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 Newton-Raphson iteration formula  g[x]  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    we can explore the relationship    for  k  sufficiently large.

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

k

Evaluate the quantity   at the root  .

Which is close to the computed value    

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    

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

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

k

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