Example 3.  Use Powell's method to find the minimum of   [Graphics:Images/PowellMethodMod_gr_75.gif].  

Solution 3.

[Graphics:../Images/PowellMethodMod_gr_17.gif]

[Graphics:../Images/PowellMethodMod_gr_18.gif]

[Graphics:../Images/PowellMethodMod_gr_19.gif]
[Graphics:../Images/PowellMethodMod_gr_20.gif]

[Graphics:../Images/PowellMethodMod_gr_21.gif]

[Graphics:../Images/PowellMethodMod_gr_22.gif]
[Graphics:../Images/PowellMethodMod_gr_23.gif]

[Graphics:../Images/PowellMethodMod_gr_24.gif]

[Graphics:../Images/PowellMethodMod_gr_25.gif]

Then perform the iterations using the subroutine Powell.

[Graphics:../Images/PowellMethodMod_gr_82.gif]



[Graphics:../Images/PowellMethodMod_gr_83.gif]

Observe that in each iteration of the taxi-cab method only one variable in the point is changing.  Convergence proceeds a polygonal path where the segments are parallel to the coordinate axes.  If the geometry of level curves is sufficiently complicated it might not be feasible to zig-zag to the minimum in this manner.

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

[Graphics:../Images/PowellMethodMod_gr_84.gif]



[Graphics:../Images/PowellMethodMod_gr_85.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2004