You need to decide how much of an error your application can tolerate. Once you know this, that tolerance can be built into a decision rule for stopping Newton's Method. Naturally, we are interested in whether the equation f(x) = 0 is close to being true. Considering this, a common stopping rule is to stop computing estimates and use the current estimate, xn, as your estimate for x*, when
|f(xn)| is less than the tolerance in the application.
When the estimates are approaching x*, they will tend to crowd together, so an indication that it would be a good time to check |f(xn)| would be when |xn - xn-1| is less than some preset tolerance level. This, however, can be deceiving, and in any case the size of |f(xn)| should be checked before the process is stopped.
Copyright c 1996 by Steve Hetzler
and Bob Tardiff Department of Mathematics
and Computer Science, Salisbury State University, Salisbury, MD 21801.