Add an example to perform a QR factorization without storing Q

[amontoison]

It's relevant when we want to solve AA'x = b <=> R'Rx = b with A' = QR.

[dpo]

When you solve either AA'x = b or A'Ax = A'b with Q-less QR, you have to perform one step of iterative refinement for the method to be backward stable. Do you think it would make sense to add functions in this module to perform the whole procedure (factorization + solve + iterative refinement)?

[amontoison]

Yes, we should add new functions for that.

I also think we should incorporate a generic API (cholesky, qr, lq, \, ldiv!) for the spmat type.
I implemented this for CuSparseMatrixCSR in CUDSS.jl, which greatly simplifies tasks for the user: https://github.com/exanauts/CUDSS.jl/blob/main/src/generic.jl

We can introduce a keyword argument seminormal (or something similar) when creating a spmat to indicate that we want to solve systems with AA' or A'A.
The \ operation will then handle the entire procedure for us (factorization + solve + iterative refinement).

[amontoison]

@abuttari
Alfredo, do you have an option to perform iterative refinement during the solve phase in qr_mumps?

[abuttari]

@abuttari Alfredo, do you have an option to perform iterative refinement during the solve phase in qr_mumps?

nope, no iterative refinement in qr_mumps currently but shouldn't take too long to add.

[dpo]

We could simply do it in Julia. It’s not strictly necessary to add it to QRMumps itself, unless you want to.

[MaxenceGollier]

I added said documentation in PR #105, I don't use iterative refinement though, let me know if I should add it to the tutorial.