Seems that LinearAlgebra.jl does not work with sparse matrices, and the libraries that do (e.g. Arpack.jl, ArnoldiMethod.jl, KrylovKit.jl) all use iterative methods, which are not efficient to find all the eigenvalues. Short of using LinearAlgebra.jl with a regular matrix, do I have an alternative?

To elaborate a bit on @Oscar_Smith ’s answer: The main reason why there is in general in no efficient algorithm is because the matrix of eigenvectors is generally not sparse even if your original matrix is. Thus if you compute all eigenvectors you are computing a dense matrix anyways and there is no …

I don’t think it’s necessary to get rancorous. As others have said, the general answer to your question is that it currently seems impossible (or at least impractical) to avoid dense methods for general sparse matrices if you want precise values for all the eigenvalues. (Unless your sparsity pattern…

For many problems in applied mathematics that at first seem to require the whole spectrum, people have worked hard to find sparse-compatible workarounds that compute a final result in a different way while avoiding computing the explicit spectrum as an intermediate step. This is a good point. For…

In a grad-school class based on Trefethen’s text I also came across a need to compute all eigenvalues/eigenvectors of sparse matrices. I ended up just using my own hand-written Arnoldi method, but in my literature review I came across a figure: [image] (Google Image search finds this presentatio…

[image] GregVernon: I also came across a need to compute all eigenvalues/eigenvectors of sparse matrices. I ended up just using my own hand-written Arnoldi method If you run Arnoldi long enough to compute all of the eigenvalues (i.e. m steps without restarting for an m \times m matrix), isn’t…

Do tell: if some eigenvalues of a VERY large matrix (what is very large in your neck of the woods?) are not enough for some people, what do they do?

IIRC at the time there was some shaking going on in the Julia ecosystem regarding eigenvalue solvers? In any case, the class was numerical linear algebra and I’d just written an Arnoldi solver in Matlab for an assignment (which I then used to make this animation on the Arnoldi Iteration page in Wiki…

[image] stevengj: Don’t those compute continuous approximations for the spectral density / DOS (as well as other functions of the spectrum), rather than individual eigenvalues per se? Admittedly I don’t know much about the method(s), but found the graphic at least interesting on the surface -…

FEAST and others build efficient solvers for large numbers of eigenpairs by separating the spectrum into pieces and distributing the pieces over compute nodes. Although they take more operations than the standard (also iterative!) dense-matrix methods, they would win out for the full spectrum of big…

[image] Ralph_Smith: Although they take more operations than the standard (also iterative!) dense-matrix methods, they would win out for the full spectrum of big enough problems because of the cleaner parallelism and memory use. I’m not aware of any implementations or wrappers of these in Julia …

Another possible option is the Eigenvalues Slicing Library: [1802.05215] The Eigenvalues Slicing Library (EVSL): Algorithms, Implementation, and Software (No wrapper in Julia that I am aware)

Finding all the eigenvalues of a sparse matrix?

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Finding *all* the eigenvalues of a sparse matrix?

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Finding all the eigenvalues of a sparse matrix?