Exercise 5.  Use the Powell's method to find the minimum of  [Graphics:Images/PowellMethodMod_gr_105.gif].  
Looking at your graphs, estimate the location of the local minima.  

Solution 5.

[Graphics:../Images/PowellMethodMod_gr_106.gif]

[Graphics:../Images/PowellMethodMod_gr_107.gif]

[Graphics:../Images/PowellMethodMod_gr_108.gif]

[Graphics:../Images/PowellMethodMod_gr_109.gif]

[Graphics:../Images/PowellMethodMod_gr_110.gif]
[Graphics:../Images/PowellMethodMod_gr_111.gif]

[Graphics:../Images/PowellMethodMod_gr_112.gif]

[Graphics:../Images/PowellMethodMod_gr_113.gif]
[Graphics:../Images/PowellMethodMod_gr_114.gif]

[Graphics:../Images/PowellMethodMod_gr_115.gif]

[Graphics:../Images/PowellMethodMod_gr_116.gif]

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 [Graphics:../Images/PowellMethodMod_gr_117.gif]

Enter the starting point and perform the iterations.  

[Graphics:../Images/PowellMethodMod_gr_118.gif]



[Graphics:../Images/PowellMethodMod_gr_119.gif]

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

[Graphics:../Images/PowellMethodMod_gr_120.gif]


[Graphics:../Images/PowellMethodMod_gr_121.gif]


Case (ii)  Go for the minimum near [Graphics:../Images/PowellMethodMod_gr_122.gif]

Enter the starting point and perform the iterations.  

[Graphics:../Images/PowellMethodMod_gr_123.gif]



[Graphics:../Images/PowellMethodMod_gr_124.gif]

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

[Graphics:../Images/PowellMethodMod_gr_125.gif]


[Graphics:../Images/PowellMethodMod_gr_126.gif]


Case (iii)  Go for the minimum near [Graphics:../Images/PowellMethodMod_gr_127.gif]

Enter the starting point and perform the iterations.  

[Graphics:../Images/PowellMethodMod_gr_128.gif]



[Graphics:../Images/PowellMethodMod_gr_129.gif]

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

[Graphics:../Images/PowellMethodMod_gr_130.gif]


[Graphics:../Images/PowellMethodMod_gr_131.gif]

Observation.  Even Mathematica is having a hard time finding the minimum of  [Graphics:../Images/PowellMethodMod_gr_132.gif].
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.

[Graphics:../Images/PowellMethodMod_gr_133.gif]


[Graphics:../Images/PowellMethodMod_gr_134.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2004