How do computer algebra packages like Sagemath implement rank of a matrix I am not sure if this is the right place to ask this question, but I believe there will be people here who do computations on computer algebra packages like Sage in their work. 
I have been using Sagemath to perform some matrix rank computations. It turns up a few bizarre results occasionally.
For example, I had to find the rank of a matrix ($100 \times 150$) with large integer entries (entries of magnitude in the range of $1$ to $10^{15}$). When I wrote the code with the matrix M declared as matrix(ZZ, R, C), or as matrix(QQ, R, C), it returns a rank of around 90 (which I believe is correct), whereas if I declare the matrix over the reals as matrix(RR, R, C), it returns a rank of around 50, which I believe is too low based on some conjectures I have.
So, overall I am curious, what are the standard way(s) to implement rank computation (and does it differ based on reals, or rationals) and where can I read more about these? If these issues arise due to precision errors, how can I get around them? And more importantly, how do I know beforehand that my computation is susceptible to precision errors?
(I tried looking up the source code of Sagemath a couple of times, but I was quickly lost, so I hope someone can point me to the precise documentation/source code)
 A: I don't know what algorithm Sage actually uses, but computing rank over the integers is fun and easy: Complexity of computing matrix rank over integers . It is NOT so easy if you want good running time, and for that there are a number of papers of Arne Storjohann, which show that one can do it asymptotically as fast as for real matrices (a surprising result, in view of coefficient blow-up). Storjohann has actually implemented his algorithms, and I am sure Sage uses this or something like it.
Chen, Zhuliang; Storjohann, Arne, A BLAS based C library for exact linear algebra on integer matrices, Kauers, Manuel (ed.), Proceedings of the 2005 international symposium on symbolic and algebraic computation, ISSAC’05, Beijing, China, July 24--27, 2005. New York, NY: ACM Press (ISBN 1-59593-095-7). 92-99 (2005). ZBL1360.65086.
As for the reals, the fastest way to compute rank is to compute the singular value decomposition, and throw away singular values below some cutoff (probably around $10^{-6}$). As pointed out by many in the comments, rank is very unstable, so unless you want the "pseudo-rank" (as above, defined by the size of the singular values), don't go there. 
