Rafael Sabino COT 4210 October 5th, 2004 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?

The Method: How does it work?

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

In Three Dimensions

Practical Considerations Guess work It may never converge Algorithm problems

Example

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