Add caching for QME doubling

[thorek1]

Summary

• add qme_caches type and constructor
• allow solve_quadratic_matrix_equation(..., Val(:doubling)) to reuse LU factorizations via LinearSolve caches
• handle QME cache RHS vectors by solving column slices individually

Testing

• apt-get update -y
• apt-get install -y julia (fails: package not found)

---

https://chatgpt.com/codex/tasks/task_e_68406fa57dc0832fa5f23b0b69075011

[codecov-commenter]

Codecov Report

Attention: Patch coverage is 2.98507% with 65 lines in your changes missing coverage. Please review.

Project coverage is 4.64%. Comparing base (3c070fd) to head (0e297e7).

Files with missing lines	Patch %	Lines
src/algorithms/quadratic_matrix_equation.jl	0.00%	38 Missing ⚠️
src/options_and_caches.jl	22.22%	7 Missing ⚠️
src/MacroModelling.jl	0.00%	6 Missing ⚠️
src/perturbation.jl	0.00%	6 Missing ⚠️
src/get_functions.jl	0.00%	4 Missing ⚠️
src/moments.jl	0.00%	2 Missing ⚠️
src/filter/inversion.jl	0.00%	1 Missing ⚠️
src/filter/kalman.jl	0.00%	1 Missing ⚠️

❗ There is a different number of reports uploaded between BASE (3c070fd) and HEAD (0e297e7). Click for more details.

HEAD has 10 uploads less than BASE

Flag	BASE (3c070fd)	HEAD (0e297e7)
	11	1

Additional details and impacted files

@@                Coverage Diff                 @@
##           rethink_parser    #109       +/-   ##
==================================================
- Coverage           86.09%   4.64%   -81.45%     
==================================================
  Files                  18      18               
  Lines               10523   10495       -28     
==================================================
- Hits                 9060     488     -8572     
- Misses               1463   10007     +8544     

☔ View full report in Codecov by Sentry.
📢 Have feedback on the report? Share it here.

🚀 New features to boost your workflow:

• ❄️ Test Analytics: Detect flaky tests, report on failures, and find test suite problems.