1. Rafael Sabino COT 4210 October 5th, 2004
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