The Modified Newton Method

Again, the idea is to reduce the number of derivative computations without hurting the convergence of the method.

Modified Newton uses formula () with

that is

This means that only one derivative computation is required, regardless of the number of iterations involved.

If after, say, iterations the speed of convergence decreases, we need to update the derivative; set

and continue with

Modified Newton is a method extensively used in scientific and engineering computations.

To obtain the order of convergence, we do the following calculations:

This means that and we say that the Modified Newton method converges linearly. Of course, for convergence we need .