There are some methods to solve problem. 1. Gauss. It really has troubles with precision. 2. Method of iterations. It works good, but your implementation should be fast enough and do about 2^50 iterations :)
We tried both methods but failed... It seems that using iterations could lead to precision problem too..we calc something like mat,and do mat^(2^60),but it turns out that the value in mat overflows... Sorry for my poor english-_-
I think that there exists a good alternative to Gaussian elimination - QR - decomposition of the matrix. It's precision, I think, would be good enough because it doesn't change the condition number of the matrix.
These methods have less calculation errors than Gauss method. But there is modification of Gauss method which more precise than QR-decomposition methods. In this modification we choose main element from all remaining elements in matrix.
You can read about methods for solving systems of linear algebraic equations in the book: А.А.Амосов "Вычислительные методы для инженеров". Of course, there are a lot of other sources.