Basis number system swap

[mateuszbaran]

• Adapt Euclidean
• Adapt ProjectiveSpace (I'm not sure the old code is even correct for real basis of complex projective space?)
• Adapt SymmetricMatrices
• Adapt quaternionic unitary matrices?

[mateuszbaran]

I've made the tests for ProjectiveSpace pass but I haven't verified if the code is indeed correct (it's just the same logic as on master). Notably, sphere only has the real coefficient basis.

[kellertuer]

I am also not sure the old Euclidean code was correct. But then we should check that. Just that today I am busy (a bit behind with my lecture for tomorrow, and a bit of grading). So I'll probably only tomorrow have time to take a look.

[mateuszbaran]

I haven't found any issues with Euclidean -- do you have a specific concern? Anyway, this can wait until tomorrow :).

[kellertuer]

The original example

using Revise
using Manifolds, ManifoldDiff, RecursiveArrayTools, FiniteDifferences

r_backend = ManifoldDiff.TangentDiffBackend(
    ManifoldDiff.FiniteDifferencesBackend()
)

f(p) = 1
grad_f(M,p) = ManifoldDiff.gradient(M, f, p, r_backend)
M = ℂ^3 × PowerManifold(Rotations(6), NestedPowerRepresentation(),3)
p = rand(M)
# causes the same error
grad_f(M,p)

also threw errors in Euclidean, but I have to check whether your fix in ManifoldsBase already fixed that then.

[mateuszbaran]

No, that wasn't a problem with Euclidean but thanks for reminding me -- there was another issue in a different place that I had to fix to make the example work.

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 99.68%. Comparing base (150e7e0) to head (4b24118).
Report is 1 commits behind head on master.

Additional details and impacted files

@@            Coverage Diff             @@
##           master     #754      +/-   ##
==========================================
- Coverage   99.69%   99.68%   -0.01%     
==========================================
  Files         123      123              
  Lines       11418    11444      +26     
==========================================
+ Hits        11383    11408      +25     
- Misses         35       36       +1     

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[kellertuer]

This looks good to me, if code coverage agrees of course.