What is the best known algorithm for exactly solving a large sparse system of linear equations? The system I'm working on is not symmetric, not positive definite and integer. The only benefit is being sparse. I also need to point out that the matrix is not square. The dimension is m×n and it is not generally either underestimate or overestimate.
The exact solution of linear equations with rational coefficients belongs to the field of computer algebra. For an entry to the literature, see
You can do a literature search based on this and the citation facilities of http://scholar.google.com .
Krylov Iterative methods are a usual choice.
If you happen to have access to Mathematica, it offers a good way to test for different method: if A is your matrix, write B=SparseArray[A]; Then use the LinearSolve function with Method->"Krylov". You can also test to see if there are advantages to retaining integer digits. Converting to real numbers may yield faster results, possibly at the cost of accuracy.