I have a large sparse matrix, for example, 128000×128000 SparseMatrixCSC{Complex{Float64},Int64} with 1376000 stored entries.
How to quickly get all eigenvalues of the sparse matrix ? Is it possible ?
I tried eigs for 128000×128000 with 1376000 stored entries but the kernel was dead.
I use a mac book pro with 16GB memory and Julia 1.3.1 on jupyter notebook.
As far as I'm aware (and I would love to be proven wrong) there is no efficient way to get all the eigenvalues of a general sparse matrix.
The main algorithm to compute the eigenvalues of a matrix is the QR algorithm. The first step of the QR algorithm is to reduce the matrix to a Hessenberg form (in order to do the QR factorisations in O(n) time). The problem is that reducing a matrix to Hessenberg form destroys the sparsity and you just end up with a dense matrix.
There are also other methods to compute the eigenvalues of a matrix like the (inverse) power iteration, that only require matrix vector products and solving linear systems, but these only give you the largest or smallest eigenvalues, and they become expensive when you want to compute all the eigenvalues (they require storing the eigenvectors for the "deflation").
So that was in general, now if your matrix has some special structure there may some better alternatives. For example, if your matrix is symmetric, then its Hessenberg form is tridiagonal and you can compute all the eigenvalues pretty fast.
TLDR: Is it possible ? — in general, no.
P.S: I tried to keep this short but if you're interested I can give you more details on any part of the answer.
