Symplectic Grassmann

[kellertuer]

this PR is a start to implement the (real) Symplectic Grassmann, which one obtains from Symplectic Stiefel the same way one obtains Grassmann from Stiefel.
It will probably also revise a few symplectic Stiefel functions.

• Introduce Hamiltonian, they allow for H^+ to be nicely defined for the symplectic inverse
• We should introduce HamiltonianMatrices? Maybe not 100% necessary but nice for completeness
• Define SymplecticGrassmann
• implement checks
• inner (after (4.10) / Lemma 4.8, BZ21)
• the test/utils.jl functions complain since Julia 1.10 to be multiply included. We should at least move those, if not also the constants to the Manufolds.jl/Test.jl extension?
• unify notation concerning J_2n and Q_2n for the symplectic element (and refer to SymplecticMatrix).
• unify/simplify notation to refer more often to symplectic_inverse when writing p^+ instead of repeating the formula.
• implement the geodesic
• implement the Cayley retraction and its inverse
• refactor rand_hamiltonian to be rand(Hamiltonian(2n))
• implement the gradient conversion (after Lemma 4.8, BZ21)
• (maybe) check for Symplectic and SymplecticStiefel, so they get rand! methods as well. They currently just have rand methods
• read docs carefully again
• check test coverage

Both symplectic Stiefel and Grassmann stem from this paper https://arxiv.org/abs/2108.12447

[codecov]

Codecov Report

Attention: 12 lines in your changes are missing coverage. Please review.

Comparison is base (64ae41a) 99.54% compared to head (9500bd5) 94.87%.

Files	Patch %	Lines
src/manifolds/SymplecticGrassmannStiefel.jl	90.47%	4 Missing ⚠️
src/manifolds/Hamiltonian.jl	95.31%	3 Missing ⚠️
src/manifolds/Symplectic.jl	97.90%	3 Missing ⚠️
src/manifolds/SymplecticGrassmann.jl	92.85%	1 Missing ⚠️
src/manifolds/SymplecticGrassmannProjector.jl	96.55%	1 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##           master     #700      +/-   ##
==========================================
- Coverage   99.54%   94.87%   -4.67%     
==========================================
  Files         108      112       +4     
  Lines       10715    10871     +156     
==========================================
- Hits        10666    10314     -352     
- Misses         49      557     +508     

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[kellertuer]

With the last few methods now I would consider this feature-complete (just not yet test&docs complete), so I moved it from draft to ready (but did not yet set the label).

[mateuszbaran]

I haven't read the actual new stuff but here are some small technical comments.

[kellertuer]

This now covers all I wanted to be included. I already bumped the version, but I would also like to include the Multinomial PR (#702) in this version I think, since that one is basically finished as well.

[mateuszbaran]

It looks like tolerance in one test for SymplecticStiefel is too low.

[kellertuer]

I am already running this locally. I am a bit surprised by that since I did not change that.

[mateuszbaran]

A few comments.

[mateuszbaran]

LGTM 👍