Example 1.  Find the minimum of the unimodal function    on the interval    using the Fibonacci search method.  Use the tolerance of    and the distinguishability constant  

Solution 1.

The smallest Fibonacci number satisfying  

        

A Mathematica function for generating the Fibonacci numbers is  

By trial and error we find that and  .

Thus we must choose  , and the first ratio we must use is

Let    and  , and start with the initial interval  .   Formulas (9) and (10) are used to compute   and   as follows:  

Thus, the minimum of f[x] will occur in the subinterval  .  We set  ,  ,  and  .  The new subinterval containing the abscissa of the minimum of   f[x]  is  .  Now use formulas (9) to calculate the interior point   as follows:  

Now compute and compare and   to determine the new subinterval  ,  and continue the iteration process.  The iterations are obtained by calling the subroutine.

In the subroutine, we have used  , and hence .  If this is acceptable then we have found an approximation to the minimum.  

If we use the distinguishability constant    in the final computation, then a plausible computation would be

(c) John H. Mathews 2004