Minimum of a Function Using the Fibonacci Sequence

Consider the function , , where is a parameter. This Demonstration approximates the minimum of using an algorithm based on the Fibonacci sequence, shown by a magenta point on the plot of . For comparison, the blue point is the minimum found by Mathematica's built-in function NMinimize. When is sufficiently small, there is good agreement.
You can vary the values of and (see the Details section for the definition of ). The Demonstration plots and you can see a table of points in the iteration of the algorithm.


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Steps involved in a Fibonacci search:
1. Calculate , where is the interval in which is defined.
2. Identify such that where is the Fibonacci sequence.
3. Set and .
4. Calculate and .
5. If , set and ; otherwise set and .
6. Set and go to step 3.
7. Stop when .
[1] J. H. Mathews. "Module for the Fibonacci Search." (Apr 30, 2013)
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