Palestine Journal of Mathematics Vol. 10(1)(2021) , 128–134 © Palestine Polytechnic University-PPU 2021 NEW ALGORITHMS FOR COMPUTING A ROOT OF NON-LINEAR EQUATIONS USING EXPONENTIAL SERIES Srinivasarao Thota, Tekle Gemechu and P. Shanmugasundaram Communicated by Suheel Khoury (S. Khuri) MSC 2010 Classifications: Primary 65Hxx; Secondary 65H04. Keywords and phrases: Root finding algorithms, Non-linear equations, Exponential series, Real root. The authors are thankful to the editor and reviewer for providing valuable inputs to improve the present format of manuscript. Abstract In this paper, we present new algorithms/methods to find a non-zero real root of the transcendental equations using exponential series. The new proposed method is based on the exponential series, which produces better approximate root than some existing methods. MATLAB and Maple implementation of the proposed method is discussed. Certain numerical examples are presented to validate the efficiency of the proposed algorithm. The method will help to implement in the commercial package for finding a real root of a given transcendental equation. 1 Introduction The non-linear problems solving in science, engineering and computing are playing important role to compute roots of transcendental equations. 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 analytically. The root-finding algorithms provide us approx- imations to the roots, these approximations are expressed either as small isolating intervals or as floating point numbers. Most of the algorithms in the literature use iteration, producing a sequence of numbers that hopefully converge towards the root as a limit. They need one or more initial guesses of the root as starting values, then each new iteration of the method produces a successively more accurate approximate root in comparison to previous iteration. The purpose of existing methods is to provide higher order convergence with guaranteed root. The existing methods may 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, (for example, Bisection, Secant, Regula-Falsi, Newton-Raphson, Muller’s methods etc.) to find an approximate root of algebraic or transcendental equations, see for example [1, 2, 4–7, 9–12, 14–22, 25–27]. If the equation f (x)= 0 is an algebraic equation, then there are many algebraic formulae available to find the roots. However, if f (x) is a polyno- mial of higher degree or an expression involving transcendental equations such as trigonometric, exponential, algorithmic etc., then there are no algebraic methods exist to express the root. In this work, the proposed new methods are based on exponential series, which provides faster roots in comparison with existing algorithms. The new proposed algorithms will be useful for computing a real root of transcendental equations. The Newton-Raphson method can be derived as a special case of the proposed method, so we select a non-zero initial approximation a for a given transcendental function f (x) such that f 0 (a) 6= 0. The rest of the paper is as follows: Section 2 describes the proposed method, their mathemat- ical formulation, calculation steps and flow chart; implementation of the proposed algorithm in Maple is presented in Section 3 with sample computations; and Section 4 discuss some numeri- cal examples to illustrate the algorithm and comparisons are made to show efficiency of the new algorithm.