Problem 26: 1998-1999
Given f(x)=x3 - x - 1, use Newton's Method to approximate the root (with initial guess x=2) such that the error < .00001. How many iterations does this take?

Difficulty rating: 

Solution:

        To begin, you need to define your important functions: f(x) = x3 - x - 1 and  f '(x) = 3x2 - 1. We now use our initial guess of x = 2 to plug into Newton's Method; getting the result from it will result (hopefully) in a better guess at what the root actually is. Each time you repeat the process, it is called an iteration, and the process is continued until the difference between consecutive approximations is less that the error stated. So, we begin:

Thus, after one iteration, we get x = 1.5454545. Use this, now, as the "old" value and repeat the process. After five iterations, the root is close enough. You should use your calculator do do the repetitive calculations, and you should get: 1.54545, 1.35961, 1.32580, 1.324719, 1.324718. As the final two have a difference of less than 0.00001 (the stated allowable error in the problem), you are finished.

Using Newton's Method, you can get as close as you want to the actual root, as long as you're willing to number crunch.


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