Abstract
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.
1. 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. Considerwhere is a nonlinear operator defined on a convex subset 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 asThis method and Newton-like methods have been studied well by many authors (see [1–12]).
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.
Let denote the space of the bounded linear maps from to . If the following equality holds, then, we call the operator , at the points and , a divided difference of order one of the nonlinear operator .
By the above definition, secant method can be generalized to Banach spaces, it is described as the following scheme:
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). Denote with an open ball with center and radius . The open ball is called a convergence ball of an iteration, if the sequence generated by the iterative method converges with any initial value in the ball. Under the assumption that the nonlinear operator has Fréchet derivatives satisfying the Hölder condition, Ren and Wu [13] have given the radius of the convergence ball which is .
The convergence ball, the semilocal convergence of secant method, and secant-like method have been studied by many other authors (see [13–18]). 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, is called the relaxed parameter. When , 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 of the relaxed method is proved to be when ; and the radius of the relaxed method is proved to be when . The error estimate is also given.
2. Convergence Ball
Theorem 1. Suppose , where the nonlinear operator is Fréchet differentiable on , exists, the Lipschitz condition (8) holds, and . DenoteWhen , starting from any two initial points in ball , the sequence generated by the relaxed secant method (7) converges to the solution . When , the sequence generated by the relaxed secant method (7) converges to the solution , with any two initial points in ball . is the unique solution in ball , that is bigger than ball and ball . Moreover, we have the following error estimate:where .
Proof. We will prove the above theorem by induction. Firstly, when , by Lipschitz condition, it is easy to get By Banach lemma, we can know is invertible. Since is well defined and we can conductThen, we can give the estimate of when . From , we have Using Lipschitz condition with (12) and (13), we have Obviously, we haveFrom , together with (15), (16), and , we haveThis means .
Similar to the procession above, when , we can get that By (13) and (18) and Lipschitz condition we can get For , This means that when .
Now, suppose is well defined, , when ; is well defined, , when . Similar to the argumentation about and , when , By the Banach lemma, it is obviously known that is invertible. Hence, is well defined. We also get When , And when , we have By the assumptions that when and when , similar to the discussions about , it is known that when and when .
Therefore, starting from any two initial points , , the sequence , generated by the relaxed secant method, is well defined when , , and when , . It means that the following holds:Denote When , from (14) we can get Then, by (27), we have By (23), we know for all . Then by (29) and (30), we can induct Then we can see Obviously, . The sequence converges to the exact solution from (32).
When , from (24), Then, by (27), we have By (24) and (26), we know for all . Then by (26) and (34), when , we can induct So we have It is easy to proof that . So the sequence converges to the solution .
Now we show the uniqueness. Assume that there exists another solution . Consider the operator . Because , we can get if the operator is invertible. From (4), we get So, we can tell that operator is invertible by Banach lemma. From the definition of and (9), it is easy to verify that ball is bigger than ball and ball . Proof completes.
Remark 2. When , the radius of the convergence ball is . We denote . From (9), we know when , , and when , . So we have the biggest convergence ball when .
3. Numerical Examples
In this section, we applied the convergence ball result given in Section 2 to solve some numerical problems.
Example 1. Let us consider Then . has a root and . It is easy to obtain Set , , . Then the radius of the convergence balls is , , . Choose the initial points , and they are in the convergence ball of the relaxed secant method. From Table 1, we can see the sequence converges to with different .
As we know, when , the relaxed secant method reduces to normal secant method. From Table 1, we can see that relaxed secant method in the case of 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 , , and
Similar to the process in [17], we know . Then, For and , we can get So in this problem. Set , , . Then, the radius of the convergence balls is , , . Set the initial points , , and they are in the convergence ball of the relaxed secant method. From Table 2, we can see the sequence converges to the solution .
From Table 2, we can know that the relaxed scant method () 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 such that . We take , , . Then, notice ; it is easy to know F is Fréchet differentiable in and we getLet and . The corresponding norm on is It can be verified easily that 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 . Set , , . Then the radius of the convergence ball is , , . Set the two initial points , 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.
Example 4. Consider the nonlinear conservative system given in [15]:Applying the centered finite difference scheme, we can get the nonlinear system:where is the step-size and is a prescribed positive integer. , are vectors with forms ofand the matrix has the form
Take the same parameters used in [15], , , and the initial points and , . Then, we can solve this problem by our relaxed secant method, and we compare it with normal secant method. For the results, see Table 4.
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 in Table 5.
Competing Interests
The authors declare that they have no competing interests.
Acknowledgments
This work is supported by National Natural Science Foundation of China (Grant nos. 11371320, 11632015), Zhejiang Natural Science Foundation (Grant no. LZ14A010002), and Scientific Research Fund of Zhejiang Provincial Education Department (Grant no. FX2016073).