Golden Section Search By Leon Gradisar 5312010 Golden

Golden Section Search By Leon Gradisar (531/2010)

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

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

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: x 1, x 2 and x 3. Since f 2 is smaller than either f 1 or f 3, it is clear that a minimum lies inside the interval from x 1 to x 3 (since f is unimodal). -The next step is evaluating it at a new value x 4. -if the function yields f 4 a: then a minimum lies between x 1 and x 4 and the new triplet of points will be x 1, x 2 and x 4 If the function yields f 4 b: then a minimum lies between x 2 and x 3, and the new triplet of points will be x 2 and x 3. -By this logic we can construct a new narrower search interval that is guaranteed to contain the function's minimum. 4/11

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(x 4) -In case f(x 4) = f 4 a our new triplet of points is x 1, x 2, x 4 then we want: -In case f(x 4)= our new triplet of points is x 2, x 4 , then we want: -Eliminating c from these two equations yields: or -Where φ is the golden ratio: 5/11

Golden Section Algorithm 6/11

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

Golden Section Algorithm Code in Makseler 8/11

Golden Search Compile Results 9/11

Golden Section References v. Milutinovic, V. , editor, “Advances in Computers: Data. Flow” , Elsevier, 2015. v. Milutinovic, V. , Salom, J. , Trifunovic, N. , Giorgi, R. “Guide to Data. Flow Super. Computing”, Springer, 2015 v. Milutinovic, V. , editor, “High-Level Language Computer Architecture, “ (Chapter 9, Data. Flow Machines, Gaudiot, J. -L. , ), Computer Science Press, 1989. v“Golden Section”, https: //en. wikipedia. org/wiki/Jack_Kiefer_(statistician), 2016 v“Golden Section”, https: //en. wikipedia. org/wiki/Golden_section_search, 2016 10/11

Questions? Than you for your attention! 11/11