Example 6.  Use Powell's Method to find  [Graphics:Images/PowellMethodMod_gr_135.gif]  and  [Graphics:Images/PowellMethodMod_gr_136.gif]  for the function  [Graphics:Images/PowellMethodMod_gr_137.gif].   Use the initial point  [Graphics:Images/PowellMethodMod_gr_138.gif].

Solution 6.

[Graphics:../Images/PowellMethodMod_gr_139.gif]

[Graphics:../Images/PowellMethodMod_gr_140.gif]


[Graphics:../Images/PowellMethodMod_gr_141.gif]

[Graphics:../Images/PowellMethodMod_gr_142.gif]

[Graphics:../Images/PowellMethodMod_gr_143.gif]
[Graphics:../Images/PowellMethodMod_gr_144.gif]

[Graphics:../Images/PowellMethodMod_gr_145.gif]

[Graphics:../Images/PowellMethodMod_gr_146.gif]
[Graphics:../Images/PowellMethodMod_gr_147.gif]

[Graphics:../Images/PowellMethodMod_gr_148.gif]

[Graphics:../Images/PowellMethodMod_gr_149.gif]



[Graphics:../Images/PowellMethodMod_gr_150.gif]

[Graphics:../Images/PowellMethodMod_gr_151.gif]

Enter the initial point  [Graphics:../Images/PowellMethodMod_gr_152.gif]  and set  [Graphics:../Images/PowellMethodMod_gr_153.gif].  

[Graphics:../Images/PowellMethodMod_gr_154.gif]

For  k=1  the construct the function  [Graphics:../Images/PowellMethodMod_gr_155.gif].
Find  [Graphics:../Images/PowellMethodMod_gr_156.gif]  where  [Graphics:../Images/PowellMethodMod_gr_157.gif]  is a minimum.  
And construct  [Graphics:../Images/PowellMethodMod_gr_158.gif].  

[Graphics:../Images/PowellMethodMod_gr_159.gif]


[Graphics:../Images/PowellMethodMod_gr_160.gif]

 

 

For  k=2  the construct the function  [Graphics:../Images/PowellMethodMod_gr_161.gif].  
Find  [Graphics:../Images/PowellMethodMod_gr_162.gif]  where  [Graphics:../Images/PowellMethodMod_gr_163.gif]  is a minimum.  
And construct  [Graphics:../Images/PowellMethodMod_gr_164.gif].  

[Graphics:../Images/PowellMethodMod_gr_165.gif]


[Graphics:../Images/PowellMethodMod_gr_166.gif]

 

 

Set   [Graphics:../Images/PowellMethodMod_gr_167.gif]   and   [Graphics:../Images/PowellMethodMod_gr_168.gif]  

[Graphics:../Images/PowellMethodMod_gr_169.gif]


[Graphics:../Images/PowellMethodMod_gr_170.gif]

Find  [Graphics:../Images/PowellMethodMod_gr_171.gif]  where  [Graphics:../Images/PowellMethodMod_gr_172.gif]  is a minimum.  
And construct  [Graphics:../Images/PowellMethodMod_gr_173.gif]  

[Graphics:../Images/PowellMethodMod_gr_174.gif]


[Graphics:../Images/PowellMethodMod_gr_175.gif]

 

 

Set [Graphics:../Images/PowellMethodMod_gr_176.gif].  

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.

[Graphics:../Images/PowellMethodMod_gr_177.gif]



[Graphics:../Images/PowellMethodMod_gr_178.gif]

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

[Graphics:../Images/PowellMethodMod_gr_179.gif]


[Graphics:../Images/PowellMethodMod_gr_180.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2004