Exercise 5.  Use the Powell's method to find the minimum of  .  
Looking at your graphs, estimate the location of the local minima.  

Solution 5.

Looking at your graphs, estimate the location of the local minima.  
Hint. The contour plot should be most useful.

Case (i)  Go for the minimum near

Enter the starting point and perform the iterations.  

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

Case (ii)  Go for the minimum near

Enter the starting point and perform the iterations.  

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

Case (iii)  Go for the minimum near

Enter the starting point and perform the iterations.  

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

Observation.  Even Mathematica is having a hard time finding the minimum of  .
Since the function is "flat" near the minimum, the best way to achieve better accuracy is to increase the  WorkingPrecision,  i.e. use extended precision in the numerical computations.

(c) John H. Mathews 2004