The Convergence Ball and Error Analysis of the Relaxed Secant Method

A relaxed secant method is proposed. Radius estimate of the convergence ball of the relaxed secant method is attained for the nonlinear equation systems with Lipschitz continuous divided differences of first order. The error estimate is also established with matched convergence order. From the radius and error estimate, the relation between the radius and the speed of convergence is discussed with parameter. At last, some numerical examples are given.


Introduction
Many scientific problems can be concluded to the form of nonlinear systems.Finding the solutions of nonlinear systems is widely required in both mathematical physics and nonlinear dynamical systems.In this paper, we will establish the convergence ball and error analysis of the relaxed secant method of nonlinear systems.Consider where  is a nonlinear operator defined on a convex subset Ω of a Banach space  with values in another Banach space .
When  is nonlinear, iterative methods are generally adopted to solve the system: The most widely used iterative method is Newton's method which can be described as  +1 =   −  (  ) −1  (  ) ,  0 is given.
This method and Newton-like methods have been studied well by many authors (see [1][2][3][4][5][6][7][8][9][10][11][12]).Newton's method requires that  is differentiable.Thus, when  is nondifferentiable, Newton method cannot be applied on it.We have to turn to other methods that do not need to evaluate derivatives.In their algorithms, instead of derivatives, divided differences are always used.The classical method of this type is the secant method.
By the above definition, secant method can be generalized to Banach spaces, it is described as the following scheme: ( > 0)  0 ,  −1 ∈ Ω. ( An interesting issue here is to estimate the radius of the convergence ball of an iterative method.Suppose  * is a solution of the nonlinear system (1) Ren and Wu [13] have given the radius of the convergence ball which is   =  √(1 + )/(1 + 2).The convergence ball, the semilocal convergence of secant method, and secant-like method have been studied by many other authors (see [13][14][15][16][17][18]). In this paper, similar to the relaxed Newton's method in [7], we considered the relaxed secant method which can be written as the following form: here,  ∈ (0, 2) is called the relaxed parameter.When  = 1, it will be the normal secant method.
In this paper, we will study the convergence ball of (7) under the assumption that the nonlinear operator  has Fréchet derivatives satisfying the following Lipschitz condition: Under the Lipschitz condition, the radius   1 of the relaxed method is proved to be /4 when 0 <  ⩽ 1; and the radius   2 of the relaxed method is proved to be (2 − )/4 when 1 <  < 2. The error estimate is also given.

Numerical Examples
In this section, we applied the convergence ball result given in Section 2 to solve some numerical problems.
As we know, when  = 1, the relaxed secant method reduces to normal secant method.From Table 1, we can see that relaxed secant method in the case of  = 1.1 outperforms the normal secant method in the sense of iteration number and CPU time.
Example 2. Let us consider the following numerical problem which has been studied in [3,17,18]: Then   () =   ,  * = 0, and   ( * ) = 1.Similar to the process in [17], we know |  −  | ⩽ |−|.Then,          ∫  From Table 2, we can know that the relaxed scant method ( = 0.999) performs the same as the normal secant method in the sense of the iteration number and CPU time, while the solution gotten by the relaxed secant method is closer to the exact solution than that by the normal secant method.
Example 3. Let us consider the nonlinear system: It comes from the following nonlinear boundary value problem of second order: which has been studied by many authors [5,13,16].Now, define the operator  :  2 →  2 such that  = ( 1 ,  2 ).We take ) .
( It can be verified easily that  * = (9, 9) is a solution of (24) and from (26) we get () is invertible.Similar to [13], we can deduce that Lipschitz continuous condition is satisfied for  = 1/9.Set  1 = 0.9999,  2 = 1,  3 = 1.06.Then the radius of the convergence ball is  1 = 2.499975,  2 = 9/4,  3 = 326/212.Set the two initial points  −1 = (9.5, 9.5),  0 = (8.5, 8.5) and they are in the convergence ball.For results, see Table 3. Table 3 shows the sequence {  } generated by the relaxed secant method.From this table, it is known that the sequence {  } converges, and also the error estimation holds.Moreover, relaxed secant method has more choices than secant method, and optimal parameter  makes the presented method outperforms the normal secant method.
From the results, we can know that, in this example, the relaxed secant method performs better.And we list the approximation solution which is gotten by the relaxed secant method in the situation  = 0.99 in Table 5.

Table 1 :
Relaxed secant method with different .

Table 3 :
Relaxed secant method with different .
, () are vectors with forms of

Table 4 :
Numerical results for nonlinear conservative systems.