Roadmap: basic linear algebra framework for generic eltypes

[jiahao]

This issue continues on from #11 and JuliaLang/julia#5381 as a placeholder to track what is left to put into place a useful skeleton of basic linear algebra for generic element types.

Routines for special matrices (#3688)

• Tridiagonal linear solver (#39701)
• `Tridiagonal eigensystem solver (@andreasnoack mentioned he had some code for this?)

Factorizations

• LL' Cholesky
• pivoted LU
• QR
  • Should we just pick one of: modified Gram-Schmidt, Givens and Householder algorithms?
  • Want both the basic and pivoted (rank-revealing) variants?
• SVD
• Check that with the necessary factorizations in place, that inv and eig work.

[jiahao]

cc: @stevengj

[stevengj]

For dense QR, it seems like the priority should be on Householder; I'm not sure if there is any point in implementing other algorithms. I don't have any strong opinion on pivoting of QR.

[simonbyrne]

Code for generic SVD is available in GenericSVD.jl. Do we want to polish this up and merge into Base?

[ViralBShah]

Yes, that sounds like a good idea.

[0joshuaolson1]

Here's one more vote for arbitrary-precision eig (eigvals).

[andreasnoack]

@0joshuaolson1 Some of this work has been done in https://github.com/andreasnoack/LinearAlgebra.jl but it still needs to be better tested.

[stevengj]

Some of this functionality has been implemented to packages, e.g. https://github.com/JuliaLinearAlgebra/GenericLinearAlgebra.jl … I suspect that we should close this issue in favor of more specific issues or packages?