Example 4.  Use Powell's method to find the minimum of   [Graphics:Images/PowellMethodMod_gr_86.gif],
This example is referred to as Rosenbrock's parabolic valley, circa 1960.

Solution 4.

Enter the function  [Graphics:../Images/PowellMethodMod_gr_87.gif] and graph the surface [Graphics:../Images/PowellMethodMod_gr_88.gif].

[Graphics:../Images/PowellMethodMod_gr_89.gif]

[Graphics:../Images/PowellMethodMod_gr_90.gif]

[Graphics:../Images/PowellMethodMod_gr_91.gif]
[Graphics:../Images/PowellMethodMod_gr_92.gif]

[Graphics:../Images/PowellMethodMod_gr_93.gif]

[Graphics:../Images/PowellMethodMod_gr_94.gif]
[Graphics:../Images/PowellMethodMod_gr_95.gif]

[Graphics:../Images/PowellMethodMod_gr_96.gif]

[Graphics:../Images/PowellMethodMod_gr_97.gif]

Perform the Powell method search for the minimum.

[Graphics:../Images/PowellMethodMod_gr_98.gif]



[Graphics:../Images/PowellMethodMod_gr_99.gif]

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

[Graphics:../Images/PowellMethodMod_gr_100.gif]


[Graphics:../Images/PowellMethodMod_gr_101.gif]

Observation.  Even Mathematica is having a hard time finding the minimum of  [Graphics:../Images/PowellMethodMod_gr_102.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_103.gif]


[Graphics:../Images/PowellMethodMod_gr_104.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2004