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Title: Newton's Method
Book Description: The Newton's Method iteration is a well-known method for solving equations of the type f(x)=0. When it converges, it does so quadratically, and so it is often an efficient procedure. The catch is that, even when there is a well-defined zero in the interval of interest, it may not converge. What is worse, the iteration may enter into a stable periodic orbit, or it might actually become "chaotic". Both can happen, for example, and this is illustrated in this WorkBook, when the function has a vertical tangent at the zero. In the workbook the player may watch the itinerary of any Newton's method calculation he sets up. He may view the graph and/or see the numeric values. There is a button that automatically sets up the iteration for our pathological function. It comes with a parameter like the parameter for the logistic function iteration. Initially, that parameter is c = 3.84 and one sees a stable periodic orbit. When the parameter is set to 4, then the player may set any initial condition and watch the chaotic behavior.
Author: James White
Suggested Use: experiment with Newton's method and chaos
Topics: chaos, Newton's method
Number of Pages: 2
Animation: Yes
Grade
Level: 