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Numerical Methods Golden Section Search Method - Theory nm.mathforcollege

Numerical Methods Golden Section Search Method - Theory http://nm.mathforcollege.com. For more details on this topic Go to http://nm.mathforcollege.com Click on Keyword Click on Golden Section Search Method . You are free. to Share – to copy, distribute, display and perform the work

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Numerical Methods Golden Section Search Method - Theory nm.mathforcollege

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  1. Numerical MethodsGolden Section Search Method - Theoryhttp://nm.mathforcollege.com

  2. For more details on this topic • Go to http://nm.mathforcollege.com • Click on Keyword • Click on Golden Section Search Method

  3. You are free • to Share – to copy, distribute, display and perform the work • to Remix – to make derivative works

  4. Under the following conditions • Attribution — You must attribute the work in the manner specified by the author or licensor (but not in any way that suggests that they endorse you or your use of the work). • Noncommercial — You may not use this work for commercial purposes. • Share Alike — If you alter, transform, or build upon this work, you may distribute the resulting work only under the same or similar license to this one.

  5. f(x) x a b Equal Interval Search Method • Choose an interval [a, b] over which the optima occurs. • Compute and • If • then the interval in which the maximum occurs is otherwise it occurs in Figure 1 Equal interval search method. http://nm.mathforcollege.com

  6. f2 f1 fu fL XL Xu X1 X2 Golden Section Search Method • The Equal Interval method is inefficient when  is small. Also, we need to compute 2 interior points ! • The Golden Section Search method divides the search more efficiently closing in on the optima in fewer iterations. Figure 2. Golden Section Search method http://nm.mathforcollege.com

  7. f2 f1 f1 fL fL fu fu a-b b X2 XL XL Xu Xu X1 X1 a b a Golden Section Search Method-Selecting the Intermediate Points b Determining the first intermediate point a Determining the second intermediate point Let ,hence Golden Ratio=> http://nm.mathforcollege.com

  8. Golden Section Search Method Hence, after solving quadratic equation, with initial guess = (0, 1.5708 rad) =Initial Iteration Second Iteration Only 1 new inserted location need to be completed! http://nm.mathforcollege.com

  9. f2 f1 f2 f1 fL fL fu fu X2 X2 XL XL Xu Xu X1 X1 Golden Section Search-Determining the new search region • Case1: If then the new interval is • Case2: If then the new interval is Case 2 Case 1 http://nm.mathforcollege.com

  10. Golden Section Search-Determining the new search region • At each new interval ,one needs to determine only 1(not 2)new inserted location(either compute the new ,or new ) • Max. Min. • It is desirable to have automated procedure to compute and initially. http://nm.mathforcollege.com

  11. Golden Section Search-(1-D) Line Search Method 0.382(αU -αl) • • Min.g(α)=Max.[-g(α)] 1st jth j j - 2 αL = Σδ(1.618)v αU = Σδ(1.618)v • • V = 0 j-2th V = 0 _ αb αU αL αa = αL j-1th Figure 2.5 Golden section partition. Figure 2.4 Bracketing the minimum point. α 2nd 0 δ 2.618δ 5.232δ 9.468δ 1.6182δ δ j - 2 j - 2 1.618δ αa = Σδ(1.618)v + 1δ(1.618)j-1 = Σδ(1.618)v = already known ! αa = αL + 0.382(αU – αl) = Σδ(1.618)v + 0.382δ(1.618)j-1(1+1.618) 3rd V = 0 V = 0 j - 1 4th V = 0 http://nm.mathforcollege.com

  12. Golden Section Search-(1-D) Line Search Method • If, Then the minimum will be between αa & αb. • Ifas shown in Figure 2.5, Then the minimum will bebetween&and . Notice that: And Thusαb (wrtαU & αL ) plays same role as αa(wrtαU & αL ) !! _ _ http://nm.mathforcollege.com

  13. Golden Section Search-(1-D) Line Search Method Step 1 : For a chosen small step size δ in α,say,let j be the smallest integer such that. The upper and lower bound on αi areand. Step 2: Compute g(αb) ,whereαa= αL+ 0.382(αU- αL) ,and αb = αL+ 0.618(αU- αL). Note that, so g(αa) is already known. Step 3: Compare g(αa) and g(αb) and go to Step 4,5, or 6. Step 4: If g(αa)<g(αb),thenαL ≤ αi ≤ αb. By the choice of αaandαb, the new pointsandhave . Compute , whereand go to Step 7. http://nm.mathforcollege.com

  14. Golden Section Search-(1-D) Line Search Method Step 5: If g(αa) > g(αb),then αa≤ αi ≤ αU. Similar to the procedure in Step 4, put and . Compute ,where and go to Step 7. Step 6: If g(αa) = g(αb) put αL = αaand αu = αb and return to Step 2. Step 7: If is suitably small, put and stop. Otherwise, delete the bar symbols on ,and and return to Step 3. http://nm.mathforcollege.com

  15. The End http://nm.mathforcollege.com

  16. Acknowledgement This instructional power point brought to you by Numerical Methods for STEM undergraduate http://nm.mathforcollege.com Committed to bringing numerical methods to the undergraduate

  17. For instructional videos on other topics, go to http://nm.mathforcollege.com This material is based upon work supported by the National Science Foundation under Grant # 0717624. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

  18. The End - Really

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