Example 6.  Consider the function  .  
6 (b).  Use the accelerated Newton's method to find the multiple root  .  

Solution 6 (b).

Graph the function.

How would we determine the order of the root ?
In practice, this must be known in advance.
For this example, the function was  .

Since the multiple root    has order  double root  ,  the accelerated Newton-Raphson iteration formula  g[x]  is

Investigate quadratic convergence at the double root  ,  using the starting value  

First, do the iteration one step at a time.  
Type each of the following commands in a separate cell and execute them one at a time.

Notice that convergence is much faster than the standard Newton-Raphson iteration.

At the double root    we can explore the ratio .

Therefore, the accelerated Newton-Raphson iteration is converging quadratically.

(c) John H. Mathews 2004