Eigenvalue decomposition of very large matrices

I'm doing a spectral decomposition of symmetric positive matrices. I have a 16Gb machine and can compute the evalues and evectors for a 30k x 30k matrix in reasonable time (~2hr). I need to scale up to a matrix 20 times that size. As eig is cubic this would take a couple of years.

I have access to a cluster that has a matlab parallel licence but am not sure how to proceed. All advice is much appreciated.

Thanks in advance.

[Matt J]

I'm naturally suspicious of anything that looks like it requires that much brute force compute power.

Are you sure your matrix has no special structure to it that could be exploited other than symmetry and positivity? Is the matrix really "positive" or did you mean "positive definite"?

Answer 1

[Paul]

If you have a sparse matrix (lots of zeros), you should use eigs (determines the eigenvalues of a sparse matrix). First declare the matrix sparse by doing:

      M=sparse(M);
      eigval = eigs(M,30000);

Normally eigs returns the 6 largest eigenvalues. The second argument of eigs in this case is the number of eigenvalues you want. If you want them all, put 30000 there.

[Matt J]

If you want them all, length(M) = #eigenvalues.

If you do this, eigs() will pre-convert the matrix from sparse to full. You may as well just use EIG and start with non-sparse M.

[Paul]

Hmm you are right, if you actually want all eigenvalues Matlab uses the eig alghorithm. If you dont want them all though, but only the largest k eigenvalues, eigs should be faster.

[Matt J]

Conor commented

Unfortunately my matrix starts out as sparse but I need to transform it by subtracting it from 1.