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Minimum of a Function Using the Fibonacci Sequence

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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.

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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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Contributed by: Housam Binous, Brian G. Higgins, and Ahmed Bellagi (May 2013)
Open content licensed under CC BY-NC-SA

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Details

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 .

Reference

[1] J. H. Mathews. "Module for the Fibonacci Search." (Apr 30, 2013) mathfaculty.fullerton.edu/mathews/n2003/FibonacciSearchMod.html.

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