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Using any iterative scheme on a computer to estimate roots of an
equation requires some condition to be satisfied so that the
algorithm `knows'when to stop. There are various
possibilities and these are listed below.

sufficiently small. If the absolute
value of two succesive rounded iterates agree to the same number of
decimal places then , the last estimate, is correct to that
number of decimal places. Always take the last estimate as it is
almost always more accurate. This raises an important point. Always
keep two more decimal places in your calculations than the final
answer needs. Thus, if the final result must be correct to 4
decimal places, then you should keep 6 decimal places in your
workings.
 sufficiently small in some sense for some
. We are
looking for the value of that makes so when
is sufficiently small we must be close to the root.
 A certain number of iterations have been performed. Stopping
after, say 10 iterations, prevents the situation where the method has
failed to converge and is aimlessly looping. This is not a problem
when you use the method by hand as you will soon see if something
has gone wrong but can cause problems when the
method is implemented on a computer.
 Stop if
. This is extremely unlikely
but if it does happen then computers do not like dividing by zero.
It means the method has located a turning point of the function.
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Up: NewtonRaphson Method
Previous: NewtonRaphson Method
Prof. Alan Hood
20000201