Add iterative refinement function

[MaxenceGollier]

I added a Julia function that given an approximate solution $$x$$ refines a solution of the system $$R^TR x \approx z$$.
Let me know what you think @dpo @amontoison when you have some time.

[amontoison]

Can you add entries for qrm_refine! and qrm_refine in the documentation?

For the docstring of qrm_refine!, I suggest to also specify the length of each vector.
You say at the top of the docstring that (m, n) is the size of A and explain that each vector is of size n or m.

Good job!

[dpo]

@MaxenceGollier thank you. Because qrm_refine is so closely tied to the seminormal equations, it would make sense to add functions specifically to solve those equations. They would call qrm_refine.

[MaxenceGollier]

You say at the top of the docstring that (m, n) is the size of A and explain that each vector is of size n or m.

I prefer to avoid specifying the dimension of A. The dimensions of the inputs should depend on the dimension of R instead, if the user calls qrm_factorize! with transp = 't', then it is the dimension of R that is relevant because we will still solve $$R^TRx = z$$ but for the problem $$AA^T x = z$$ instead of $$A^TAx = z$$ in that case.

@MaxenceGollier thank you. Because qrm_refine is so closely tied to the seminormal equations, it would make sense to add functions specifically to solve those equations. They would call qrm_refine.

I agree, I will add a separate PR for that.
In all cases, thank you for your quick feedback @amontoison, @dpo !

[amontoison]

Could you add an option to refine RR'x = b or R'Ry = c?
I agree to keep the PR small and focused.

It will require more work to address the request of Dominique:
#88 (comment)

[MaxenceGollier]

Could you add an option to refine RR'x = b or R'Ry = c? I agree to keep the PR small and focused.

It will require more work to address the request of Dominique: #88 (comment)

I don't see in what situation a problem like RR' x = b would arise. Whether the problem is A' A x = b or AA' x = b, we would still get R'R x = b

[amontoison]

You're right, except if A is square we don't need it because qr_mumps requires the QR of A' is A has more columns than rows.
This corner case is too specific to justify spending time on.