1. By Leon Gradisar (531/2010) Golden Section Search

2. Golden Section Method Introduction -Golden Section Method uses constant Interval of reduciton -Which can be seen in different aspects of proportion from geometry to architecture. 2/11

3. Golden Section Search Introduction The Golden section search is a technique for finding the extrem um (minimum or maximum) of a strictly unimodal function by succe ssively narrowing the range of values inside which the extremum is known to exist. -The technique derives its name from the fact that the algorithm ma intains the function values for triples of points whose distances for m a golden ratio. -Fibonacci search and Golden section search were discovered by K iefer (1953) 3/11

4. 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. 4/11

5. 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) = f 4a 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: 5/11

6. Golden Section Algorithm 6/11

7. Maxeler MAXELER IN A NUTSHELL  Dataflow paradigm  The write to the memory is postponed until the data processing is finished  Decreases cost of reading and writing temporary result  Tokens on the entry points of vertices in a graph are a condition for operation completition  Loop oriented, big data  As less data dependences as possible 7/11

8. Golden Section Algorithm Code in Makseler 8/11

9. Golden Search Compile Results 9/11

10. Golden Section References  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 Architectu re,“ (Chapter 9, DataFlow Machines, Gaudiot, J.-L.,), Computer Science Press, 1989.  “Golden Section”, https://en.wikipedia.org/wiki/Jack_Kiefer_(statistician), 2016https://en.wikipedia.org/wiki/Jack_Kiefer_(statistician)  “Golden Section”, https://en.wikipedia.org/wiki/Golden_section_search https://en.wikipedia.org/wiki/Golden_section_search, 2016 10/11

11. Questions? Than you for your attention! 11/11