Example 1.  Use the Nelder-Mead method to find the minimum of   .  

Solution 1.

Enter the function   and graph the surface .

Start with the three vertices  

The function takes on the values  

The function values must be compared to determine , and ;  

The vertex    will be replaced.  The points and are constructed.  

The function value    is less than  ,  so the situation is Case (i).  
Since  ,  we have moved in the right direction, and the vertex must be constructed:

The function value    is less than  ,  and we replace with and the new triangle has vertices

The process continues and generates a sequence of triangles that converges down on the solution point  .  The computer implementation of the Nelder-Mead algorithm is given below

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

(c) John H. Mathews 2004