I'm trying to find a few targeted eigenvalues of a large sparse (N=1e6,nnz=4e6) non-symmetric matrix. Currently I'm using MATLAB's 'eigs' function with the 'sigma' option and this uses the Shifted Inverse method with Arnoldi iterations. The problem is that I'm running out of memory with the LU matrix factorisation, even with 32GB of RAM. I'm also trying incomplete LU factorisations as preconditioners to iterative solvers (bicg), which seem to work, but are very slow. I can do the iterative solve without a preconditioner, but this is even slower.
Is there any software or methods that can accomplish this task without matrix factorisations? Any other suggestions/strategies would be appreciated.