Three-point iterations derived from exponential curve fitting

Three-point Iterative methods make use of an approximating function,
p(x)
of
F(x)
which functions have three
x
and
y
values in common. Some functions
p(x)
give an analytic expression in
x
when
p(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.