Three-point iterations derived from exponential curve fitting

Abstract

Three-point Iterative methods make use of an approximating function, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p(x)</tex> of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F(x)</tex> which functions have three <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">y</tex> values in common. Some functions <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p(x)</tex> give an analytic expression in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</tex> when <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p(x) = O \cdot p(x)</tex> 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.