I am finding the eigenvalues of dense NxN Hermitian matrix which is calculated from a density operator in quantum physics. All the eigenvalues are needed as I need to calculate the sum of the absolute value of them. I would like to know whether there are good solvers for this problem that can run in parallel?
The problem size I am interested should be at least N=40000. The one I am currently using is the gsl library, which has the advantage that it takes no extra working memory except the matrix itself. However, the running time is close to 5 hours for N=20000 because of no parallelization. I am going to run the problem in computing cluster that has walltime limit of 3 days so my program is likely to be killed.
I have tried SLEPc, but it does not scale well for the dense matrix. The computation time with 8 cores is almost the same as single core gsl! Not to mention it takes more memory and less accurate result in my case.