I am looking for preconditioners which don't assume anything about the matrix or its origins. I basically want to be able to type in the following in MATLAB and have quick solving time:
a = rand(5000,5000); b = rand(5000,1); precond_a= my_precond_algorithm(a); qmr(a,b,1e-8,100,precond_a)
Needless to say, $a$ is dense.
I have looked into:
LU works well. But that's no surprise.
I am still to find a good algorithm for ILU for dense matrices but I reckon that should work relatively well.
Sparse Inverse Approximators (Benzi et. al.) .
A paper by Prakash and Mittra discusses the use of Multifrontal Preconds for solving dense Maxwell Equations discretization.
Other than LU, I am still a little concerned about the viability of using them as effective preconditioners for large dense matrices. Any resources/comments would be much appreciated!