ENH: optimize: Add TFQMR method to approximate the inverse of the Jacobian in "Newton-Krylov" and "root"

[zhaog6]

Reference issue

What does this implement/fix?

In this PR, just like GMRES etc., TFQMR method can be used to approximate the inverse of the Jacobian in the process of solving nonlinear equations using Newton-Krylov method and finding root.

Additional information

[zhaog6]

Dear Andrew @andyfaff , Could you review the PR? Thanks.

[andyfaff]

I'm not familiar with that code (or the maths behind it), so I'm not best placed to review this PR. @mdhaber, @pv you'd be better placed than me.

(I note there's no tests with this PR).

[mdhaber]

Thanks for the contribution, @zhaog6 , but I am not familiar with it either. Maybe it would help find a reviewer if you would point out where (in code and primary source documents, like the original papers or a texbook) supports this.

[tylerjereddy]

A bit of a fishing expedition, but @ev-br and @AtsushiSakai also show up in some flanking lines of the git blame so could try asking them.

[zhaog6]

Thanks all @andyfaff @mdhaber @tylerjereddy for your suggestions, I added some test cases and I will contact Pauli@pv

[zhaog6]

Dear Pauli @pv ,Could you review this PR? Thanks.

[rgommers]

Closed because this PR is continued in gh-15981.