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numerical methods golden section search method theory http nm mathforcollege com
Numerical MethodsGolden Section Search Method - Theoryhttp://nm.mathforcollege.com
slide2
For more details on this topic
  • Go to http://nm.mathforcollege.com
  • Click on Keyword
  • Click on Golden Section Search Method
you are free
You are free
  • to Share – to copy, distribute, display and perform the work
  • to Remix – to make derivative works
under the following conditions
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.
equal interval search method

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.

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golden section search method

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

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golden section search method selecting the intermediate points

f2

f1

f1

fL

fL

fu

fu

a-b

b

X2

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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=>

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golden section search method1
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!

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golden section search determining the new search region

f2

f1

f2

f1

fL

fL

fu

fu

X2

X2

XL

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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

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golden section search determining the new search region1
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.

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golden section search 1 d line search method
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

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golden section search 1 d line search method1
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 ) !!

_

_

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golden section search 1 d line search method2
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.

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slide14

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.

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the end
The End

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acknowledgement
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

slide17

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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