**Exercise 5.** Use the
Powell's method to find the minimum of .

Looking at your graphs, estimate the location of the local
minima.

**Solution 5.**

Looking at your graphs, estimate the location of the local
minima.

Hint. The contour plot should be most useful.

**Case (i)** Go for the
minimum near

Enter the starting point and perform the iterations.

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

**Case (ii)** Go for
the minimum near

Enter the starting point and perform the iterations.

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

**Case (iii)** Go for
the minimum near

Enter the starting point and perform the iterations.

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