improve performance of stream cauchy invert

[mantepse]

📚 Description

Replace loops by sums with lazy assignments and provide a dedicated get_coefficient for Stream_cauchy_invert

Fixes #34553

📝 Checklist

• I have made sure that the title is self-explanatory and the description concisely explains the PR.
• I have linked an issue or discussion.
• I have created tests covering the changes.
• I have updated the documentation accordingly.

⌛ Dependencies

[mantepse]

Warning: there is another commit, which removed Stream_cauchy_invert.get_coefficient: 814aa7c

I am not sure what to make of this. Also, in the get_coefficient here, it seems to me that at least isinstance(self._series, Stream_exact) and not self._series._constant could be done in __init__.

We would have to check whether the modifications here are really an improvement.

[codecov-commenter]

Codecov Report

Patch coverage: 93.08% and project coverage change: -0.01 ⚠️

Comparison is base (5330147) 88.61% compared to head (3ef7684) 88.61%.

❗ Current head 3ef7684 differs from pull request most recent head 31ca686. Consider uploading reports for the commit 31ca686 to get more accurate results

Additional details and impacted files

@@             Coverage Diff             @@
##           develop   #35338      +/-   ##
===========================================
- Coverage    88.61%   88.61%   -0.01%     
===========================================
  Files         2148     2148              
  Lines       398855   398913      +58     
===========================================
+ Hits        353438   353488      +50     
- Misses       45417    45425       +8     

Impacted Files	Coverage Δ
src/sage/rings/lazy_series_ring.py	89.77% <ø> (ø)
src/sage/rings/lazy_series.py	92.35% <84.21%> (-0.08%)	⬇️
src/sage/data_structures/stream.py	96.96% <94.28%> (-0.51%)	⬇️

... and 28 files with indirect coverage changes

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[tscrim]

We got rid of it because we realized that the Cauchy inverse always had to compute things in order, so it was always dense. Thus, any changes need to be in iterate_coefficients().

[mantepse]

Ah, excellent that you remember / are pointing this out! I don't know yet when I can continue working on this, but all your comments so far are very helpful, thank you!

[tscrim]

I will try to look at this again soon too to try and see what speedups can be done, but it will probably not be a high priority for me.

[github-actions]

Documentation preview for this PR (built with commit 1d52328; changes) is ready! 🎉

[tscrim]

LGTM.