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Golden Section Search Method

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Golden section search method l.jpg

Golden Section Search Method

Major: All Engineering Majors

Authors: Autar Kaw, Ali Yalcin

http://nm.mathforcollege.com

Transforming Numerical Methods Education for STEM Undergraduates

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Golden Section Search Methodhttp://nm.mathforcollege.com


Equal interval search method l.jpg

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

(a+b)/2

Figure 1 Equal interval search method.

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f2

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Xl

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Golden Section Search Method

  • The Equal Interval method is inefficient when  is small.

  • 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 l.jpg

f2

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a-b

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Golden Section Search Method-Selecting the Intermediate Points

Determining the first intermediate point

Determining the second intermediate point

Golden Ratio=>

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Golden Section Search-Determining the new search region

  • If then the new interval is

  • If then the new interval is

  • All that is left to do is to determine the location of the second intermediate point.

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Example l.jpg

Example

.

2

2

2

The cross-sectional area A of a gutter with equal base and edge length of 2 is given by

Find the angle  which maximizes the cross-sectional area of the gutter. Using an initial interval of find the solution after 2 iterations. Use an initial .

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Solution l.jpg

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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:

X1=?

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Solution Cont

To check the stopping criteria the difference between and is calculated to be

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X2

Xl

Xu

Solution Cont

Iteration 2

X1

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Theoretical Solution and Convergence

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.

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Additional Resources

For all resources on this topic such as digital audiovisual lectures, primers, textbook chapters, multiple-choice tests, worksheets in MATLAB, MATHEMATICA, MathCad and MAPLE, blogs, related physical problems, please visit

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THE END

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