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Newton Raphson Method

Newton Raphson Method. Rafael Sabino COT 4210 October 5 th , 2004 . Introduction . History of Newton Raphson’s Method The Method Example Practical Considerations Summary. History . Discovered by Isaac Newton and published in his Method of Fluxions, 1736

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Newton Raphson Method

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  1. Newton Raphson Method Rafael Sabino COT 4210 October 5th, 2004

  2. Introduction • History of Newton Raphson’s Method • The Method • Example • Practical Considerations • Summary

  3. History • Discovered by Isaac Newton and published in his Method of Fluxions, 1736 • Joseph Raphson described the method in Analysis Aequationum in 1690 • Method of Fluxions was written earlier in 1671 • Today it is used in a wide variety of subjects, including Computer Vision and Artificial Intelligence

  4. Why is it useful? • Suppose a dealer would like to sell you a car, for $18,000 dollars or for payments of $375 per month • What is the monthly interest rate they are charging you?

  5. The Method: How does it work?

  6. Correction Term • Correction Term: • The term used to get an estimate closer to root r:

  7. Correction Term: Continued • If the sequence of x’s becomes closer and closer to r as n becomes large, then:

  8. Algorithm • The limit describes the algorithm • In actual use, the Newton-Raphson method is also used with certain special tests

  9. In Three Dimensions

  10. Practical Considerations • Guess work • It may never converge • Algorithm problems

  11. Example

  12. Example: Cont’ • As the method iterates, the x sequences start getting closer and closer to the root

  13. Summary • The Newton-Raphson method is an alternative to solve roots for equations • It uses a correction factor • Used in computer programs today to solve extremely complicated equations

  14. Reference • Calculus, Fourth Edition, James Steward, 1999 • The New Turing Omnibus, AK Dewdney, 1993 • www.somath.com

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