In real problems when the size is large ($n>100000$) the best approach is the numerical solution of the system by an iterative method like GMRES, CG, BiCG, etc. All those algorithms depends strongly on matrix-vector products, so the complexity is as much as $O(n^2)$ which performs better than a direct solution by a LU decomposition with complexity of $O(n^3)$.
Sometimes the scale of the problem is not large enough to the proposed methods to be an improvement compared to LU decomposition, so there is a work you have to do in balance the applicability of the methods.
