Example 2.  Use the taxi-cab method to find the minimum of   [Graphics:Images/PowellMethodMod_gr_30.gif].  

Solution 2.

Enter the function [Graphics:../Images/PowellMethodMod_gr_31.gif].

[Graphics:../Images/PowellMethodMod_gr_32.gif]


[Graphics:../Images/PowellMethodMod_gr_33.gif]

Then perform the iterations using the subroutine TaxiCab.

[Graphics:../Images/PowellMethodMod_gr_34.gif]



[Graphics:../Images/PowellMethodMod_gr_35.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.  
The
3D plot of the points  [Graphics:../Images/PowellMethodMod_gr_36.gif]  involved in the iteration is given below.  
    

[Graphics:../Images/PowellMethodMod_gr_37.gif]

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

[Graphics:../Images/PowellMethodMod_gr_38.gif]


[Graphics:../Images/PowellMethodMod_gr_39.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2004