A NEW ROOT-FINDING ALGORITHM USING EXPONENTIAL SERIES

In this paper, we present a new root-finding algorithm to compute a non-zero real root of the transcendental equations using exponential series. Indeed, the new proposed algorithm is based on the exponential series and in which Secant method is special case. The proposed algorithm produces better approximate root than bisection method, regula-falsi method, Newton-Raphson method and secant method. The implementation of the proposed algorithm in Matlab and Maple also presented. Certain numerical examples are presented to validate the efficiency of the proposed algorithm. This algorithm will help to implement in the commercial package for finding a real root of a given transcendental equation.


Introduction
The root finding algorithms are the most relevant computational problems in science, engineering. The applications of root finding algorithms arise in many practical applications of Biosciences, Physics, Engineering, Chemistry etc. As mentioned in [1], the finding of any unknown appearing implicitly in engineering or scientific formulas, gives rise to root finding problem. A root of a function f (x) is a number 'α' such that f (α) = 0. Generally, the roots of transcendental functions cannot be expressed in closed form or cannot be computed exactly. The root-finding algorithms give us approximations to the roots, these approximations are expressed either as small isolating intervals or as floating point numbers. In the literature, there are several root finding algorithms are available, see for example, [1][2][3][4][5][6][7][8][9][10][11]. The basic root-finding methods are Bisection, False position, Newton-Raphson, Secant methods etc. Most of the algorithms use iteration, producing a sequence of numbers that hopefully converge towards the root as a limit. The rates of converge of different algorithms are different. That is, some algorithms are faster converges to the root than others algorithms. The purpose of existing algorithms is to provide higher order convergence with guaranteed root. Many existing algorithms do not guarantee that they will find all the roots; in particular, if such an algorithm does not find any root, that does not mean that no root exists. There are many well known root finding algorithms available to find an approximate root of algebraic or transcendental equations., see for example, [1,5,6,8,9,11].
In this work, we propose a new algorithm based on exponential series and secant method becomes a special case of the proposed algorithm. This algorithm provides faster roots in comparison of the previous methods. The new proposed algorithm will be useful for computing a real root of transcendental equations. The rest of the paper is as follows: Section 1 describes the proposed method, their mathematical formulation, calculation steps and flow chart; implementation of the proposed algorithm in Maple is presented in Section 2 with sample computations; and Section 3 discuss some numerical examples to illustrate the algorithm and comparisons are made to show efficiency of the new algorithm.

New algorithm using exponential series
The new iterative formula using exponential series is proposed as follows, for any two initial approximations x 0 , x 1 of the root, By expanding this iterative formula, one can obtain the standard secant method as in first two terms, and many methods are obtained based on series truncation. Indeed, This is shown in the following theorem.
Using the standard expansion of e x as exp(x) = 1 + x + 1 2 and from equations (1.1) and (1.5), we have Since f (x n ) ≈ 0, when we neglect higher order terms, then the above equation gives secant method in first two terms. Indeed, we have the following formulae obtained from first two terms, three terms and four terms of the expansion respectively as given in equations (1.2)-(1.4).
In the above equations, we obtained secant method having (1 +  II. Compute f (x 0 ) and f (x 1 ).
III. Compute the next approximate root using formula given in (1.1).

IV. Repeat
Step II and III until we get desired approximate root.
Flow chat of the proposed algorithm is presented in Figure 1.

Proposed algorithm in MATLAB
In this section present MATLAB implementation of the proposed algorithm as follows.

Proposed algorithm in MAPLE
In this section, we provide the implementation of the proposed method in Maple as follows. To execute the maple code, one should enter the required date at type text. Sample computations using the implementation of the proposed algorithm are presented in Section 3.

Numerical examples
This section provides some numerical examples to discuss the algorithm presented in Section 1 and comparisons are taken into account to conform that the algorithm is more efficient than other existing methods.
Example 3.1. Consider an equation This equation has two real roots −0.7780895987 and 1.134724138. The following Table 1 shows the comparison between various existing methods and proposed method at accurate to within ǫ = 0.00001 with initial approximations x 0 = 1 and x 1 = 1.5. One can observe from the Table 1 that the proposed algorithm gives approximate root quicker than the other existing methods.
Example 3.2. Consider the following transcendental equations. We compare the number of iterations required to get approximation root. The numerical results are provided in Table 2.
The numerical results given in Table 2 shows that the proposed method is more efficient than other methods. f (x) = x 6 − x − 1 One can use the implementation of the proposed algorithm to speed up the manual calculations.

Conclusion
In this work, we presented a new algorithm to compute an approximate root of a given transcendental function better than previous existing methods as illustrated. The proposed new algorithm was based on exponential series having better convergence than previous existing methods (for example, Bisection, Regula-Falsi, Secant methods etc.). This proposed algorithm is useful for solving the complex real life problems. Implementation of the proposed algorithm in Matlab and Maple is also discussed presented sample computations.