Make LinearAlgebra._zeros use similar

[dkarrasch]

Fix #40899 (comment)

[maleadt]

Thanks! Disregarding the CI failures here, this does make the relevant CUDA.jl code work again.

[andreasnoack]

The problem is that similar(::SparseMatrixCSC) is a SparseMatrixCSC. We need the output of _zeros to be a dense array.

[dkarrasch]

So we need to specialize _zeros(::Type{T}, b::AbstractSparseVector, n::Integer) to give a dense vector, right? I'll add that in SparseArrays and see how it goes.

[andreasnoack]

The problem is that _zero was supposed to be a single simple helper function. With these changes, it almost becomes part of the interface. I'm not sure what the right solution is. This wouldn't be a problem if similar(::SparseMatrixCSC) returned a Matrix.

[dkarrasch]

How did it work earlier? How did we use similar yet avoided the *Diagonal and sparse issues?

[andreasnoack]

How did it work earlier?

I think that we effectively called zeros or Matrix for these cases. For some reason it didn't cause issues for CUDA. In my PR, I made all \ use the same code path so a few extra \ might now use zeros/Matrix instead of similar. Maybe Cholsky\... since @maleadt mentioned potrs. Maybe a better solution is to add \ method to CUDA similar to the one here that initializes the rhs with a CuArray. We don't really have the right generic function to initialize the right hand side of a solve. At least that would require changing similar for sparse arrays in a way that probably isn't feasible at this stage.

[maleadt]

Maybe Cholsky\... since @maleadt mentioned potrs.

Yup, that was the one. I'll duplicate a version of \ for CuArrays in CUDA.jl.

[dkarrasch]

So we can close this, right?

[andreasnoack]

I'd think so. @maleadt can you please confirm that you were able to work around the issue by adding a method to CUDA.jl?

[maleadt]

Sorry for being unclear, yes this can be closed. It would be nice to revisit this at some point though, because code like https://github.com/JuliaGPU/CUDA.jl/blob/8ba1fe4f1c2b94e309eb4c51583783a43e7f297d/src/linalg.jl#L17-L48 is pretty fragile.