Quadratic Convergence Iterative Algorithms of Taylor Series for Solving Nonlinear Equations

Solving the root of algebraic and transcendental nonlinear equation f 0 ( x ) = 0 is a classical problem which has many interesting applications in computational mathematics and various branches of science and engineering. This paper examines the quadratic convergence iterative algorithms for solving a single root nonlinear equation which depends on the Taylor’s series and backward diﬀerence method. It is shown that the proposed iterative algorithms converge quadratically. In order to justify the results and graphs of quadratic convergence iterative algorithms, C++/MATLAB and EXCELL are used. The eﬃciency of the proposed iterative algorithms in comparison with Newton Raphson method and Steﬀensen method is illustrated via examples. Newton Raphson method fails if f 0 ( x ) = 0 , whereas Steﬀensen method fails if the initial guess is not close enough to the actual solution. Furthermore, there are several other numerical methods which contain drawbacks and possess large number of evolution; however, the developed iterated algorithms are good in these conditions. It is found out that the quadratic convergence iterative algorithms are good achievement in the ﬁeld of research for computing a single root of nonlinear equations.


Introduction
T he estimation of a single root of nonlinear equations leads to wide range of applications in the field of numerical analysis, which emerges in applied sciences and engineering, for instance: distance, rate, time problems, population change and trajectory of a ball [1][2] [3]. We consider the following nonlinear equation.
f (x) = 0 (1.1) To estimate the roots of Equation 1.1, several approaches have been established, see for instance [4] [5]. Among them, the Taylor series is an important and basic concept: By using Taylor series, one of the most important technique Newton Raphson's method has been developed [6][7] [8] which is given as follows.
By using Newton forward difference, we get Steffensen method [9]: Moreover, to improve the efficiency and order of convergence, several approaches have been developed, see for instance [10] [11], given as follows.
In addition, this study suggests Algorithm 3.1 and Algorithm 3.2 of Taylor series and backward difference method for computing a root of nonlinear equations. The proposed algorithms have second order of convergence. Some illustrations are given to express the applicability of the proposed iterative algorithms. Quadratic convergence iterative algorithms are compared with Newton Raphson method and Steffensen method. In literature, it is observed that there are various numerical techniques which have crucial weakness, for example, if the initial guess is chosen far from the actual solution. Additionally, such techniques also take large number of evolutions.
This study aims to work on these types of conditions and ascertain that the proposed iterated algorithms are decent attainment in these conditions in comparison with Newton Raphson method and Steffensen method for estimating a root of nonlinear equations.

Iterated Method
We consider the following nonlinear equation.
To generalize Equation 2.2 with f (x r ) = 0 and taking only three terms, we obtain: By using backward difference operator, we obtain

By substituting Equation 2.3 into equation (2.4), we get
This implies that This equality proposes the iterative methods mentioned below in order to solve the nonlinear equations under certain conditions.

New Iterative Methods
Suppose f : X → R is a scalar function of Taylor series expansion, we get Equation 2.6.

Algorthim 3.1
For certain x 0 , the following iterative method is considered to compute the approximate solution x n+1 ,

Algorithm 3.2
For a given x 0 , the iterative schemes given below are used to approximate the solution x n+1 : 2f (xn) By using above condition in Equation 2.6, we get

Rate of Convergence
In this section, the order of convergence for Algorithm 3.1 and Algorithm 3.2 will be proved as quadratic with the help of theorems.

Theorem 4.1
then the convergence of Algorithm 3.1 is of order two.

Proof of the concept
To examine the convergence of Algorithm 3.1, we suppose x is a root of f (x) = 0 and e n be the error at the n th iteration, then e n = x n − a. By expand-

Theorem 4.2.
Assume x is a root of f (x) = 0. If f (x) is neighborhood of x, then the convergence of Algorithm 3.1 is of order two.

Proof of the Concept
To investigate the convergence of Algorithm 3.1, suppose x is a root of f (x) = 0, and e n be the error at n th iteration, then e n = x n − a, then by expanding and f x n − f (xn) 2f (xn) from Taylor's Series about a point a, we have f (x n ) = f (a) (e n + ce 2 n ) (4.7) f (x n ) = f (a) (1 + 2ce n + 3ce 2 n ) (4.8) From equation (4.7) and equation (4.8), we have From equation (4.9), we get 2f (xn) and again using Taylor's series about a with the help of Equation 4.10, we have .13 shows that the Algorithm 3.2 is quadratically convergent. Hence, it is proved that the convergence order for Algorithm 3.1 and Algorithm 3.2 is 2.

Numerical Results
This section discusses the following nonlinear equations to demonstrate the efficiency of proposed Algorithm 3.1 and Algorithm 3.2 by equating with the Newton Raphson method and Steffensen method Following tables and graphs depict the efficiency of proposed Algorithms.
The results are obtained with the help of C++/MATLAB and EXCEL. All tables and Figure  1-4 exhibit the numerical comparisons of the proposed Algorithm 3.1 and 3.2 with Newton Raphson method and Steffensen method.

Conclusion
Root calculating problem of algebraic and transcendental nonlinear equations occurs frequently in applied mathematics and scientific computing. In this study, the new methods are established for solving algebraic and transcendental nonlinear equations. The proposed methods are based on the Taylor's series and backward difference method. It turned out that the proposed numerical techniques are quadratically convergent. The efficiency of the proposed methods is illustrated by a few examples of nonlinear equations and is compared with the existing numerical methods such as Newton Raphson method and Steffensen method. The well-known software C++/MATLAB and EXCEL are used to compute the numerical results. The numerical outcomes have shown that the proposed algorithms perform well and are easily applicable to the nonlinear equations than the existing numerical techniques..