Example 5.  Use the modified Newton's method to find the double root  ,  and triple root   ,  of the cubic polynomial  .  

Solution 5.

Graph the function.

The modified Newton-Raphson iteration formula  g[x]  is found.

Investigate quadratic 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 much faster than the standard Newton-Raphson iteration.

At the double root    we can explore the ratio .

Therefore, the modified Newton-Raphson iteration is converging quadratically.

Now investigate the other root.
Investigate quadratic convergence at the triple 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 much faster than the standard Newton-Raphson iteration.

At the triple root    we can explore the ratio .

Therefore, the modified Newton-Raphson iteration is converging quadratically.

(c) John H. Mathews 2004