Newton Raphson Method

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Newton Raphson Method

• Rafael Sabino
• COT 4210
• October 5th, 2004

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Introduction

• History of Newton Raphsons Method
• The Method
• Example
• Practical Considerations
• Summary

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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

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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?

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The Method How does it work?
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Correction Term

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

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Correction Term Continued

• If the sequence of xs becomes closer and closer
  to r as n becomes large, then

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Algorithm

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

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In Three Dimensions
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Practical Considerations

• Guess work
• It may never converge
• Algorithm problems

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Example
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Example Cont

• As the method iterates, the x sequences start
  getting closer and closer to the root

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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

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Reference

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