DOC: updated preconditioner doc for iterative linear solver

[Afleloup]

Closes #5818

Reference issue

Closes #5818

What does this implement/fix?

The argument M in iterative linear solver
'bicg', 'bicgstab', 'cg', 'cgs' is not standard notation
in comparison to the literature. It is the inverse of the
preconditioner usually denoted M^-1. Their should be a
precondition on M for the preconditioned conjugate gradient.
It is now explicited in the documentation and linked
to relevant references.

Additional information

References

.. [1] TREFETHEN, Lloyd N. et BAU, David. Numerical linear algebra. 
       Society for Industrial and Applied Mathematics, 2022.
.. [2] SAAD, Yousef. Iterative methods for sparse linear systems. 
       Society for Industrial and Applied Mathematics, 2003.
.. [3] "Conjugate Gradient Method, Wikipedia, 
       https://en.wikipedia.org/wiki/Conjugate_gradient_method
.. [4] "Preconditioner", 
       Wikipedia, https://en.wikipedia.org/wiki/Preconditioner
.. [5] KATRUTSA, Alexandr, BOTCHEV, Mike, OVCHINNIKOV, George, et al. 
       How to optimize preconditioners for the conjugate gradient method: 
       a stochastic approach. arXiv preprint arXiv:1806.06045, 2018.
.. [6] CARABA, Elena. Preconditioned conjugate gradient algorithm. 2008.
.. [7] "Biconjugate gradient stabilized method", 
       Wikipedia, https://en.wikipedia.org/wiki/Biconjugate_gradient_stabilized_method
.. [8] "Conjugate gradient squared", Wikipedia,
       https://en.wikipedia.org/wiki/Conjugate_gradient_squared_method
.. [9] "Biconjugate gradient method", Wikipedia, 
       https://en.wikipedia.org/wiki/Biconjugate_gradient_method

[thalassemia]

Thanks for the contribution! These are just typo fixes. Someone familiar with the math still needs to review this.

[ilayn]

Thank you @Afleloup

I think this is conflating the point here a bit too much. Just because the argument name is M is not meant to be that we find a preconditioner and invert it and then pass here. We can correct the docs saying simply $MA$ should improve the conditioning of the problem as an extra sentence, and that should be enough.

The theoretical treatment is irrelevant here since the operator name can be anything. It's utility is important.

The point of all the inverses attached to M is to emphasize that it kinda sorta inverts A. But in fact it does a more important thing which is, ideally, improves the condition number of the LHS. Hence inverse is a red herring here. The existing docs read correct to me, but can be clarified for the mismatch with the literature.

Note that the preconditioner definition is all over the place, as noted with the wiki page you added as a reference

It is also common to call $T=P^{-1}$ the preconditioner, rather than $\displaystyle P$, since $P$ itself is rarely explicitly available.

[Afleloup]

Thanks, I will change it this way then. I didn't see the sentence you mentioned in the wikipedia article and thought the literature I had was consistent with the other notation.

[ilayn]

No problem, thank you for tackling this issue regardless.