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I have a complex Hermitian matrix of size about $70000\times 70000$. I want about 100 eigenvalues near 0. However, I know that every eigenvalues are two-fold degenerate. I found out that the running time of eigsh is extremely slow (more than 5 times) compared to the situation with no degeneracy.

Also, I found out from the following link Computation time of eigenvalues with ARPACK depends on what? that degeneracy is bad to eigsh algorithm.

What can I do to make my code run faster? In the answer of the link suggests to increase the size of the Krylov subspace, but what size is the adequate one?

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    $\begingroup$ Are you using the shifting method? $\endgroup$
    – nicoguaro ♦
    Jun 2, 2021 at 13:35
  • $\begingroup$ @nicoguaro Yes, I am using the shift invert method with sigma=0. $\endgroup$
    – Laplacian
    Jun 3, 2021 at 14:30

1 Answer 1

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ARPACK recommends to use ncv > 2 nev. The default value of ncv would fit that constraint. A couple of suggestions:

  • increase ncv to 300?
  • increase the tolerance tol to $1.0e-14$.
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  • $\begingroup$ Thanks! I will try it soon and see what happens. $\endgroup$
    – Laplacian
    Jun 1, 2021 at 5:32
  • $\begingroup$ Sadly, increasing ncv does not help. Reducing tolerence helps a little bit, as it should be. $\endgroup$
    – Laplacian
    Jun 2, 2021 at 5:11

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