A simple modification of Newton’s method to achieve convergence of order 1+2

A simple modification to the standard Newton method for approximating the root of a univariate function is described and analyzed. For the same number of function and derivative evaluations, the modified method converges faster, with the convergence order of the method being 1+2≈2.4 compared with 2 for the standard Newton method. Numerical examples demonstrate the faster convergence achieved with this modification of Newton’s method. This modified Newton–Raphson method is relatively simple and is robust; it is more likely to converge to a solution than are either the higher order (4th order and 6th order) schemes or Newton’s method itself.