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

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 themodified secantmethod, respectively. For that, we use a technique based on Fibonacci series. At last, some numerical examples are given.


Introduction
A large number of nonlinear dynamic systems and scientific engineering problems can be concluded to the form of nonlinear equation where 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][4][5][6][7][8][9][10][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 firstorder differentiable convex subset can be open or closed, suppose * is the root of the equation ( ) = 0, an open area ( * , ) is called the convergence ball of the iterative algorithm. Authors [13][14][15][16][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.
That completes the proof of Theorem 1.

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

Example 1. Let us consider
It is obviously that ( ) = 2 . ( ) = 0 has a root * = 1 and ( * ) = 2. 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 = 2/5 = 2/5 = 0.4 at least. Advances in Mathematical Physics 5 Example 2. Let us consider the following numerical problem which has been studied in [4,11,13]: 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 = 2/5 = 2/5 ≈ 0.1472 at least.

Data Availability
No data were used to support this study.

Conflicts of Interest
The authors declare that they have no conflicts of interest.