Example 4.  Use Newton's method and Steffensen's acceleration method to find numerical approximations to the multiple root    of the function  .  
Show details of the computations for the starting value  .  Compare the number of iterations for the two methods.

Solution 4.

Graph the function.

Starting with  , use the Newton-Raphson method to find a numerical approximation to the root.

We can use Mathematica's Solve procedure to determine some of the roots.

For Newton's method, how far away is the ninth iteration    from the root   ?
Note. The last iteration is actually stored in  .

Starting with  , use Steffensen's acceleration method to find a numerical approximation to the root.

For Steffensen's acceleration method, how far away is the ninth iteration from the root   ?
Note. The last iteration is actually stored in  .

This is closer than     which was obtained with Newton's method.

(c) John H. Mathews 2004