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On the Convergence Ball and Error Analysis of the Modified Secant Method.

1. Introduction

A large number of nonlinear dynamic systems and scientific engineering problems can be concluded to the form of nonlinear equation

f(x) = 0, (1)

where f is a nonlinear operator defined on a convex subset D of a complex dimension space C. Hence, finding the roots of the nonlinear (1) is widely required in both mathematical physics and nonlinear dynamic system. Iterative methods are considerable methods. There are many iterative methods for solving the nonlinear equation.

Secant method [1, 2], which uses divided differences instead of the first derivative of the nonlinear operator, is one of the most famous iterative methods for solving the nonlinear equation. Secant method reads as follows:

[x.sub.n+1] = [[x.sub.n] - [[x.sub.n-1], [x.sub.n]; f].sup.-1] f([x.sub.n]), (n [greater than or equal to] 0) ([x.sub.0], [x.sub.-1] [member of] D) (2)

where the operator [x, y; f] is called a divided difference of first-order for the operator f on the points x and y(x [not equal to] y) if the following equality holds:

[x, y; f] (x - y) = f(x) - f(y). (3)

Due to the well performance of the secant method, secant method and secant-like methods have been widely studied by many authors [3-11]. The authors [12] proposed a new method for solving the nonlinear equation.

Convergence ball is a very important issue in the study of the iterative procedures. When nonlinear operator f is first-order differentiable convex subset D can be open or closed, suppose [x.sub.*] is the root of the equation f(x) = 0, an open area B([x.sub.*], R) is called the convergence ball of the iterative algorithm. Authors [13-17] have discussed the convergence of the iterative methods using a convergence ball B([x.sub.*], R) with center [x.sub.*] and radius R. For example, Ren and Wu [15] discussed the convergence of the secant method under Holder continuous divided differences using a convergence ball.

In this study, we consider the modified secant method with the below form based on [12]

[x.sub.n+1] = [x.sub.n] - f([x.sub.n]) ([x.sub.n] - [x.sub.n-1])/ 3 f ([x.sub.n]) + f ([x.sub.n-1]) - 4 f (([x.sub.n] + [x.sub.n-1])/2 (n [greater than or equal to] 0), (4)

and we will establish the convergence ball and give the error analysis of the modified secant method for the nonlinear equation.

2. Convergence Ball Study

Theorem 1. Suppose [x.sub.*] is the root of the equation f(x) = 0 and f'([x.sub.*]) [not equal to] 0. f is first-order differentiable, where the derivative of f satisfies the Lipschitz condition: [absolute value of (f'[([x.sub.*]).sup.-1] (f'(x) - f'(y)))] [less than or equal to] K[absolute value of (x - y)] for all x, y [member of] D and K > 0. Then, the sequence {[x.sub.n]} generated by the modified secant method (4), starting from any two initial points [x.sub.0], [x.sub.-1] [membe rof] B([x.sub.*], R), converges to the solution [x.sub.*]. [x.sub.*] is the unique solution in B([x.sub.*], 2/ K), where B([x.sub.*], R) [subset] B([x.sub.*], 2/K). Moreover, the following error estimate holds:

[mathematical expression not reproducible] (15)

Here, R = 2/5K; {[F.sub.n]} is a Fibonacci series, [F.sub.0] = [F.sub.1] = 1, [F.sub.n+1] = [F.sub.n] + [F.sub.n-1], (n [greater than or equal to] 1).

Proof. From the condition of Theorem 1, we know [x.sub.0], [x.sub.-1] [member of] B([x.sub.*], R). Assume [x.sub.1], [x.sub.2], ..., [x.sub.n] are generated by the modified secant (4) and [x.sub.k] [member of] B([x.sub.*], R)(0 [less than or equal to] k [less than or equal to] n). Following, we will prove that [x.sub.n+1] [member of] B([x.sub.*], R); we have

[mathematical expression not reproducible] (6)

f is first-order differentiable, so in convex domain D, first-order difference of f can be written in the following integral form:

f[x, y] = f(x) - f(y)/x - y = [[integral].sup.1.sub.0] f' (tx + (1 - t)y) dt (x, y [member of] D). (7)

Now, we give the estimate of

[mathematical expression not reproducible] (8)

Obviously

[mathematical expression not reproducible] (9)

and

[mathematical expression not reproducible] (10)

Using Lipschitz condition with the above (9) and (10), we have

[mathematical expression not reproducible] (11)

and

[mathematical expression not reproducible] (12)

We divide above inequality (12) number 2, so

[mathematical expression not reproducible] (13)

According to the definition of R and [x.sub.n], [x.sub.n-1] [member of] B([x.sub.*], R), we get K[absolute value of ([x.sub.*] - [x.sub.n])] + (K/2)[absolute value of ([x.sub.*] - [x.sub.n-1])] < 3/5 < 1, by Banach Lemma, so 3 f[[x.sub.n], ([x.sub.n] + [x.sub.n-1])/2] - f[[x.sub.n-1], ([x.sub.n] + [x.sub.n-1])/2] is reversible and also

[mathematical expression not reproducible] (14)

Dividing (14) inequality number 2, we can get

[mathematical expression not reproducible] (15)

and with (11) and (15), we get following estimate formula:

[mathematical expression not reproducible] (16)

and with (6) and (16) and [x.sub.n], [x.sub.n-1] [member of] B([x.sub.*], R), we get following estimate formula:

[mathematical expression not reproducible] (17)

This means that [x.sub.n+1] [member of] B([x.sub.*], R). So, from any [x.sub.0], [x.sub.-1] [member of] B([x.sub.*], R), the sequence {[x.sub.n]} of the modified secant method is convergent, the root [x.sub.*] [member of] B([x.sub.*], R), and by mathematical induction [x.sub.n] [member of] B([x.sub.*], R)(n [greater than or equal to] 1).

In the following, we will derive the estimate of the modified secant method. Denote [e.sub.n] = [x.sub.*] - [x.sub.n], [[rho].sub.n] = [absolute value of ([e.sub.n])]/R, (n [greater than or equal to] -1); from the above proof we can get 0 [less than or equal to] [[rho].sub.n] < 1(n [greater than or equal to] -1); from inequality (17), it is known that [absolute value of ([e.sub.n+1])] [less than or equal to] [absolute value of ([e.sub.n])] and

[mathematical expression not reproducible] (18)

Hence, [absolute value of ([[rho].sub.n+1])] [less than or equal to] [absolute value of ([[rho].sub.n])], (n [greater than or equal to] -1); moreover, we have

[mathematical expression not reproducible] (19)

so, we obtain the inequality

[[rho].sub.n] [less than or equal to] [[rho].sub.n-1] [[rho].sub.n-2], (n [greater than or equal to] 1). (20)

Now, we use mathematical induction to proof that the inequality [mathematical expression not reproducible] is correct.

[mathematical expression not reproducible] (21)

Suppose the inequality [mathematical expression not reproducible], (k [greater than or equal to] 1) is correct when 3 [less than or equal to] k [less than or equal to] n - 1; here {[F.sub.k]} is Fibonacci sequence, [F.sub.0] = [F.sub.1] = 1, [F.sub.k+1] = [F.sub.k] + [F.sub.k-1], (k [greater than or equal to] 1). So, when k = n, we have

[mathematical expression not reproducible]. (22)

That means the inequality [mathematical expression not reproducible],(k [greater than or equal to] 1) has been proved. So

[mathematical expression not reproducible]. (23)

From the definition of [[rho].sub.n] and above formulation, we can get

[mathematical expression not reproducible] (24)

At last, we show the uniqueness of the solution in the area B([x.sub.*], 2/K). Assume that there exists another solution [y.sub.*] [member of] B([x.sub.*], 2/K), [y.sub.*] [not equal to] [x.sub.*]. We consider the operator A = [[integral].sup.1.sub.0] f'([tx.sub.*] + (1 - t) [y.sub.*])dt. Since A[[y.sub.*] - [x.sub.*]] = f([y.sub.*]) - f([x.sub.*]) = 0, if operator A is invertible, then [y.sub.*] = [x.sub.*]. Indeed from (24), we have

[mathematical expression not reproducible] (25)

Then, by Banach lemma, we can tell that operator A is invertible. From the definition of radius R, it is easy to verify that the ball B([x.sub.*], 2/K) is bigger than B([x.sub.*], R).

That completes the proof of Theorem 1.

3. Numerical Examples

In this section, the convergence ball results were applied to numerical examples.

Example 1. Let us consider

f (x) = [x.sup.2] - 1, x [member of] [0, 2]. (26)

It is obviously that f'(x) = 2x. f(x) = 0 has a root [x.sub.*] = 1 and f'([x.sub.*]) = 2. It is easy to know

[absolute value of (f' [([x.sub.*]).sup.-1] (f' (x) - f' (y)))] [less than or equal to] [absolute value of (x - y)]. (27)

According to Theorem 1, we can obtain the fact that the radius of the convergence ball of the modified secant method is R = 2/5K = 2/5 = 0.4 at least.

Example 2. Let us consider the following numerical problem which has been studied in [4, 11, 13]:

f (x) = [e.sup.x] - 1,

D =[-1, 1]. (28)

f'(x) = [e.sup.x], [x.sub.*] = 0, and f'([x.sub.*]) = 1?

We know [absolute value of ([e.sup.x] - [e.sup.y])] [less than or equal to] e[absolute value of (x - y)]; hence,

[absolute value of (f' [([x.sub.*]).sup.-] (f' (x) - f' (y)))] [less than or equal to] e ([absolute value of (x - y)]). (29)

So K = e in this problem.

By Theorem 1, we can obtain the fact that the radius of the convergence ball of the modified secant method is R = 2/5K = 2/5e [approximately equal to] 0.1472 at least.

Example 3. Let us consider the nonlinear equation

f' (x) = 2/5 sin x + x, x [member of] [0, 2], (30)

Here, f' (x) = (2/5) cos x + 1, [x.sub.*] = 0, and f'([x.sub.*]) = 7/5.

We know that [absolute value of (cos x - cos y)] [less than or equal to] [absolute value of (x - y)]; then it is obvious that

[absolute value of (f' [([x.sub.*]).sup.-] (f' (x) - f' (y)))] [less than or equal to] 2/7 ([absolute value of (x - y)]). (31)

In this case, the radius of the convergence ball of the modified secant method is R = 2/5K = 7/5 = 1.4 at least, according to Theorem 1.

https://doi.org/10.1155/2018/2704876

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is supported by National Natural Science Foundation of China (Grant nos. 11771393, 11371320, and 11632015), Zhejiang Natural Science Foundation (Grant nos. LZ14A010002, LQ18A010008), Scientific Research Fund of Zhejiang Provincial Education Department (Grant no. FX2016073), and the Science Foundation of Taizhou University (Grant no. 2017PY028).

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Rongfei Lin (iD), (1) Qingbiao Wu (iD), (2) Minhong Chen, (3) and Xuemin Lei (2)

(1) Department of Mathematics, Taizhou University, Linhai 317000, Zhejiang, China

(2) Department of Mathematics, Zhejiang University, Hangzhou 310027, Zhejiang, China

(3) Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou 310012, Zhejiang, China

Correspondence should be addressed to Qingbiao Wu; qbwu@zju.edu.cn

Received 21 January 2018; Revised 14 March 2018; Accepted 11 April 2018; Published 2 July 2018

Academic Editor: Kaliyaperumal Nakkeeran
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Title Annotation:Research Article
Author:Lin, Rongfei; Wu, Qingbiao; Chen, Minhong; Lei, Xuemin
Publication:Advances in Mathematical Physics
Article Type:Report
Geographic Code:1USA
Date:Jan 1, 2018
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