I have a simple (and small) linear homogeneous system $Ax=0$, where the entries of the $N\times M$ matrix $A$ are small integers. I do not need fancy methods which efficiently solve almost singular matrices and treat roundoff errors etc. But I need to know if a system is singular and also the general solution for underdetermined systems. It must be rock-solid. Any reference to such an algorithm is welcome. I am coding in C. Is SVD the way to go?
Numeric solution of simple but possibly singular linear system
linear solver
The Integer Matrix Library is a C library that claims to be able to compute the null space of an integer matrix. See also this answer on MathOverflow to the exact same question, which gives a list of libraries (including PARI, which also can be called from C and is still being updated). You could also take a look at LinBox, even though it's written in C++.