LinearAlgebra: move alg to a keyword argument in symmetric eigen

[jishnub]

Currently, eigen for symmetric matrices accepts an alg as the second positional argument, but this departs from the pattern of eigen(A, B) representing a generalized eigenvalue problem. This PR moves alg to a keyword argument instead. This prevents packages from specializing on alg, but I'm unsure if there's demand for that.

Incidentally, alg as a keyword argument in eigen is also consistent with how it is passed to the sort family of functions, although there's no direct relation between these.

[LilithHafner]

Losing the ability to specialize on a custom alg type seems like a pretty big loss. How would one write EigenAlgorithms.jl? FWIW, specialization on algorithms in sorting using the public API is a mess.

[ViralBShah]

Aren't eigenvalue computations large and slow enough that there wouldn't be gains from specialization? Is the argument that it would make the rest of the computation type unstable in some way?

[jishnub]

Perhaps we may have an internal function that specialises on alg? It doesn't need to be eigen. This function may be public, and methods may be added by other packages.

[LilithHafner]

@ViralBShah, the argument is that the extension API is not as discoverable. Performance and type stability should be fine.

# LinearAlgebra.jl
public eigen
eigen(::AbstractMatrix, ::Algorithm) = ...

---

# EigenAlgorithms.jl
eigen(::AbstractMatrix, ::MyFancyAlgorithm) = ...

vs

# LinearAlgebra.jl
public eigen, eigen_dispatch
eigen(x::AbstractMatrix; alg::Algorithm) = eigen_dispatch(x, a)
eigen_dispatch(::AbstracMatrix, ::Algorithm) = ...

---

# EigenAlgorithms.jl
eigen_dispatch(::AbstractMatrix, ::MyFancyAlgorithm) = ...

[jishnub]

I don't think this needs to be particularly discoverable, as this isn't user-facing. We may add a note to developers in the docstring of eigen to point to this function.

[DilumAluthge]

We have moved the LinearAlgebra stdlib to an external repo: https://github.com/JuliaLang/LinearAlgebra.jl

@jishnub If you think that this PR is still relevant, please open a new PR on the LinearAlgebra.jl repo.