1. Numerical Methods Newton’s Method for One - Dimensional Optimization - Theory http://nm.mathforcollege.com http://nm.mathforcollege.com

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5. Newton’s Method-Overview Open search method A good initial estimate of the solution is required The objective function must be twice differentiable Unlike Golden Section Search method Lower and upper search boundaries are not required (open vs. bracketing) May not converge to the optimal solution 5 http://nm.mathforcollege.com

6. 6 Newton’s Method-How it works The derivative of the function,Nonlinear root finding equation, at the function’s maximum and minimum. The minima and the maxima can be found by applying the Newton-Raphson method to the derivative, essentially obtaining Next slide will explain how to get/derive the above formula. http://nm.mathforcollege.com

7. 7 Slope @ pt. C ≈ We “wish” that in the next iteration x i+1 will be the root, or. Thus: Slope @ pt. C = F(x) xixi x i+1 x i – x i+1 A B E F C D F(x i ) - 0 F(x i+1 ) F(x i ) x Or Hence: N-R Equation Newton’s Method-To find root of a nonlinear equation http://nm.mathforcollege.com

8.  If,then.  For Multi-variable case,then N-R method becomes 8 Newton’s Method-To find root of a nonlineat equation http://nm.mathforcollege.com

9. Newton’s Method-Algorithm Initialization: Determine a reasonably good estimate for the maxima or the minima of the function. Step 1. Determine and. Step 2. Substitute (initial estimate for the first iteration) and into to determine and the function value in iteration i. Step 3.If the value of the first derivative of the function is zero then you have reached the optimum (maxima or minima). Otherwise, repeat Step 2 with the new value of 9 http://nm.mathforcollege.com

10. THE END http://nm.mathforcollege.com

11. This instructional power point brought to you by Numerical Methods for STEM undergraduate http://http://nm.mathforcollege.com Committed to bringing numerical methods to the undergraduate Acknowledgement

12. For instructional videos on other topics, go to http://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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14. Numerical Methods Newton’s Method for One - Dimensional Optimization - Example http://nm.mathforcollege.com http://nm.mathforcollege.com

15. For more details on this topic  Go to http://nm.mathforcollege.comhttp://nm.mathforcollege.com  Click on Keyword  Click on Newton’s Method for One- Dimensional Optimization

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

17. 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.

18. 18 Example 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. 2 2 2   http://nm.mathforcollege.com

19. 19 Solution The function to be maximized is Iteration 1: Use as the initial estimate of the solution http://nm.mathforcollege.com

20. 20 Solution Cont. Iteration 2: Iteration  10.78542.8284-10.82841.04665.1962 21.04660.0062-10.39591.04725.1962 31.04721.06E-06-10.39231.04725.1962 41.04723.06E-14-10.39231.04725.1962 51.04721.3322E-15-10.39231.04725.1962 Summary of iterations Remember that the actual solution to the problem is at 60 degrees or 1.0472 radians. http://nm.mathforcollege.com

21. THE END http://nm.mathforcollege.com

22. This instructional power point brought to you by Numerical Methods for STEM undergraduate http://http://nm.mathforcollege.com Committed to bringing numerical methods to the undergraduate Acknowledgement

23. For instructional videos on other topics, go to http://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.

24. The End - Really