Add forward rules for dense linear solves

[jlperla]

Summary

• add forward-mode support for dense A \ b and A \ B
• reuse the existing compute_lu_cache path used by the reverse rule
• support active or constant RHS, active square A, and batched forward mode
• keep active rectangular solves unsupported for now

Tests

• julia --project=test test/runtests.jl --verbose rules/internal_rules/linear_algebra_rules

[jlperla]

@wsmoses This one I felt more confident implementing. Seems to pass tests/runic/etc. and I think it follows the patterns of others in that file.

Let me know if you want anything changed.

[codecov]

Codecov Report

❌ Patch coverage is 26.66667% with 22 lines in your changes missing coverage. Please review.
✅ Project coverage is 67.39%. Comparing base (51a2569) to head (a13965e).

Files with missing lines	Patch %	Lines
src/internal_rules/linalg.jl	26.66%	22 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##             main    #3106      +/-   ##
==========================================
- Coverage   70.04%   67.39%   -2.65%     
==========================================
  Files          66       66              
  Lines       21840    21803      -37     
==========================================
- Hits        15297    14694     -603     
- Misses       6543     7109     +566     

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[wsmoses]

same comment of fine for now, but we should tablegen likely here

[jlperla]

Yes on tablegen. I asked an AI to come up with tablegen but it ended up looking very complicated because of managing the pivots. But it got the dense LU stuff working and passing tests. Happy to post that up as a PR for you guys to look at and suggest changes/take it over. But not sure it is worth your time.