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

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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 http://nm.mathforcollege.com

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=? http://nm.mathforcollege.com

Solution Cont To check the stopping criteria the difference between and is calculated to be http://nm.mathforcollege.com

Solution Cont Iteration 2 XL X2 Xu X1 http://nm.mathforcollege.com

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 1.0472 radians and gives a maximum cross-sectional area of 5.1962. http://nm.mathforcollege.com

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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.

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