A family of Newton-like methods based on an exponential model

Abstract

We derive a one-parameter family of second-order methods for solving f(x)=0. The approximation process is carried out by fitting the model m(x)=e px (ax+b) to the function f(x) and its derivative f′(x) at a point x i and solving m(x)=0. Numerical examples show that the method works when Newton's method fails. A composite method with cubic convergence, based on the new scheme, is also presented. Error analysis providing the order of convergence is given. Theoretical comparisons of computational efficiency are made and some rules to assist in the selection of an iteration formula are derived.