Numerical Methods Golden Section Search Method Theory http

Numerical Methods Golden Section Search Method Theory http: //nm. mathforcollege. com

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

Golden Section Search Method n n 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. f 2 f 1 fu f. L XL X 2 X 1 Xu Figure 2. Golden Section Search method 6 forcollege. com ht

Golden Section Search Method. Selecting the Intermediate Points f 2 f 1 f. L fu f. L f 1 fu a-b b b XL a X 1 b Xu Determining the first intermediate point Let X 2 XL a Xu X 1 a Determining the second intermediate point , hence Golden Ratio=> forcollege. com 7

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! forcollege. com 8

Golden Section Search. Determining the new search region f 2 f. L XL n If 9 X 2 Case 1 X 1 fu f. L Xu XL f 1 fu X 2 X 1 Case 2 Xu Case 1: If n f 2 f 1 then the new interval is Case 2: then the new interval is forcollege. com ht

Golden Section Search. Determining the new search region n 10 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. forcollege. com ht

Golden Section Search(1 -D) Line Search Method Min. g(α)=Max. [-g(α)] jth 1 st j-2 th j-1 th • j v αU = VΣδ(1. 618) =0 j-2 αL αL = Σδ(1. 618)v V=0 0 δ 2. 618δ 5. 232δ • α 9. 468δ 0. 382(αU - αl) • • _ αa= αL αb αU Figure 2. 5 Golden section partition. Figure 2. 4 Bracketing the minimum point. δ 2 nd 1. 6182δ 1. 618δ j-2 αa = αL + 0. 382(αU – αl) = Σδ(1. 618)v + 0. 382δ(1. 618)j-1(1+1. 618) V=0 j-2 j-1 V=0 αa = Σδ(1. 618)v + 1δ(1. 618)j-1 = Σδ(1. 618)v = already known ! 11 forcollege. com 3 rd 4 th ht

Golden Section Search(1 -D) Line Search Method n n If , Then the minimum will be between αa & αb. If as shown in Figure 2. 5, Then the minimum will be between & and. Notice that: And _ _ Thus αb (wrt αU & αL ) plays same role as αa(wrt αU & αL ) !! 12 forcollege. com ht

Golden Section Search(1 -D) Line Search Method Step 1 : For a chosen small step size δ in α, say smallest integer such that The upper and lower bound on αi are , let j be the. and . Step 2: Compute g(αb) , where αa= 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), then αL ≤ αi ≤ αb. By the choice of αa and αb, the new points and have. Compute , where and go to Step 7. 13 forcollege. com ht

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 = αa and αu = αb and return to Step 2. Step 7: If is suitably small, put Otherwise, delete the bar symbols on to Step 3. 14 and stop. , and return forcollege. com ht

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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.

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