1 / 24

For more details on this topic Go to http://nm.mathforcollege.com Click on Keyword

Numerical Methods Newton’s Method for One -Dimensional Optimization - Theory http://nm.mathforcollege.com. For more details on this topic Go to http://nm.mathforcollege.com Click on Keyword Click on Newton’s Method for One-Dimensional Optimization. You are free.

hachi
Download Presentation

For more details on this topic Go to http://nm.mathforcollege.com Click on Keyword

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Numerical MethodsNewton’s Method for One -Dimensional Optimization - Theoryhttp://nm.mathforcollege.com

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

  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. 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 http://nm.mathforcollege.com

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

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

  10. The End http://nm.mathforcollege.com

  11. 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

  12. 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.

  13. The End - Really

  14. Numerical MethodsNewton’s Method for One -Dimensional Optimization - Examplehttp://nm.mathforcollege.com http://nm.mathforcollege.com

  15. For more details on this topic • Go to http://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. 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. http://nm.mathforcollege.com

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

  20. Solution Cont. Iteration 2: 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. 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

  23. 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.

  24. The End - Really

More Related