I am trying to solve a very sparse linear system Ax = b. A is very sparse – if A is of size N^2 x N^2, all nontrivial elements for any row k are located in between (k-2N, k+2N). The overall the number of nontrivial elements in A is bounded by 13*N^2 (for a matrix with N^4 elements)
Now back to the original question: Currently I am using mldivide to solve the system. On a 1e6 x 1e6 matrix, the program takes around 60-75s 64 bit machine. Now it is possible to beat this performance using iterative methods? The ones I have tried (built into MATLAB) do not seem to offer any advantages; however, I think that may be due to the fact that I am not using a preconditioner. The problem is, how do I effectively get a preconditioner? I tried using ilu, but that is also pretty slow (and there is no guarantee that it will do the trick).