implement test_approx for Eigen

[simeonschaub]

This is needed for JuliaDiff/ChainRules.jl#431.

[simeonschaub]

@sethaxen Is this strong enough for the eigen tests in ChainRules? Or do we actually want to require the eigenvectors to be approximately equal as well?

[oxinabox]

I wonder if we should do this for all subtypes of Factorization ?

[simeonschaub]

Hm, yes, maybe? Although I am not really familiar with those, so perhaps we should wait until an actual use case comes up?

[oxinabox]

I will let @sethaxen give final approval on this, as I don;t know how things work with Eigen

[sethaxen]

This is to be used to test that the primal part of frule and rrule returns the same decomposition as the primal function, right? In that case, I think we do care whether the eigenvalues and eigenvectors match. But also, it should not be hard to make them match, right, since the eigen rules call eigen internally? Perhaps I'm missing something.

[simeonschaub]

Hmm, yes you might be right. Looking at the failures in https://github.com/JuliaDiff/ChainRules.jl/runs/2769420821?check_suite_focus=true#step:6:282 for example again, it looks like the values shown do actually match precisely, any ideas what could cause this? There shouldn't be any NaN's occuring in those tests, right? Even if there would be degeneracy, that shouldn't cause the results to be non-deterministic no? I can look into this some more if you don't have any ideas either.

[sethaxen]

it looks like the values shown do actually match precisely, any ideas what could cause this? There shouldn't be any NaN's occuring in those tests, right?

I'm not certain. But I do see some undef's in that failed test. Is it possible a matrix has undefined values?

[simeonschaub]

I only see that for the arrays with 0 elements, but I could be wrong.

[simeonschaub]

Ah, wait! It seems like Eigen now also contains condition numbers which are just random noise in this instance. That's probably why tests fail.