Two-variable iterative method:
This method is applicable for finding the square root of and converges best for . This, however, is no real limitation for a computer based calculation, as in base 2 floating point and fixed point representations, it is trivial to multiply s by an integer power of 4, and therefore by the corresponding power of 2, by changing the exponent or by shifting, respectively. Therefore, one can move s to the range. Moreover, the following method does not employ general divisions, but only additions, subtractions, multiplications, and divisions by powers of two, which are again trivial to implement. A disadvantage of the method is that numerical errors accumulate, in contrast to single variable iterative methods such as the Babylonian one.
The initialization step of this method is
While the iterative steps read
Then, (while ).
Note that the convergence of , and therefore also of , is quadratic.
The proof of the method is rather easy. First, we rewrite the iterative definition of as
Then it is straightforward to prove by induction that
And therefore the convergence of to the desired result is ensured by the convergence of to 0, which in turn follows from.