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

Solution 6 (a).

Graph the function.

It appears that round off error in the calculation of f[x] is introducing quite a bit of round-off error or noise.
Let's see what happens!

The Newton-Raphson iteration formula  g[x]  is

Investigate linear 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 slow, but the sequence is converging to the multiple root  .  

Observe.  The function value is very close to zero, and the approximation   is a long way from the root !

At the multiple root    we can explore the relationship    for  k  sufficiently large.

This will be done by investigating the ratio    for  k  sufficiently large.

(c) John H. Mathews 2004