Numerical Methods Golden Section Search Method Theory http
- Slides: 31
Numerical Methods Golden Section Search Method Theory http: //nm. mathforcollege. com
For more details on this topic Ø Ø Ø Go to http: //nm. mathforcollege. com Click on Keyword Click on Golden Section Search Method
You are free n n to Share – to copy, distribute, display and perform the work to Remix – to make derivative works
Under the following conditions n n n 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.
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
THE END http: //nm. mathforcollege. com
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
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.
The End - Really
Numerical Methods Golden Section Search Method Example http: //nm. mathforcollege. com
For more details on this topic Ø Ø Ø Go to http: //nm. mathforcollege. com Click on Keyword Click on Golden Section Search Method
You are free n n to Share – to copy, distribute, display and perform the work to Remix – to make derivative works
Under the following conditions n n n 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.
Example. 2 2 2 The cross-sectional area A of a gutter with equal base and edge length of 2 is given by (trapezoidal area): Find the angle which maximizes the cross-sectional area of the gutter. Using an initial interval of find the solution after 2 iterations. Convergence achieved if “ interval length ” is within 23 forcollege. com ht
Solution The function to be maximized is Iteration 1: Given the values for the boundaries of we can calculate the initial intermediate points as follows: f 2 f 1 X 1=? XL 24 X 2 X 1 Xu XL=X 2 X 2=X 1 Xu forcollege. com ht
Solution Cont To check the stopping criteria the difference between and is calculated to be 25 forcollege. com ht
Solution Cont Iteration 2 XL 26 X 2 X 1 Xu forcollege. com ht
Theoretical Solution and Convergence Iteration 1 2 3 4 5 6 7 8 9 xl 0. 0000 0. 6002 0. 8295 0. 9712 1. 0253 xu 1. 5714 1. 2005 1. 1129 1. 0794 1. 0588 x 1 0. 9712 1. 2005 0. 9712 1. 0588 1. 1129 1. 0588 1. 0794 1. 0588 1. 0460 x 2 0. 6002 0. 9712 0. 8295 0. 9712 1. 0588 1. 0253 1. 0588 1. 0460 1. 0381 f(x 1) 5. 1657 5. 0784 5. 1657 5. 1955 5. 1740 5. 1955 5. 1908 5. 1955 5. 1961 f(x 2) 4. 1238 5. 1657 4. 9426 5. 1657 5. 1955 5. 1937 5. 1955 5. 1961 5. 1957 1. 5714 0. 9712 0. 6002 0. 3710 0. 2293 0. 1417 0. 0876 0. 0541 0. 0334 The theoretically optimal solution to the problem happens at exactly 60 degrees which is 1. 0472 radians and gives a maximum cross-sectional area of 5. 1962. 27 forcollege. com ht
THE END http: //nm. mathforcollege. com
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
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.
The End - Really
- Golden search method example
- Secant method example problems with solution
- Successive approximation method in numerical methods
- Golden-section search
- Images.search.yahoo.com
- 1http
- Golden search method
- Graphical method numerical analysis
- Interpolation definition in numerical analysis
- Taylor series numerical methods
- Different types of errors in numerical methods
- Backward euler scheme
- Cfd numerical methods
- Chronicle of higher education
- Numerical methods of descriptive statistics
- Bessel's interpolation formula proof
- Numerical methods
- Relative true error formula
- Interpolation in numerical methods
- Pde project topics
- Numerical methods for partial differential equations eth
- Linear differential equations
- Golden ratio in renaissance art
- Golden ratio and golden rectangle
- Trapezoidal riemann sum
- Numerical method
- Direct wax pattern technique
- Heuristic search methods
- Effective internet research
- Uninformed search examples
- Http //mbs.meb.gov.tr/ http //www.alantercihleri.com
- Http //pelatihan tik.ung.ac.id