Example 1.  Use Newton's method and Muller's 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 1.

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 eighth iteration    from the root   ?
Note. The last iteration is actually stored in  .

Starting with  , , and , use Muller's method to find a numerical approximation to the root.

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

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

We are done.

Aside.  Compare with Mathematica's built in routine.

Mathematica's answer is not so good, need to adjust the number of iterations and the working precision.

(c) John H. Mathews 2004