Example 2.  Use the value    and compare Methods A,B and C for finding the triple root    of the equation  .

Solution 2.

First, we will compute the iterations for each method, and afterward a table comparing the methods is given.  For the method C, all the iterations in the linear search are included.  

Using formula (2), the standard Newton-Raphson method.  

Using formula (9) Method A, the accelerated Newton-Raphson method with  m=3.  

Using formula (11) Method B, the modified Newton-Raphson method.

Using formula (13) Method C, the adaptive Newton-Raphson method.

The details for obtaining   are:

Since    we reject    and set    and continue the iteration using formula (13).  The subroutine makes all these choices automatically.
        

Caveat.  Another shortcoming of Method A, the accelerated Newton-Raphson method is the wrong choice for  m.  Suppose that one chose    instead of  ?  Let's see what happens!

If the wrong choice for  m  is made then convergence will be linear (or slow).  

(c) John H. Mathews 2004