Three-point iterations derived from exponential curve fitting

Three-point Iterative methods make use of an approximating function,p(x)ofF(x)which functions have threexandyvalues in common. Some functionsp(x)give an analytic expression inxwhenp(x) = O \cdot p(x)can be a hyperbolic function, a quadratic polynomial, or an exponential function. In this paper it is demonstrated that the hyperbolic function is a special case, derived from the exponential function. The condition for convergence and the rate of convergence are discussed. A comparison by means of some examples is made with the hyperbolic function and with the method of Jarratt and Nudds which is a modification of the hyperbolic approximation.