.

The n-Dimensional Case
In this section, we consider the -dimensional case of the method, and we also study these iterative methods' order of convergence. Consider the system of nonlinear equations The Scientific World Journal where each function ( 1 , 2 , . . . , ), = 1, 2, . . . , , maps a vector = ( 1 , 2 , . . . , ) of the -dimensional space into the real line . The system (30) of nonlinear equations in unknowns can also be represented by defining a function mapping into as ( ) = ( 1 ( ), 2 ( ), . . . , ( )) . Thus, the system (30) can be written in the form Let : ⊆ → be a sufficiently differentiable function on a convex set ⊆ and let * be a real zero of the nonlinear mapping ( ); that is, ( * ) = 0. For any , ∈ , we may write Taylor's expansion for as follows (see [17]): for = 1, we have If we estimate ( + ( − )) in the interval [0, 1] by its value in = 0, that is, by ( ), then we have By using (34) in (33), we have We can get an iterative method from (35) to solve the system (31); it is known as Newton's method The method (36) has already been proved that it has quadratic convergence. Now we begin to deduce a higher order iterative method. If we estimate ( + ( − )) in the interval [0, 1] by its value in = 1, that is, by ( ), then we have By using (37) in (33), and cutting off the error [ ( )] (still use "="), we have Let +1 be the solution of (38); we can obtain a new iterative method On the other hand, from the easy Newton's method (see [4]), we obtain Now we consider the convex combination of (39) and (40). Let ≥ 0, ≥ 0, and + = 1; then we can deduce from (39) and (40) that the following iterative formula holds: Since the iterative method (41) is implicit-type method, we use Newton's method as predictor and then use the new method (41) as corrector; in this way, we can get a workable iterative method.

Remark 4.
If we take = 0, = 1, our algorithms (42) and (43) can be written in the following form: Notice that, at each iteration, the number of functional evaluations is 2 2 + 2 .
In this section, we consider the convergence and convergent rate of Algorithm 1. We obtain a convergence theorem as follows.
The Scientific World Journal 5

(52)
By applying (50) we have After some manipulations we obtain The Scientific World Journal Now, applying Taylor's formula for ( ) at the point , we have From (42), (50), and (55), we obtain And after some manipulations we can get Therefore, we have From (49) and (54), we get By substituting (57), (58), (59), and (54) into (47) we have that is, The Scientific World Journal 7 Equation (61) can be written as follows: which proves the conclusion of our theorem. The proof is completed.

Numerical Examples
In this section we present some examples to illustrate the efficiency and the performance of the newly developed method (42)-(43) (present study HM). This new method was compared with Newton's method (NM), the method of Aslam Noor and Waseem [12] (NR1), the method of Cordero et al. [13] (NAd1), the method of Darvishi and Barati [4] (DV), and the method of Cordero and Torregrosa [10] (CT) in the number of iterations, CPU time, error, and convergence order. All computations were done using the PC with Pentium(R) Dual-Core CPU T4400 @2.20 GHz. All the programming is implemented in MATLAB 7.9. The convergence order is computed approximately by the following formula: As the iterative formula (43) contains parameters and , we make the numerical examples based on = 1 and = 0. Example 1. The test function is as follows (see [10]): This problem has a solution * = 2. We test this problem by using 0 = 4 as a starting point. The test results are listed in Table 1.
The test function is as follows (see [9,10,13,14]): This problem has a solution * = (0, 0) . We test this problem by using initial value 0 = (0.3, −0.3) as a starting point. The test results are listed in Table 2.
Example 3. The test function is as follows (see [4]): We test this problem by using 0 = (0.5, 0.5, 0.5) . The test results are listed in Table 3.

Conclusion
From the seven examples in Section 4, we can see that the newly developed method (42)-(43) has the advantages of fast convergence speed (we can get from the CPU time), small number of iterations. Especially, the value of convergence order that appears in Tables 2-7 is the highest compared to the other four methods. Although our method's convergence order is not always higher than the method of Cordero and Martínez and Torregrosa (NAd1), ours is superior in the number of iterations and CPU time to the other four methods. In a word, our method (42)-(43) is quite robust and effective.