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

Abstract

We aim to study the convergence properties of a modification of secant iteration methods. We present a new local convergence theorem for the modified secant method, where the derivative of the nonlinear operator satisfies Lipchitz condition. We introduce the convergence ball and error estimate of the modified secant method, respectively. For that, we use a technique based on Fibonacci series. At last, some numerical examples are given.

1. Introduction

A large number of nonlinear dynamic systems and scientific engineering problems can be concluded to the form of nonlinear equationwhere is a nonlinear operator defined on a convex subset of a complex dimension space . 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:where the operator is called a divided difference of first-order for the operator on the points and if the following equality holds:

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 is first-order differentiable convex subset can be open or closed, suppose is the root of the equation , an open area is called the convergence ball of the iterative algorithm. Authors [13–17] have discussed the convergence of the iterative methods using a convergence ball with center and radius . For example, Ren and Wu [15] discussed the convergence of the secant method under Hölder continuous divided differences using a convergence ball.

In this study, we consider the modified secant method with the below form based on [12]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 is the root of the equation and . is first-order differentiable, where the derivative of satisfies the Lipschitz condition: for all and . Then, the sequence generated by the modified secant method (4), starting from any two initial points , converges to the solution . is the unique solution in , where . Moreover, the following error estimate holds:Here, ; is a Fibonacci series, .

Proof. From the condition of Theorem 1, we know . Assume are generated by the modified secant (4) and . Following, we will prove that ; we have is first-order differentiable, so in convex domain , first-order difference of can be written in the following integral form:Now, we give the estimate of ObviouslyandUsing Lipschitz condition with the above (9) and (10), we haveandWe divide above inequality (12) number 2, soAccording to the definition of and , we get , by Banach Lemma, so is reversible and alsoDividing (14) inequality number 2, we can getand with (11) and (15), we get following estimate formula:and with (6) and (16) and , we get following estimate formula:This means that . So, from any , the sequence of the modified secant method is convergent, the root , and by mathematical induction .
In the following, we will derive the estimate of the modified secant method. Denote ; from the above proof we can get ; from inequality (17), it is known that andHence, ; moreover, we have so, we obtain the inequality Now, we use mathematical induction to proof that the inequality is correct. Suppose the inequality is correct when ; here is Fibonacci sequence, . So, when , we have That means the inequality has been proved. SoFrom the definition of and above formulation, we can getAt last, we show the uniqueness of the solution in the area . Assume that there exists another solution , . We consider the operator . Since , if operator is invertible, then . Indeed from (24), we have Then, by Banach lemma, we can tell that operator is invertible. From the definition of radius , it is easy to verify that the ball is bigger than .
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 It is obviously that . has a root and . It is easy to know According to Theorem 1, we can obtain the fact that the radius of the convergence ball of the modified secant method is at least.

Example 2. Let us consider the following numerical problem which has been studied in [4, 11, 13]: , and
We know ; hence, So in this problem.
By Theorem 1, we can obtain the fact that the radius of the convergence ball of the modified secant method is at least.

Example 3. Let us consider the nonlinear equationHere, , and .
We know that ; then it is obvious thatIn this case, the radius of the convergence ball of the modified secant method is at least, according to Theorem 1.

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).