Example 5.  Use Newton's method to construct a linearly convergent sequence    which converges slowly to the multiple root    of  .  
Then use the Aitken   process to construct     which converges faster to the root  .  

Solution 5.

Graph the function.

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

Since we know the root is   ,  we can determine the error for each iteration.

Newton's method is converging linearly (or slowly), the error at each step is being reduced by approximately one-half.  Let us apply Aitken's acceleration process to a sequence    of iterations generated by Newton's method.

Again, we can determine the error for each term.

The sequence    is converging to  p  faster than the sequence    converges to  p.  

(c) John H. Mathews 2004