Added hseqr! LAPACK function

[albertomercurio]

This fixes JuliaLang/LinearAlgebra.jl#962

[albertomercurio]

Ok, I think that's all.

[dkarrasch]

Nice. Is that copy-paste from @Jutho's package? We definitely need tests for this. Also, could/should we add dispatch rules for the eigen function to call this whenever possible and meaningful? Like norm calls BLAS for strided arrays with the usual floating point numbers etc.?

[albertomercurio]

Nice. Is that copy-paste from @Jutho's package?

Yes, but with a bit of improvement.

Also, could/should we add dispatch rules for the eigen function to call this whenever possible and meaningful?

The hseqr! function computes the eigenvalues and, optionally, the Schur decomposition. It never gives you the eigenvectors. Thus, I think the correct comparison should be with the schur function, which calculates the factorization for a general matrix.

I will do it soon.

[dkarrasch]

One last gentle request: can you test that the promotion works as desired? Like apply your tests to a small integer UpperHessenberg matrix.

[dkarrasch]

I'm not sure if we announced added LAPACK features in the news, but it seems like so.

Pinging @Jutho so that he can take into account that this is "now" available through LinearAlgebra.

[Jutho]

I have been following and I am very happy to see this. That takes away a burden of the maintenance of KrylovKit.

[albertomercurio]

Ok, I think that's all.