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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

Newton Raphson Method

Rafael Sabino

COT 4210

October 5th, 2004

introduction
Introduction
  • History of Newton Raphson’s Method
  • The Method
  • Example
  • Practical Considerations
  • Summary
history
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
why is it useful
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?
correction term
Correction Term
  • Correction Term:
      • The term used to get an estimate closer to root r:
correction term continued
Correction Term: Continued
  • If the sequence of x’s becomes closer and closer to r as n becomes large, then:
algorithm
Algorithm
  • The limit describes the algorithm
  • In actual use, the Newton-Raphson method is also used with certain special tests
practical considerations
Practical Considerations
  • Guess work
  • It may never converge
  • Algorithm problems
example cont
Example: Cont’
  • As the method iterates, the x sequences start getting closer and closer to the root
summary
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
reference
Reference
  • Calculus, Fourth Edition, James Steward, 1999
  • The New Turing Omnibus, AK Dewdney, 1993
  • www.somath.com