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For large and matrices you can use the (restarted) Arnoldi method to compute a partial Schur decomposition AQ=QR where Q is tall and skinny and R has a few dominant eigenvalues on the diagonal (i.e. eigenvalues on the boundary of the convex hull).
MATLAB uses ARPACK's implementation of this when you call `eigs`
I wrote my own implementation ArnoldiMethod.jl in julia, which unlike MATLAB/ARPACK supports arbitrary number types, and also should be more stable in general, and equally fast.
[1] https://github.com/JuliaLinearAlgebra/ArnoldiMethod.jl
I guess it is a rite of passage to rewrite it. I'm doing it for SciPy too together with Propack in [1]. Somebody already mentioned your repo. Thank you for your efforts.
[1]: https://github.com/scipy/scipy/issues/18566