Example 2.  Use Newton's method to find the roots of the cubic polynomial  .  
2 (a) Fast Convergence.  Investigate quadratic convergence at the simple root  ,  using the starting value  
2 (b) Slow Convergence.  Investigate linear convergence at the double root  ,  using the starting value  

Solution 2.

Graph the function.

The secant iteration formula    is

2 (a) Fast Convergence.  Investigate quadratic convergence at the simple 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 fast and the sequence is converging to the simple root    

At the simple root  , the order of convergence is known to be .  We can explore the ratio    for  k  sufficiently large.

k

2 (b) Slow Convergence.  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 double root    

Compare our result with Mathematica's built in numerical root finder.

This can also be done with Mathematica's built in symbolic solve procedure.

(c) John H. Mathews 2004