Peculiar Behavior of the Newton Method

Consider the function , which has the three obvious roots 1, 2, and 3. These roots can be obtained using the Newton technique
.
This iterative numerical technique requires an initial guess, .
This Demonstration computes the solution obtained by the Newton method (indicated by a green dot) for user-set value of the initial guess (indicated by the blue line). It turns out that when the initial guess is not properly chosen, one can get an unexpected solution: the solution furthest away from the initial guess is obtained.
This can be clearly seen from the first two snapshots where one finds, for the following initial guesses and , the roots and , respectively. A plot of the solution obtained for any value of the initial guess is given and the peculiar behavior of the Newton method is indicated by the two blue circles. This phenomena is readily explained if one plots versus , which presents a very pronounced increase or decrease when is in the region either around or near .

SNAPSHOTS

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DETAILS

Reference
[1] K. J. Beers, Numerical Methods for Chemical Engineering, Cambridge: Cambridge University Press, 2007.
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