MATLAB: How to find all the eigenvectors of a very large, sparse, symmetric, positive-definite matrix in MATLAB 7.8 (R2009a)

MATLAB

I am trying to get all the eigenvectors and eigenvalues for a very large (22500×22500) sparse, symmetric, positive-definite real matrix A.

I am able to get all the eigenvalues of this matrix using the function EIG. However when I try to find the eigenvectors using EIGS, I run into out of memory problems.

[V D] = eigs(A, 22000);

Best Answer

You are receiving the out of memory error because EIGS tries to return all of the eigenvectors at once and it is not possible to store all the 22500 eigenvectors on a 32-bit machine.

As a workaround, knowledge of the eigenvalues can be used to access the eigenvectors in chunks using the function EIGS as follows:

[V D] = eigs(A,100,0);

The above code will give 100 eigenvectors corresponding to eigenvalues closest to 0. For better results, never use an eigenvalue as the shift (third argument in EIGS).

When retreiving the eirgenvectors, note that although the matrix is sparse the eigenvectors are most likely dense.

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MATLAB: How to make sparse, group association

" I would prefer the matrix to be symmetric." Why should it be? If person 1 and 2 are associated with group 4, it does not mean that person 4 is associated with group 1 and 2. There's not even a guarantee your matrix will be square unless you have the same number of groups as persons.

"Don't send me the link to matlab sparse page, please. I don't get it." Maybe you should try getting it, otherwise you won't get very far.

demodata = [1 5
            1 3
            2 4
            1 4
            2 1
            3 2];
connections = sparse(demodata(:, 1), demodata(:, 2), 1)

Also note that your original storage is actually more efficient in term of memory than the sparse matrix since you don't have to store the '1' that carries no information.

MATLAB: How should I compute the eigenvectors of a sparse, real, symmetric matrix

Just convert the matrix to a full one, and use eig. Unless less you want just a few eigenvectors, then the decomposition using the sparse matrix will generally be slower anyway. E.g.:

>> A = sprand(2e3, 2e3, 2e-3);
>> A = A' + A;
>> tic; [V, D] = eigs(A, 2e3-2); toc;
Elapsed time is 21.727999 seconds.
>> tic; [V, D] = eig(full(A)); toc
Elapsed time is 2.488241 seconds.

Moral of the story: sparse isn't always faster!

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