Fast calculation of eigenvalues


Now I have a symmetric matrix A. I need to find the first n eigenvalues and eigenvectors. (I have not actually started to write code, but in the algorithm process I observed the need to solve the eigenvalues and eigenvectors of matrix A.)

On the details of matrix A:

  • A should be a Laplace matrix.
  • The size of A may be 100,000 × 100,000.
  • A is a symmetric positive semidefinite matrix.
  • The matrix A that I get from calculation is usually dense, and it is not a sparse matrix.
    (If you have some suggestions for solving sparse matrix in the same situation, it is also possible)


My question is which method can be used for quick solution ? (including parallel computing with GPU or multi-core processor)

So far as I know, there are the following ways to achieve my goal in Julia:

  • using LinearAlgebra: eigvals(A) and eigvecs(A)
  • using Arpack: eigs(A)
  • using KrylovKit: eigsolve(A)

At present, I don’t know which of the three methods is the fastest (for large symmetric matrix). Or is there any faster calculation method besides the above methods?

Any reply is highly appreciated! :grinning:

1 Like

Properties of the matrix?

Sorry for the lack of detailed information :sweat_smile:, I’ve updated the questions so far.

The important point is: is it sparse or dense? If dense, you can’t easily beat eigen. If it’s sparse and you only need a couple of eigenpairs, use Lanczos from Arpack/KrylovKit, or LOBPCG from IterativeSolvers.


Also, “large” is somewhat subjective, so it would be useful to know the typical size.

An MWE with a function that generates a matrix that reflects the structure of your matrix may result in more concrete answers.


You have the option ishermitian in KrylovKit.jl. It should not speed things up too much though.

Thank you for your advice, I have updated the question. :grinning: