Calculating a SQRT digit-by-digit? I need to calculate the SQRT of  $x$ to $y$ decimal places. I'm dealing with $128$-bit precision, giving a limit of $28$ decimal places. Obviously, if $\,y > 28$, the Babylonian method, which I'm currently using, becomes futile, as it simply doesn't offer $y$ decimal places.
My question to you is, can I calculate the Square Root digit-by-digit? For example, calculating the $n$-th digit, assuming I already have digit $\,n - 1$. I'm well aware this may be impossible, as I've not found any such method through Google yet, just checking here before I give up in search of arbitrary precision methods.

According to https://en.wikipedia.org/wiki/Computational_complexity_of_mathematical_operations#Mathematical_constants, square root of 2 can be computed as fast as the optimal multiplication algorithm. According to https://en.wikipedia.org/wiki/F%C3%BCrer%27s_algorithm, multiplication can be done in time n log n 2^(log * n), where log * n is the super-logarithm of n, the inverse tetration of n to the base 2.
