1. Numerical Methods Golden Section Search Method - Theory http://nm

2. For more details on this topic
Go to
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. 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 x
f(x) a b
Figure 1 Equal interval search method.

6. 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. X2 XL X1 Xu fu f2 f1 fL
Figure 2. Golden Section Search method

7. Golden Section Search Method-Selecting the Intermediate Pointsb XL X1 Xu fu f1 fL
a-b b X2 a XL X1 Xu fu f2 f1 fL b
Determining the first intermediate point a
Determining the second intermediate point
Let ,hence
Golden Ratio=>

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

9. Golden Section Search- Determining the new search regionX2 XL X1 Xu fu f2 f1 fL X2 XL X1 Xu fu f2 f1 fL
Case 1
Case 2
Case1:
If then the new interval is
Case2:

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.

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

12. Golden Section Search- (1-D) Line Search Method
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 ) !! _ _

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 are and . Step 2: Compute g(αb) ,where αa= αL (αU- αL) ,and αb = αL (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.

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 = αa and 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.

15. The End

16. Acknowledgement
This instructional power point brought to you by
Numerical Methods for STEM undergraduate
Committed to bringing numerical methods to the undergraduate

17. For instructional videos on other topics, go to
This material is based upon work supported by the National Science Foundation under Grant # 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

19. Numerical Methods Golden Section Search Method - Example http://nm

20. For more details on this topic
Go to
Click on Keyword
Click on Golden Section Search Method

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

22. 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.

23. 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

24. 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:
XL=X2
X2=X1 Xu X2 XL X1 Xu f2 f1
X1=?

25. Solution Cont
To check the stopping criteria the difference between and is calculated to be

26. Solution Cont
Iteration 2 XL X2 Xu X1

27. Theoretical Solution and Convergence
Iteration xl xu x1 x2
f(x1)
f(x2)  1
0.0000
1.5714
0.9712
0.6002
5.1657
4.1238 2
1.2005
5.0784 3
0.8295
4.9426 4
1.0588
5.1955
0.3710 5
1.1129
5.1740
0.2293 6
1.0253
5.1937
0.1417 7
1.0794
5.1908
0.0876 8
1.0460
5.1961
0.0541 9
1.0381
5.1957
0.0334
The theoretically optimal solution to the problem happens at exactly 60 degrees which is radians and gives a maximum cross-sectional area of

28. The End

29. Acknowledgement
This instructional power point brought to you by
Numerical Methods for STEM undergraduate
Committed to bringing numerical methods to the undergraduate

30. For instructional videos on other topics, go to
This material is based upon work supported by the National Science Foundation under Grant # 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.

31. The End - Really