Integer square root
For example, because and .
One way of calculating and is to use Newton's method to find a solution for the equation , giving the iterative formula
to ensure that
Using only integer division
For computing for very large integers n, one can use the quotient of Euclidean division for both of the division operations. This has the advantage of only using integers for each intermediate value, thus making the use of floating point representations of large numbers unnecessary. It is equivalent to using the iterative formula
By using the fact that
one can show that this will reach within a finite number of iterations.
However, is not necessarily a fixed point of the above iterative formula. Indeed, it can be shown that is a fixed point if and only if is not a perfect square. If is a perfect square, the sequence ends up in a period-two cycle between and instead of converging.
Domain of computation
Although is irrational for many , the sequence contains only rational terms when is rational. Thus, with this method it is unnecessary to exit the field of rational numbers in order to calculate , a fact which has some theoretical advantages.
One can prove that is the largest possible number for which the stopping criterion
ensures in the algorithm above.
In implementations which use number formats that cannot represent all rational numbers exactly (for example, floating point), a stopping constant less than one should be used to protect against roundoff errors.