Newton Raphson Method

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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

Rafael Sabino

COT 4210

October 5th, 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
• 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?
• 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:
• The term used to get an estimate closer to root r:
Correction Term: Continued
• If the sequence of x’s becomes closer and closer to r as n becomes large, then:
Algorithm
• The limit describes the algorithm
• In actual use, the Newton-Raphson method is also used with certain special tests
Practical Considerations
• Guess work
• It may never converge
• Algorithm problems
Example: Cont’
• As the method iterates, the x sequences start getting closer and closer to the root
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
• Calculus, Fourth Edition, James Steward, 1999
• The New Turing Omnibus, AK Dewdney, 1993
• www.somath.com