I am currently researching on the viability of using KS methods for solving large dense systems. What I wish to prove (or disprove) is that methods like CG, BiCG and QMR are as good (if not better) than the generic LU or QR decomposition methods in place today.
Till now I have tried non-preconditioned versions of CG on symmetric positive definite matrix (duh!) and compared it against LU(DGESV). (I have written so in Fortran and MATLAB as well) The results are not pleasing. CG loses by orders of magnitude.
I have written (or found and optimized) codes for QMR, BiCG, CGNE, CGNR and others as well but their results are worse.
Now, I have two choices
Try improve BLAS Level 2 routines to yield a better convergence. ( I use Intel MKL in Fortran). But, it turns out, there is nothing I can do to make this better. Or can I? Profiling shows bad use of threads and matrix-vector performance. Which is not surprising for BLAS 1/2 operations.
Work on preconditioners and hope that I come across a good one which solves a linear system faster than LU/QR etc.
(Bonus Option) Leave this idea and try something else for my thesis. (I have time left to do this). Maybe something else in the domain of Krylov Subspace Methods.
Any help is much appreciated. P.S. I had no clue this group existed. This is awesome!