Example 4.  Use Powell's method to find the minimum of   ,
This example is referred to as Rosenbrock's parabolic valley, circa 1960.

Solution 4.

Enter the function   and graph the surface .

Perform the Powell method search for the minimum.

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

Observation.  Even Mathematica is having a hard time finding the minimum of  .
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.

(c) John H. Mathews 2004