International Journal of Science and Research (IJSR) ISSN (Online): 2319-7064 Index Copernicus Value (2013): 6.14 | Impact Factor (2014): 5.611 Volume 4 Issue 10, October 2015 www.ijsr.net Licensed Under Creative Commons Attribution CC BY Fastest Iteration Method for Estimation of Solution of Nonlinear Function  =0 (Bennie’s Method) Benedictor Alexander Nguchu Tianjin University of technology and Education, School of Electronics Engineering Abstract: There has been a considerate progress and achievement in development of mathematical models and different formulas of which they are used to describe behavior of different systems. Several Numerical methods were used for estimations of fixed points, roots, and series expansion. However in many cases when roots were evaluated using the fixed point method, the number of iteration required to end process by inferring the final answer was very long, sometimes it took more than 1000 iterations inferring the final answer .Fixed point method has been found not to converge in some functions,  =  thus disclosed failure. In this paper we introduced the fastest method of iteration (Bennie’s iteration) for Estimation of roots of a functions and square roots with zero error- bounds and high convergence speed. Fixed point iteration and slop finding had accelerated the establishment of the algorithm. With this approach mathematicians found easier and straightforward to evaluate roots of a function manually. Moreover the algorithm resolved all problems of which fixed point method found to diverge. In addition to that this paper deduced the newton’s method from the proposed Bennie’s method (scheme). Keywords: Numerical methods, fastest iteration method, high convergence speed, small bound error, fixed point method 1. Introduction The numerical methods are powerful algorithms used for estimation of solutions for many nonlinear single variable systems or functions, but still we have found that to some of functions these methods were unable to work due to theoretical limitations. [1]And therefore mathematicians have been suggesting the alternative methods for high accuracy. This paper was firstly aimed to solve the problem of slow convergence speed revealed in fixed point method. Secondly this paper is intended to find solutions of all functions where the condition    <1 on [a b] is not met for a chosen starting point of  0 ∈   , for which a fixed point ∈ [, ] . Those functions such that   ≥ 1 , (for which fixed point failed to converge) Thirdly this paper is intended to give a scheme which gives small error bound estimated to zero, further more in this research paper we will develop a new scheme calledfastest iteration (Bennie’s method) for finding square rootsand deduce newton’s method 2. Related Works Despite having different works related to methods ofsolving function  =0, for nonlinear function of single variable, we have common methods which are frequently used, these are fixed point method, newton’s method, regular falsi method, secant method, bisection method, Muller’s method, and deflation method [2] . Figure 1: bisection, Regula falsi, Newton Method, Secant Method demonstration [2]. Figure 2: Fixed Method Demonstration [2] However in this research we analyze fixed point method and Newton’s Method as related works. And these works will be used as the reference, motivating factors for foundation of Bennie’s method. A fixed point Method can be defined as an algorithm of the form   +1 =    (1) Which used for solving ()=0 , where n is the number of iteration we can go. The technique used is first to change ()=0 to a form  = [2]. Paper ID: SUB158660 256