eigen value problem for sparse matrices

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amirhossein amirabadi
评论： Andrew Knyazev 2018-9-21
Hi, I need to calculate all eigen values and eigen vectors of a very large sparse matrix(above 20k*20k) but an out of memory error will occure! how can I overcome this problem? thank you.
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amirhossein amirabadi
amirhossein amirabadi 2018-8-28
编辑：amirhossein amirabadi 2018-8-28
and by the way I find a way to solve my problem, I figured that only small eigenvalues(and corresponding eigenvectors) are needed. so I used eigs for 1000 eigenvalues: eigs(my matrix,1000,'sm') thank you for your attention matt j.
Andrew Knyazev
Andrew Knyazev 2018-9-21
If the matrix is real symmetric or Hermitian, you may also want to try https://www.mathworks.com/matlabcentral/fileexchange/48-lobpcg-m

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采纳的回答

Matt J
Matt J 2018-8-22
Just use
eig(full(yourMatrix))
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amirhossein amirabadi
amirhossein amirabadi 2018-8-23
编辑：amirhossein amirabadi 2018-8-23
I tried single and it worked for one case(270k*270k), but by encrease the size of matrix(for example 350k), out of memory came again! my english is not very good I am sorry, thanks for your attention
amirhossein amirabadi
amirhossein amirabadi 2018-8-23
I tried this kinds of solutions before, I seeking for an algorithm to calcualate eigen values/vectors one by one, because I only need two members of the each eigen vectors! but eig and eigs give all of them. if I can calculate eigen vectors one by one, I save two members that I need, then clearing the others can save lots of memory.

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更多回答（2 个）

Christine Tobler
Christine Tobler 2018-8-22
编辑：Christine Tobler 2018-8-22
The problem is that the eigenvectors of a sparse matrix are dense (in all practically relevant cases), so to store them would require 3GB. Additionally, the algorithm requires some workspace, which will also be several GB.
One thing you can try is to use the 'vector' option, which saves memory because the eigenvalues are returned as a vector instead of a diagonal matrix (will need 3GB less memory):
[U, D] = eig(full(yourMatrix), 'vector');
I tried on my computer with this formula, and it took about 10GB of memory (and ran for 8 minutes), so you'll need a machine with more than this to compute all eigenvalues and eigenvectors of this matrix.
Note I'm assuming your matrix is symmetric; a non-symmetric matrix may require more memory than this.
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amirhossein amirabadi
amirhossein amirabadi 2018-8-23
Hi, thank you for your attention, yes my matrices are symmetric and also hermitian,
amirhossein amirabadi
amirhossein amirabadi 2018-8-23
I tried this kinds of solutions before, I seeking for an algorithm to calcualate eigen values/vectors one by one, because I only need two members of the each eigen vectors! but eig and eigs give all of them. if I can calculate eigen vectors one by one, I save two members that I need, then clearing the others can save lots of memory.

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Christine Tobler
Christine Tobler 2018-8-23
There aren't any practical algorithms for computing eigenvalues one by one. You can compute a subset of eigenvalues around a given value using eigs (assuming that you can solve a linear system with your matrix without going out of memory). This way, you could run through the spectrum of the matrix, and try to get all the eigenvalues, but this is quite a messy approach.
If the matrix is too large to solve a linear system with in memory, there is no way to compute anything except a set of its extreme eigenvalues using eig and eigs in MATLAB.
There are some iterative methods available that, instead of solving a linear system, would require a preconditioner (a matrix that is very similar to the inverse of the original matrix, but can be applied more quickly). It's typically quite tricky to find a good preconditioner, and very dependent on the structure of the matrix coming in.
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amirhossein amirabadi
amirhossein amirabadi 2018-8-23
编辑：amirhossein amirabadi 2018-8-23
I know that there is no algorithm for computing eigenvalues one by one and my problem is not about eigenvalues, its about eigenvectors, eig and eigs give eigenvectors together in a matrix in the same size of the original one and it cost lots of memory. if an algorithm exists that capable to calculate the eigenvectors by using the corresponding eigenvalues(eigenvalues can be reached by eig or eigs) it would be very helpful. as I said I only need two members of each eigenvector and if I can reach eigenvectors one by one, after calculating one of them I save that two members and clear the others and then I go after the next one and ... . thank you by the way.
Christine Tobler
Christine Tobler 2018-8-24
So if you have enough memory to compute all the eigenvalues (can you really do this for your 350k problem?), you could use the fact that A*x - d*x = 0, and solve the linear system (A - d*I) * x = 0. This has the problem that the matrix on the left-hand side matrix is singular - but you can instead choose a number a bit off from d, (maybe 1e-5? Depends on the other sizes), and calling eigs to compute the closest few eigenvalues and eigenvectors to that shift.
This will not be cheap, because the shifted matrix A - d*I has to be factorized for each shift, but it does compute each eigenvector separately (assuming that you can factorize A - d*I i memory|.

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