Stopping Criterion

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.

1. $\vert x_{n+1} - x_{n}\vert$ sufficiently small. If the absolute value of two succesive rounded iterates agree to the same number of decimal places then $x_{n+1}$, 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.

2. $\vert F(x_{n})\vert$ sufficiently small in some sense for some $x_{n}$. We are looking for the value of $x$ that makes $F(x) = 0$ so when $F(x)$ is sufficiently small we must be close to the root.

3. 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.

4. Stop if $F^{\prime}(x_{n}) = 0$. 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.