Tool to diagonalize large matrices

I want to compute a diffusion kernel, which involves taking exp(b*A) where A is a large matrix. In order to play with value开发者_高级运维s of b, I'd like to diagonalize A (so that exp(A) runs quickly).

My matrix is about 25k x 25k, but is very sparse - only about 60k values are non-zero. Matlab's "eigs" function runs of out memory, as does octave's "eig" and R's "eigen." Is there a tool to find the decomposition of large, sparse matrices?

Dunno if this is relevant, but A is an adjacency matrix, so it's symmetric, and it is full rank.

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Have you tried SVD, svds for sparse matrix in matlab.

EDIT: one more thing, don't do full rank SVD since the dimension is big, use a small rank, say 500, so that your solution fits in the memory. This cuts the small eigenvalues and their vectors out. Thus it does not hurt your accuracy much.

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Have you considered the following property: exp(A*t) = L^(-1) {(sI-A)^(-1)} where L^(-1) the inverse Laplace transform? - provided that you can invert (sI-A)

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If you have access to a 64 bit machine and octave compiled with 64 bit support, you might be able to get around this problem.

Also, I don't know what platform you are running all of this on, but in UNIX based systems you can use ulimit to increase the maximum allowed stack size for user processes.

For example, you can run

ulimit -u unlimited

and this will ensure that there are no memory limits etc on your processes. This is not a good idea in general, since you have have runaway processes that will completely bog down your machine. Try instead

ulimit -s [stacksize]

to increase the stack size limit.

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Octave has splu which does lu decomposition for sparse matrices. I am not sure whether it can handle 25k x 25k but worth a shot.

Alternatively if your matrix is structured like so: A = [B zeros;zeros C] then you can diagonalize B and C separately and put them together into one matrix. I am guessing you can do something similar for eig.

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In R you could check igraph package and function arpack which is interface to ARPACK library.