Example 6.  Use Powell's Method to find    and    for the function  .   Use the initial point  .

Solution 6.

Enter the initial point    and set  .  

For  k=1  the construct the function  .
Find    where    is a minimum.  
And construct  .  

For  k=2  the construct the function  .  
Find    where    is a minimum.  
And construct  .  

Set      and     

Find    where    is a minimum.  
And construct    

Set .  

If the minimization is done with an application of the golden ratio or Fibonacci search then slightly different results might be obtained.
Recall that we used Mathematica's built in procedure FindMinimum to perform the line searches in the above computations.

Perform the iteration using the subroutine Powell.

Let us compare this answer with Mathematica's built in procedure FindMinimum.

(c) John H. Mathews 2004