**Example 1.** **
**Find the minimum of the unimodal
function on
the interval .

**Solution 1.**

**Solution by
solving**** **

A root-finding method can be used to
determine where the derivative is
zero.

Since and have
opposite signs, by the intermediate value
theorem a root of
lies in the interval . The
results obtained by using the secant method with the initial
values and are
given by the following computations.

The conclusion
from applying the secant
method is that

The second derivative is and
we compute

Hence, by the second
derivative test, the
local
minimum
of on
the interval
is

**Solution using the
golden ratio search for the minimum of**** **

Let and ,
and start with the initial interval . Formulas
(1) and (2) yield

Since ,
the new subinterval is .

To continue the iteration we set ,
,
,
and compute .

Now compute and compare and to determine the new subinterval and continue the iteration process. The list of computations are obtained by using the GoldenSearch subroutine are:

At the twenty-fifth iteration the
interval has been narrowed down to

.

This interval has
width .

However, the computed function values at the endpoints agree to ten
decimal places

hence the algorithm is terminated. A difficulty in using search
methods is that the function may be "flat" near the minimum, and this
limits the accuracy that can be obtained. The secant
method was able to find the more accurate answer

and

*
*Although the golden ratio search
is the slower
in this example, it has desirable feature that it can be applied in
cases where
is not
differentiable.

Let us compare these answers with *Mathematica*'s subroutine
FindMinimum.

This is close to the values obtained with the golden ratio search. If more decimal places are needed, we can print them out.

However, they are not as accurate as those computed with the secant method applied to solving .

(c) John H. Mathews 2004