1. Introduction

There are cases where thousands of nonlinear equations depending on some independent variables must be solved effectively. Consider the nonlinear system

(1)

where , that is, a system with equations and unknowns. Finding roots of systems of nonlinear equations efficiently is of major importance and has widespread applications in numerical and applied mathematics. There are many approaches to solve (1). The second order Newton's method is one of the most common iterative methods adopted for finding approximate solutions of nonlinear system (1) (for more details see[1]). The Newton's method to solve (1) is an important and basic method which converges quadratically. Recently, some third and fourth order iterative methods have been proposed and analyzed for solving systems of nonlinear equations that improve some classical methods such as the Newton's method and Chebyshev-Halley methods. It has been demonstrated that the methods are efficient, and can compete with Newton's method, for more details see[2-10]. To solve system (1), Darvishi and Barati[11-13] presented some high order iterative methods, free from second order derivative of function . Darvishi[14,15] presented some multi-step iterative methods, free from second order derivative of function . Frontini and Sormani[16] obtained a third-order method based on a quadrature formula to solve systems of nonlinear equations. Babajee et al.[17] proposed a fourth order iterative method to solve system (1).

One-step SOR-Newton method to solve nonlinear system (1) which presented in[1] is as follows:

where , is the relaxation parameter in SOR method and

It is obvious that we have

The aim of this paper is to introduce the SOR-SN algorothm. To achieve this, we follow the two-step Steffensen type method which presented by Ren et al[18]. They have presented a class of one-parameter Steffensen type methods with fourth-order convergence to solve nonlinear equations

(2)

where f is a real function. We extend their method to solve systems of nonlinear equations and then combine the new method and SOR-Newton method to obtain the mixed method SOR-SN to solve (1). The convergence analysis of method will be presented. Also, we compare CPU time and number of iterations of SOR-SN method and another SOR methods to solve some nonlinear systems.

2. SOR- SN Algorithm

In this section we introduce the SOR-SN algorithm, to solve systems of nonlinear equations. The SOR-SN method to solve for

is defined by solving the following equation for

(3)

We set

Then we obtain from (3) as follows

where

whereas Finally, we set

or

(4)

2.1. Convergence Analysis

In this part, we state and prove convergence theorem for our SOR-SN method. The proof is similar to proof of convergency of SOR algorithm in[19]. First, to follow[19], we need some assumptions. These assumptions are as follows

a) D is a convex set in .

b) For system , is gradient of , namely, and function is strictly convex.

c) On domain , function is twice continuously differentiable.

d) Set for some is nonempty and compact. Note that exists and set is nonempty and compact.

e) for and , unless is the point at which attains its minimum, where for is Hessian matrix of at .

Assumptions a) and b) show that the Hessian matrix of is positive definite. It is obvious that sets are convex for all . Assumption d) shows that attains its minimum at some point . If assumptions a), b) and c) hold and function attains its minimum at some point then assumption d) is nontrivially satisfied, that is, there exists such that and is compact also is a non-increasing function. From b) the minimum point is unique. At last, a point is the minimum point of function if and only if .

Now, we can present the following theorem on convergency of SOR-SN method.

Theorem 1. If sequence be generated by

and

where

in fact, we have

(5)

where

Also if ,, be defined by

and be defined by where

and for some which satisfies in

then for any sequence is well-defined and converges to .

Proof. We first prove that sequence is well-defined. By consideration of structure of sets and we have

Note that we have

this shows that . But sequence is non-increasing, thus , from this we have . In a similar manner and by mathematical induction we can prove that all terms of sequence are contained in hence the sequence is well-defined. We now prove that converges. By using Taylor's theorem on function for (where denotes the open line segment joining and ) we have

(6)

Note that in the Steffensen's method we have

thus from (5) we have

(7)

setting (7) in (6) yields

(8)

From the convergence of the Steffensen's method for which satisfies in

we have

Consequently, from this and (8) we have

Thus

where . Since the Hessian matrix of is positive semidefinite, hence . We consider two cases for :

Case 1.

By assumption e) there exists some subsequence converges to and we know that and as is a non-increasing sequence, we have Therefore, by our assumption, as is a continuous function, we have .

Case 2.

We know that satisfies in

we consider two possible cases as follows:

Case 1.

.

We know that the sequence is non-increasing and bounded from below, hence it is a convergent sequence. Then , we suppose that is a limit point of and we consider the following sets and . From (7) we have

from the convergency of the Steffensen's method we have

thus

.

As and is bounded, hence we have

Thus

This shows that is a nonempty set. If be an empty set, hence and as is a non-increasing sequence, . Else if be a nonempty set, thus

For we define the following sets

Clearly sets are closed and nonempty sets, because and , hence .

Also we define . Since is a closed set then there exists such that for , we have Remember that hence, there is an integer such that for we have Let . From the continuity of there exists such that implies that

We define set such that for all the following relations hold

(9)

and

(10)

Note that for any we may select such that if then , otherwise for every sequence which reduces to zero, there exists such that and . All such are contained in which is a compact set. This results that, there exists a limit point of which is such that . We have from (9) that

And also from (10) we have

hence

where denotes the inner product of vectors and . This leads to a contradiction, because is a strictly convex function. Thus there exists such that implies that Also from , there exists such that for we have Also there exists such that hence there is an such that thus . If for , then we have which is a contradiction. Hence for . This shows that for there is not any such that , therefore there is not any in thus is an empty set, which is a contradiction. Thus .

Case 2.

We can state a similar discussion for this case.

3. Numerical Results

In this section we solve some nonlinear systems by SOR-SN method. We compare the CPU time and number of iterations of our method with SOR-Newton(SOR-N), SOR-Steffensen(SOR-St) and SOR-Secant(SOR-Se) methods. Numerical computations have been carried out in MATLAB. The stopping criteria is , where shows the Infinity norm.

Note that in SOR-secant method we need to two initial guesses, namely and , while for another methods we only need to one initial guess. In the following parts, we present some examples to compare these methods:

Example 1 . Consider the following nonlinear system

The exact solution of the above system is

.

For an initial guess we set and also for SOR-Secant method we set . The numerical results are given in Table 1.

Table 1. Results for Example 1 Method err () CPU time (s) Iter. SOR-SN 4.8738 14.087318 21 SOR-N 7.6309 22.913211 65 SOR-St 6.3949 35.742580 65 SOR-Se 6.7503 26.924918 65

Example 2 . We consider the following nonlinear system

Its exact solution is . For an initial guess we set and also for SOR-Secant method we set . The numerical results are given in Table 2.

Table 2. Results for Example 2 Method err () CPU time (s) Iter. SOR-SN 4.2633 6.978867 10 SOR-N 5.6567 15.557740 35 SOR-St 6.4758 24.313022 34 SOR-Se 6.4758 18.594764 34

Example 3. Consider the following system

Its exact solutions are and for different values of . For an initial guess we set and also for SOR-Secant method we set . The numerical results are given in Table 3.

Table 3. Results for Example 3.

Example 4. Consider the following nonlinear system

One of the exact solutions of the system is for different values of . For an initial guess we set also and for SOR-Secant method we set .

The numerical results are given in Table 4.

Table 4. Results for Example 4.

In the following example, we apply the new method to solve a boundary value problem. The stopping criteria used in this case is , where is the th approximation of the solution.

Example 5 . We consider the following boundary value problem, which is given in[3]:

The exact solution of this problem is. To discretize the problem we use the second order finite difference method. The zeros of the following nonlinear functions will provide us an estimation of the solution of the problem:

,where

and for we have , such that

In the above system, the second order approximations are used for and by step . By this step, we set the nodes as:

.

Also, denotes the unknown . In Table 5 some results can be observed. For any initial estimation, we analyze the number of iterations and CPU time needed to converge to the solution. The initial estimations used are:

For SOR-secant in Table 5 we set and in Table 6 we set And in both cases. As we can see from Tables 5 and 6 for all cases the number of iterations and CPU time for our new method are less than number of iterations and CPU time for the other methods.

Table 5. Results for       in Example 5

Table 6. Results for       in Example 5.

4. Conclusions

In this paper, we have presented SOR-SN algorithm to solve systems of nonlinear equations. We have shown that our method is convergent. In comparison with another SOR type methods, such as, SOR-Newton, SOR-Steffensen and SOR-Secant methods, SOR-SN algorithm were better in CPU time and number of iterations.