By Leon Gradisar (531/2010)

Golden Section Search

Golden Section MethodIntroduction

-Golden Section Method uses constantInterval of reduciton

-Which can be seen in different aspects of pro-portion from geometry to architecture.

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Golden Section SearchIntroduction

The Golden section search is a technique for finding the extremum (minimum or maximum) of a strictly uni-modal function by successively narrowing the range of val-ues inside which the extremum is known to exist.

-The technique derives its name from the fact that the al-gorithm maintains the function values for triples of points whose distances form a golden ratio.

-Fibonacci search and Golden section search were discov-ered by Kiefer (1953)

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Golden Section

Idea

-The diagram above illustrates a single step in the technique for finding a minimum.-The value of has already been evaluated at the three points: x1,x2 and x3. Since f2 is smaller than either f1 or f3, it is clear that a minimum lies inside the interval from x1 to x3 (since f is unimodal).

-The next step is evaluating it at a new value x4.-if the function yields f4a: then a minimum lies between x1 and x4 and the new triplet of points will be x1,x2 and x4 If the function yields f4b: then a minimum lies between x2 and x3, and the new triplet of points will be x2 and x3.-By this logic we can construct a new narrower search interval that is guaranteed to contain the function's minimum.

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Golden Section

Workflow

-From the diagram, it is seen that the new search interval will be either between and with a length of a+c , or between and with a length of b .-To ensure that the spacing after evaluatingf(x4)

-In case f(x4) = f4a our new triplet of points is x1,x2,x4

then we want:

-In case f(x4)= our new triplet of points is x2,x4,x4 , then we want:

-Eliminating c from these two equations yields: or

-Where φ  is the golden ratio:

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Golden SectionAlgorithm

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Maxeler

MAXELER IN A NUTSHELL

Dataflow paradigmThe write to the memory is postponed until the data processing is finishedDecreases cost of reading and writing temporary resultTokens on the entry points of vertices in a graph are a condition for operation completition

Loop oriented, big dataAs less data dependences as possible

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Golden Section

Algorithm Code in Makseler

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Golden Search

Compile Results

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Golden SectionReferences

Milutinovic, V., editor, “Advances in Computers: DataFlow” ,Elsevier, 2015.

Milutinovic, V., Salom, J., Trifunovic, N., Giorgi, R. “Guide to DataFlow SuperComputing”, Springer, 2015

Milutinovic, V., editor, “High-Level Language Computer Architecture,“ (Chapter 9, DataFlow Machines, Gaudiot, J.-L.,), Computer Science Press, 1989.

“Golden Section”, https://en.wikipedia.org/wiki/Jack_Kiefer_(statistician), 2016“Golden Section”, https://en.wikipedia.org/wiki/Golden_section_search, 2016

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Questions?

Than you for your attention!

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