Good approximation is the main aim of any iteration method so that required roots can
achieved efficiently in less number of steps. In this paper a new type of iteration is
introduce for approximation for finding a roots for transcendental equations. And it is
observed that it can be extend for Non-Smooth functions also. It is an approximation
method to find out the root of a given equation and we approximate the graph of the
given equation by two vertical ellipses under conditions of (i) touching externally, and
(ii) intersecting orthogonally under certain condition. This approximation works well
even when the slope of the function is parallel to the real line where usually Newton’s
method fails. In this process under each of the above condition we discuss the following: