Implement iterative eigenvalue finder for eigs

[cshaa]

The current implementation of eigs relies on usolveAll to find eigenvectors. This often fails because of numerical errors. A more robust approach would be to use an iterative algorithm. The easiest one in this case would be the inverse iteration algorithm. The gist of the algorithm is this:

1. Take a random vector b.
2. Let A be a matrix and k an approximation of its eigenvalue.
3. The expression ( (A − kE)⁻¹ )ⁿ b converges to an eigenvector as n → ∞.
4. Instead of actual inversion, which is a costly operation, we can perform an implicit inversion using usolve bₖ = (A − kE) bₖ₊₁.

The relevant code is here.

[josdejong]

Thanks for the clear explanation, makes sense. Are you planning on picking this up yourself @m93a?

[cshaa]

Are you planning on picking this up yourself?

When I have the time and mood, yes, definitely. (I expect this means in a month or so, but I can't tell with certainty.) However, if anyone wants to tackle this issue earlier, feel free to :)

[josdejong]

👍

[josdejong]

Published now in v9.4.2