If I use scipy.linalg.solve (which, I believe, calls the gesv LAPACK function) on an unknown problem of ~ 12000 (with size of ~ 12000 sq., Dense, asymmetric matrix) on my workstation, I get a good answer in 10-15 minutes .
Just to explore the limits of the possible (note that I do not say “useful”), I doubled the resolution of my main problem, which leads to the need to solve for ~ 50,000 unknowns. Although this will technically work on my workstation, as soon as I add a few dozen more swap copies, it would be wiser to use some HWs with ample RAM, and so I released it to the AWS EC2 quad-equalizer ... where it has been looking for the last 14 hours (hey, spotty specimens are cheap) and it's impossible to tell how far it goes.
Unfortunately, I don't know what the time complexity of the solvers is (my google-fu didn't help me with this). If it is O (N ^ 2), I would expect it to be done in about 4 hours; if it is O (N ^ 3), then perhaps it will be done in 16 hours. Of course, the interpretation of N as the number of unknowns - which is four times - the number of elements in the matrix increased by 16 times!
And tips that will help me determine if this has a chance to complete (project) in my life or did not get a grateful result!
Additional Information:
Sparse matrices are of little interest here (my matrix is dense, and in any case scipy does not work with more than 2**31 non-zero elements, even on a 64-bit basis).
I am using Debian / Squeeze scipy on a Ubuntu 12.04 workstation on EC2. Obviously, both are 64-bit.
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